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Teaching Through Problems Worth Solving Resource Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 8
Teaching through problems worth solving resource inquiry-based, curriculum-linked, differentiated math problems for grade 2, teaching through problems worth solving resource inquiry-based, curriculum-linked, differentiated math problems for grade 3, peter liljedahl's research on teaching through problem solving, our thinking school, my thinking classroom..., reflections of students learning in a thinking classroom..., teachers learning through problem solving - numeracy leadership group.
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- Date Jun 10, 2016
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Peter Liljedahl's Research on Teaching Through Problem Solving
Developing a thinking classroom.
"What was missing for these students, and their teachers, was a central focus in mathematics on thinking. The realization that this was absent in so many classrooms that I visited motivated me to find a way to build, within these same classrooms, a culture of thinking, both for the student and the teachers. I wanted to build, what I now call, a thinking classroom – a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion."
Peter Liljedahl, Simon Fraser University, Canada
DISTRICT MATH NIGHT PRESENTATIONS
Developing Additive and Multiplicative Reasoning
Additive reading extras.
Multiplicative Reasoning Extras
Professional Development. Workshops. Presentations.
Celebrating 100 Years with Big Beautiful Problems (Grades 6–8) - Alicia Burdess and Jessie Shirley 100 Days of Professional Learning, Online with the National Council of Teachers of Mathematics
Big Beautiful Problems can be life-changing for teachers and students alike as they show how math becomes alive and is connected to our world. See how much fun math can be as we get caught up in the challenge, excitement, and flow of learning. Deepen your skills, confidence, and joy as we experience and explore some of our favorite problems.
Teaching Math Through Problem Solving in a Thinking Classroom Online 2020
Mighty peace 2020, mathematical playground problems, skyscrapers, where are all of the beautiful problems mcata 2019, big beautiful problems geeks unite teaching middle school math 2018, la numeratie. st. gerard - francais, numeracy in gpcsd 2007-2018, teaching through problems worth solving mighty peace 2018, early numeracy 2017, afternoon at holy cross a conversation about math 2018.
Friday afternoon MATH fun with Kateri 2016
Final Cass Math 2015
Problem Solving in STM 2018
Problem Solving at Kateri 2016
Grade 123 New Teachers Day 2018
Mathematics Teaching and Learning at St. Mary Beaverlodge 2016
Admin. Assoc. Mtg. 2015
Diehard Jug Problem
Additive and Multiplicative Reasoning
Additive Reasoning
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PD Friday 2016
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Mathematics Teaching and Learning at Ste. Marie 2017
PD Friday St. Mary Beaverlodge 2016
Numeracy with Admin GPCSD 2017
Problem Solving at Mother Teresa 2016
Mathematical challenges for able pupils all ages in Key Stages 1 and 2
Teaching Through Problems Worth Solving In A Thinking Classroom
Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 2
Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 3
Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 8 Version 3.0
Numeracy leadership group - building capacity in our schools, more learning and problem solving with the numeracy leadership group.
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Visibly random groups, building thinking classroom, four purposes of assessment, peter liljedahl research - now you try one, peter liljedahl research -homework, learning, teaching, and loving math - books to read.
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Hundreds of Challenging Math Problems Worth Solving
The Mathematical Practices that are in the College and Career Readiness Standards for Adult Education define what it means to be a mathematically proficient student. As adult education instructors, our job is to help our students navigate over the swells in the tempest of their angst until they get to the point where they have enough confidence in their own abilities to weather the sea of mathematics.
Consider the first standard, MP1: Make sense of problems and persevere in solving them. It explains that our students should be able to approach a math problem or situation — something a bit challenging that they might never have seen before — and find a way into solving it. They should be able to develop flexibility with numbers and the ability to approach problems with the understanding that there are multiple pathways that can lead to a solution. They should have confidence that if they stick with it long enough, they can find an appropriate strategy that will solve the problem. That means that we, as their teachers, need to provide intriguing math problems that catch our students’ attention and then nourish and celebrate the different methods of problem solving that our students intuitively gravitate towards. ( Mathematical Mindsets, Jo Boaler , 2016)
One resource I strongly recommend for these kinds of math problems is Open Middle . As of this writing, there are over 300 high-interest problems housed at the website. They are organized by grade level and topic that make it easy to zero in on a particular skill level and area of focus that you want to bring into your classroom.
