math problem solving routine

Enhance Learning with Number Sense Routines

You might ask, “What are Number Sense Routines, and how can they revolutionize the teaching process?” Well then, let’s delve into what they are and the immense benefits they can bring to your classroom.

What are Number Sense Routines?

Number sense routines are a series of activities, targeted tasks, and engaging exercises aimed at helping learners develop a deep understanding of numbers and arithmetic. Designed to be fun and engaging and promote critical thinking, these routines make math seem less like a rigid set of rules and more like an exciting puzzle to solve. Think of it as your secret weapon for making mathematics engaging and enjoyable for your students!

These routines come in a variety of formats. Some might be short, 5 to 15-minute tasks curled into your lesson, while others might be standalone activities. The key is to make them consistent and regular fixtures in your lessons so your students become increasingly comfortable with making sense of numbers.

Benefits of Implementing Number Sense Routines in the Classroom

But why should you incorporate number sense routines in your teaching? Well, simply put, they are game-changers! Here are just a few of the benefits:

• Building a Strong Foundation:  Number sense routines can help your students build a strong foundation for understanding more complex mathematical concepts down the line. By getting a solid grasp of numbers and their relationships, learners can better tackle problems involving arithmetic, algebra, geometry, etc., with great confidence and ease.
• Boosting Problem-Solving Skills: These routines help learners think critically about numbers and arithmetic operations . As they routinely encounter different problems and challenges, their problem-solving skills become sharper.
• Encouraging Active Learning: Number sense routines go beyond mere rote learning. Instead, they foster an environment of active engagement where learners are actively investigating, reasoning, and questioning. They also provide opportunities for learners to discuss their thinking and strategies with their peers, promoting collaboration and teamwork.
• Building Confidence: Regular engagement in number sense routines helps students gain confidence in their mathematical abilities. They become comfortable with numbers, simple arithmetic operations, and problem-solving strategies, encouraging them to tackle more challenging problems.
• Flexibility:  These routines can be adapted to cater to different learners’ abilities and needs. This flexibility ensures every student is engaged and benefits from the activities regardless of their skill level.

Revamping your math lessons with number sense routines might appear daunting at first, but trust us, the benefits are immense. They help transform your maths lessons into engaging, active learning sessions that can improve students’ understanding of numbers, increase their problem-solving skills, and boost their confidence in tackling mathematics. So why not give it a go? Give your learners this invaluable tool to thrive in their mathematical journey and beyond. You’ll be amazed at the difference it makes!

The Number Bounce Routine

Now, you’re probably asking, “What exactly is the Number Bounce Routine?” It’s a powerful number sense routine that, when incorporated into your daily syllabus, can significantly enhance students’ counting skills and broaden their understanding of numerical relationships.

How Does the Number Bounce Routine Work?

Imagine a game of ping pong but with numbers. That’s pretty much how the Number Bounce Routine operates! You start by setting up a numerical range, say from 1 to 50. Then, using a random number generator, generate a number within this range. This will be your “bounce” number.

The student’s task in this routine is to bounce up and down to or from that number. For example, suppose the bounce number is 20. A student might “bounce” down by counting backward from 20 to 1. Another student might “bounce” up by listing numbers from 20 to 50.

Additionally, you could introduce a “bounce pattern,” like bounce by twos or fives. This way, students get to exercise their growing skills in skip counting.

Encourage your students to be creative! They can bounce at a slow pace, fast pace, high pitch, low pitch, even in a singing voice! This not only makes the routine more engaging but also stimulates their auditory learning pathways.

Benefits of the Number Bounce Routine

Let’s explore some of the benefits that the Number Bounce Routine brings to your classroom.

• Enhances Counting Skills:  This routine provides a fun way for students to practice counting. By bouncing up and down different number scales, they hone their counting skills, both forward and backward, and can easily recognize the order of numbers.
• Reinforces Number Relationships: The activity promotes a working knowledge of number relationships. When bouncing to and fro a number, students visualize the numerical scale, understand sequences, and see how numbers relate.
• Boosts Memory:  Bouncing aloud and in various voices stimulates auditory learning, helping students retain knowledge more effectively.
• Engages Active Participation:  The unpredictability of the bounce number piques students’ curiosity and introduces a play element that motivates them to participate actively.
• Encourages Peer Interaction: As students take turns bouncing numbers, they engage with each other, providing an excellent platform for them to learn from their peers.

Like any routine, the key to the Number Bounce Routine’s success lies in regular practice. Commit to it, and gradually, you’ll witness significant improvements. Your students will not only gain a firmer grasp of numbers and their relationships but also find math far more enjoyable.

To close off, remember, as Confucius once said, “Tell me, and I will forget. Show me, and I may remember. Involve me, and I will understand.” So, involve your students in routines like the Number Bounce, and they will understand and relish the magic of numbers!

Counting Sequences Routine

You may have dabbled with number routines before, but have you tried your hand at the Counting Sequences Routine? It’s an addition to your mathematical toolbox that can dramatically help boost your number insight and the overall comprehension of numerical relationships in your classroom.

What is the Counting Sequences Routine?

Now, here’s what you need to know. The Counting Sequences Routine consists of simply counting aloud but with a twist. Instead of counting in the usual sequence of one, two, three, and so on, you count in various set numeral sequences.

For instance, you can count by twos, threes, fives, or even tens. What’s more, don’t limit yourself to starting from one. Initial numbers could be something else like seven, then counting up by threes: seven, ten, thirteen. You see, the possibilities are limitless!

The aim is to expose you to different sequences, making you comfortable and familiar with numbers. In an interactive environment, everyone counts in a particular sequence. It illuminates the patterns hidden within numbers and develops a deeper understanding of number relationships.

How the Counting Sequences Routine improves number relationships

Counting Sequences Routines provide an engaging way to bolster your knowledge of numbers. They’re more than just easy-breezy counting; they’re a trove of benefits that can dramatically improve your understanding of number relationships.

