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September 4, 2023

How Drawing Unlocks Creativity and Problem-Solving

Drawing is a universal form of expression that transcends language barriers and taps into the innate human desire to communicate visually. Beyond its artistic allure, drawing possesses a remarkable ability to enhance problem-solving skills and unleash creativity. The act of problem solving drawing by putting pencil to paper isn’t just reserved for artists; it’s a cognitive tool that can spark innovation and lead to novel solutions.

In this article, we will explore how drawing contributes to problem-solving and the ways in which it can be harnessed as a powerful thinking tool.

When faced with complex problems, the act of problem solving drawing forces you to externalize your thoughts and visually map out connections, making abstract concepts more tangible. This visual representation often reveals patterns, relationships, and gaps that might not be as evident through traditional verbal or text-based approaches.

Problem solving drawing enables you to approach challenges from a fresh perspective, breaking down barriers that hinder innovative thinking. ‘

You might also like: 151 Drawing Ideas with Examples for Inspiration

A problem-solving chart is a visual framework that organizes information and ideas related to a specific problem or challenge. It can take various forms, such as flowcharts, mind maps , or concept diagrams.

These charts serve as dynamic tools to capture and analyze the various elements of a problem, providing a clear overview and aiding in identifying potential solutions.

The process of creating a problem-solving chart encourages structured thinking and encourages you to explore different angles of a problem.

Basics of drawing and sketching for beginners

Drawing for Beginners

The Art of Public Speaking

When faced with complex problems, the act of problem solving drawing forces you to externalize your thoughts

Using problem solving drawing to create a diagram involves several steps:

Define the Problem: Clearly articulate the problem you’re trying to solve. This will serve as the central focus of your diagram. Gather Information: Collect relevant data, ideas, and concepts related to the problem. Jot down keywords, phrases, and concepts that come to mind.
Choose a Format: Depending on the complexity of the problem, choose a suitable format for your diagram. Mind maps are great for brainstorming, while flowcharts can illustrate processes and decision trees.
Organize the Information: Begin placing the collected information in the diagram. Connect related ideas with lines or arrows, indicating relationships and dependencies.
Analyze and Iterate: Step back and assess the diagram. Does it highlight potential solutions? Are there gaps or missing connections? Make adjustments as needed.
Identify Solutions: Use the insights gained from the diagram to identify possible solutions. The visual representation can help you see connections that lead to innovative ideas.

Problem solving drawing offers a myriad of additional benefits. It enhances focus and concentration, allowing you to immerse yourself in the creative process.

Drawing also serves as a stress-relieving activity, providing a means to unwind and detach from the demands of everyday life.

Moreover, it boosts visual thinking skills, enabling you to grasp complex concepts and information more readily.

Absolutely not.

The beauty of drawing as a problem-solving tool is that it transcends artistic skill.

You don’t need to create intricate masterpieces; your goal is to externalize your thoughts and ideas visually. Simple sketches, doodles, or diagrams can effectively convey your thinking process.

The act of drawing is more about engaging with the problem and unleashing creativity than producing gallery-worthy art.

You might also like: 21 Best Drawing Courses

If you’re new to drawing but eager to enhance your creativity, there are several exercises that can help kickstart your artistic journey.

Begin with basic shape doodling.

Set aside time to doodle various shapes, lines, and patterns. This exercise not only gets your hand accustomed to the motion of drawing but also encourages your brain to think visually.

Another exercise is blind contour drawing.

Choose an object and draw it without looking at the paper. This exercise improves hand-eye coordination and forces you to observe details.

Additionally, try drawing from memory.

After studying an object or scene, put it out of sight and attempt to draw it from memory. This exercise sharpens your observation skills and encourages you to focus on essential details.

Lastly, experiment with negative space drawing .

Instead of drawing the object itself, draw the spaces around and within it. This exercise challenges your perception and helps you see objects in a new light.

By engaging in these drawing exercises regularly, you’ll not only improve your artistic skills but also ignite your creativity in various aspects of problem-solving and thinking.

Drawing isn’t solely reserved for those with an affinity for art; it’s a potent tool that can unlock creativity and enhance problem-solving skills. Problem solving drawing enables you to approach challenges from a fresh perspective, providing a visual representation of intricate ideas and connections.

As you craft problem-solving charts, you organize information, identify patterns, and explore innovative solutions. Beyond its cognitive benefits, drawing offers stress relief, improved concentration, and heightened visual thinking skills.

Embrace drawing as a thinking tool, regardless of your artistic prowess, and harness its potential to transform the way you approach problems and ignite your creative spark.

Design Thinking

Design is endlessly trying, refining, improving until slowly something begins to emerge that is so ingenious that it looks like magic if you don’t know what went on before: that’s what evolution does. – Designer, Joris Laarman

Design is all around us, whether it takes the form of objects and spaces, images and interactions, or systems and processes. Beyond meeting a need, design can experiment with forms, techniques, or materials to be an expression of a concept or beauty.

In this Web Quest, you will explore notions of design and learn about a creative problem-solving process called Design thinking. How do artists solve problems? Use these activities and videos at home, online, or in the classroom to spark curiosity, conversation, and critical thinking.

Web Quest includes:

A scavenger hunt activity, where students identify how objects in the Denver Art Museum collection speak to both aesthetics and function
A lesson plan inspired by Basket Chair (includes two videos, a facilitator’s guide and supportive materials for kids to dive into dive into design related concepts)

Print out this scavenger hunt or insert it into the learning management system of your choice to create an interactive activity prompting kids to look closely at the artworks found in the object gallery below.

Object Gallery

Web Quest Resources

Design Thinking: Facilitator's Guide (web)
Design Thinking: Facilitator's Guide (PDF)
Design Thinking: Instructions for Kids (PDF)
Scavenger Hunt Worksheet (PDF)
Slides for Elementary (PowerPoint)
Slides for Elementary (Google)
Slides for Middle and High School (PowerPoint)
Slides for Middle and High School (Google)
Library Resources

The DAM established Creativity Resource thanks to a generous grant from the Morgridge Family Foundation. Featured activities are supported by funding from the Tuchman Family Foundation, The Freeman Foundation, The Virginia W. Hill Foundation, Sidney E. Frank Foundation – Colorado Fund, Colorado Creative Industries, Margulf Foundation, Riverfront Park Community Foundation, Lorraine and Harley Higbie, an anonymous donor, and the residents who support the Scientific and Cultural Facilities District (SCFD). Special thanks to our colleagues at the University of Denver Morgridge College of Education. The Free for Kids program at the Denver Art Museum is made possible by Scott Reiman with support from Bellco Credit Union.

Design Thinking: Artists Solve Problems is supported by Herman Miller Cares.

How to solve your problems through drawing (w/ Liana Finck) (Transcript)

Listen along.

How to Be a Better Human How to solve your problems through drawing (w/ Liana Finck) October 9, 2023

[00:00:00] Chris Duffy: You're listening to How to Be a Better Human. I'm your host, Chris Duffy. When I think about the artists, comedians, writers, when I think about the people that I admire the most, they run the gamut when it comes to the topics that they cover and their styles and the formats that they work in, but the thing that ties everyone who I admire together, the thing that they all share is that they found a way to be distinctly and uniquely themselves. That is the trait that I admire the most.

And, and that is why I am so excited about today's episode of our podcast, because our guest, the cartoonist Liana Finck, makes work that could only be made by her. You can see her cartoons in places like The New Yorker, but her style is so distinct from what a classic New Yorker cartoon is. You would never mistake her work for something by anyone else. Because Liana's drawings, they're much simpler, but they're also more impressionistic, and they convey so much emotion.

And they're also very, very, very, very funny. Liana is hilarious and her humor comes from this deeply honest confessional place. It's some of the best proof I've ever seen that the more specific and the more true to yourself you go, the more universally your jokes will connect. Whether it is Liana overthinking what to say at a party or analyzing the pressures of adulthood or motherhood, she is always so funny, and she is always 100 percent herself. If you can't already tell, I am a huge fan and I'm so excited to talk to her. Here's a clip from Liana's TED Talk.

[00:01:26] Liana Finck: When I first started making cartoons for The New Yorker about a decade ago, I kept my ideas light and quirky. I didn't draw anything too personal. I figured I was too specific, too hard to relate to, and read possibly too female.

It took a breakup to get me to start drawing more autobiographically. The pain I was feeling, although objectively pretty run of the mill, was impossible to ignore. I knew that drawing was my strongest problem solving tool, so I decided to diagram what I was going through.

By making these drawings, I could see how my ex and I had hurt each other and move on. Drawing from my own life was a revelation to me, not only because it helped me understand myself better, but because it made me see for the first time that people could relate to me. Now that I had this amazing tool, there were so many problems I wanted to solve with it.

[00:02:39] Chris Duffy: We're going to be right back with much more from Liana after these quick ads. Don't go anywhere.

[00:02:47] Chris Duffy: Today we're talking about humor, creativity, and cartoons with Liana Finck.

[00:02:53] Liana Finck: Hi, I'm Liana Finck. I'm a cartoonist, which in my case means that I make New Yorker cartoons and graphic novels and a lot of random freelance stuff.

[00:03:03] Chris Duffy: One of the things that is kind of the driving idea behind the interview, which is also a big part of your TED talk, is how drawing and how art more generally can help you to figure out who you actually are and be more yourself. So could you talk to us a little bit about that journey of how your cartoons let you be more yourself or figure out who that self really is?

[00:03:28] Liana Finck: I think for me, drawing coincided with wanting to figure out who I was. I'm not sure that they need to go together for everyone. I think some people draw, and some people are on a desperate quest to figure out why they are the way they are, and some people are both. And some people are neither and there's room to be creative in all, in each of those quadrants.

[00:03:56] Chris Duffy: Where do you put yourself in those quadrants between, uh, just drawing and just really trying to figure out who you are?

[00:04:02] Liana Finck: I'm both of those things for now. I, lately I feel like I've done enough figuring out, and I don't really need to do that more, so I'm trying to figure out… I, I really hope drawing is a throughline in my life. So I'm trying to figure out new reasons to draw.

[00:04:19] Chris Duffy: One of the challenges of being in the public eye, especially the way you are, I imagine, is that you get known for one particular thing, and people feel really connected to you. But then when you change, it’s hard to figure out if you're allowed to be a different person than your audience expects if they got to know you a long time ago.

[00:04:44] Liana Finck: Yeah, I, I feel that, and it's hard to talk about ‘cause, like, everyone has a different, like, journey with their art career. So yeah, mine is that I was known in this very flimsy way as, uh, I’m not sure this was the main thing, but I felt like the main thing was that I was an Instagram personality, and I posted a lot of drawings every day on Instagram, and they were somewhat autobiographical, and I feel like that is ending for me to some extent. I don't, I don't know if I chose for it to end without admitting it or if it ended and I was forced for it to end, but I feel like I'm being forced to choose something new like, like I can't go on the way I was going on, even if I wanted to. I'm hoping there is something new that I'm allowed to choose. I'm not sure yet.

[00:05:39] Chris Duffy: Does it feel like it was a choice in the first place? Like, did you choose cartoons and drawing as your medium?

[00:05:45] Liana Finck: I chose drawing. I drew as a little kid and I was always kind of praised for it. And I think as a sad teenager, I wanted to be a writer, but like, I used the tiny shred of canniness that I had to realize that I had a better chance at getting into art school than getting into school to write.

Um, because I had nothing to show for writing, so I, I like chose, I chose drawing, even though it was something that I had done as a younger and not sad kid. And I'm, I think I'm glad I did, although graphic novels are kind of cumbersome.

[00:06:24] Chris Duffy: Yeah, it's also interesting because the drawings in your cartoons, I think sometimes people get really hung up on the idea that everything has to be, like, photorealistic, and you have to draw, like, a hand that perfectly resembles the photo of a hand and, and—

[00:06:37] Liana Finck: Yeah.

[00:06:37] Chris Duffy: It’s clearly not what you're going for when you draw your cartoons.

[00:06:40] Liana Finck: No. Yeah, I'm against that ideologically, but I think also I can't do it, so it's lucky that I don't want to.

[00:06:49] Chris Duffy: That's my favorite kind of ideological stances.

[00:06:51] Liana Finck: Yeah.

[00:06:51] Chris Duffy: I both technically can't do this, and I am ideologically opposed to it.

[00:06:55] Liana Finck: It's a little self serving. But also, who has the time to draw like that? Why? Why would you draw like that?

[00:07:03] Chris Duffy: Like, because I think for a lot of people, like me included, I think if you draw something that doesn't look, I'm just going to put this in quotes, right, “perfect,” it feels like, well, I'm not a good artist.

[00:07:12] Liana Finck: Yeah. I really disagree. So, I don't remember actually, if I read the new Temple Grandin book about different ways of thinking, but I do know maybe from reading about the book or maybe from reading it without realizing I was reading it, that she puts visual thinking in two categories.

One category is, like, people who think in very involved photorealistic pictures, and she puts herself in that category. And then she has another visual category of people who think in symbols and kind of boil that, like they're like idea people, but they put visuals to each idea, and, like, the simpler and more direct the better.

And I resonate; that, that category resonated with me. I don't know if the categories are real, but I like that. And, um, I think that's what I use drawing for. And that's a really good thing to use drawing for if that's your bent.

[00:08:09] Chris Duffy: It also makes your work feel approachable, I think? Because there's this sense of, like, people can relate to not just the ideas that you're conveying, but the way that you're conveying it also feels approachable. I don't know if that’s, that's something you consciously think about or not.

[00:08:27] Liana Finck: I think consciously about drawing the thing you want to get across and not cluttering up your drawing with things that you're not caring much about. So, like, I care a lot about emotion, so if I'm using body language or facial expression or some kind of symbol to convey the emotion or the point that I'm trying to get across, that's all I need for my drawing.

I think, though, there’s a lot to be said for using drawing to explore a little bit. And I think you need diff—a slightly different mindset for that. You need to be able to just kind of chill and not know where you're going with something, which has never been good for me. But, um, 'cause I freak out and I overwork things. But, um, I've been reading Goodnight Moon several times a day for the past year.

[00:09:23] Chris Duffy: Uh huh.

[00:09:24] Liana Finck: There’s just more and more details that you see in that book every time you read, every time you look at it. And, and that's like a very novelistic way to draw, just where, like, there's all these details and the reader can learn more and invent more every time they see them.

So I, I think neither of these is a photorealistic thing. I just think of photorealistic drawing as, like, really different. It's more like some kind of engineering or something where you’re, like, crafting something. And it's not, it's very foreign to me. I admire it, but it's not, to me, you don't need that to get ideas across.

[00:10:06] Chris Duffy: So, if someone is listening, and they have this sense, like, “It would be fun to try to make cartoons. It would be fun to draw. It would be fun to express myself in that way.” What are three things that they should keep in mind as they get started? What would you tell someone who's just getting started, even if they were like a kid?