So what makes a problem open-middle?
The website explains:
- They have a “closed beginning” meaning that they all start with the same initial problem.
- They have a “closed end” meaning that they all end with the same answer.
- They have an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem .
Problems with Multiple Solution Methods
As an example, consider the following problem. At first glance this may seem pretty straightforward, requiring only a knowledge of the rules of subtraction in order to solve it
Each problem comes with a hint. For this problem the hint is: Remember that zeros are not allowed. Because you can only use each digit once, the digit in the hundreds place can’t be the same.
Try solving it yourself before looking at the student solutions listed below. Keep in mind that there is one optimal answer to this problem albeit multiple ways to arrive at this solution.
(To find this and other subtraction problems – Open Middle Subtraction )
This is listed under Grade 2, so keep this in mind when navigating the website. Don’t necessarily start with high-school level if your students need the basics in algebraic thinking and are just beginning to learn how to be flexible with numbers . If you’ve already worked out a solution to the problem, read on to see examples of how my students approached this.
Notice that one student did not arrive at the optimal answer on his first try. He was a tad disappointed when he saw that a classmate had found a smaller difference until I asked him if there might be another combination of numbers, besides what his peer had already found, that might lead to the optimal answer of 14. This led to a flurry of renewed interest in the problem as all students, even those whose calculations had already led to 14, busied themselves in trying to discover what other number combinations would work. Two of those solutions are pictured above, which clearly indicate a pattern that can lead to additional solutions.
To wrap up this activity, have each student who discovered the optimal answer of 14 explain how they arrived at their answer. When all solutions that students found are placed on the board, ask students what they noticed about the pattern that is evidenced in the solutions. If they have not already found and commented on the pattern in the combinations of numbers that work, ask students if they see any patterns in the solutions and discuss how the pattern changes with each equation. Recognizing patterns and developing flexibility with numbers go a long way in helping our students to become mathematically proficient.
Problems with Multiple Solutions
Similarly, there is a problem called Table of Values: Not a Function .
These activities can reinforce the basic definition of a function. Many times we focus our instruction on finding the rule of a function when we present a table of values to our students, but it’s also important that they are able to look at an input/output table, a graph, a range and domain set with arrow notation, or sets of ordered pairs and correctly identify which ones represent a function and which ones do not.
(To find these and other function problems – Open Middle Functions )
There may be no readily apparent rule for a set of values that our students might see on an HSE exam, other than it is or is not a function based on the definition of a function: a relationship between two values where each input value has only one unique output. While having an understanding of this rule is a procedural skill, requiring students to create their own set of values using their choice of an input/output table, graph, range and domain set with arrow notation, or sets of ordered pairs, requires a higher depth of knowledge, especially if students are subsequently tasked to create a real-life scenario that their function represents. Students are more apt to remember the basic definition of a function if they have engaged in this kind of an open-ended activity.
Whenever you are planning a unit on a particular math topic, it is well worth visiting Open Middle. There is such a wide range of problems, you are sure to find something to engage your students and teach them that they are capable of creative problem-solving.
Explore the site and then take a minute or two to add a comment below. Let us know what problems you use and how they go with your students.
The Math Practitioner is a newsletter published by The Adult Numeracy Network (ANN) for teachers of adult numeracy and high school equivalency math.
If you are interested in receiving this newsletter on a regular basis, please join our national community of adult education math practitioners at the ANN website .
ANN is a community dedicated to quality mathematics instruction at the adult level. We encourage collaboration and leadership, and would like to hear about what you are doing in your classroom to help your students reach their goals in mathematics. We’re also interested in your “aha” moments as an adult education instructor. To that end, we encourage submissions of articles, activities, and other items of interest related to math for adult learners for consideration for our newsletter. Please direct all correspondence regarding The Math Practitioner to: Patricia Helmuth, [email protected] .
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1 thought on “ Hundreds of Challenging Math Problems Worth Solving ”
Thank you so much for introducing this fantastic resource.
I really love how the Open Middle problems engage students, especially in a mixed-level math class.