• Encourages Number Familiarity:  By counting in various sequences, you familiarize yourself with numbers and their order. You’ll be surprised at how soon you start feeling comfortable with numbers that once seemed distant or unrelated.
• Helps Identify Number Patterns: As you partake in various sequences, you notice specific patterns or principles. For instance, when you count by twos or tens, the last digit of each number follows a distinct pattern. Noticing such patterns can boost your number sense significantly.
• Enhances Mental Math Skills: Counting sequences promotes mental math proficiency. Regular practice can help you master different addition patterns, making computations quicker and more accurate.
• Boosts Confidence in Mathematics:  As your familiarity and understanding of numbers grow, so does your confidence. By recognizing patterns and improving mental computation, you build confidence in your mathematical abilities—a surefire ticket to excelling in future mathematical endeavors.
• Build Peer Support and Interaction:  As everyone participates in the routine, it fosters a supportive, accepting environment. It can help you learn collaboratively by listening to each other count, correcting miscounts, and applauding successes.

In conclusion, the Counting Sequences Routine is a dynamic and beneficial strategy that provides the platform for a deep understanding of how numbers relate. Keep in mind that the fundamental ingredient in the recipe for success with these routines is consistency. Make them a part of your regular schedule, and before you know it, you’ll be a number maven!

Remember, as Benjamin Franklin said, “Tell me, and I forget. Teach me, and I remember. Involve me, and I learn.” So, get involved, immerse yourself in the Counting Sequences Routine, and unlock the joy of numbers!

Problem-Solving Routine

Alright, let’s dive into another powerful routine – the Problem Solving Routine. These practices equip you with strategies to face mathematical problems head-on and solve them efficiently. By making this routine a habit, you will build a solid foundation of skills that help foster critical thinking.

Introducing the Problem-Solving Routine

The Problem Solving Routine , as the name suggests, focuses on revising and improving your problem-solving abilities. It’s not solely about the right or wrong answers. Instead, it’s majorly about the journey there – the process. This routine encourages you to understand the problem deeply and explore multiple strategies and solutions.

Engaging in these routines frequently is an integral part of developing your mathematical prowess and enhancing your overall mathematical literacy. As you work through these routines, you’ll notice a gradual increase in your confidence and ability to approach problems from various angles.

Now, what does the process look like in practice?

Consider a mathematical word problem given to you. Instead of jumping in to solve immediately, take a moment to understand the problem’s context fully. Ask questions like: What is the problem asking? What information is provided? What strategies can I apply?

This phase, known as the exploration phase, boosts your understanding of the problem and provides a clear path to solving it. The illustration phase follows next, where you can visually represent the problem to deepen your understanding further. After these phases, you can confidently approach solving the problem, verifying your answers, and discussing different tactics and solutions with your peers.

Enhancing Critical Thinking and Problem-Solving Skills

Here’s the good news: the benefits of engaging in a problem-solving routine extend far beyond the math classroom. They’re essential skills that will equip you for real-life situations as well!

• Fosters Critical Thinking: Problem-solving provides plenty of opportunities to think critically. How do you interpret the problem? Are there multiple ways to solve it? What approach seems most efficient? Your ability to answer these questions effectively improves your critical thinking skills, not just in math but in every aspect of your life.
• Boosts Confidence: As you work through problems, apply different strategies, and find successful solutions, your confidence in your mathematical abilities surges. This confidence can seep into your approach to other areas of life, encouraging you to tackle challenges head-on.
• Encourages Persistence: Problems are often complex and demand time and effort to solve. By engaging in a Problem-Solving Routine, you learn the valuable skill of persistence – sticking with a problem until you figure it out.
• Enhances Mathematical Literacy: Finally, consistent problem-solving enhances your mathematical literacy – your ability to reason, analyze, and communicate about math.

Go ahead and incorporate this routine into your math practices . Remember, the key to seeing the benefits is consistency. Embrace the Problem-Solving RoutProblem-Solvingence the gratification of unraveling mathematical puzzles, all while boosting your critical thinking skills and becoming a more confident problem solver. Your journey into becoming a problem-solving whiz starts here!

Using Number Sense Routines in the Classroom

At this point, you are fluent with the concept of problem-solving Routines. Now, let’s talk about another impactful practice: Number Sense Routines. You will find these routines especially beneficial in building students’ confidence, understanding, and love for mathematics.

Importance of Using Number Sense Routines in the Classroom

Number Sense Routines  are quick, engaging activities that help students develop a deep understanding of numbers, their relationships, and the number system, a fundamental aspect of their mathematical curriculum. These routines are not just about rote memorization of numbers or math facts. Instead, they provide a robust foundation for numerical fluency and flexible thinking, thereby enhancing students’ ability to solve complex math problems with ease.

Let’s breakdown why you break into implementing these routines in your classroom:

• Building Numerical Fluency: Numerical fluency doesn’t mean quickly rattling off multiplication tables . It denotes an in-depth understanding and capacity to manipulate and play with numbers. Number Sense Routines aid in strengthening this numeracy skill.
• Fostering Flexible Thinking:  Remember, just as there can be multiple routes to a destination, there may be different methods to solve a math problem. Amid activities like composing and decomposing numbers, students learn to approach problems flexibly and creatively.
• Boosting Confidence: As students get comfortable with numbers and start understanding their properties, their confidence shoots up. They become enthusiastic about learning and face new math concepts enthusiastically.

Remember, as an educator, your goal isn’t just to teach math. You aim to build lifelong learners who can look at math to see intriguing puzzles, not intimidating hurdles. Incorporating Number Sense Routines in your classroom is one step towards achieving that goal!

Further Resources for Implementing Number Sense Routines

You might be wondering, “These wonders are promising, but how do I implement them in my class?” No worries. Let’s explore resources to help seamlessly implement Number Sense Routines in your teaching practice.