[00:10:28] Liana Finck: Okay. I think there are a lot of different ways to do this, but the ways that I do are one, when I have something that I want to get across, and two, when I just want to make a cartoon and I don't have anything in mind. So for the first way, when there's something that's just eating you up and you need to solve it. And, and you want to maybe solve it by drawing. It could be a drawing, or it could be words. If there's a problem you're trying to solve in your life, to tell yourself, “I'm going to solve this by summing it up and understanding it and how can I diagram it for myself.” And, I used to draw them the minute I had those ideas, but lately, I put them in words just because it's more convenient for me, so I email myself an idea when there's something I'm trying to figure out, and I'm like, “Why is this stressing me out?”And then I'll think about why, and I'll, I'll write it down in a line, and maybe I'll turn it into a drawing later.

The other way that I make cartoons is when you want to be a cartoonist, so you have to make cartoons. Um, I like to be somewhere where I don't, where I feel like I can just sit still and doodle.

And in my case, I love to sit in a cafe or on a train or somewhere where people are just allowed to sit and kind of observe the scenery without getting out of people's way all the time. And then I'll just doodle. And, like, maybe after an hour or two of doodling, something will happen and the ideas will start to make sense.

Um, and then those ideas that make sense, they, it takes a lot more, like, pushing and pulling before it's actually a good cartoon. But I like to do the ideas all, all in one day and do the pushing and pulling another day.

[00:12:18] Chris Duffy: I love that. I’ve had I think about the idea first and then try and sum up the this thing that is troubling you in your life or that you're struggling with trying to sum that up in its simplest form. It, by the, by the drawing, those feel like there's so much more doable than when someone is like, “Start by drawing three orbs and then connect the orbs with an oval. And eventually you'll have a horse.”

How would you describe your visual style? Do you have like a, a words that you use to describe it?

[00:12:48] Liana Finck: I have a few different styles, but lately they all, they're all really the same style. It's just a question of whether I magnify it before presenting it or not. My more complicated and illustration-y drawings are not magnified, so they're drawn at the same scale that someone would see them. And my New Yorker drawings and my graphic novel drawings are usually like that.

But my Instagram drawings, the, um, really essential, really, really simple drawings, I draw them like let's say one to two inches tall, and then I magnify them either by photographing them up close with my phone or by scanning them in and blowing them up so it looks like they're drawn much more confidently than they are. It looks like they're drawn with, like, a big, a thick line, but they're actually drawn with this, like, thin pen, or like a Muji gel pen.

[00:13:46] Chris Duffy: Nice.

[00:13:46] Liana Finck: But everyone is different. I don't feel like pen recommendations really transfer from person to person.

[00:13:53] Chris Duffy: Yeah, it also feels like, um, I feel like it's kind of counter to the whole thing that we've been talking about, which is the idea that, like, if you get the perfect tool, then you'll be the perfect artist.

[00:14:02] Liana Finck: Yeah.

[00:14:02] Chris Duffy: As opposed to just, what do you like and what is the thing you're trying to express?

[00:14:06] Liana Finck: Yeah.

[00:14:06] Chris Duffy: And that is, the closer you can get to that is the closer you get to perfect.

[00:14:09] Liana Finck: Yeah, I think so. I think if there's something that makes you feel like you're speaking in your normal voice in a way that people can understand you, but with drawing, then that's a good tool for you.

[00:14:27] Chris Duffy: Okay. Well, we're going to take a quick break for me to speak in my not normal voice. Instead, I'm going to switch over to my ad voice and we will be right back with more from Liana after that.

[00:14:43] Chris Duffy: And we are back. I want to go back to something you said earlier, which is that if there's something that you want to explore through drawing, you send yourself an email. I'm always really interested in how people, especially people who are doing work that is comedy or, or funny, which many of your cartoons are, although some of your work is really serious too.

But as a comedian myself too, one of the things that I, I tell people when they ask, like, “Oh, I want to do this. How do you do it?” is that the biggest thing is actually, like, keeping track of the things that make you laugh or that you think are funny. ‘Cause—

[00:15:16] Liana Finck: Yeah.

[00:15:16] Chris Duffy: You think that they'll stick with you, and they just disappear. Like, I have a, a notes app on my phone that I just, like, have a running doc and I always start a new one, and it gets too long because just in the course of a day, there'll be, you know, 30 things that I, are the seed of something. And when I look back, most of them are worthless and don't even make sense, but a few of them are. So is that how you keep track of your own ideas or the little seeds? You send an email to yourself?

[00:15:40] Liana Finck: Yeah, I send an email to myself. I, I’m experimenting with having a separate email address just for ideas so that I don't like have to see—

[00:15:48] Chris Duffy: Oh.

[00:15:50] Liana Finck: And what emails I have to write back to.

[00:15:50] Chris Duffy: I love that. I’m, I'm so tempted to ask what that email address is, but that would be a horrible thing to do 'cause then people start emailing you ideas to that email address.

[00:15:57] Liana Finck: That's so funny. Yeah, I've never, I've actually never told anyone that I ha—even have this email address,

[00:16:04] Chris Duffy: But I think that the important part is that it's like something that is, like, clean and separate, where it's like you go in and it's just ideas. It feels like it is a way of keeping it almost like sacred, like that sacred space for just the ideas and the possibilities.

[00:16:18] Liana Finck: Do you have seasons when you have a lot of ideas and then seasons when maybe you're too busy and you don't?

[00:16:23] Chris Duffy: Absolutely.

[00:16:23] Liana Finck: And like, how on earth do you save your ideas for that, the time when you need ideas?

[00:16:32] Chris Duffy: Oh yeah, well I actually feel like right now I am in one of those seasons where I have fewer ideas and I'm just trying to execute, and so these days when I'm like I have, should find something funny to put up online, I’m, like, scrolling back so far and it—

[00:16:46] Leana Finck: Yeah. Yeah.

[00:16:46] Chris Duffy: It’s not a good feeling to me.

[00:16:48] Liana Finck: No, it’s not.

[00:16:48] Chris Duffy: I’m like, "Oh my god, I'm not… I’m not creative at all. I have to go years back before I find something good that I haven't put out yet. And the reason I didn't put it out then is because it wasn't good enough. And, but now I think it is.”

[00:16:58] Liana Finck: It's really confusing for me. I wish I had, like, I could find a math person to, like, really diagram it.

[00:17:06] Chris Duffy: Yeah. What type of season do you think you're in right now?

[00:17:09] Liana Finck: Eugh. I, I just took, like, you caught me at the end of I think the longest period I've ever taken without doing any work because I just, I have a residency in Berlin and I have a little kid, and I'm alone here with him. And so I just spent one week being a tourist with my husband and kid, and then one week getting my kid acclimated to his daycare here, which is this whole process in Berlin that usually takes eight weeks. But they fast-tracked me, and then one week of, like, a lot of formal meetings at the residency. They’re really nice. But also it's been too, too many reasons to not work.

[00:17:52] Chris Duffy: Yeah.

[00:17:52] Liana Finck: So, I can’t, and I have all these different streams of work that I'm supposed to be doing. Like, technically I'm here to be working on a book that isn't serious, but I'm pretending it's serious. It's supposed to be like a funny, a funny, light book, but also I make cartoons every week. So I really want to have ideas for my cartoons every week. And those aren't coming. I wish they would come. So I'm not, I'm in a no phase right now. There's nothing happening.

[00:18:22] Chris Duffy: How do you think of the balance between, like, finding that perfect inspiration and using it for something versus making the inspiration come ‘cause it's that day of the week where the inspiration has to come?

[00:18:36] Liana Finck: I'm very much in the camp of making it come, but I have really, really mixed feelings about making people read it.

[00:18:44] Chris Duffy: Uh-huh.

[00:18:44] Liana Finch: I think I'm like, I'm pretty weak, and I don't think I would make anything unless I knew I had an audience. So, I end up, like, roping ra—nice people into looking at my stuff when it's not good.

And I feel so bad about that, and it kind of spirals into badness. And I also, like, feel that I, I owe The New Yorker a weekly batch of cartoons. And, so I feel less guilty about sending in bad cartoons every week because I know they're asked, they, they like that I send things in every week, and if they're bad, at least I made them.

But then, if they don't buy a cartoon because they were bad, that makes me spiral also because I'm like, “I'm worthless. I was supposed to send them good cartoons and they were bad and everyone knows.” And yeah, a lot of mixture of guilt and shame.

[00:19:39] Chris Duffy: And I think it feels like sometimes when you get to the end of the line and you don't have the cool, impressive idea and you just have to be honest about where you are, those are the most authentic ones.

Um, you wrote one a few weeks ago that literally is just called “If I'm Being Honest” and it's like, “I've been feeling really blank lately,” and then you have, you don't even have the word. You’re like, “I can't quite figure it out. Is it empty? Is it closed? Is it dense, internal, not verbal?”

[00:20:06] Liana Finck: Yeah.

[00:20:06] Chris Duffy: And you talk about how when you feel like that, it's really hard to come up with anything to draw. And so this is, it's kind of a meta thing where you're sending out a drawing that is of how hard it is to—I couldn't relate to this more. This is so funny, but also so clearly you being honest about, like, sitting in front of the computer and going like, “What am I going to put out? How do you get yourself over that hump of like, “I'm going to put out a cartoon that's about not knowing what to write in a cartoon”?

[00:20:34] Liana Finck: For me, the hump with that kind of cartoon is not, like, admitting failure, but it is this fear of repeating myself. I'm pretty, n—I've been, that, the one you described was on my Substack newsletter. So it's like a three-page car comic, let's say, as opposed to a single panel cartoon. And I find, I think one can really repeat oneself with single panel cartoons, but it takes a while. Like, I feel like I'm just starting to, like, hit that wall after making cartoons for over 10 years, but with the longer comics, you get to kind of explore a feeling you have. And I don't know about you, but I think I have about 10 feelings ever.

[00:21:22] Chris Duffy: Yeah. Yeah.

[00:21:22] Liana Finck: So I'm like, I think, like, I'm sure I've made that cartoon, that comic several times already because it's a strong feeling; it makes for a good comic. But like, that, like that kind of, kind of essential feeling comic gets really repetitive if you're making a new comic every week And I've been making these for, like, a year, I think, and I'm already really repeating myself. So that's really interesting, and that's an argument for making like an even longer comic because then you can kind of sink into one idea instead of pretending that you have a new idea each week.

[00:21:58] Chris Duffy: It's interesting because even that sense of repeating yourself is also a really relatable idea. I… Even someone who's not an artist in any way.

[00:22:05] Liana Finck: I think in therapy, like, when you, or with a diary or something, it's really freeing to realize that the problem you have is the same problem you've always had. And then, like, recognizing that can help you just sidestep it. But if the problem is your art, like, what are you going to do? Like, sidestep, what does sidestepping it mean?

[00:22:24] Chris Duffy: But then also, yeah, if you put the pressure on the art of to always have a revelation, right?

[00:22:32] Liana Finck: Yeah.

[00:22:32] Chris Duffy: Like sometimes it's not necessarily going to be revelatory, it's just going to be a drawing.

[00:22:36] Liana Finck: Yeah, I do feel a little bit like I could be at the point where I just stop drawing because I've solved the problems that I wanted to solve through drawing.

[00:22:43] Chris Duffy: And what, what was the problem that you were originally trying to solve?

[00:22:46] Liana Finck: One was social anxiety, just like figuring out the, I don't know, the hundred forms of social anxiety that torment me. And then one other thing had to do with, like, gender stuff that I think I've had, like, a thread of, like, ra—like, feminist rage in me that has… I, I’d like to make a longer project exploring this, but I think, um, I think it has a few aspects.

One is, um, like being, it, like feeling expected to live for others and then one is eating issues. Like I had, I was kind of ruled by eating stuff for a decade there, and I think I'm still dealing with it. So, I talk about it like it's universal. I think it's pretty personal. One was dating. Like, how are you supposed to have a partnership with another person that you love?

Like, if the history of such partnerships are really, really unequal, like, if you're in a heterosexual relationship, it wasn't working smoothly for me, and I was trying to figure it out. And I'm not saying I figured any of these things out, but, like, I think I outlined the problems for myself to the point where I don't need to keep outlining them.

I think, like, comedy and honesty probably come from the same source. But like, okay, I think here's a really wonky theory that, um, comedy is a, maybe all comedians actually know this, and it's boring, but that comedy is a one-two punch, and honesty is a one punch.

I always think of, like, a New Yorker cartoon with a picture that's a setup, and then the caption that's usually the punchline as a one two punch, but then when you just make a single, simple drawing, it's just honesty, it's not funny, because it's all, it’s all there, and there's no saying one thing and then shifting it.

[00:24:54] Chris Duffy: We talked about, like, how sometimes, um, sometimes drawings will come to you from like an idea or an emotion that you're trying to express or, or something that you're processing. You put a cartoon out that has, it's basically a, a chart or, no, a graph and on the—I’m really pushing my math understanding to the test—but on the Y axis is rage.

[00:25:18] Liana Finck: Oh. Mm-hm.

[00:25:18] Chris Duffy: And on the x axis is age, and you have it going down. Rage is decreasing with age.

[00:25:24] Liana Finck: Yeah. So I've been told that I do all my axes backwards, so... Perhaps it's, it should be age on the Y and rage on the X, but I can't fathom why.

[00:25:33] Chris Duffy: You are certainly asking the wrong person for that one, but it seems right to me. I think time is supposed to go on the, on the X. Yeah, that seems right to me.

[00:25:42] Liana Finck: Yeah.

[00:25:42] Chris Duffy: Do you feel like your rage has been decreasing just as a natural product of getting older or also as, like, moving and all of those other things?

[00:25:49] Liana Finck: I know it's really different for everyone, but I think I had some things that I wasn't dealing with that were bad, and then I had a crisis and started dealing with them.

So I had this like, like glowing pinnacle of clarity and rage for a while, and that's where my Instagram drawings came from. And I think I've worked through that thing and it's, I think I'm seeing a bit more broadly, but a bit less, like, arrow-sharp. So for me, rage has decreased with age. Maybe it'll be back. Like, there's lots and lots to be enraged about.

[00:26:26] Chris Duffy: Yeah, it makes sense too, right? I think sometimes the more that you know about life and situations, the more rage there is to have, but—

[00:26:33] Liana Finck: I think sadness has increased. So it's not like I'm like just very Pollyanna. But yeah, I think I feel more helpless about the things that I'm angry about right now.

[00:26:44] Chris Duffy: Yeah. Well, we crossed paths at the TED conference, and at the, the in-person TED conference, which I don't know about for you, but for me is always like, uh, I've only been twice, but it was extremely, it's extremely, like, intimidating and overwhelming.