Consider an alternative to the subtraction example you wrote about, where a teacher gives their students a handout with a bunch of problems requiring the subtraction of three digit numbers.
Faster students who are comfortable with the procedure will race through the handout, some completing it before you’ve even finished handing out the sheet. And then you have to give them something else to do. This problem gives students a lot of practice with calculation (think about how many different subtraction problems they’ll do on their way to solving this problem) but it requires more than that. Students have an opportunity to reason, look for patterns and structures, which deepens their learning and extends it beyond just this one problem. There is a puzzle-like quality to the problems on Open Middle. Students persevere because they are not just doing all of these one-off calculations. They are doing a series of calculations that connect to each other and are moving towards the overall goal of finding the smallest difference. It is also very easy to extend the problems here for students who do finish early. If a students finds what they think is the smallest difference you can ask them to prove that their difference is the smallest. You could also ask them for the largest possible difference between two 3-digit numbers, given the same conditions.
I love that this site has challenging problems for exploring number sense and operations and challenging problems in functions, algebra and geometry. One thing I really appreciate about the algebra problems is that because they are “open middle” – meaning they have more than one way to solve them – they are accessible to a wider range of students. When students work on problems where there is only one solution method, it often curtails their sense-making and perseverance because for those kinds of problems, you either know how to solve them or you might as well put your pencil down. These open middle problems keep students in the game and give them an opportunity to do a lot of great mathematical reasoning.
Also, I had a lot of fun working on this great Open Middle function problem posted on Twitter by Graham Fletcher (@gfletchy) – https://pbs.twimg.com/media/C9VkpFwVwAAV5BH.jpg:large
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Visual Models for Problem Solving in 1st Grade
May 10, 2020
As students enter 1st grade, they continue to work on math comprehension using early structures, like the Kindergarten journal we introduced last week, but now we begin to add visual models to the mix!
Let’s recap a child’s developmental journey through problem solving:
- In the early childhood years, a child needs lots of developmentally appropriate experience interacting with real objects in a physical world .
- The physical world is captured in a quantitative picture , which young children observe and use as a springboard for mathematical conversations.
- We transition into a more structured math work mat to help young students be able to connect numbers to words and words to numbers, still using familiar situations from real life.
- The math work mat gives way to a formal math journal in Kindergarten that makes use of math comprehension skills. It provides a structure for students to explain their understanding of numbers within real world situations that will carry on throughout elementary school.
Each of the stages of development builds on the skills developed in the previous step, so it is important that students aren’t rushed through these stages. The goal is to teach students the why behind the how so they aren’t just memorizing procedures but truly understand what is happening as they solve problems.
This 1st grade year is the last stage in the Math4Littles progression , in my opinion. After this, there isn’t much scaffolding, so we really want to carefully implement all the previous stages of problem solving before we turn the students loose, because we don’t want them to start guessing and checking. In taking students on this developmental journey, we are trying to build them a solid foundation for visual models to help them to understand problem solving.
How We Used to Teach Problem Solving
When I was teaching 1st grade, I remember a strategy that we used for problem solving called the C.U.B.S. method. Each of those letters stood for a step in the problem solving process so students could remember what to do: C – circle the numbers, U – underline the word, B – box the operation, S – solve the problem. Seems like a simple process that gives kids a really great structure to start to understand what words problems are asking, right? But what I realized is that this strategy doesn’t hold up long term.
“Shannon has 5 lollipop and Scott has 4 more lollipops than Shannon. How many do they have all together?”
I watched students follow that procedure with this type of problem. They circled the 4 and 5, underlined important information and put a box around the words all together , which means add because we’ve all seen the T-charts of addition/subtraction vocabulary – it says difference , it means we’re going to subtract, if you see all together , we’re going to add. But that strategy gives me 4, 5, and all together . If you go back to the question, you’ll realize the answer isn’t 9.
As I often do, I asked myself why ? Why isn’t it 9? A little more reading comprehension is required to decode that answer. The problem says I had 5 lollipops. Scott had 4 more than me, which means he also had 5. Adding that up, he had 9 and I had 5, so there were 14 all together.
Why are we teaching kids procedures with concepts they don’t understand? Sometimes the strategies that we teach in math are conditional, meaning they only work for a certain amount of kids or a certain length of time. Then you have to worry about teaching them when to apply it and the rules for applying it, and what was meant to make things easier for students ends up being more complicated.