• Books: Jessica F. Shumway and Dr. Nicki Newton have written excellent resources on Number Sense Routines that provide useful insights, lessons, and strategies.
• Online Resources: Websites like Illuminations, YouCubed or the National Council of Teachers of Mathematics website provide a wealth of free, readily accessible Number Sense Routine activities and ways to implement them effectively.
• Educational Apps: Applications like Math Expressions , Number Sense Trajectories, and Number Talks can be fun, interactive tools to practice these routines.

As you start using these resources, remember that consistency is key. Make Number SencriticalRoutines a regular part of your math instructions, and watch your students develop into confident, flexible, and creative problem solvers!

Building a strong foundation in solid math is not just about solving problems; it’s also about understanding them, developing a relationship with numbers, and becoming flexible, confident thinkers. By incorporating the Problem-Solving RoutProblem-Solving Sense Routines in your classroom, you’re nurturing these skills in your students, helping them become adept in math and beyond! The journey might seem challenging sometimes, but remember, the end result is beyond. Start today and watch your students transform into mathematicians of morrow!

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5 Ways to Build Math Problem Solving Skills (based on brain research)

Whether talking about state tests or meeting with your team to plan the next math unit, the conversation inevitably turns to word problems. But knowing how to build math problem-solving skills without resorting to pages of boring story problem practice can be hard.

These days word problems aren't the basic one-step wonders that many of us dealt with as students. Instead, multi-step story problems that require students to apply multiple concepts and skills are incorporated into instruction and state assessments.

Understanding brain research can help simply the process of teaching this challenging format of math problem-solving to students, including those who struggle.

What research says about building master problem solvers in math

Have you seen how many math skills we must teach these days? No teacher has enough time to build critical math skills AND effectively teach problem-solving…or do they?

Research would argue we are going about these tasks all wrong. They say there are many reasons students struggle with math word problems , but one big one is that we aren't doing what's best for the brain. Instead, here's what the brain research says about the must-have elements for building step-by-step math problem-solving mastery.

Finding #1: Becoming a master problem solver requires repetition.

Duh, right? Any good teacher knows this…but what's the best recipe for repetition if you want students to master math word problems? How much practice? How often?

Let's start with the concept of mastery.

How do you develop math problem solving skills?

In the 1990's, Anders Ericsson studied experts to explore what made some people excel. Findings showed a positive correlation between the amount of deliberate practice (activities that require a high level of concentration and aren't necessarily inherently fun) and skill level.

In other words, the more practice someone gets, the more they improve. This became the basis of Malcolm Gladwell's 10,000-hour rule, which stated that it takes 10,000 hours to make you an expert in a field.

But what should that practice look like for students who struggle with word problems? Is it better to have a deep dive into story problems, or do short bursts of practice do more for long-term understanding?

Designing Better Word Problem Activities: Building Step-by-step Math Problem-Solving Practice

We can look at Ebbinghaus' work on memory & retention to answer that.  He found spacing practice over time decreased the number of exposures needed. In other words, small amounts of practice over several days, weeks, or even months actually means you need LESS practice than if you try to cram it all in at once.

For over 80 years, this finding has stood the test of time. While research has shown that students who engage in mass practice (lots of practice all at once) might do better on an assessment that takes place tomorrow, students who engage in repeated practice over a period of time retain more skills long-term (Bloom & Shuell, 1981; Rea & Modigliani, 1985).

And how long does the research say you should spend reviewing?

How long should problem-solving practice really be?

Shorter is better. As discussed earlier, peak attention required for deliberate practice can only be maintained for so long. And the majority of research supports 8-10 minutes as the ideal lesson length (Robertson, 2010).

This means practice needs to be focused so that during those minutes of discussion, you can dive deep – breaking down the word problem and discussing methods to solve it.

Teacher Tip: Applying this finding to your classroom

Less is actually more as long as you plan to practice regularly. While students who struggle with word problems may need a great deal of practice to master word problems, ideally, this practice should be provided in short, regular intervals with no more than 8-10 minutes spent in whole group discussion.

Here are a few simple steps to apply these findings to your math classroom:

• Find 8-12 minutes in your daily schedule to focus on problem-solving – consider this time sacred & only for problem-solving.
• Select only 1-2 word problems per day. Target step-by-step math problem-solving to build math problem-solving skills through a less-is-more approach using Problem of the Day .

Finding #2: Students who are challenged & supported have better outcomes.

Productive struggle, as it is called in the research, focuses on the effortful practice that builds long-term understanding.

Important to this process are opportunities for choice, collaboration, and the use of materials or topics of interest (which will be discussed later).

This productive struggle also helps students build flexible thinking so that they can apply previously learned skills to new or unfamiliar tasks (Bransford, Brown, & Cocking, 2000).

“Meaningful learning tasks need to challenge ever student in some way. It is crucial that no student be able to coast to success time after time; this experience can create the belief that you are smart only if you can succeed without effort.” -Carol Dweck

It is also critical to provide support and feedback during the challenging task (Cimpian, Arce, Markman, & Dweck, 2007). This prevents frustration and fear of failure when the goal seems out of reach or when a particularly challenging task arises.

Simple ways to build productive struggle into your math classroom

Giving students who struggle with word problems a chance to struggle with challenging word problems is critical to building confidence and skills. However, this challenge must be reasonable, or the learner's self-esteem will falter, and students need support and regular feedback to achieve their potential.

Here are a few simple things to try:

• Select problems that are just at the edge of students' Zone of Proximal Development.
• Scaffold or model with more challenging problems to support risk-taking.
• Give regular feedback & support – go over the work and discuss daily.

Finding #3: Novelty & variation are keys to engagement.

When it comes to standardized testing (and life in general), problems that arise aren't labeled with the skills and strategies required to solve them.

This makes it important to provide mixed practice opportunities so students are focused on asking themselves questions about what the problem is asking and what they are trying to find.

This type of variation not only supports a deeper level of engagement, it also supports the metacognitive strategies needed to analyze and develop a strategy to solve (Rohrer & Taylor, 2014).