[00:27:02] Liana Finck: Yeah.

[00:27:02] Chris Duffy: Because I checked into my hotel and genuinely, this was the line checking into the hotel. The person in front of me is an astronaut and the person behind me is—

[00:27:10] Liana Finck: Where they rang the thing?

[00:27:12] Chris Duffy: No, he was not wearing his bubble suit, but I, someone said like, “Oh, that's an astronaut.” No like some, like they were like pointing it out. And then the person behind me was Esther Perel, the famous therapist.

[00:27:23] Liana Finck: Yeah. I love her.

[00:27:23] Chris Duffy: And then I'm in the middle and I, you know, it's like impossible to be in that situation and not be like, “Why am I here?” And the thing that I'm, I guess I'm asking is like, you sometimes end up in situations where it's like everyone else kind of is, feels big, and some of the power of your work is like admitting that you feel small and not confident.

[00:27:52] Liana Finck: Yeah, totally.

[00:27:52] Chris Duffy: And I wonder how you navigate the like, the disconnect between, like, being in a place where it's like everyone is impressive and confident and then you're like, but that's kind of not my thing.

[00:28:03] Liana Finck: There's not much of a precedence for it for be, for, like, maintaining a career on speaking a lot, like saying a lot of things in kind of a small, normal voice. I want to outline the problems and the benefits. I think together, I'm going to add something to the benefits category. And that is that I don't have too much, like, imposter syndrome because I've, I take pride in being an imposter.

[00:28:32] Chris Duffy: Mm. Tell me more about that. How do you take pride in that?

[00:28:33] Liana Finck: Like when people tell me I draw badly, I’m like, “What kind of a bougie are you who wants to, like, who cares about drawing?That's so weird.” Like, like, it, it, it wouldn’t even cross my mind to try to draw better. And like, be—I think because of that, I believe that I could draw really, really well if I wanted to, which isn't true.

[00:28:56] Chris Duffy: Uh-huh.

[00:28:56] Liana Finck: Yeah. I think you could be burdened by too much, by too much skill in something. Although I really wish, like… I don't have an MFA in writing, I have no idea how to structure a longer piece, and I do wish I had a little more knowledge without all of it, but I don't know how to ask for it.

[00:29:16] Chris Duffy: You have a cartoon that, that's kind of got me thinking about that, that, um, idea of like being in spaces where people are really impressive, which is it's, uh, it's two people talking and one says to the other, “Why aren't you bragging? Are you some kind of loser?” That's really relatable to me, and I bet you it's relatable to a lot of other people is the idea that, like, if you're not bragging about yourself, that you might as well, like, it means that there's nothing to brag about.

[00:29:44] Liana Finck: Yeah.

[00:29:44] Chris Duffy: Or that like the only way to be out in the world is to be like loud and impressive and big. How do you find, how do you balance those? Or, how do you find the spaces where you don't have to feel like you're a loser for not bragging? Or what do you do when you feel like you have to brag?

[00:29:56] Liana Finck: I'm really bad at bragging. I was really bad at dating. That was part of why I was bad at dating is that I think on a first date, people just inher—or even on their, like, dating profile, people inherently want to show their best side. And I inherently want to show my worst side.

[00:30:13] Chris Duffy: Uh-huh.

[00:30:13] Liana Finck: Because I think it’s, like, more connect-y, but it does not work. And I did feel that at the TED conference, I don't present myself well. And, like, yeah, my husband's a good photographer, and he is good at being photographed, and I find, like, when I compare myself to him, I find that I'm absolutely the opposite.

And I almost hope to look terrible and to photograph things so that they look terrible. And it's some kind of defense mechanism. And don't look at me. I look terrible. So don't even judge me. Like, don't even compare me to other people.

[00:30:49] Chris Duffy: Mm. I think there's the defense mechanism part that makes total sense to me. We've talked a lot about your cartoons that have been about, like, social situations or emotions, but you also have done a lot of work that is about faith. And your most recent book, Let There Be Light, centers around, uh, a God that's a woman, but also an artist with self doubt. And you talked a lot about that. I love that.

Um, and I just, I guess I'd wonder how the idea of believing in something bigger than yourself and struggling with that belief, how does that tie into your work and into things that make you laugh?

[00:31:28] Liana Finck: Ooh, that's a really good question. I've never believed in God. I'm of the Jewish sector that I don't, I don't even know how to pinpoint it. I think it's like a specific part of Eastern Europe that we come from, or at least that like some of our Jewish souls come from, um, where it's not an issue. And like, we don't talk about that. It's like in bad taste to like even consider God, but, um, we believe in custom, and we talk about God, but we don’t think about whether we believe in them or not, but, um, it was important to me to not believe in a female god rather than a male god.

[00:32:09] Chris Duffy: My mom is Jewish, so I have a, an Eastern European Jewish background as well. And, um, my wife was like, got really interested in Judaism. And so she took like a Judaism 101 class, which I've never seen my mom and aunt happier in their lives. Two Jewish women being like, “Oh, she's going to learn about Judaism.”

[00:32:24] Liana Finck: Yeah.

[00:32:24] Chris Duffy: And something that the rabbi said in the Zoom class that I just, like, happened to walk by as he goes, “In Judaism, we believe there is at most one God.” And I was like, I love that that's what the rabbi said.

[00:32:35] Liana Finck: That is so funny.

[00:32:35] Chris Duffy: He’s like, “Look, there's definitely not two. There might not be one, but there's certainly no more than one.”

[00:32:42] Liana Finck: That's amazing.

[00:32:44] Chris Duffy: Yeah.

[00:32:44] Liana Finck: I want to take that class.

[00:32:46] Chris Duffy: I love the idea of it being important to you to not believe in a female god. That's fantastic. That's a, that’s a perfect sentence.

[00:32:51] Liana Finck: I think there have been moments in my life where I believe in, like, art and eternity and things, and I think I'm not in one of those moments, for better or for worse. I think when one is, like, sad, one is more likely to believe in something m-more mystical. But, yeah, it does affect your art when you're believing in something.

[00:33:15] Chris Duffy: Well, Liana, it's been such a pleasure talking to you. Thank you so much for making the time to be on the show. I really appreciate it.

[00:33:19] Liana Finck: Thank you so much. This has been an honor and a huge joy.

[00:33:26] Chris Duffy: That is it for today's episode of How to Be a Better Human. Thank you so much to today's guest, Liana Finck. I am your host, Chris Duffy, and you can find more from me, including my weekly newsletter and upcoming live shows, at chrisduffycomedy.com.

How to Be a Better Human is brought to you on the TED side by Daniella Balarezo, Cloe Shasha Brooks, and Joseph DeBrine, who are, all kidding aside, genuinely each creative and unique artists in their own right. This episode was fact checked by Julia Dickerson and Matheus Salles, who frequently reject my excuses that I am not going for, quote, “hyper-realistic audio facts”, but instead trying to make impressionistic line drawings for the ears. They do not accept that, and neither should they.

On the PRX side, our show is put together by a team that is almost mystical in their God-like powers to create podcasts. Morgan Flannery, Noor Gill, Patrick grants, and Jocelyn Gonzales.

And of course, thanks to you for listening to our show and making this all possible. We could not do it without you. If you are listening on Apple, please leave us a five star rating and review.

And if you're listening on the Spotify app, answer the discussion question that we've put up there on mobile. I love to see your answers. We are always so excited to see what you have to say. So please put up your thoughts there. We will be back next week with even more How to Be a Better Human.

Fun teaching resources & tips to help you teach math with confidence

Math Strategies: Problem Solving by Drawing a Picture

I am a very visual learner . Whenever I am facing a word problem of any kind, my initial reaction is to draw a picture. Even if it is a fairly simple problem and I think I already know how to solve it (or even already know the answer), I will almost always still draw a picture . but even if you don’t think of yourself as a “visual learner,” drawing pictures (or other visuals) is still one of the most powerful strategies. Brain research shows that when solving math problems, the ‘visual’ parts of our brain light up, even when we don’t draw a picture! So rest assured, problem solving by drawing a picture is helpful for any student!

Things can become so much clearer when you have a visual! I love this helpful post for teaching problem solving by drawing a picture!

–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Solve Math Problems by Drawing a Picture:

Maybe I’m drawn to this strategy because I’m such a great artist…no, that’s definitely not it! I believe it is because seeing a visual representation of the problem can put things in perspective, help organize the information, and enable students to make connections that may not have been otherwise seen.

Because of the impact of visuals on our brain and our learning, this is an important and helpful problem solving strategy . Especially if you are stuck and don’t know where to go or what to do. Then you have nothing to lose, right?

When I was teaching high school, I would often encourage students to draw a picture when working on distance/rate/time problems .

It is very easy to get bogged down in all the details and numbers, especially if the problem includes unnecessary information (details that you don’t really need to know in order to solve it). Wading through everything you’re given and making sense of what’s important can be easier when you draw a picture!

It’s also incredibly important to draw a picture when working on geometry tasks, such as Pythagorean theorem problems or similar triangles and indirect measurement . Even if you know how to solve it without a picture, you will greatly increase you chances of a careless mistake if you don’t take the extra five seconds to draw a picture.

One important thing to remember, however, is that the picture does not need to be pretty . In fact, in some cases it may not even be a picture, just a visual representation of the information.

And that’s ok! The point is to help you solve the math problem , not to win an art award. ( Thank goodness, because seriously, I’m no artist! ).

If you would like to discuss this strategy with your students and help encourage them to use it when appropriate, I’ve created a short set of problems to do just that!

These word problems could be used with grades 2-4 and include a page that specifically states, “Draw a picture…” and then another page of problems were it would be useful to draw a picture, but it is not explicitly stated.

The goal is to get students used to organizing the information in a meaningful way to help them better think about and/or solve the problem.

{Click HERE to go to my shop and download the Problem Solving by Drawing a Picture Practice Problems !}

What do you think? Do you use this problem solving strategy or encourage your students to try it? Do you think it’s helpful?

Here are the other articles in this series on problem solving:

Problem Solve using Guess and Check
Problem Solve by Finding a Pattern
Problem Solve by Making a List
Problem Solve by Solving an Easier Problem
Problem Solve by Working Backwards

Thanks so much for your Math freebie. Drawing pictures is a great way to access student understanding. Arlene LMN Tree

Thanks Arlene! Yes, I agree! Students have to show what they know to be able to draw an appropriate picture and solve. Thanks for stopping by! 🙂

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Problem Solving: Draw a Picture

Problem-solving is a critical 21st Century and social-emotional skill

Looking for more resources on 21st Century skills and social-emotional learning? Find them in our FutureFit resources center .

What Is It?

The draw a picture strategy is a problem-solving technique in which students make a visual representation of the problem. For example, the following problem could be solved by drawing a picture:

A frog is at the bottom of a 10-meter well. Each day he climbs up 3 meters. Each night he slides down 1 meter. On what day will he reach the top of the well and escape?

Why Is It Important?

Drawing a diagram or other type of visual representation is often a good starting point for solving all kinds of word problems . It is an intermediate step between language-as-text and the symbolic language of mathematics. By representing units of measurement and other objects visually, students can begin to think about the problem mathematically . Pictures and diagrams are also good ways of describing solutions to problems; therefore they are an important part of mathematical communication.

How Can You Make It Happen?

Encourage students to draw pictures of problems at the very beginning of their mathematical education. Promote and reinforce the strategy at all subsequent stages. Most students will naturally draw pictures if given the slightest encouragement.

Introduce a problem to students that will require them to draw a picture to solve. For example:

Marah is putting up a tent for a family reunion. The tent is 16 feet by 5 feet. Each 4-foot section of tent needs a post except the sides that are 5 feet. How many posts will she need?

Demonstrate that the first step to solving the problem is understanding it. This involves finding the key pieces of information needed to figure out the answer. This may require students reading the problem several times or putting the problem into their own words.

16 feet by 5 feet 1 post every 4 feet, including 1 at each corner No posts on the short sides

Choose a Strategy

Most often, students use the draw a picture strategy to solve problems involving space or organization, but it can be applied to almost all math problems. Also students use this strategy when working with new concepts such as equivalent fractions or the basic operations of multiplication and division.

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TeacherVision Editorial Staff

The TeacherVision editorial team is comprised of teachers, experts, and content professionals dedicated to bringing you the most accurate and relevant information in the teaching space.

Problem Solving Strategy: Draw a Picture

Make problem solving easier for students be teaching multiple strategies . Here is an explanation of how and why to encourage students to draw pictures for solving math word problems . Teaching students to draw pictures illustrating the details has many benefits.

Draw Pictures Strategy – Problem Solving

No matter what grade, drawing pictures can make solving problems a go-to strategy. This strategy can generally help students improve their accuracy, draw on their strengths as visual learners and determine importance in details. Read about how differing grades can benefit from this strategy and download free printable problems that encourage practicing this math strategy for second grade, third grade and fourth grade.

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Solving Word Problems – Strategy Posters for Kids
Problem Solving Printables for “Math Curse”
Math Problems Made Real: Kids Lemonade Stand

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Leslie {aka the original Teach Junkie} loves learning new things to make teaching easier and more effective. She enjoys featuring creative classroom fun when she's not designing teacher shirts, making kindergarten lesson plans or planning her family's next trip to Disney World.

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Can art make us better problem solvers?

In the search for novel ways to hone our problem-solving skills , spending time with a work of art may be the simplest and most effective training, according to the art historian Amy Herman.

Herman has been teaching professionals —homicide detectives, medical students, lawyers, and engineers—to read paintings as a way to improve their analytical faculties. “Art provides a safe space outside of ourselves to analyze our observations and convert those observable details into actionable knowledge,” Herman writes in the introduction to her new book, Fixed. How to Perfect the Fine Art of Problem-Solving .

Doing so can help us understand how and why things go wrong and, more importantly, how to fix them, she explains.

Putting the lesson into practice

In her book, Herman explains how to navigate a complex composition, step by step.

Consider Théodore Géricault’s grisly painting, The Raft of the Medusa .

Herman writes:

Take in its scope, notice its details, count things, catalog what you think might be going on. Then, take a breath and let your mind wander. What did the chaos of the preceding scene bring to mind? A natural disaster? A human-made catastrophe? The current state of your country? Maybe you were reminded of more personal scenarios: office drama, an argument at home that got out of hand, Zoom Thanksgiving.

No matter who you are or where you live, chances are you can relate to the desperation depicted above.

A crucial skill in Herman’s approach is the art of noticing—the ability to quell the impulse to pick up our mobile devices and to pause long enough to ruminate on the details of a visual spectacle before us. This is particularly salient in the age of short attention spans , where the average museum-goer spends less than 30 seconds looking at a work of art .

Looking at art also attunes us to nuance and ambiguity, explains Herman. It’s a skill that’s critical for hostage negotiators to managers trying to read the room.