When we start working with strategies, I want to be able to find that vertical zip, meaning if I show you how this strategy might work in first grade, it has to work as the child gets older too so that they don’t have to learn a whole new set of strategies every year because every teacher teaches it differently. Honestly, the CUBS method would probably work for 75% of the problems in first grade. Students are doing more advanced part-whole addition problems, part-whole subtraction, part-whole missing addends, and they’ll start doing a few multi-step problems, all of which fit in the part-whole family, for which the CUBS method works well. But when you move out of that genre of problems, it falls apart.
In the Kindergarten journal, we featured part-whole addition, part-whole subtraction, part-whole missing addend, a few problems with teen numbers, and a mixed review. The journal is very structured because it is intended to start students thinking about what they’re reading in the story problem: We have a story, a sentence form, a quick draw area, a number bond, a 10-frame, and a computation area. As they transition to 1st grade, how do we remove some of that scaffolding while still keeping it developmentally appropriate?
We have to be really careful with the way we make this transition, because very quickly, students can jump to the “circle the numbers, box the word” strategy and many times they just appeal to us because they don’t know what to do. It’s a word problem and it’s confusing, so they just add because we’re talking about adding that week.
Additive Comparison Problems
Additive comparison problems, where I have an amount and you have the same amount but you may have more or less than I do, are introduced after students have spent some time working on multi-step part-whole problems.
This type of problem is really a play on language, in my opinion, which makes it really confusing for kids to understand exactly what it is asking. So, we really want kids to take a step back to understand the additive comparison problems, which are coded AC in our journals. I find that building these problems with unfix cubes is a good way to start.
Let’s take this problem: Shannon has 10 pet rocks and Sherry has 4 pet rocks. How many more rocks does Shannon have than Sherry?
In some ways it seems like this might be a missing addend problem, but in fact we’re really comparing my pet rocks to Sherry’s pet rocks and we’re asking how many more does one have than the other. This really requires students to take it to the concrete level and make a bar model with unifix cubes.
I put 10 cubes to represent Shannon’s pet rocks, and then I’ll use different color cubes to show Sherry’s 4. Then, I want to compare the lengths of those two bars and figure out what the problem is really asking, which is the gap between where Sherry’s bar stops and Shannon’s bar stops. The question mark is asking for how many more does Shannon have?
Sometimes, the language of an additive comparison problem might be reversed and say how many less does Sherry have? Since it is a play on words, which sometimes becomes confusing for students, we really need to put thought into how we go about teaching kids to do a problem like this.
Visual Models for Additive Comparison Problems
If I were to line up all the programs we work with, every one of them has bit of a different name for visual models: model drawings, tape diagrams, bar models, unit bars. We’re going to universally call them visual models for word problems.
These aren’t the little quick draws we’ve been doing in Kindergarten because, as students get older and the problems get more complex, I’m not going to be able to draw 13 ducks and then 9 more because it will take too long! Instead, I want to put it into a visual model that has these units.
This first grade year is a transitional time where kids are going from the quick draw to what I’m going to call proportional bars, which have a length of individual cubes that are representative of the quantities we’re talking about in the problem.
I just was working with a first grade teacher last week on a Zoom call, and this teacher had not been able to attend our workshop on their campus about visual models. She, like most teachers I work with, didn’t understand why visual models were so important. She thought her students should be able to do quick draws and didn’t understand why they had to do boxes. She told me she was a big proponent of encouraging students to solve problems in different ways, so why would she possibly want to teach students a procedure like this and make them solve word problems in this way.
After I took her through the same progression of problem solving we’ve been going through in our blog the past few weeks, she was sold! I took her up through fifth grade to help her see why it is that, in 1st grade, we’re asking students to stop doing quick draws and start to use a visual model that has a unit bar with different pieces. This proportional model is also a great transition into using a non-proportional bar.
Let’s say I had 92 pet rocks and Sherry has 45 pet rocks. A quick draw clearly won’t work for this problem, and I don’t have enough room on my paper to draw a proportional model for those numbers. But I can draw a longer bar that represents Shannon’s rocks, write in 92 rocks, and draw a shorter bar to show Sherry’s 45 rocks so I could see the proportionality.