The benefits of novelty in learning

A 2013 study also supports the importance of novelty in supporting reinforcement learning (aka review). The findings suggested that when task variation was provided for an already familiar skill, it offered the following benefits:

• reduced errors due to lack of focus
• helped learners maintain attention to task
• motivated and engaged student

Using variety to build connections & deepen understanding

In addition, by providing variations in practice, we can also help learners understand the skills and strategies they are using on a deeper level.  

When students who struggle with word problems are forced to apply their toolbox of strategies to novel problem formats, they begin to analyze and observe patterns in how problems are structured and the meaning they bring.

This requires much more engagement than being handed a sheet full of multiplication story problems, where students can pull the numbers and compute with little focus on understanding.

Designing word problems that incorporate variety & novelty

Don't be afraid to shake things up!

Giving students practice opportunities with different skills or problem formats mixed in is a great way to boost engagement and develop meta-cognitive skills.

Here are a few tips for trying it out in the classroom:

• Change it up! Word problem practice doesn't have to match the day's math lesson.
• Give opportunities to practice the same skill or strategy in via different formats.
• Adjust the wording and/or topic in word problems to help students generalize skills.

Finding #4: Interest and emotion increase retention and skill development.

Attention and emotion are huge for learning. We've all seen it in our classroom.

Those magical lessons that hook learners are the ones that stick with them for years to come, but what does the research say?

The Science Behind Emotion & Learning

Neuroscientists have shown that emotions create connections among different sections of the brain (Immordino-Yang, 2016) . This supports long-term retrieval of the skills taught and a deeper connection to the learning.

This means if you can connect problem-solving with a scenario or a feeling, your students will be more likely to internalize the skills being practiced. Whether this is by “wowing” them with a little-known fact or solving real-world problems, the emotional trigger can be huge for learning.

What about incorporating student interests?

As for student interests, a long line of research supports the benefits of using these to increase educational outcomes and student motivation, including for students who struggle with word problems (Chen, 2001; Chen & Ennis, 2004; Solomon, 1996).

Connecting classwork with student interests has increased students' intentions to participate in future learning endeavors (Chen, 2001).

And interests don't just mean that love of Pokemon!

It means allowing social butterflies to work collaboratively. Providing students with opportunities to manipulate real objects or create models. Allowing kids to be authentic while digging in and developing the skills they need to master their learning objectives.

What this looks like in a math class

Evoke emotion and use student interests to engage the brain in deep, long-lasting learning whenever possible.

This will help with today's learning and promote long-term engagement, even when later practice might not be as interesting for students who struggle with word problems.

Here's how to start applying this research today:

• Find word problems that match student interests.
• Connect real-life situations and emotions to story problem practice.
• Consider a weekly theme to connect practice throughout the week.

Finding #5: Student autonomy builds confidence & independence.

Autonomy is a student's ability to be in control of their learning. In other words, it is their ability to take ownership over the learning process and how they demonstrate mastery.

Why students need to control their learning

Research shows that providing students a sense of control and supporting their choices is way to help engage learners and build independent thinking. It also increased intrinsic motivation (Reeve, Nix, & Hamm, 2003).

However, this doesn't mean we just let kids learn independently. Clearly, some things require repeated guidance and modeling. Finding small ways that students can take control of the learning process is much better in these instances.

We know that giving at least partial autonomy has been linked to numerous positive student learning outcomes (Wielenga-Meijer, Taris, Widboldus, & Kompier, 2011).

But how can we foster this independence and autonomy, especially with those students who struggle to self-regulate behavior?

Fostering independence in students who struggle to stay on task

Well, the research says several conditions support building toward independence.

The first (and often neglected) is to explain unappealing choices and why they are one of the options.

When it comes to word problems, this might include explaining the rationale behind one of the strategies that appears to be a lot more work than the others.

It is also important to acknowledge students' negative feelings about a task or their ability to complete it. While we want them to be able to build independence, we don't want them to drown in overwhelm.

By providing emotional support, we can help determine whether a student is stuck with the learning or with the emotions from the cognitive challenge.

Finally, giving choices is recommended. Identifying choices you and your students who struggle with word problems can live with is an important step.

Whether this is working in partners, trying an alternative method, or skipping a problem and coming back, students need to feel like they have some ownership over the challenge they are working through.

By building in opportunities for autonomy, and choice, teachers help students build a sense of self-efficacy and confidence in their ability to be successful learners across various contexts (McCombs, 2002,2006).

We know this leads to numerous positive outcomes and has even been linked to drop-out prevention (Christenson & Thurlow, 2004).

Fostering autonomy in your classroom

You're not going to be able to hold their hands forever.

Giving opportunities to work through challenges independently and to feel ownership for their choices will help build both confidence and skills.

Here's how to get started letting go:

• Give students time to tackle the problem independently (or in partners).
• Don't get hyper-focused on a single method to solve – give opportunities to share & learn together.
• Provide appropriate support (where needed) to build autonomy for all learners – like reading the problem orally.

Finding #6: Students need to be taught how to fail & recover from it.

Despite Ericsson's findings discussed early on in this post, talent does matter, and it is important to teach students to recover from failure because those are the moments when they learn the most.

A 2014 study by Brooke Macnamara analyzed 88 studies to determine how talent factored into deliberate practice.

Her findings show what we (as teachers) already know, students may require different amounts of practice to reach the same skill level…but how do we keep those struggling students from keeping up?

Growth mindset research gives us insight into ways to support students who struggle with word problems, encourage all students in math problem-solving, and harness the power of failure through “yet.”

You might not be able to do something yet, but if you keep trying, you will. This opens the door for multiple practice opportunities where students learn from each other.

And what about the advanced students?

Many of these students have not experienced failure, but they may have met their match when it comes to complex word problems.

To support these students, who may be experiencing their first true challenge, we need to have high standards and provide constructive, supportive feedback on how to grow.