“The optimal way to look at art, whether alone or with others, is to look at the object first, speak after looking, and only then, read the label,” Herman tells Quartz. “My hope is that by learning to look at art in a structured way inspires and refreshes critical inquiry and that same model will be applied to when confronting problems in need of solutions.”

Herman, who once led the education department at the Frick Collection in New York City, insists that there’s no shame in “using art to study ourselves and the problems we face every single day.” “Art can be many different things to many different viewers,” she argues. “If the power of a work of art can be channeled to enable a viewer who does not have a background in art or art history to approach their vocation in a different and more expansive way, why not unleash that?”

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Problem Solving Series: Introduction to Model Drawings

Oct 13, 2017

When students are looking at story problems, they’ll typically do one of two things.

They might scan through the problem, circle the numbers and underline the key words like all together or difference . If the class is working on adding that week, then they will add the numbers, if they’re subtracting that week, they’ll subtract the numbers. In this situation, students don’t know mathematically what the problem is actually asking them so they do a lot of guessing and checking.

In another scenario, students might just give up or appeal to a teacher for help. They just wait for someone to help them or just give up because it’s too complex. Honestly, who can blame them? Sometimes when you’re reading a story problem, there are so many words they just start swimming in front of you. Today, kids need a lot of help with comprehension of math problems, with understanding what a problem is actually asking them to do. The words of the problem circle in their heads and they don’t have a visual reference for understanding what the problem is asking.

We want to make sure that we don’t teach conditions in math, for example, you only add if you see all together . Some kids will see those key words, and just automatically add whatever digits they see in the story problem. But that doesn’t always work. Here’s an example:

The child might circle the numbers 5 and 3, see the key words all together , and think “oh! That’s on my T-chart under the plus!” so then they just add and get 8, which is incorrect. In reality, this is an additive/comparison problem where Shannon has 5, but Kathleen also has 5, as well as an additional 3. The child should add 5 + 5 + 3 to get 13 as the correct answer. So many times, students don’t completely understand what the problem is asking because they’ve been conditioned to circle numbers and underline keywords.

Even in first grade, students are asked to be doing additive/comparison and a variety of part/whole problems. By the time students get to 3rd and 4th grade, they’re doing multiplicative/comparison problems and many multi-step problems, so their key word clues aren’t going to help.

So how are we actually teaching problem solving? Are we doing it in a systematic way that can help students independently apply these concepts on their own without a teacher present helping them, like in a testing situation? Twenty-five percent of the Next Generation Assessment will be problem solving. In 21st century math, a child will encounter a lot of application problems that will require them to know how to approach the problem without any help.

Enter: model drawing. Model drawing goes by many names: some schools call it bar models, some call it tape diagrams, and some call it action model drawings. I prefer to call it model drawing because it’s pretty universal to any kind of model you could make instead of being locked into one type of diagram that really isn’t a bar. No matter what you call it, model drawings are amazing! They were adopted from a Singaporian problem solving approach for students, beginning with proportional models to look at individual squares or units being represented, and then going throughout first grade, and taking the idea and bringing it into a actual non-proportional manipulative like a unit bar.

There are five different types of model drawings in elementary school. Over the next five blog posts, we will break down each of the type of problems to help you give a model for what they would look like. In middle school, we start to add in model drawings using ratios and proportions as well.

Resources for learning how to incorporate model drawings:

Classroom poster. You will definitely need a step-by-step model drawing poster . This is a great poster that should be hung up in every classroom. Our project schools are required to hang one in every classroom from 1st grade up so that students see a systematic way that we do problem solving, not just in their class, but throughout the building. When students are in second grade and they’ve learned problem solving this way, it’s a quick connection to see how they’ll be problem solving in 3rd grade too. Same set of steps, different content. The idea of model drawing actually grows with students from first grade all the way through eighth grade, so you can use it as a sustained idea.

Reference Books. Step by Step Model Drawing and More Challenging Model Drawings are two great resources to support your journey into problem solving as you learn to do model drawings. Both authors do a great job of really teaching you step by step, how to get model drawings integrated into your class. These stand-alone books are intended to help you get these kinds of problems into your classroom, while the others are more of a resource for using with students.
Resource Books. We need to become good at model drawing ourselves. Teachers are generally book-followers, we go by the directions it gives us in the books. Sometimes, however, the book doesn’t really give us a systematic way to handle story problems. Many of our teachers have to teach themselves how to do model drawings. A great way to do this at the beginning is to use our Word Problems books (one for each grade level, 1-6). We typically recommend the book for the teacher’s grade level and the one below, so if you were teaching 3rd grade, you’d get the 3rd grade book, and 2nd grade book, partly because the students are going to be as new to model drawings as you are. So, if I’m working on additive/comparison problems, I might grab a few problems from the 2nd grade book, then go into the 3rd grade book. Each book has approximately 180 story problems and is really great for kids to use and for teachers to get started.

Manipulatives

What manipulatives go with model drawings?

Let’s revisit our story problem:

Cuisenaire Rods are really great for part/whole and additive/comparison problems. On the proportional side, a student could grab a Cuisenaire rod of five to represent what Shannon had, and the same color could be used to represent Kathleen’s five pieces. But then they could grab an additional three to tack onto the end of Kathleen’s five. We particularly like the connecting Cuisenaire rods because they allow kids to be able to interlock some of them, but in many schools, they have the block style Cuisenaire rods, and those will work just fine too.

The 1” square tiles are mainly used for multiplicative/comparison problems.

$problem solving drawings$

You’ll also need a Math Work Mat . Kids really need this Math Work Mat when they’re using the manipulatives because they can build their model on this space instead of in their actual journal paper. Many kids, when using manipulatives as well as pen/paper, will build their model on the paper they’re supposed to draw or put their computation on. You don’t have to use our official document, any piece of paper that students are conditioned to use as a space for building their model will work.

ThinkingBlocks.com (via the Math Playground) is another excellent resource that we like to use in our math workshop in the Math with Technology station. (We blogged about it here: Thinking Blocks ) If you have Chromebooks or laptops, or even iPads (with downloadable apps), you can use this program. There are several sections available in ThinkingBlocks.com – addition problems, part/whole problems, comparison bars, multiplication, proportions, etc. One of the problems we used to have with this tool was that many of the students we were using it with couldn’t actually read the problem. But Thinking Blocks just added a new feature where you can click on a speaker icon and hear the text of the story read to you, which is huge!

Setting up for Success

As you begin to implement model drawings in your class, we recommend using model drawing and the journal templates at the minimum of 2-3 times per week. Kids need consistent structure with understanding the comprehension strategy to help them attack problems. We have seen huge growth with kids that come to us with their problems.

The most important thing is for the teacher to be patient, it will take time to learn this new language because most of us did not learn this way. We did not have a pictorial representation or even think about using any kind of concrete objects to help us master harder story problems.

Ultimately we want you to feel comfortable enough with the different types of story problems that you could look at your math series and to know which type of problem you were facing and how to do the required diagram. Not every math series uses these types of problems, but it is my strong recommendation that, even if it doesn’t, you apply these five types of problems to it. Your math book isn’t your bible, it’s a resource for you to teach your curriculum. Your curriculum is your standards. So we want to be able to apply this approach to whatever we’re using. It’s challenging to be able to look at a story problem and just know that, “Oh! This is an additive/comparison problem, so I’m going to help my kids get set up for it this way.”

Model drawing does take a lot of modeling, and it also might take you some time to go a grade level back in order to break down the problems before you bring them up in a way that students can understand.

The next few weeks on our blog, we will spend some time diving deep into each of the five areas that you would see in a classroom and some different ways to introduce them into your classroom. We hope you’ll check out our resources as well!

To celebrate our new series, we would like to offer you 10% off of the items that support model drawing with the code: SISMODEL

Check out our Model Drawing category to see all the products in one place. We also have Problem Solving Kits that are already created and ready to go for each grade level! Just click on the kit, add it to your cart, and you’ll have everything you need!

*Curriculum , *Word Problems , Audience - K-5 , Series - Problem Solving | 0 comments

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Drawings of mathematical problems predict their resolution

Solving arithmetic problems, even simple subtractions, involves mental representations whose influence remains to be clarified. Visualizing these representations would enable us to better understand our reasoning and adapt our teaching methods. A team from the University of Geneva (UNIGE), in collaboration with CY Cergy Paris University (CYU) and University of Burgundy (uB), analyzed drawings made by children and adults when solving simple problems. The scientists found that, whatever the age of the participant, the most effective calculation strategies were associated with certain drawing typologies. These results, published in the journal Memory & Cognition , open up new perspectives for the teaching of mathematics.

Learning mathematics often involves small problems, linked to concrete everyday situations. For example, pupils have to add up quantities of flour to make a recipe or subtract sums of money to find out what's left in their wallets after shopping. They are thus led to translate statements into algorithmic procedures to find the solution. This translation of words into solving strategies involves a stage of mental representation of mathematical information, such as numbers or the arithmetic operation to be performed, and non-mathematical information, such as the context of the problem.

The cardinal or ordinal dimensions of problems

Having a clearer idea of these mental representations would enable a better understanding of the choice of calculation strategies. Scientists from UNIGE, CYU and uB conducted a study with 10-year-old children and adults, asking them to solve simple problems with the instruction to use as few calculation steps as possible. The participants were then asked to produce a drawing or diagram explaining their problem-solving strategy for each statement. The contexts of some problems called on the cardinal properties of numbers -- the quantity of elements in a set -- others on their ordinal properties -- their position in an ordered list.

The former involved marbles, fishes, or books, for example: ''Paul has 8 red marbles. He also has blue marbles. In total, Paul has 11 marbles. Jolene has as many blue marbles as Paul, and some green marbles. She has 2 green marbles less than Paul has red marbles. In total, how many marbles does Jolene have?''. The latter involved lengths or durations, for example: ''Sofia traveled for 8 hours. Her trip started during the day. Sofia arrived at 11. Fred leaves at the same time as Sofia. Fred's trip lasted 2 hours less than Sofia's. What time was it when Fred arrived?"

Both of the above problems share the same mathematical structure, and both can be solved by a long strategy in 3 steps: 11 -- 8 = 3; 8 -- 2 = 6; 6 + 3 = 9, but also in a single calculation: 11 -- 2 = 9, using a simple subtraction. However, the mental representations of these problems are very different, and the researchers wanted to determine whether the type of representations could predict the calculation strategy, in 1 or 3 steps, of those who solve them.

''Our hypothesis was that cardinal problems -- such as the one involving marbles -- would inspire cardinal drawings, i.e. diagrams with identical individual elements, such as crosses or circles, or with overlaps of elements in sets or subsets. Similarly, we assumed that ordinal problems -- such as the one mentioning travel times -- would lead to ordinal representations, i.e. diagrams with axes, graduations or intervals -- and that these ordinal drawings would reflect participants' representations and indicate that they would be more successful in identifying the one-step solution strategy,'' explains Hippolyte Gros, former post-doctoral fellow at UNIGE's Faculty of Psychology and Educational Sciences, associate professor at CYU, and first author of the study.

Identifying mental representations through drawings

These hypotheses were validated by analyzing the drawings of 52 adults and 59 children. ''We have shown that, irrespective of their experience -- since the same results were obtained in both children and adults -- the use of strategies by the participants depends on their representation of the problem, and that this is influenced by the non-mathematical information contained in the problem statement, as revealed by their drawings,'' says Emmanuel Sander, full professor at the UNIGE's Faculty of Psychology and Educational Sciences. ''Our study also shows that, even after years of experience in solving addition and subtraction, the difference between cardinal and ordinal problems remains very marked. The majority of participants were only able to solve problems of the second type in a single step''.

Improving mathematical learning through drawing analysis

The team also noted that drawings showing ordinal representations were more frequently associated with a one-step solution, even if the problem was cardinal. In other words, drawing with a scale or an axis is linked to the choice of the fastest calculation. "From a pedagogical point of view, this suggests that the presence of specific features in a student's drawing may or may not indicate that his or her representation of the problem is the most efficient one for meeting the instructions -- in this case, solving with the fewest calculations possible," observes Jean-Pierre Thibaut, full professor at the uB Laboratory for Research on Learning and Development.

''Thus, when it comes to subtracting individual elements, a representation via an axis -- rather than via subsets -- is more effective in finding the fastest method. Analysis of students' drawings in arithmetic can therefore enable targeted intervention to help them translate problems into more optimal representations. One way of doing this is to work on the graphical representation of statements in class, to help students understand the most direct strategies,'' concludes Hippolyte Gros.

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Materials provided by Université de Genève . Note: Content may be edited for style and length.

Journal Reference :

Hippolyte Gros, Jean-Pierre Thibaut, Emmanuel Sander. Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults . Memory & Cognition , 2024; DOI: 10.3758/s13421-024-01523-w

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ORIGINAL RESEARCH article

When can making a drawing hinder problem solving effect of the drawing strategy on linear overgeneralizations and problem solving.

$\r\nJanina Krawitz*$

Department of Mathematics, University of Münster, Münster, Germany

The strategy of making a drawing has been claimed to facilitate mathematical problem solving. However, De Bock et al. (2003) surprisingly found that drawing negatively affected performance in solving non-linear geometry problems, in which the area or volume of similar figures or solids had to be determined by a given scaling factor. The authors suggested that making a drawing increased the number of overgeneralizations and negatively affected students’ performance. Our study involves a partial replication and also an important validation and extension of this study by addressing two factors: low-quality drawing strategy and poor visual monitoring, both of which might explain the negative effect of drawing. First, we expected that improving the quality of the drawing strategy by prompting students to highlight important information in their drawings would diminish the negative effect of the drawing strategy. Second, we expected that fostering visual monitoring while drawing, by offering problems with small scaling factors, would diminish the negative effect of the drawing strategy. We conducted a randomized controlled trial with 180 students (ninth- to eleventh-graders) to investigate the effects of drawing and visual monitoring on solving non-linear geometry problems. Our results replicated the previous finding that drawing negatively affects performance. We confirmed that linear overgeneralizations are a prevalent reason for this finding. Elaborating on previous findings revealed that the quality of the drawing strategy but not visual monitoring was responsible for the effect of the drawing strategy on linear overgeneralizations. Furthermore, an exploratory analysis of students’ awareness of linear overgeneralizations indicated that improving the quality of drawing strategy and enhancing visual monitoring did not lead to a greater awareness of the mistakes learners made because of linear overgeneralizations. We conclude that the way the drawing strategy is used determines whether it is useful or damaging, and more efforts are essential to enable students to apply it appropriately.