The hardest thing to remember when we do visual models for word problems is that it actually has nothing to do with math! We’re not actually solving the problem on the model; we are solely using a reading comprehension strategy.
One of the biggest misconceptions we addressed when we started rolling out the 1st grade journal samples that I’ll be using in this video, was that the total doesn’t go on the line. If the problem asks for a total, we represent that in the visual model with a question mark.
We also want to make sure that we label the visual model. For example, putting a B above the books that Erin had and an L above the books she got at the library.
The whole point of this process is to provide a systematic way for students to work through problems that doesn’t stop working after 1st grade or when you start working on a different type of story problem. In fact, this strategy carries through multiplicative comparison problems and fractions, all the way into ratios and proportions in middle school.
Step-by-Step Problem Solving
Read the problem. Then, have someone read it and repeat it, and every time a new piece of math information is presented, we’re going to put a chunk. So, as kids are reading the problem, they start to learn how to dissect what’s being asked.
Not all first grade students will be able to read the story problem, but this process is modeled day after day after day in the first grade classroom, so eventually the child will become independent.
I’m going to read a story problem: Mark has 9 strawberries, 6 of them are small. The rest are large. How many strawberries are large?
Then, I’ll go back and read it in chunks: Mark has 9 strawberries . This is a new piece of mathematical information, so students will repeat that statement back and highlight or put a line there. The students also like to say chunk! Then we continue reading: Six of them were small. I’ll stop, repeat it, and the students say chunk! as they mark that chunk in their journals. Now we have two pieces of mathematical information. Let’s continue: The rest were large . Repeat and then chunk! So, we’ve got three sections of information that the problem has given us that we need to replicate in our visual model. Finally, How many strawberries are large? Repeat that and then chunk!
By going through the problem slowly and methodically, students can really see these sections that they’re reading, and, as they’re going on to the subsequent steps of solving the problem, they can actually check off that they’ve included all the chunks of information in their visual model.
In our problem, it asked me how many strawberries are large? To put it in a sentence form, I would say: Mark has ____ large strawberries. I like to say Hmm for the ____ as we’re reading it out loud.
In Kindergarten, we provide the sentence for students, leaving the blank space for their answer. But in 1st grade, we take some of the scaffolding away. It might say “There were _____ large ____” and the students have to fill in the blanks.
The sentence form is a great way to make sure that kids are comprehending what they’re reading. Generally, students in first grade have a difficult time trying to create a sentence form, because they aren’t yet developmentally ready to give you a complete answer in reading. But students will be required to do a sentence form in 2nd through 5th grade so we can be sure they understand the problems being asked, so it’s really great practice to start in 1st grade with the scaffolding.
Proportional model. We start the 1st grade year with a proportional model. We may scaffold here for the who or the what, and students will eventually start to learn what goes in that visual model. In this case, we’re talking about all of Mark’s strawberries, even though the question itself is only asking about how many of them are large.
In a proportional model, you might see the 9 squares. This is a missing addend problem so that title is going to have PWMA at the top, and there will be exactly nine squares. Some people might think that’s giving it away, but remember the goal of visual models? It’s not to solve the problem but understand what’s going on in the problem, so we’re more concerned about whether or not the student can label the drawing correctly.
In this example, the student would total the bar at 9 and check off the first chunk of the problem that we read earlier – Mark has nine strawberries.
The next part says “6 of them are small.” In 6 of my boxes, I’ll make six Xs, or I might make small circles, and at the top I can either write small or abbreviate with an s .
Then it says “the rest are big.” I could label that other section of the boxes B for big, or write the whole word if I wanted. Then, I need to put a question mark above that section between 9 (the total number of strawberries) and 6 (the number of small strawberries). That section represents the large strawberries, which is what my sentence form reminds me that I’m looking for.
Technically, a student could just look at this easy proportional model and say there are 3 large strawberries because it’s right there in front of them. So some people might think this journal is just too easy, but at the end of the day, students are solidifying the process. They’re going back up to the problem and putting a check when they add Xs or circles for the six small strawberries. They’re putting in a check when they’ve talked about putting in the large strawberries. Then they put a question mark to show what we’re looking for. There’s a lot of detail that we’re looking for kids to have to interact with the text in math to show the comprehension.