Then we need to give them space to try again.

There is great power in allowing students to revise and try again, but our grading system often discourages being comfortable with failure.

Building the confidence to fail in your classroom

Many students feel the pressure always to have the right answer. Allowing students to fail safely means you can help them learn from these failures so they don't make the same mistake twice.

Here's how you can safely foster growth and build math problem solving skills through failure in your classroom:

• Build in time to analyze errors & reflect.
• Reward effort & growth as much as, if not more than, accuracy.
• ​At least initially, skip the grading so students aren't afraid to be wrong.

Getting started with brain-based problem solving

The brain research is clear.

Spending 45 minutes focused on a sheet of word problems following the same format isn't the answer.

By implementing this research, you can save yourself time and the frustration from a disengaged class.

Based on this research, I've created Daily Problem Solving bundles to save you time and build math problem-solving skills. You can get each month separately or buy the full-year bundle at a major discount.

Currently, I offer these bundles for several grade levels, including:

Try Daily Problem Solving with your Learners

Of course, you do! Start working to build step-by-step math problem-solving skills today by clicking the button below to sign up for a free set of Daily Problem Solving.

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

Simple Daily Math Warm-Ups to Build Number Sense

Looking for a way to start math each day to get kids’ brains working and also build number sense and problem solving skills? Here are simple, daily math warm-ups you can implement in 10-15 minutes.

When I was in school, most of my middle and high school classes had a daily ‘math warm up’ time to get us going. The drill was simple: come to class, get out a blank piece of paper and answer the 5 math problems on the chalkboard. Was this helpful? Maybe to an extent-we got a bit of spiral skills review and got our brains thinking about math instead of science or history or lunch. But for the most part, it felt like busywork to me. So today I want to share some alternatives to the typical ‘math warm-ups’ that will wake up kids’ brains and get them thinking about math, but also deepen their number sense and problem solving skills . Plus, most kids think these math routines are super FUN!

Math Warm-Ups to Build Number Sense: Getting Started

There are literally books upon books of number sense routines that you can use and implement in your classroom. But I wanted to keep things super simple for you. I don’t want you to have to spend hours scouring the internet looking for ideas or spend hours prepping complex games or activities with lots of materials and parts.

So although this list is not exhaustive in terms of ways you can start math class each day, I hope it gives you enough ideas that you have a simple daily warm-up with about 10-15 minutes of prep (or less).

I also chose number sense routines that won’t take a lot of time to complete with your students. This is simply meant to be a 10-15 minute exercise to challenge students’ thinking and develop number sense. They’re not intended to be the bulk of your lesson for the day.

Ok, let’s dig in! Below you will find 7 daily math warm up ideas , as well as why I think they are beneficial for students ( i.e. not busywork or time wasters! )

1. Daily Number Talks

I talk about number talks quite often, because I think this is a wonderful way to start your math time. Number talks focus on mental math and computation strategies , so if your students need more meaningful math fact practice or struggle to use efficient strategies, number talks is the way to go.

>> Read more : 5 Reasons Number Talks Should be a Regular Part of Your Math Routine

>> Get Started : Grab FREE Number Talk Planning Pages HERE

2. Mental Math Challenges

This set of mental math challenges is like a single number talk on steroids . These pages give a starting number and then a series of computation challenges for kids to solve mentally until they reach the end.

These are beneficial because you get to choose the starting number, meaning you can easily differentiate for different ability levels.

In addition, these can provide some quiet, independent practice, which is sometimes a good break from group number talks.

>> Get Started : Learn more and grab this FREE set of challenges HERE

3. Esti-Mysteries

Challenging kids with an esti-mystery is a unique kind of number talk. These were created by Steve Wyborney, and he has lots of additional sets available on his site here .

This allows you to weave in a variety of different skills practice and vocabulary review, along with estimation practice, an often neglected skill. Plus, kids LOVE these! It’s such a fun challenge.

>> Read More : Benefits of Esti-Mystery Challenges | FREE Esti-Mystery Recording Pages

>> Get Started : Grab a FREE set of Esti-mystery Challenges HERE

4. What’s the Question? Prompts

Again, this type of math prompt is sure to get your kids thinking, since it’s not what they’re used to. Way too often, kids see math as nothing but computation problems to solve. One problem that has one solution with one correct method to arrive at that solution.

So open ended challenges that force kids to think flexibly and consider math as  more than a closed, single solution are a win in my book!

A ‘what’s the question?’ prompt flips a typical math problem on its head . Instead of giving kids a math problem or word problem and asking for the solution, you give them a ‘solution’ and ask, “What’s the question?” In other words, what might someone be looking for to arrive at this solution? What is a context that might provide this solution?

Sample Solutions:

• The solution is 42 square inches. What’s the question?
• The solution is 1 hour and 30 minutes. What’s the question?
• The solution is 67.7 meters. What’s the question?

Another way to use this type of prompt is to provide the beginning of a story problem and omit the given question .

For example, say your textbook includes this problem as practice:

Sarah and Ben each had $100. Sarah spent $57 on clothes and Ben spent $44 on a video game. How much more did Sarah spend than Ben?

Instead of presenting the entire problem to students, just give them the information to form a context. Leave out the “How much more did Sarah spend than Ben?” and instead ask, “What’s the question?”

You could even provide a numberless word problem and leave off the question . This adds an additional layer of things you can discuss as a group as students make sense of the situation and also think about what the question could be. For example,

Sarah and Ben each had some money. Sarah spent some on clothes and Ben spent some on a video game. What’s the question?

This opens it up more, because we don’t know who spent more, we don’t know how much they had at first, we don’t know if there’s money leftover, etc. This will force kids to think about different scenarios and the math implications .

5. Number or Fraction of the Day Challenges

These simple daily prompts will help kids to think about numbers multiple ways , a key to developing a strong number sense.

How exactly you implement this will depend on your grade level.