Introduction

Making a drawing is considered a powerful strategy in mathematical problem solving ( Pólya, 1945 ). According to the theory of external representations ( Cox, 1999 ), it can support problem solving by helping problem solvers organize the information, and it can make missing and implicit information (e.g., relations between objects) explicit. Therefore, it deepens understanding and facilitates self-explanatory activities. Empirical evidence for the benefits of drawing for problem solving has been found in various studies (e.g., Van Essen and Hamaker, 1990 ; Hembree, 1992 ; Zahner and Corter, 2010 ; Rellensmann et al., 2016 ). However, the drawing strategy does not seem to be helpful for solving some types of problems, and surprisingly, it can even be disadvantageous by decreasing students’ performance in solving non-linear geometry problems ( De Bock et al., 2003 ). It seems that drawing leads to an increase in students’ well-known tendency to engage in linear overgeneralizations, which means that learners tend to apply linear models to non-linear situations ( Van Dooren et al., 2005 ). From a broader perspective, this finding demonstrates the need to investigate the processes elicited by the drawing strategy and the key factors that determine the beneficial use of the strategy. On the basis of these considerations, the goals of the present study are twofold: (a) to replicate De Bock et al. (2003) surprising finding that drawing hinders students’ ability to solve non-linear geometry problems and (b) to find explanations for this unexpected phenomenon. On the basis of prior research, we suggest that the insufficient quality of the drawing strategy and a lack of opportunity to use the drawing strategy for monitoring purposes are crucial factors that have contributed to the negative effects of drawing. Our aim is to clarify whether these factors come into play while students solve non-linear geometry problems and, more specifically, whether it is possible to diminish the negative effect of drawing by addressing these factors.

Drawing Strategy and Linear Overgeneralizations

Self-generated drawing.

External visual representations are omnipresent in contexts of learning and education. Thus, they serve different functions. First, the ability to deal with external visual representations such as drawings can be considered a learning goal on its own because in many situations in class and everyday life, it is necessary to interpret, construct, and work with them ( National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010 ). Second, they have been claimed to enhance learning by relieving working memory, promoting self-explanation activities, and leading to a deeper understanding of the learning material ( Cox, 1999 ; Mayer, 2005 ; Van Meter and Garner, 2005 ). An important distinction has to be made between ready-made and self-generated external visual representations. For the latter, learners construct representations on their own, which means that they are actively involved in externalizing their mental representation, which includes the processes of organizing, selecting, and integrating the information given in the problem ( Van Meter and Garner, 2005 ). In the present paper, we focus on self-generated drawing. We define the drawing strategy as the process of constructing an external visual representation that corresponds to the mathematical problem structure and is aimed at solving the problem ( Van Meter and Garner, 2005 ).

Self-generated drawing influences the process of problem solving, as it guides the learner’s attention and directs or even determines his or her actions. Theories of cognition assume that when beginning to solve a problem, humans construct an internal representation of the problem situation called a mental model ( Johnson-Laird, 1980 ). While drawing, the mental model is transformed into an external visual representation (i.e., a drawing). This process is more than a simple translation because it involves a re-organization of the given information and dynamic iterations between the mental model and the externalized model in order to match both representations ( Cox, 1999 ). Re-organizing the information can make key elements of the problem and its relations visible so that the information can be more easily processed after a drawing is constructed (see section “Quality of Drawing Strategy”) ( Larkin and Simon, 1987 ). In order to successfully solve the problem, it is crucial that the structure of the problem be adequately presented in the external visual representation. Otherwise, drawing could cause perceptual and cognitive biases, which might guide the problem solver away from the goal ( Zhang, 1997 ; Cox, 1999 ).

Studies investigating the effect of drawing on problem solving performance have arrived at divergent findings. A number of empirical studies have found that drawing positively affects problem solving in mathematics ( Van Essen and Hamaker, 1990 ; Hembree, 1992 ; Zahner and Corter, 2010 ; Rellensmann et al., 2016 ). Strong support for the benefits of the drawing strategy were provided by the meta-analysis conducted by Hembree (1992) . Training students to draw was identified as the most effective treatment for improving problem solving performance compared with training them to use other strategies such as handling extraneous data, verbalizing concepts, or using guess-and-test procedures. However, several factors seem to determine whether the drawing strategy is helpful or not. For example, Van Essen and Hamaker (1990) found that drawing showed a positive effect for fifth-graders but not for first- and second-graders, indicating that the benefits of making a drawing depend on the specific difficulties learners encounter while solving problems. Another important factor seems to be the type of problem because, for some types of problems, drawing was shown to be beneficial [e.g., probability problems ( Zahner and Corter, 2010 ) or arithmetic word problems ( Van Essen and Hamaker, 1990 )], whereas for other types of problems, in particular non-linear geometry problems, no effect ( De Bock et al., 1998 ) or even a negative effect ( De Bock et al., 2003 ) was found. The most important factor that determined whether making a drawing was beneficial or not seemed to be the quality of the drawing strategy, which we address in the next section.

Quality of Drawing Strategy

The quality of the drawing strategy refers to the correctness and the explicit presentation of key information. Accordingly, the high-quality use of the drawing strategy means that the drawing as the product of the drawing process is correct and complete with regard to the important elements and their relations. Both criteria need to be met so that the rapid processing capabilities of a learner’s visual system can be used to make perceptual judgments instead of depending on difficult logical inferences ( Cox, 1999 ).

Research on self-generated drawing has shown that the benefits of drawing are strongly related to the quality with which the drawing strategy is applied ( Van Garderen and Montague, 2003 ; Uesaka et al., 2007 ; Schwamborn et al., 2010 ; Mason et al., 2013 ; Rellensmann et al., 2016 ). Learners who apply the drawing strategy in a high-quality way perform better on problem solving and learning outcome tests than learners who apply the drawing strategy in a lower quality manner. Problem solving research has shown that students often fail to use a high-quality drawing strategy because they tend to generate pictorial representations with merely a decorative function instead of depicting important elements and their relations ( Hegarty and Kozhevnikov, 1999 ; Van Garderen and Montague, 2003 ). For non-linear geometry problems, a qualitative analysis of students’ solutions indicated that the quality with which the drawing strategy was applied was usually too poor – regarding correctness and the explicit presentation of key information – to help students solve the problems ( De Bock et al., 1998 ). Hence, the request to draw is probably not enough, and it might be necessary to give students support that will render the drawing strategy more helpful for problem solving. Empirical indications for this claim have been provided in studies of text-based learning. In the study by Van Meter (2001) , applying the drawing strategy was more effective for conditions in which students’ drawing process was supported by providing illustrations or prompts to compare the illustrations with self-generated drawings. It was found that supporting students’ drawing activities had a positive effect on the performance of comprehensive free recall but not recognition items. These results indicate that improving the quality of the drawing strategy is essential for students’ performance if the assessment requires them to build connections between the information given in the problem, as is the case when students solve non-routine mathematical problems.

Visual Monitoring

Another important factor in the context of research on self-generated drawing is that the drawing strategy can enhance monitoring processes. Monitoring has been claimed to be essential for problem solving ( Flavell, 1979 ) and plays an important role in detecting incorrect intuitions and misconceptions such as linear overgeneralizations ( Van Dooren et al., 2004 ). The drawing strategy can be considered a tool that can be used for monitoring for the following reasons. When students use the drawing strategy, they construct a visual representation on the basis of an abstract symbolic representation. As visual representations are limited in abstraction, they aid processability and lead to the generation of new information ( Stenning and Oberlander, 1995 ). Hence, the drawing strategy can be used to detect inconsistencies. In particular, in problem solving, the drawing strategy can be applied with the goal of revealing mistakes and inaccuracies in the student’s mental model of the problem situation. In the following, when the drawing strategy is applied for monitoring purposes, we refer to this as visual monitoring.

Empirical evidence for the claim that drawing strategy can be used for monitoring purposes can be derived from the study by Stylianou (2011) . The problem solving activities of experts (mathematicians) and novices (middle school students) were analyzed by using qualitative methods in order to identify the purposes of the drawing strategy. Both experts and novices used the drawing strategy to monitor the progress of problem solving, including checking the correctness and making informed decisions about subsequent actions. However, in contrast to experts, middle school students engaged in visual monitoring only infrequently and – if at all – to verify their result at the end of the problem solving process. This finding indicates the importance of supporting school students in their visual monitoring activities.

Further indications come from text-based learning research. Van Meter (2001) analyzed the think-aloud protocols of fifth- and sixth-graders who read a science text under two conditions: Self-generated drawing compared with working with ready-made drawings. It was found that students who used self-generated drawing engaged in significantly more monitoring events, such as looking back and self-questioning, compared with students who worked with ready-made drawings. Further, monitoring events were higher when students received additional support during their drawing activity. Hence, supporting students’ drawing activities is crucial for determining the way in which the drawing strategy is used. In sum, drawing seems to fulfill monitoring purposes, and supporting the drawing activities increases visual monitoring. However, research has yet to determine the extent to which these results are valid for mathematical problem solving.

Linear Overgeneralizations

Misconceptions often emerge when learners generalize their prior knowledge by systematically activating it in contexts in which it is inappropriate ( Smith et al., 1993 ). A well-known example of such a misconception is the “illusion of linearity,” the tendency to apply linear models to non-linear situations, which will be referred to as linear overgeneralizations in the following. Linearity and especially proportionality can be considered the simplest but also the most important property of mathematical relationships (two quantities change with an equal amount of growth). Many facts of the real world are based on linear and proportional relationships. Also in mathematics education, linearity plays a central role and emerges during the time children are in school in the contexts of different mathematical topics ranging from arithmetic word problems, to linear functions, to advanced concepts such as the diameter and circumference of a circle. However, the intensive treatment of linearity might result in the disadvantage that some students will develop false conceptions, namely, the idea that linear models have a kind of universal validity. As a consequence, they might mistakenly transfer the principle of linearity to non-linear contexts.

Empirically, students’ strong tendency to engage in linear overgeneralizations has been supported by a large amount of research that has included different age groups ranging from primary school ( Van Dooren et al., 2005 ) to university students ( Esteley et al., 2010 ) and has referred to different mathematical domains such as arithmetic word problems ( Van Dooren et al., 2005 , 2010 ), algebraic patterns ( Stacey, 1989 ), geometry ( De Bock et al., 1998 , 2003 ; Ayan and Bostan, 2018 ), and probability ( Van Dooren et al., 2003 ). More specifically, linear overgeneralizations seemed to increase after linear problems were taught in class ( Van Dooren et al., 2005 ), supporting the assumption that students’ experiences with linear concepts in the mathematics classroom are responsible for their strong tendency to engage in linear overgeneralizations. However, even very young students (second- and third-graders) tend to give linear answers to non-linear problems, indicating that other factors also need to be taken into account. One of these factors could be individuals’ tendency to reduce information in their environment into structures that are as simple as possible, which is known as the “Law of simplicity” ( Chater and Vitányi, 2003 ). As linearity and in particular proportionality is the simplest form of relationship between two quantities, this bias may also occur independent of the effect of students’ experiences with linear problems in class.

One of the most investigated types of problems with regard to linear overgeneralizations is the non-linear geometry problem, in which students are asked about how enlarging or reducing a geometrical figure affects its area or volume. For example: “You need approximately 400 g of flower seed to lay out a circular flower bed with a diameter of 10 m. How many grams of flower seed would you need to lay out a circular flower bed with a diameter of 20 m?” ( De Bock et al., 1998 , p. 68). A series of studies demonstrated that students between the ages of 12 and 16 were usually not able to solve such problems ( De Bock et al., 1998 , 2002b , 2003 ). Overall, these studies reported particularly low solution rates for younger students (rates of 2% and 7% for correct solutions for 12- to 13-year-olds), but wrong answers were usually given among the older students too (23% correct solutions for 13- to 14-year-olds; 17%, 22%, 43% correct responses for 15- to 16-year-olds). Building on these findings, De Bock et al. (2002a) conducted an interview study to investigate which aspects are responsible for the frequent appearance of linear overgeneralizations. They found that some of the students had the firm conviction that any relationship between two variables could be expressed by a constant of proportionality. However, the majority of the students used linear models in an intuitive manner, without being aware of the model they chose. Students apparently do not recognize the mistakes they make on the basis of linear overgeneralizations and therefore probably perceive that their solutions to these problems are correct even when they are incorrect.

Further, a teaching experiment conducted by Van Dooren et al. (2004) showed that it is possible to decrease the number of linear overgeneralizations in the solutions to such problems. In 10 experimental lessons, major holes in students’ prior geometrical knowledge and their linearity preconceptions were addressed by eliciting cognitive conflicts. Further aims of the intervention were to facilitate students’ meta-conceptual awareness, including monitoring and enhancing a deeper understanding from the use of multiple external representations of the central mathematical contents. Although the automatic use of linear strategies was successfully reduced, some of the students in the experimental group still tended to engage in linear overgeneralizations, whereas others started to also apply non-proportional strategies to proportional problems, indicating that the intervention was not successful in terms of fostering a deep conceptual understanding of differences in linearity and non-linearity in some of the students. These results provide the first hints that external representations might be beneficial for diminishing linear overgeneralizations. Further support comes from the study by De Bock et al. (2002b) who found that providing ready-made drawings of the original and scaled figures on graph paper had a positive albeit small effect on solution rates for non-linear geometry problems. Graph paper allows comparisons to be made of the areas of the figures by counting the squares and thus facilitates the recognition of the non-linear relationship of the areas.

We view these findings as initial indications of the importance of external representations for overcoming linear overgeneralizations and performance. Further indications pointing in the opposite direction come from research on self-generated drawing.

Effects of the Drawing Strategy on Linear Overgeneralizations and Performance

A series of experimental studies investigated the impacts of making a drawing on linear overgeneralizations and performance. In one of these studies ( De Bock et al., 1998 ), students in a drawing condition were instructed to draw before solving each item. The instructions were given at the beginning of the test using an example item. Contrary to theoretical considerations, no effect of making a drawing on performance was found. The percentage of correct solutions for the group of 12- to 13-year-old students remained at only 2% and was also found to be low for 15- to 16-year-old students regardless of the drawing instructions.

Different drawing instructions were implemented in a subsequent study ( De Bock et al., 2003 ). In the drawing condition, students between the ages of 13 and 16 were given a drawing of a geometrical object for each problem (e.g., a square) and were asked to complete the drawing by supplementing a scaled copy of the object using the given scaling factor. The surprising finding was that students who received these instructions showed significantly lower solution rates than the control group (23% vs. 44%). An additional analysis of the solution processes from this study suggested that self-generated drawing did not elicit visual solution strategies such as “paving” – determining the area of a plane figure by paving it with similar Figures – and hence, the drawing strategy was apparently not applied appropriately. This is a potential reason why drawing is not beneficial, but it does not explain the negative effect. An analysis of the problems used in this study provided another reason for this result. Making a drawing might hinder students’ progress while solving non-linear problems because the process of drawing might divert their attention to unimportant elements or even to elements that could interfere with their solution process: Figures are typically depicted by their circumferences, which change linearly through scaling. In the process of drawing, learners work with linear relationships and may erroneously transfer them to the area. This might render the quality of the drawing strategy insufficient because key information (i.e., the area) is not made salient in the drawing. Increasing the quality of the drawing strategy by highlighting the area in the drawings may guide learners’ attention to the important elements of the problem, thus helping them identify non-linear properties while drawing.