In some of our schools, we will do a unit bar at the bottom of the page. In the 1st grade journal we’ve created for Math4Littles, we’re going to leave the bar off and introduce the non-proportional bar a little bit later in the year. There is nothing wrong with having a model of the proportional bar and then underneath it having the non-proportional bar. In our journal, we plan to show the proportional bar, and then bring in both types of bars so that kids could see the relationship between the two. If where about this non proportional bar, where would I slice it to put the nine in? And then where’s my question mark? is it labeled? etc.
The integral parts of visual models are: labelling the who or what, taking the bar and adjusting it based on the information that’s given, and writing in their question mark. Then it’s time to solve!
Computation. Although this step might not seem necessary because our sample problem is so simple, and to first graders after they do so many, it seems simple and both teachers and students might wonder why they’re even doing it, but I can promise that these problems will become more complex, very quickly. In our 1st grade journal, we will feature this look at the proportional bar, and then transition to having proportional and non proportional models, and then eventually just leaving it blank and having the student put in a non proportional bar to see that they can develop this progression.
1st Grade Goals
The goal is, by the end of their first grade year, students should be able to solve problems with larger numbers and a non-proportional bar. You certainly don’t want to rush that progression. 1st grade is a really nice scaffold for students to get to that point of independence, because when we get to 2nd grade, we don’t do a whole lot of scaffolding. There are more open-ended sentences, more blanks, and students are doing more of the work.
Additionally, we want to mix up the types of problems we’re solving, give students time to understand them. You might do three days of part-whole addition to see if they can get it under their belt. Then do some part-whole subtraction, then mix the two to see if students are just following a pattern where we’re adding today or subtracting today. We want to know that they can really apply what they’re learning. Multi-step problems, where students have to add and then subtract, or vice versa, are next. Give students lots of good practice, and then mix it up again to see if they’re really following the words, or if they’re just learning a procedure. The last type of problem that we would integrate in the first grade is additive comparisons.
Video Tutorials
In the video tutorials, you’ll see aspects of four different problems being displayed. Some will have the proportional bar, some will have the proportional and the non proportional and some just won’t have it just so you can get an overall idea of what this looks like as we go.
[yotuwp type=”playlist” id=”PL76vNL0J-a405ysBIwEwXfaMp5883yGh4″ ]
As you watch the videos, think about how you could set this up in your classroom, starting with some of the sample problems that we’re offering as a free download today. We will be releasing a full 1st grade journal soon, so stay tuned!
Join us next week for problem solving in 2nd grade: What are the different problems that 2nd grade is going to encounter? How are journals coded? As we start to look at how journals are coded, which you certainly could use these tutorial videos right away in your classroom or in your distance learning by thinking about story problems in a different way.
*Addition , *Subtraction , *Word Problems , Audience - Lower Elementary (K-2) , Series - Math4Littles | 0 comments
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Teaching Through Problems Worth Solving Resource Inquiry-based, Curriculum-linked, Differentiated Math Problems for Grade 8. Picture · Grade 8 Version 3.0.
Grade 4-7, 8-9 ; Subject Mathematics ; Resource type Lesson plan/Unit plan
The Alicia Burdess website is a resource site for free excellent teacher resources ... Teachers Learning Through Problem Solving - Numeracy Leadership Group.
Diehard Jug Problem · Teaching Math Through Problem Solving in a. Thinking Classroom Online 2020 · Mighty Peace 2020 · Mathematical Playground Problems
Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the
CHALLENGING MATH PROBLEMS WORTH SOLVING DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL Get Our Favorite Problems Take The Online Workshop WANT GOOGLE
Fall 2020 AMTNYS Fall ConferenceNurturing an environment where learners actively look for, and engage in finding multiple strategies for
NLPS Learns · Teaching Through Problems Worth Solving · Three Act Math Tasks · Menu Math · Open Middle Tasks · Questions about teaching, learning, resources or
As of this writing, there are over 300 high-interest problems housed at the website. They are organized by grade level and topic that make
Each of the stages of development builds on the skills developed in the previous step, so it is important that students aren't rushed through