In the early grades, it might include writing the number in expanded form, showing with base ten blocks, +/- 1 and +/1 10, etc. Older students might focus on larger numbers and some place value addition and subtraction (daily number +500, -200, etc.).

Upper elementary students can build fraction sense by using fraction of the day prompts. This can include things like ‘find 2 equivalent fractions’ or ‘fraction + ___ = 1’. You can also have students place the faction on a number line or write a story problem equal to the given fraction.

This will help students think flexibly about whole numbers, fractions or even decimals.

>> Get Started : Grab a FREE set of number of the day prompts HERE

6. Target Number Challenge

Again, this is a super simple challenge that takes virtually no prep from you. There are so many benefits for students though. This math warm up will challenge kids to think about numbers in multiple ways, see patterns and learn to decompose numbers.

In this routine, you simply provide students a ‘target’ number. This can be anything at all, depending on the age and ability of your students.

Then set a timer for 5 minutes (or 10 minutes if your students are older and working with more complex numbers) and have kids record equations that are equal to the target number.

For example, if the target number is 10, students might write:

• 2 + 4 + 4 = 10
• 20 – 10 = 10
• 30 – 20 = 10

For older students, you can use larger numbers, or fraction and decimal values.

You can also add restraints or requirements such as, each equation must include 2 operations. Or each equation must include 4 different terms (20 – 12 + 6 – 4 = 10). Or each equation must use an exponent, etc.

7. Logic Puzzles as Math Warm-Ups

Lastly, I want to mention simple logic puzzles as a way to get your kids ‘warmed up’ and ready to learn math. Although logic puzzles won’t necessarily build fluency and computation skills, they are a fun way to ‘warm up’ the brain. They will also strengthen logical and critical thinking skills, which in turn better prepare and equip kids to make sense of word problems and other math challenges.

In addition, they can be a great way for kids to develop perseverance in problem solving , as the desire to solve the puzzle pushes them on in a way that a typical math problem may not.

> Read More : How to Raise Logical Thinkers and Why it Matters

>> Get Started : Try any of the FREE sets of logic puzzles I have available at the links below, or purchase a complete set of puzzles, depending on the ages you’re looking for:

• Hands On Logic Puzzles with Pattern Blocks | ages 4-9
• A Year of Logic Puzzles ( sudoku and grid puzzles with monthly themes ) | ages 9+

FREE logic puzzles to try out:

• Set of 3 Pattern Block Puzzles
• Valentine’s Day Logic Puzzles
• Pi Day Logic Puzzles
• 4th of July Logic Puzzles
• Thanksgiving Logic Puzzles
• Christmas Logic Puzzles

Well, I hope this has given you some meaningful, easy to implement ideas for daily math warm ups and I hope you are encouraged to look for ways to strengthen number sense and build problem solving skills with the routines you use in your classroom!

Have another easy math warm-ups routine to add to this list? Feel free to share it with me via email here: mathgeekmama(at)gmail.com.

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Daily math routines that spark student curiosity

Classroom routines can build creativity and community. Find out how.

It’s the educator’s eternal question: How do I keep students engaged?

When designing daily math practice, teachers are always working to make real-world math problems fresh and relevant, find new entrance points for concepts, or simply come up with surprises. All of these approaches can be very effective.

And though it may seem counter-intuitive, so can routines.

The power of instructional routine

The word routine can connote a sense of doing something mechanically, even without thinking. But teachers know that well-placed classroom routines can open opportunities for creative thought.

Routines provide a way for you and your students to build and maintain a sense of familiarity and structure throughout the school year. They also free up time teachers would otherwise spend giving directions. When students know exactly how a certain activity should run, and understand all instructions and expectations, everything goes more smoothly.

That’s why a core set of shared routines can be a powerful, practical force for establishing an effective classroom learning community..

Bringing math routine into the classroom

We know routines can be effective in any classroom. Now, we also have research offering direct evidence that certain routines are particularly effective in math classrooms.

Think-pair-share

Do you want your students to have more time to think before solving and sharing about a problem? 

GOAL: Provides opportunities to identify, compare, and contrast multiple strategies

TIP: During partnered discussion, consider displaying sentence frames such as, “ First they… Next they…” “Their strategy was to…” or “I see a/an… in both strategies.”.

How to do it:

• Invite students to solve a problem that can be solved with multiple strategies. Then, display two or more different responses representing different strategies.
• Give students time to analyze the strategies on their own and then invite them to discuss them with a partner.
• Facilitate a class discussion to describe, compare, contrast, and connect the different strategies. Utilize open-ended questions like, “Why did different strategies lead to the same outcome?” or, “What was helpful about each strategy?”

Where to learn more

We worked with our curriculum team to develop routine cards for math teachers, so you can implement routines that are part of our math program in your classroom. Most of the routines you’ll find throughout Amplify Desmos Math have been specifically proven effective in math classrooms. All of them have been adapted from established teaching practices.

We invite you to access a sample set of some of our most popular routines and decide which ones to try out in your classroom!

Download free math instructional routine cards .

Explore Desmos Classroom .

Learn more about Amplify Desmos Math .

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Problem-based learning can put students on a path to math success. In this three-part series, you’ll learn what it is and how it looks in a classroom.

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Inside Problem Solving

The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity. The problems were developed by the Silicon Valley Mathematics Initiative and are aligned to the Common Core standards.

To request the Inside Problem Solving Solutions Guide, please get in touch with us via the feedback form .

Courtney’s Collection   Cut It Out Cutting a Cube Digging Dinosaurs Diminishing Return First Rate Friends You Can Count On Game Show Got Your Number Growing Staircases Measuring Mammals Measuring Up Miles of Tiles Movin ‘n Groovin On Balance Once Upon A Time Part and Whole Party Time Piece it Together Polly Gone Rod Trains Surrounded and Covered Squirreling It Away The Shape of Things The Wheel Shop Through the Grapevine Tri-Triangles What’s Your Angle?