Another aspect that also affects the recognition of non-linearity concerns visual monitoring. Visual monitoring should reveal that the area changes non-linearly through scaling. However, visual monitoring might potentially not come into effect if the size of the scaling factor is too large. For problems with small scaling factors (e.g., doubling the side length), the difference between the area or volume of the original and of the modified figure becomes salient while drawing so that visual monitoring should uncover the non-linear relationship. Whereas for large scaling factors (e.g., if the side length is twelve or more times larger), the difference in the area or volume cannot be easily visually estimated. Consequently, it can be expected that visual monitoring, enabled by using small scaling factors, can help learners overcome their difficulties with linear overgeneralizations so that they will demonstrate better performance in problem solving. However, even if students recognize the non-linear relationship by engaging in high-quality drawing or visual monitoring, they are not necessarily able to solve the problem. Instead, they might change the problem by imposing an inappropriate structure that enables them to apply available solution strategies ( Goos, 2002 ). It is possible that they might detect the non-linear property of the area but nevertheless use linear models to solve the problem because they lack adequate solution strategies ( Weber, 2001 ). Students who recognize the non-linear relationship of the areas are probably aware of their inappropriate application of linear overgeneralizations and will consequently perceive their lower performance in solving the problems than students who do not recognize the non-linear relationship. Thus, we assume that students’ perceptions of their performance in solving non-linear geometry problems might be an indicator of students’ awareness of the non-linear property of the problems. Data on students’ perceived performance will help us interpret the effect of drawing quality and visual monitoring on linear overgeneralizations and performance.

Research Questions and Expectations

On the basis of theoretical considerations and prior empirical findings, we posed the following research questions:

1. Does the instruction to make a drawing of the scaled figure lead to a larger number of linear overgeneralizations and have a negative effect on problem solving performance of non-linear geometry problems?

2. Does improving the quality of the drawing strategy by highlighting important information in the drawing decrease the number of linear overgeneralizations and diminish the negative effect of the drawing strategy on problem solving performance?

3. Does visual monitoring decrease the number of linear overgeneralizations and diminish the negative effect of the drawing strategy on problem solving performance?

4. Does drawing or visual monitoring affect students’ perceived performance when solving non-linear geometry problems?

Expectations for Research Question 1 (Drawing)

The first research question addresses the replication of De Bock et al.’s (2003) finding that making a drawing hinders students’ ability to solve non-linear geometry problems. Following theoretical domain-specific considerations regarding the reasons for learners’ linear overgeneralizations, we assume that self-generated drawing distracts learners and draws their attention toward elements of the problem that interfere with their solution process, for example, the linear relationship of the circumferences of the original and scaled figures in problems with rectilinear plane figures. Because of the very common tendency to engage in linear overgeneralizations ( Van Dooren et al., 2008 ), they may erroneously transfer the linear relationship of the circumferences to the non-linear relationship of the areas. The same considerations can be made for problems with non-rectilinear figures and solids regarding the linear property of the diameter and the non-linear property of the volume. Thus, we expected the drawing strategy to increase the number of linear overgeneralizations and negatively affect problem solving performance.

Expectations for Research Question 2 (High-Quality Drawing):

We expected that increasing the quality of the drawing strategy by highlighting key information would diminish the negative effect of the drawing strategy. Hence, we expected that students applying a high-quality drawing strategy and students not applying the drawing strategy would show the same number of linear overgeneralizations and performance in solving non-linear geometry problems. Further, we expected fewer linear overgeneralizations and higher performance from students who applied the high-quality drawing strategy than students who used the lower quality drawing strategy. The rationale behind these expectations is that the effects of the drawing strategy strongly depend on drawing quality. One key characteristic of a high-quality drawing is the explicit presentation of key information. For non-linear geometry problems, mistakes are made due to an inappropriate focus on the side length or the diameter of the figure or solid and its linear properties instead of considering the area or the volume, respectively. Hence, highlighting the area or volume of the drawn figure or solid will improve the quality of the drawing strategy and should lead to fewer linear overgeneralizations and a higher performance than the use of a lower quality drawing strategy.

Expectations for Research Question 3 (Visual Monitoring)

We expected that visual monitoring would diminish the negative effect of the drawing strategy. Consequently, visual monitoring while drawing should lead to a similar number of linear overgeneralizations and a similar performance in solving non-linear geometry problems in comparison with solving the problems without a drawing. In addition, we expected that visual monitoring would lead to fewer linear overgeneralizations and a higher performance than drawing without visual monitoring. We enhanced visual monitoring by using small scaling factors instead of large ones on the basis of our assumption that for small scaling factors, the non-linear relationship of the areas becomes salient while drawing. Consequently, visual monitoring could help overcome the linear overgeneralizations that were elicited by the drawing strategy.

Expectations for Research Question 4 (Effects on Perceived Performance)

The fourth research question followed an exploratory approach. Thus, we did not have specific expectations. The aim of analyzing students’ perceived performance is to increase the validity by using different indications of students’ success ( Rovers et al., 2019 ) and to gather further information that helps to explain the findings from our experimental study. Students’ perceived performance in the drawing and visual monitoring conditions will provide indications of students’ awareness of the non-linear property of the problems. Students who notice the non-linear relationship because they make a high-quality drawing or engage in visual monitoring might lack the mathematical knowledge to proceed and will therefore nevertheless stick to the application of linear models and will report lower perceived performance.

Materials and Methods

Sample and procedure.

The present study involved 198 students (57.1% female, mean age = 16.15 years) from nine classes, including ninth-graders (12.6%), tenth-graders (48.5%), and eleventh-graders (38.9%). Students came from four high-track schools (German Gymnasium) and one comprehensive school (German Gesamtschule).

Students in each class were randomly assigned to one of five groups: Students in the experimental conditions received either drawing (D) or drawing with highlighting (DQ) instructions, aimed at increasing the quality of the drawing strategy, and the test version with either large [11, 12, or 13 as used in the study by De Bock et al. (2003) ] or small scaling factors (3, 4, or 5), aimed in enhancing visual monitoring (V− and V+ groups). These conditions resulted in four combinations of experimental conditions (DV−, DV+, DQV−, DQV+). Students in the control group (CG) received no drawing instructions and the test version with large scaling factors as in the study by De Bock et al. (2003) . All groups worked on a paper-and-pencil test consisting of four experimental items, which were non-linear geometry problems, and three additional buffer items. All items were taken from the study by De Bock et al. (2003) . Drawing and drawing with highlighting instructions were embedded in each item on the test. Figure 1 shows a sample item with drawing with highlighting instructions as used in the version of the test that was administered in the DQ condition. Students in the D group received the same drawing instructions (part a) but no highlighting instructions (part b).

Figure 1. Sample item with drawing and highlighting instructions. Tasks were adopted from De Bock et al. (2003 , p. 449).

After taking the test, students completed a questionnaire. The aim of the questionnaire was to collect data on how solving non-linear geometry problems and the experimental treatment were perceived by the students. Thus, the questionnaire included four statements for measuring students’ perceived performance.

Treatment Check

To check the implementation of the treatment, we examined whether students in the experimental and control groups followed the instructions. The results confirmed that students followed the drawing instructions and the instructions to draw and highlight. As intended, the number of drawings in the D groups was significantly higher than in the CG [96.1% vs. 40.2%; t (43.636) = 7.542, p < 0.001, d = 1.903]. Further, the number of highlighted drawings in the DQ groups was significantly higher than in the CG [80.0% vs. 0.65%; t (84.756) = 22.526, p < 0.001, d = 3.608] and D groups [80.0% vs. 2.4%; t (109.960) = 15.798, p < 0.001, d = 3.033]. However, 19 of 41 participants of the control group made at least for one of the items a spontaneous drawing. These students seem to perform similar or even better than students who did not make a drawing (50.0% vs. 45.5% correct solutions; 18% vs. 29% linear overgeneralizations). To ensure that spontaneous drawings did not distort our results, we again addressed our research questions by analyzing an adjusted subsample. The adjusted subsample included only students who acted in accordance with their condition. As our analysis revealed nearly the same effect sizes for the adjusted subsample and the whole sample, we analyzed the whole sample in our study.

In addition, we examined students’ age and last mathematics grade by computing an ANOVA to ensure the comparability of the treatment conditions. As expected, no significant difference between the groups was found ( p > 0.10).

Linear Overgeneralizations and Problem Solving Performance

Linear overgeneralizations were estimated by analyzing whether the solution was based on a linear model (coded 1) or not (coded 0). Students’ performance was analyzed by assigning a score of 1 for the correct solution and a score of 0 for an incorrect solution. Two independent coders were involved in scoring the test booklets. Inter-rater reliability was calculated for each problem on a subset of 20% of the test booklets which were scored by both coders with sufficient inter-rater agreement (Cohen’s κ ≥ 0.773). Reliability was satisfactory (Cronbach’s α = 0.729 for linear overgeneralizations and 0.754 for performance). All problems were taken from De Bock et al. (2003) and are listed here in the version for the V− groups in Table 1 .

Table 1. Experimental items in the V− groups.

Perceived Performance

Students rated the statements on the questionnaire using a five-point Likert scale (from full disagreement to full agreement). The scale for measuring perceived performance was adapted from prior studies ( Hänze and Berger, 2007 ; Schukajlow and Krug, 2014 ; Schukajlow et al., 2015 , 2019a ). It included four items: “I noticed that I really understood the arithmetic problems”; “I felt able to master the arithmetic problems”; “I feel able to master similar arithmetic problems”; and “I felt confident about my knowledge about the topic today.” The scale reliability (Cronbach’s α) was 0.863.

Data Analysis

The hypotheses were tested with a 3 × 2 MANOVA with Drawing (no drawing, D, DQ) and Visual Monitoring (V− and V+) as the independent variables and Linear Overgeneralizations and Performance as the dependent variables. There was homogeneity of variance as assessed with Levene’s test ( p > 0.05). Significant main effects were further analyzed with post hoc Tukey tests. The reported p -values for Linear Overgeneralizations and Performance were one-tailed due to our directional expectations. We followed De Bock et al. (2003) procedure to ensure the comparability of the results. This included conducting our analysis with only two of four experimental items. Including all four items in the MANOVA revealed the same results because the effect sizes from the two analyses were very similar.

To analyze Perceived Performance, we conducted a 2 × 2 ANOVA with Drawing (D, DQ) and Visual Monitoring (V− and V+) as factors. Homogeneity of variance was confirmed. Because of the exploratory approach, no assumptions were made about the direction of the effects, and two-tailed p -values are reported.

An overview of the mean scores and standard deviations for all variables in the different experimental conditions is presented in Table 2 .

Table 2. Mean scores (and standard deviations) of all variables in the different experimental conditions.

Effect of Drawing Strategy on Linear Overgeneralizations and Performance

In line with our expectations, the drawing strategy increased the number of linear overgeneralizations. Students who applied the drawing strategy with a lower quality (D groups) tended to make more linear overgeneralizations than students who did not use this strategy (CG) (43.5% vs. 24.4%). The MANOVA revealed a marginally significant main effect of Drawing on Linear Overgeneralizations [ F (2,197) = 1.970, p = 0.071; η p 2 = 0.020]. Post hoc comparisons using the Tukey test indicated significant differences ( p < 0.05, d Cohen = −0.461) between students who used the drawing strategy and the control group, which did not draw.

Further, our expectation that the drawing strategy would have a negative effect on performance was confirmed. Students who applied the drawing strategy with a lower quality (D groups) achieved significantly lower performance scores than students who did not use this strategy (28.6% vs. 47.6%). The MANOVA revealed a significant main effect of Drawing on Performance [ F (2,197) = 4.323, p < 0.05; η p 2 = 0.043], and post hoc comparisons indicated significant differences ( p < 0.05 d Cohen = 0.436) between students who applied the drawing strategy and students who did not.

These findings did not interact with the use of the two test booklets (large- or small-sized scaling factors), which were administered to the D and DQ groups but were not administered to the CG group for economic reasons (large scaling factors only). We will elaborate on this point in the results for the third research question (see section “Effect of Visual Monitoring on Linear Overgeneralizations and Performance”). To ensure comparability, we conducted an additional analysis in which only the groups who received the test version with the large scaling factor were included (CG, DV−, DQV−). The results were similar with even stronger effect sizes (Linear Overgeneralizations: η p 2 = 0.022; Performance: η p 2 = 0.069).

Effect of High-Quality Drawing Strategy on Linear Overgeneralizations and Performance

We were able to partly confirm the hypothesis that the high-quality drawing strategy (DQ) would diminish the negative effect of the drawing strategy. As expected, students who used the high-quality drawing strategy engaged in linear overgeneralizations comparably as often as students who did not use the drawing strategy (CG) (35.0% vs. 24.4%). Post hoc Tukey tests confirmed that there were no statistically significant differences between students who used the high-quality drawing strategy and the control group ( p = 0.198, d Cohen = −0.261). However, contrary to our expectations, students who used the high-quality drawing strategy did not show significantly fewer linear overgeneralizations than students who used the drawing strategy with a lower quality (D) (35.0% vs. 38.9%, Tukey tests: p = 0.211, d Cohen = 0.197).

Further, we expected that students who used the high-quality drawing strategy (DQ) would show the same performance as students who did not use the drawing strategy (CG). Contrary to our expectations, Tukey tests indicated that the mean performance score for the DQ group was significantly lower ( p < 0.05, d Cohen = 0.471) than the score for the CG (27.5% vs. 47.6%). Also the comparison of the two drawing conditions yielded results that went contrary to our expectations: The use of high-quality drawing strategy (DQ) did not lead to a higher performance than a lower quality use of the drawing strategy with a lower quality (D) (27.5% vs. 28.6%; p = 0.493, d Cohen = 0.027).

Effect of Visual Monitoring on Linear Overgeneralizations and Performance

We expected that the use of a drawing strategy would not hinder problem solving when used for monitoring purposes, referred to here as visual monitoring. Visual monitoring was operationalized by the smaller-sized scaling factor because we assumed that a smaller scaling factor would make relations between objects in the drawing salient and would therefore inspire visual monitoring.

The results did not confirm our expectations. Students in the visual monitoring group did not differ in the number of linear overgeneralizations from students who could not perform visual monitoring (42.0% vs. 32.5%). There was no significant main effect of Visual Monitoring on Linear Overgeneralizations [ F (1,197) = 0.698, p = 0.202; η p 2 = 0.004], and there was also no effect of the Visual Monitoring × Drawing interaction on Linear Overgeneralizations [ F (1,197) = 0.334, p = 0.282; η p 2 = 0.002].