Cutting a Cube (K.G.B.4) Digging Dinosaurs (K.OA.A.2) First Rate (K.CC.B.5, K.CC.C.6) Growing Staircases (K.CC.B.5) On Balance (K.MD.A.2)

Cutting a Cube (1.G.A.1) Growing Staircases (1.OA.A.1) Rod Trains (1.MD.A.2, 1.OA.C.6) Measuring Mammals (1.MD.A.1) Miles of Tiles (1.OA.A.1) Movin ‘n Groovin (1.OA.A.1) Piece it Together (1.G.A.2)

Courtney’s Collection (2.MD.C.8) Digging Dinosaurs (2.MD.C.8) Got Your Number (2.OA.B.2, 2.NBT.A.1, 2.NBT.A.4, 2.NBT.B.5) Miles of Tiles (2.NBT.B.5) Part and Whole (2.G.A.3) Piece it Together (2.G.A.1) Squirreling It Away (2.OA.1) The Shape of Things (2.G.A.1) Through the Grapevine (2.MD.D.9, 2.MD.D.10) What’s Your Angle? (2.G.A.1)

Measuring Up (3.OA.A.3) Once Upon A Time (3.MD.A.1) Part and Whole (3.G.A.2, 3.NF.A.1, 3.MD.C.6) Party Time (3.OA.A.3) Piece it Together (3.MD.C.5, 3.MD.D.8) Polly Gone (3.MD.D.8) Surrounded and Covered (3.MD.C.6, 3.MD.D.8) The Wheel Shop (3.OA.A.1, 3.OA.A.2) Tri-Triangles (3.OA.A.3)

Courtney’s Collection (4.MD.A.2) Digging Dinosaurs (4.MD.A.2) Diminishing Return (4.OA.A.3, 4.MD.A.2) Friends You Can Count On (4.OA.A.3) Game Show (4.OA.C.5) Growing Staircases (4.OA.C.5) Measuring Mammals (4.OA.A.2) Measuring Up (4.OA.A.3) Once Upon A Time (4.OA.A.3) Part and Whole (4.G.A.3) Party Time (4.NF.B.4c) Piece it Together (4.G.A.2, 4.MD.C.6) Squirreling It Away (4.OA.3) The Shape of Things (4.G.A.3) The Wheel Shop (4.OA.A.3) Tri-Triangles (4.OA.C.5)

Digging Dinosaurs (5.NBT.B.7) Movin ‘n Groovin (5.NF.B.4)

Courtney’s Collection (6.NS.B.4) Cutting a Cube (6.G.A.4, 6.RP.A.3c) Diminishing Return (6.RP.A.3a, 6.RP.A.3b) First Rate (6.RP.A.3b, 6.RP.A.2) Measuring Up (6.RP.A.3c, 6.EE.A.1, 6.EE.B.7) On Balance (6.EE.B.5, 6.EE.B.6, 6.EE.B.8) Once Upon A Time (6.NS.B.2, 6.NS.B.4) Movin ‘n Groovin (6.RP.A.3d) Part and Whole (6.G.A.1) Piece it Together (6.G.A.4) Polly Gone (6.G.A.1) Surrounded and Covered (6.RP.A.2, 6.RP.A.3b) Tri-Triangles (6.EE.A.1, 6.EE.B.6, 6.EE.C.9)

Courtney’s Collection (7.SP.C.8b) First Rate (7.RP.A.2b, 7.RP.A.3, 7.EE.B.4a) Friends You Can Count On (7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b) Game Show (7.SP.C.8a, 7.SP.C.8b) Got Your Number (7.NS.A.3) Measuring Mammals (7.RP.A.2a, 7.RP.A.2b, 7.RP.A.2c 7.RP.A.1) Measuring Up (7.RP.A.2b, 7.RP.A.2c, 7.RP.A.3, 7.EE.B.4) Movin ‘n Groovin (7.RP.A.2c, 7.RP.A.3) Part and Whole (7.NS.A.1D) Piece it Together (7.G.B.6) Polly Gone (7.G.B.6, 7.G.B.4) Rod Trains (7.SP.C.8b) Squirreling It Away (7.SP.8b) Surrounded and Covered (7.G.B.4, 7.G.B.6) Through the Grapevine (7.SP.A.2)

Cutting a Cube (8.G.A.1a) Digging Dinosaurs (8.EE.C.7b, 8.F.B.4) Diminishing Return (8.EE.C.7.b) Miles of Tiles (8.EE.C.8b, 8.EE.C.8c) Movin ‘n Groovin (8.EE.B5) On Balance (8.EE.C.8b, 8.EE.C.8c) Once Upon A Time (8.EE.C.8b) Squirreling It Away (8-F.1) Through the Grapevine (8.SP.A.1, 8.SP.A.2) The Wheel Shop (8.EE.C.8b, 8.EE.C.8c)

Courtney’s Collection (A-CED.A.2) Digging Dinosaurs (A-CED.A.2) Diminishing Return (A-CED.A.1) Growing Staircases (A-CED.A.2) Measuring Mammals (A-CED.A.2, A-REI.B.3, A-REI.C.6) Measuring Up (A-CED.2) Miles of Tiles (A-APR.A.1, A-SSE.A.1a, A-SSE.A.2) On Balance (A-CED.A.2, A-REI.C.6) Once Upon A Time (A-CED.A.1) Part and Whole (A-APR.D.6) Polly Gone (A-REI.C.6) Squirreling It Away (A-CED.2, A-CED.3, A-REI.6, A-REI.8, A-REI.10) The Wheel Shop (A-REI.C.6, A-REI.D.12) Tri-Triangles (A-CED.A.1, A-REI.B.4b, A-SSE.A.2)