Our expectations were not confirmed for performance either: Visual monitoring did not diminish the negative effect of the drawing strategy on performance (32.0% vs. 32.1%). As was already found for the number of linear overgeneralizations, there was no significant main effect of Visual Monitoring on Performance [ F (1,197) = 1.312, p = 0.127; η p 2 = 0.007], and there was no effect of the Visual Monitoring × Drawing interaction on Performance [ F (1,197) = 0.337, p = 0.281; η p 2 = 0.002].

Effect of Drawing Strategy and Visual Monitoring on Perceived Performance

The results of the ANOVA showed that the quality of the drawing strategy did not affect students’ perceived performance [ F (1,153) = 0.183, p = 0.670; η p 2 = 0.001]. Students who were given the high-quality drawing strategy (DQ) perceived that their performance was the same as students who were given the lower quality strategy (D) ( M = 3.54, SD = 0.88 vs. M = 3.58, SD = 0.83).

However, the results revealed a significant effect of visual monitoring on students’ perceived performance [ F (1,153) = 16.357, p < 0.01; η p 2 = 0.097]. Students in the visual monitoring group perceived a significantly higher performance than their peers who could not easily engage in visual monitoring ( M = 3.84, SD = 0.79 vs. M = 3.32, SD = 0.83).

Further, no significant effect of the Drawing × Visual Monitoring interaction on Perceived Performance was found [ F (1,153) = 0.571, p = 0.571; η p 2 = 0.004].

The present study was aimed at replicating De Bock et al. (2003) finding that the drawing strategy hinders students’ ability to solve non-linear geometry problems. We also aimed to elaborate on the potential reasons for this finding by addressing two factors: the quality of the drawing strategy and visual monitoring. Furthermore, we performed an exploratory analysis of students’ perceived performance in order to gather information about students’ awareness of linear overgeneralizations with the hope that this would help us interpret the results of our experimental study.

Negative Effect of the Drawing Strategy

Our results replicated the previous findings of a negative effect of the drawing strategy on the performance of non-linear geometry problems and confirmed the previous assumption that linear overgeneralizations are a prevalent reason. We found that students who applied the drawing strategy (D groups) made more linear overgeneralizations than students who did not draw. Self-generated drawing seems to guide learners toward mistakenly focusing on the linear relationships depicted in the drawings. However, the effect of the drawing strategy on the number of linear overgeneralizations was smaller than the effect for performance, indicating that applying the drawing strategy may have also resulted in other mistakes, perhaps because of the cognitive cost associated with the externalization process ( Zhang, 1997 ).

Further, the replication of the negative effect of drawing on performance indicates that the findings are stable across different samples. Even the solution scores were very similar to the ones reported by De Bock et al. (2003) , with a rate of about 75% for incorrect solutions in the drawing group and 50% in the non-drawing group in both studies.

On a global level, the finding that self-generated drawing hinders students’ ability to solve non-linear geometry problems shows that drawing strategy is not a one-size-fits-all solution and stresses the need to elaborate on the factors that determine beneficial strategy use.

Following theoretical considerations about the importance of the quality of self-generated drawing that was confirmed in prior research, we expected that the drawing strategy would hinder students’ ability to solve non-linear geometry problems because it would be applied insufficiently when students solved non-linear problems. Therefore, we increased the quality of the drawing strategy by addressing its key feature by explicitly presenting information that is essential for solving the problem.

The results confirmed the importance of the quality of the drawing strategy. Improving the quality of the drawing strategy diminished the increase in linear overgeneralizations that previously resulted from the drawing strategy. In particular, we found that students who used a high-quality drawing strategy did not differ in the number of linear overgeneralizations they made from students who did not use the drawing strategy, whereas students who applied a lower quality drawing strategy made a larger number of linear overgeneralizations compared with non-drawing students. This finding helps to explain the negative effect of self-generated drawing on solving non-linear geometry problems: Applying the drawing strategy in a high quality way ensures that the area, which is a key element of the problem, will be visible in the drawing. This seems to prevent at least some of the students from being guided by their drawing toward mistakenly focusing on elements of the problem that will interfere with their ability to solve the problem, such as the linear properties of the circumference or the side length. However, more efforts are essential for investigating how we can improve drawing quality so that the drawing strategy can become beneficial.

Contrary to our expectations, we found that improving the quality of the drawing strategy did not diminish the negative effect of self-generated drawing on performance, although it did diminish the negative effect with respect to the number of linear overgeneralizations. Apparently, improving the quality of the drawing strategy did not help students solve the problems. Even the high-quality use of the drawing strategy did not seem to elicit the visual solution strategies that could help students find the right solution. In line with prior research, this finding points out the lack of visual solution strategies, such as calculating the area by paving the figure in order to solve non-linear geometry problems ( De Bock et al., 2002b , 2003 ). Future research should investigate whether training students to use visual solution strategies can lead to the beneficial use of the drawing strategy.

Another factor that we addressed in order to explain the negative effect of drawing strategy was visual monitoring, the use of the drawing strategy for monitoring purposes. Monitoring has been identified as essential for problem solving ( Flavell, 1979 ) and was found to be important for detecting linear overgeneralizations ( De Bock et al., 2002a ). For non-linear geometry problems, we assumed that visual monitoring would take place for problems with small scaling factors but would not for large ones because the non-linear relationship of the areas becomes salient while drawing when small scaling factors are used.

However, the findings did not confirm our expectation that visual monitoring diminishes the effect by which self-generated drawing hinders students’ ability to solve non-linear geometry problems. Visual monitoring did not affect the number of linear overgeneralizations or performance. One potential reason for this finding is that students’ tendency to engage in linear overgeneralizations is very strong and difficult to change by engaging in subtle actions such as visual monitoring ( De Bock et al., 2007 ). Visual monitoring may have helped students recognize the non-linear relationship between the areas while drawing, but because students lacked knowledge of how to proceed, they stuck to their use of the linear models they were familiar with to solve the problem. Another reason might be that students did not even notice the non-linear relationship of the areas while drawing because they did not use the drawing strategy for monitoring. Consequently, our assumption that visual monitoring takes place when the drawing strategy is applied to problems with small scaling factors needs to be reconsidered. Previous research has indicated that, in contrast to experts, students use the drawing strategy only infrequently to monitor their solution processes ( Stylianou, 2011 ), so they might not have engaged in visual monitoring even though the non-linear property of the area was made salient while they were drawing. We need more research on how visual monitoring affects the drawing strategy and on how to clarify the mechanisms that can improve visual monitoring in students.

Taken together, our findings confirmed the idea that applying a strategy can have negative effects on students’ performance. The use of a drawing strategy and its effects on solving non-linear problems demonstrates that more efforts are essential for clarifying which factors, apart from fostering linear overgeneralizations, affect the decrease in students’ performance. On a more general level, we argue that there is a need to also further investigate the negative effects of other strategies and identify why some students are misguided when they apply a specific strategy even when this strategy might be helpful for the majority of students. The quality of strategy use seems to be an important factor that should be addressed more often in research on strategies. In addition, research on cognitive factors such as strategic knowledge about drawing ( Lingel et al., 2014 ; Rellensmann et al., 2019 ) or on emotional and motivational factors such as enjoyment of drawing and the costs of drawing ( Uesaka and Manalo, 2017 ; Schukajlow et al., 2019b ) can contribute to clarifying the conditions under which drawing is helpful and when it has a hindering effect.

Awareness of Linear Overgeneralizations

On the basis of theoretical considerations, we assumed that increasing the quality of the drawing strategy and enhancing visual monitoring would affect students’ awareness of linear overgeneralizations even if it did not affect their performance. Learners may recognize the non-linear property of the problem but might still stick to linear models because they lack the mathematical knowledge necessary to proceed. Indications of whether students were aware of their linear overgeneralizations could be derived from their perceived performance. If students did not notice that drawing guided them incorrectly toward an inadequate use of linear models, they presumably perceived that their performance was higher than the performance of students who were aware that their solution was probably wrong because they made inappropriate linear assumptions.

In order to validate our findings, we conducted an exploratory analysis of students’ perceived performance. Our findings indicated that neither improving the quality of drawing strategy nor enhancing visual monitoring led to a greater awareness of linear overgeneralizations. This finding is in line with prior research that pointed to the intuitive nature of linear misconceptions ( Van Dooren et al., 2004 , 2008 ).

It seems that students also encountered other difficulties while solving the problems, ones that did not rely on the non-linear properties of the problem. Students in the group in which visual monitoring was enhanced by using small scaling factors perceived that their performance was even higher than students in the low visual monitoring group who worked on problems with large scaling factors, although the two groups had the same performance scores. The use of small scaling factors probably led to a higher perceived performance because it facilitated the calculations, but it did not lead to a higher performance because the learners made mistakes on the basis of the linear overgeneralizations that they were not aware of. These points indicate that we also need to investigate other difficulties students encounter in solving non-linear geometry problems with the help of a drawing strategy in order to develop a complete picture of the difficulties encountered while solving non-linear problems.

Strengths and Limitations

We investigated the effect of the drawing strategy for solving non-linear geometry problems by using an experimental design with drawing conditions and a control group that was not instructed to draw. We implemented a treatment check, which showed that students reliably followed the instructions. However, 19 of 41students in the control group spontaneously made drawings. Therefore, we additionally analyzed an adjusted subsample that included only the students in drawing conditions who actually drew and the students in the control group who did not draw. This analysis showed the same results as the previous analyses.

In order to keep the design of the study as simple as possible, the control group worked only on the test version with large scaling factors. As noted in Section “Effect of Drawing Strategy on Linear Overgeneralizations and Performance,” we conducted additional analysis to ensure the comparability of the different drawing conditions. However, the design of our study does not allow to compare students of no drawing and drawing conditions for tests with small scaling factors.

Another important limitation is that our findings are valid for the effects of instructions to make a drawing, but not for spontaneous drawing activity. Descriptive analysis of students’ solutions indicated that students’ spontaneous drawing did not have negative (or even might have slightly positive) effects on students’ achievement-related outcomes. Thus, it might be that spontaneous drawing activity is positively related to students’ achievement-related outcomes. Identifying task- and person-related factors that predict spontaneous use of drawings for non-linear problems is another open question.

A further limitation concerns the operationalization of the factors of drawing quality and visual monitoring. On the basis of theoretical considerations, we assumed that drawing quality would improve if we highlighted the key information given in the problem. Further, we assumed that visual monitoring would be enhanced by the use of small scaling factors compared with large scaling factors. Although both assumptions are plausible, our manipulation might address other factors in addition to these two factors. For example, using small scaling factors decreases the difficulty of the calculations.

As our study was a partial replication of the study by De Bock et al. (2003) , we decided to use the same material to render the results as comparable as possible and therefore included only two items in the analyses. As reported in the method section, additional analyses based on all of the four experimental items showed the same results, but future research should increase the number of items to strengthen the validity of these findings.

Data Availability Statement

The datasets generated for this study are available on request to the corresponding author.

Ethics Statement

Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

Author Contributions

SS and JK designed the study. JK analyzed the data and drafted the manuscript. SS revised the manuscript critically for important intellectual content.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Van Garderen, D., and Montague, M. (2003). Visual-spatial representation, mathematical problem solving, and students of varying abilities. Learn. Disabil. Res. Pract. 18, 246–254. doi: 10.1111/1540-5826.00079

Van Meter, P. (2001). Drawing construction as a strategy for learning from text. J. Educ. Psychol. 93, 129–140. doi: 10.1037/0022-0663.93.1.129

Van Meter, P., and Garner, J. (2005). The promise and practice of learner-generated drawing: literature review and synthesis. Educ. Psychol. Rev. 17, 285–325. doi: 10.1007/s10648-005-8136-3

Weber, K. (2001). Student difficulty in constructing proofs: the need for strategic knowledge. Educ. Stud. Math. 48, 101–119. doi: 10.1023/a:1015535614355

Zahner, D., and Corter, J. E. (2010). The process of probability problem solving: use of external visual representations. Math. Think. Learn. 12, 177–204. doi: 10.1080/10986061003654240

Zhang, J. (1997). The nature of external representations in problem solving. Cogn. Sci. 21, 179–217. doi: 10.1207/s15516709cog2102_3

Keywords : drawing strategy, geometry problems, illusion of linearity, linear overgeneralizations, monitoring, problem solving, self-generated drawing

Citation: Krawitz J and Schukajlow S (2020) When Can Making a Drawing Hinder Problem Solving? Effect of the Drawing Strategy on Linear Overgeneralizations and Problem Solving. Front. Psychol. 11:506. doi: 10.3389/fpsyg.2020.00506

Received: 29 October 2019; Accepted: 02 March 2020; Published: 27 March 2020.

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Copyright © 2020 Krawitz and Schukajlow. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Janina Krawitz, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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Drawings of mathematical problems predict their resolution

by University of Geneva

A team from the University of Geneva (UNIGE), in collaboration with CY Cergy Paris University (CYU) and University of Burgundy (uB), have analyzed drawings made by children and adults when solving simple problems. The scientists found that, whatever the age of the participant, the most effective calculation strategies were associated with certain drawing typologies.

These results, published in the journal Memory & Cognition , open up new perspectives for the teaching of mathematics.

This translation of words into solving strategies involves a stage of mental representation of mathematical information, such as numbers or the arithmetic operation to be performed, and non-mathematical information, such as the context of the problem.

The cardinal or ordinal dimensions of problems

The participants were then asked to produce a drawing or diagram explaining their problem-solving strategy for each statement. The contexts of some problems called on the cardinal properties of numbers—the quantity of elements in a set—others on their ordinal properties—their position in an ordered list.

The former involved marbles, fishes, or books, for example: "Paul has 8 red marbles. He also has blue marbles. In total, Paul has 11 marbles. Jolene has as many blue marbles as Paul, and some green marbles. She has 2 green marbles less than Paul has red marbles. In total, how many marbles does Jolene have?"

The latter involved lengths or durations, for example: "Sofia traveled for 8 hours. Her trip started during the day. Sofia arrived at 11. Fred leaves at the same time as Sofia. Fred's trip lasted 2 hours less than Sofia's. What time was it when Fred arrived?"

Both of the above problems share the same mathematical structure, and both can be solved by a long strategy in 3 steps: 11–8 = 3; 8–2 = 6; 6 + 3 = 9, but also in a single calculation: 11–2 = 9, using a simple subtraction. However, the mental representations of these problems are very different, and the researchers wanted to determine whether the type of representations could predict the calculation strategy, in 1 or 3 steps, of those who solve them.

'"Our hypothesis was that cardinal problems—such as the one involving marbles—would inspire cardinal drawings , i.e., diagrams with identical individual elements, such as crosses or circles, or with overlaps of elements in sets or subsets.

"Similarly, we assumed that ordinal problems—such as the one mentioning travel times —would lead to ordinal representations, i.e., diagrams with axes, graduations or intervals—and that these ordinal drawings would reflect participants' representations and indicate that they would be more successful in identifying the one-step solution strategy," explains Hippolyte Gros, former post-doctoral fellow at UNIGE's Faculty of Psychology and Educational Sciences, associate professor at CYU, and first author of the study.

Identifying mental representations through drawings

These hypotheses were validated by analyzing the drawings of 52 adults and 59 children. "We have shown that, irrespective of their experience—since the same results were obtained in both children and adults—the use of strategies by the participants depends on their representation of the problem, and that this is influenced by the non-mathematical information contained in the problem statement, as revealed by their drawings," says Emmanuel Sander, full professor at the UNIGE's Faculty of Psychology and Educational Sciences.

"Our study also shows that, even after years of experience in solving addition and subtraction, the difference between cardinal and ordinal problems remains very marked. The majority of participants were only able to solve problems of the second type in a single step."

Improving mathematical learning through drawing analysis

"From a pedagogical point of view, this suggests that the presence of specific features in a student's drawing may or may not indicate that his or her representation of the problem is the most efficient one for meeting the instructions—in this case, solving with the fewest calculations possible," observes Jean-Pierre Thibaut, full professor at the uB Laboratory for Research on Learning and Development.

"Thus, when it comes to subtracting individual elements, a representation via an axis—rather than via subsets—is more effective in finding the fastest method. Analysis of students' drawings in arithmetic can therefore enable targeted intervention to help them translate problems into more optimal representations. One way of doing this is to work on the graphical representation of statements in class, to help students understand the most direct strategies," concludes Gros.

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Maths with David

Problem solving. draw diagram.

In mathematics, diagrams are often a useful way of organising information and help us to see relationships. A diagram can be a rough sketch, a number line, a tree diagram or two-way table, a Venn diagram, or any other drawing which helps us to tackle a problem.

Labels (e.g. letters for vertices of a polygon) are useful in a diagram to help us be able to refer to items of interest.

A diagram can be updated as we find out new information.

Examples of using a diagram to tackle a problem

First we will read all three examples and have a quick think about them and then we will look at how a diagram can help us with each one:

Restaurant Example

A restaurant offers a “business lunch” where people can choose either fish or chicken or vegetables for their main course, accompanied by a side portion of rice, chips, noodles or salad. How many different combined meals can they choose between?

Rectangle Area Example

To the nearest centimetre, the length and width of a rectangle is 10cm and 8cm.

What are the limits of accuracy for the area of the rectangle?
the lengths of the sides?

Prime Numbers Example

Masha says that if she writes out numbers in rows of six then all of the prime numbers will either be in the column that has 1 at the top, or they will be in the column that has 5 at the top. How can you find out if she is correct?

Worked Solutions to Examples

One way to tackle this would be to write out a list, being systematic to ensure that all combinations are considered.

Another is to draw out a diagram like the one below. As shown, you actually don’t need to finish the diagram in order to conclude how many combinations there are:

You could also use a 2-way table as shown below:

Drawing a rough sketch of the rectangle labelled with the boundaries of its side lengths can really help us to visualise the situation here:

It can then be helpful to draw sketches of the smallest possible rectangle and the largest possible rectangle:

We can now answer the questions, so (a) the smallest possible area is 7.5 x 9.5 = 71.25cm 2 and the largest “possible” area is 8.5 x 10.5 = 89.25cm 2 . So the limits of accuracy are [71.25,89.25) cm 2 .

For (b), we can see from the sketches that the difference between the minimum and the maximum values is 1cm in the case of both the width and the lenght. For part (ii) we simply subtract the numbers above to give 89.25-71.25 = 18cm 2 .

Here, listing out numbers, especially for the first few is going to be helpful. We should list them as specified in the question, and we can highlight the prime numbers:

Because we know that no even numbers other than 2 are prime, we know that further prime numbers cannot be in the second, fourth or sixth column. The third column keeps adding 6s, so it is adding multiples of 3 to multiples of 3, so the numbers will always be divisible by 3, so further numbers in this column cannot be prime. So she is correct that the prime numbers must be in the first or the fifth column.

31 Questions of increasing difficulty

1.) In a cement factory, cement bags are placed on pallets made of planks of wood and bricks. The number of bricks needed to make a pallet is calculated as being one more than the length of the plank in metres (as shown below):

a.) What length of pallet uses five bricks?

b.) If the pallet is 7m long, how many bricks are used in it?

The factory needs pallets with a total length of 15m for the next batch of cement. It has planks of wood that are 4m long and 3m long.

c.) What combinations of planks can they have?

d.) How many bricks would be needed for each combination?

2.) Sonia wants to plant an apple tree in her garden. She needs to make sure that there is a circular area of lawn with diameter 3m around the base of the tree, so that all of the fruit will fall onto the lawn area.

Below is a (not to scale) sketch of Sonia’s garden:

Where could the tree be placed to meet her requirements?

3.) The diagram below represents towns A and B in a mountainous region:

The mountain rescue helicopters from both towns will always be sent to rescue any casualty within a 25km radius of town A or town B. The fire and rescue team from town B will travel to any accident scene closer to town B than town A.

Shade the region that the helicopters and town B’s fire an rescue team will both cover.

4.) A rectangle has length (2x+3) and width (x-1).

a.) Write an expression for the perimeter of the rectangle.

b.) Write an expression for the area of the rectangle.

The area of the rectangle is 250cm 2 .

c.) How long is the longest side?

d.) What is the perimeter of the rectangle?

5.) The probability that Hannah catches the 6.30am train to the city is 0.7.

If she misses the train, she will be late for work.

The probability that the train will be late is 0.15.

If the train is late, she will be late for work.

What is the probability that Hannah will be on time for work on a particular day?

Worked Solutions to Questions

How to Solve Scale Drawings Word Problems

Scale drawings word problems involve the application of proportion and scale in real-life scenarios. These problems require a solid understanding of how to interpret and work with scale drawings, as well as the ability to set up and solve equations. In this article, we will guide you through the step-by-step process of solving scale drawings word problems. Whether you are a student studying for a math exam or an individual looking to enhance your problem-solving skills, this article will help you master the art of solving scale drawings word problems.

A Step-by-step Guide to Solving Scale Drawings Word Problems

Let’s break it down step by step:

Step 1: Understand the Problem

The first step is to carefully read the problem and make sure you understand what you’re being asked to find. You need to identify the actual size of the object or distance and the scale used in the drawing. This is typically represented as a ratio.

Step 2: Identify the Scale

The scale should be given in the problem. It might be something like $1:100$, which means that $1$ unit on the drawing corresponds to $100$ of the same units in real life. If the scale isn’t given directly, you may have to calculate it from information in the problem.

Step 3: Set Up a Proportion

Once you’ve identified the scale, you can set up a proportion to solve for the unknown. A proportion is an equation that shows that two ratios are equivalent.

For example, if a $1$-inch drawing represents $4$ feet of a building, and you have a $3$-inch drawing, your proportion might look like this:

$ \frac{1 in}{4 ft}=\frac{3 in}{X ft} $

Step 4: Solve for the Unknown

Next, you solve the proportion for the unknown variable. To do this, you can cross-multiply and solve for the variable.

Using the above example:

$1 in \times X ft = 3 in \times 4 ft$

$ X = \frac{(3 in \times 4 ft)}{1 in}= 12 ft $

So, a $3$-inch drawing of the building would represent a $12$-foot actual height.

Step 5: Check Your Answer

Finally, always check your answer to make sure it makes sense in the context of the problem.

In our example, does it make sense that the building is $12$ feet tall if the $3$-inch drawing is scaled up? Since our scale was $1$ inch for every $4$ feet, and we had a $3$-inch drawing, it indeed makes sense that the actual height is $3$ times the $4$ feet, which is $12$ feet.

Remember, practice makes perfect. The more problems you work through, the better you’ll get at solving these types of problems.

by: Effortless Math Team about 10 months ago (category: Articles )

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COMMENTS

Drawings Shape Math Problem-Solving Skills
Abstract. Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults. There is an ongoing debate in the scientific community regarding the nature and role of the mental representations involved in solving arithmetic word problems. In this study, we took a closer look at ...
Art of Problem Solving
Art of Problem Solving offers two other multifaceted programs. Beast Academy is our comic-based online math curriculum for students ages 6-13. And AoPS Academy brings our methodology to students grades 2-12 through small, in-person classes at local campuses. Through our three programs, AoPS offers the most comprehensive honors math pathway ...
How Drawing Unlocks Creativity and Problem-Solving
Drawing is a universal form of expression that transcends language barriers and taps into the innate human desire to communicate visually. Beyond its artistic allure, drawing possesses a remarkable ability to enhance problem-solving skills and unleash creativity. The act of problem solving drawing by putting pencil to paper isn't just reserved for artists; it's a cognitive tool that can spark ...
Design Thinking: Artists Solve Problems
Artists Solve Problems. Design is endlessly trying, refining, improving until slowly something begins to emerge that is so ingenious that it looks like magic if you don't know what went on before: that's what evolution does. - Designer, Joris Laarman. Design is all around us, whether it takes the form of objects and spaces, images and ...
How to solve your problems through drawing (w/ Liana Finck ...
The pain I was feeling, although objectively pretty run of the mill, was impossible to ignore. I knew that drawing was my strongest problem solving tool, so I decided to diagram what I was going through. By making these drawings, I could see how my ex and I had hurt each other and move on. Drawing from my own life was a revelation to me, not ...
Problem Solving Strategy: Draw a Picture
These word problems could be used with grades 2-4 and include a page that specifically states, "Draw a picture…" and then another page of problems were it would be useful to draw a picture, but it is not explicitly stated. The goal is to get students used to organizing the information in a meaningful way to help them better think about ...
Problem Solving: Draw a Picture
The draw a picture strategy is a problem-solving technique in which students make a visual representation of the problem. For example, the following problem could be solved by drawing a picture: A frog is at the bottom of a 10-meter well. Each day he climbs up 3 meters. Each night he slides down 1 meter. On what day will he reach the top of the ...
Hone your problem-solving skills with fine art
KEVIN: After reading your book, it seems art assists problem-solving in two distinct ways. First, studying art develops perceptive skills that we can use to gather the insights and information ...
Resources
Art of Problem Solving offers free resources for avid problem solvers, including games, Alcumus, math videos, the AoPS Wiki, and a LaTeX tutorial. Art of Problem Solving AoPS Online. Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ ...
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Make problem solving easier for students be teaching multiple strategies. Here is an explanation of how and why to encourage students to draw pictures for solving math word problems. Teaching students to draw pictures illustrating the details has many benefits. Draw Pictures Strategy - Problem Solving No matter what grade, drawing pictures can make solving problems a go-to strategy. This ...
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Published February 10, 2022. In the search for novel ways to hone our problem-solving skills, spending time with a work of art may be the simplest and most effective training, according to the art ...
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Elementary Math Problem Solving - Draw a pictureIn this video, we explore one of eight problem-solving strategies for the primary math student. Students are ...
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Drawings of mathematical problems predict their resolution
Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults. Memory & Cognition , 2024; DOI: 10.3758/s13421-024-01523-w Cite ...
When Can Making a Drawing Hinder Problem Solving? Effect of the Drawing
Introduction. Making a drawing is considered a powerful strategy in mathematical problem solving (Pólya, 1945).According to the theory of external representations (), it can support problem solving by helping problem solvers organize the information, and it can make missing and implicit information (e.g., relations between objects) explicit.. Therefore, it deepens understanding and ...
Draw your problem. Your hand, a pen and some paper are…
Drawing benefits the brain in many ways, according to Cara Bean's cute illustrated 'Why Draw?' booklet: Drawing can instigate the development of critical thinking and problem-solving. The ...
Drawings of mathematical problems predict their resolution
More information: Hippolyte Gros et al, Uncovering the interplay between drawings, mental representations, and arithmetic problem-solving strategies in children and adults, Memory & Cognition ...
Problem Solving. Draw Diagram
A diagram can be a rough sketch, a number line, a tree diagram or two-way table, a Venn diagram, or any other drawing which helps us to tackle a problem. Labels (e.g. letters for vertices of a polygon) are useful in a diagram to help us be able to refer to items of interest. A diagram can be updated as we find out new information.
94,100+ Problem Solving Drawing Illustrations, Royalty-Free ...
Creative concept for infographic, presentation, report. Can be used for topics like insurance, strategy, production. of 100. Choose from 94,184 Problem Solving Drawing stock illustrations from iStock. Find high-quality royalty-free vector images that you won't find anywhere else.
How to Solve Scale Drawings Word Problems
A Step-by-step Guide to Solving Scale Drawings Word Problems. Let's break it down step by step: Step 1: Understand the Problem. The first step is to carefully read the problem and make sure you understand what you're being asked to find. You need to identify the actual size of the object or distance and the scale used in the drawing.
Art of Problem Solving
Art of Problem Solving offers two other multifaceted programs. Beast Academy is our comic-based online math curriculum for students ages 6-13. And AoPS Academy brings our methodology to students grades 2-12 through small, in-person classes at local campuses. Through our three programs, AoPS offers the most comprehensive honors math pathway ...
Scale drawing word problems (article)
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. ... Solving a scale drawing word problem. Relate scale drawings to area. Math > 7th grade > Scale copies ...

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ORIGINAL RESEARCH article

Introduction

Drawing Strategy and Linear Overgeneralizations

Quality of Drawing Strategy

Visual Monitoring

Linear Overgeneralizations

Effects of the Drawing Strategy on Linear Overgeneralizations and Performance

Research Questions and Expectations

Expectations for Research Question 1 (Drawing)

Expectations for Research Question 2 (High-Quality Drawing):

Expectations for Research Question 3 (Visual Monitoring)

Expectations for Research Question 4 (Effects on Perceived Performance)

Materials and Methods

Treatment Check

Linear Overgeneralizations and Problem Solving Performance

Perceived Performance

Data Analysis

Effect of Drawing Strategy on Linear Overgeneralizations and Performance

Effect of High-Quality Drawing Strategy on Linear Overgeneralizations and Performance

Effect of Visual Monitoring on Linear Overgeneralizations and Performance

Effect of Drawing Strategy and Visual Monitoring on Perceived Performance

Negative Effect of the Drawing Strategy

Awareness of Linear Overgeneralizations

Strengths and Limitations

Data Availability Statement

Ethics Statement

Author Contributions

Conflict of Interest

Drawings of mathematical problems predict their resolution

The cardinal or ordinal dimensions of problems

Identifying mental representations through drawings

Improving mathematical learning through drawing analysis

Liquid droplets shape how cells respond to change, shows study