Cut It Out (F-BF.A.1a) Digging Dinosaurs (F-IF.C.7b, F-IF.C.7e) Diminishing Return (F-BF.A.1a) First Rate (F-IF.B.6, F-BF.A.1a) Growing Staircases (F-LE.A.2, F-BF.A.2, F-BF.A.1a) Movin ‘n Groovin (F.BF. A.1a) Rod Trains (F-BF.A.1a) Squirreling It Away (F.LE.2, F-BF.1a, F-BF.2) Surrounded and Covered (F-BF.A.1a) Tri-Triangles (F-BF.A.1a) What’s Your Angle? (F-BF.A.1a)

Cut It Out (G-CO.B.6) Growing Staircases (G-MG.1) First Rate (G-SRT.C.8) Measuring Mammals (G-SRT.B.5) Miles of Tiles (G-MG.A.3) Once Upon A Time (G-C.A.2) Piece it Together (G.MG.A.1, G-MG.A.3, G.GMD.A.1, G.SRT.C.8) Polly Gone (G-CO.B.7, G-GPE.B.7, G-MG.A.3, G-GPE.B.4) The Shape of Things (G-C.A.2, G-CO.C.10, G-CO.C.11, G-SRT.B.5, G-MG.A.1) What’s Your Angle? (G-MG.A.3, G-C.A.2)

Digging Dinosaurs (S-ID.6.a) Diminishing Return (S-CP.A.2, S-CP.B.8) Friends You Can Count On (S-CP.A.4, S-CP.A.5, S-CP.B.6) Game Show (S-MD.A.1, S-MD.A.2, S-MD.A.3) Growing Staircases (S-ID.6a) Party Time (S-CP.A.1, S-CP.B.9, S-CP.B.8) Squirreling It Away (S-ID.6a) Through the Grapevine (S-IC.B.4, S-ID.A.1, S-ID.A.2, S-ID.A.3, S-ID.B.5, S-ID.B.6c) The Wheel Shop (S-CP.A.1)

Why Problem Solving?

Problem solving is the cornerstone of doing mathematics. George Polya, a famous mathematician from Stanford, once said, "A problem is not a problem if you can solve it in 24 hours." His point was that a problem that you can solve in less than a day is usually a problem that is similar to one that you have solved before, or at least is one where you recognize that a certain approach would lead to the solution. Bu t in real life, a problem is a situation that confronts you and you don’t have an idea of where to even start. Mathematics is the toolbox that solves so many problems. Whether it is calculating an estimate measure, modeling a complex situation, determining the probability of a chance event, transforming a graphical image or proving a case using deductive reasoning, mathematics is used. If we want our student s to be problem solvers and mathematically powerful, we must model perseverance and challenge students with non-routine problems.

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1.6: Problem Solving Strategies

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• Michelle Manes
• University of Hawaii

Think back to the first problem in this chapter, the ABC Problem. What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solve them), you learn strategies and techniques that can be useful. But no single strategy works every time.

How to Solve It

George Pólya was a great champion in the field of teaching  effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, How to Solve it . Pólya died at the age 98 in 1985. [1]

George Pólya, circa 1973

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 ( http://creativecommons.org/licenses/by/2.0 )], via Wikimedia Commons ↵

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

• First, you have to understand the problem.
• After understanding, then make a plan.
• Carry out the plan.
• Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

A Problem Solving Strategy: Try Something!

If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

Note that being "good at mathematics" is not about doing things right the first time. It is about figuring things out. Practice being okay with having done something incorrectly. Try to avoid using an eraser and just lightly cross out incorrect work (do not black out the entire thing). This way if it turns out that you did something useful, you still have that work to reference! If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what was left after paying Brianna. Finally, Alex saw David and gave him 1/2 of the remaining money. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem? This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

A Problem Solving Strategy: Draw a Picture

Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

A Problem Solving Strategy: Make Up Numbers

Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

Try this: Assume (that is, pretend) Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person.

Or try working backward: suppose Alex has some specific amount left at the end, say $10. Since he gave David half of what he had before seeing David, that means he had $20 before running into David. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

(Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64... It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

Most people want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. Instead of asking the teacher, “Is this right?”, you should be ready to justify it and say, “Here’s my answer, and here is how I got it.”

A Problem Solving Strategy: Try a Simpler Problem

Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said, “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

The ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

A Problem Solving Strategy: Work Systematically

If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

A Problem Solving Strategy: Use Manipulatives to Help You Investigate

Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

A Problem Solving Strategy: Look for and Explain Patterns

Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table. If possible, actually describe these to a friend.
• Explain and justify any of the patterns you see (if possible, actually do this with a friend). If you don't have a partner to work with, imagine they asked you, "How can you be sure the patterns will continue?"
• Expand this to find what calculation(s) you would perform to find the total number of squares on a 100 × 100 chess board.

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

(Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner if possible (even if you have not solved it). What did you try? What progress have you made?

A Problem Solving Strategy: Find the Math, Remove the Context

Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

A Problem Solving Strategy: Check Your Assumptions

When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

Math Problem Solving: Why I Changed My Format

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Today’s post tells the story of how I evolved my math problem solving program in the classroom and provides tips for how to build powerful math students who can not only analyze word problems but also solve non-routine problems efficiently using effective math problem solving tools. I also share some of the tools that I found to be successful for me in the classroom. Enjoy!

How I Used to Do Math Problem Solving

When I first started teaching 16 years ago, I knew it was important to teach math problem solving in the classroom, so I had a daily/weekly activity where I picked a strategy and “taught” my students how to use that strategy to solve specific problems. I was so young and misguided that I didn’t realize I had completely missed the mark. Even when my district’s math curriculum coordinator stopped by for a visit and encouraged me to allow the students to solve the problems without giving them a specific strategy to use, I struggled to really “get it.”

Franky, it wasn’t until year two or three that I really started to see math problem solving differently. At that time, I was using a myriad of problem solving challenges sporadically throughout the week as time allowed. While I certainly think the time was well-spent, I don’t believe my students were really developing their math problem solving toolboxes in the way that I wanted them to.

When I fast forward a couple of years, I find myself with a new way to infuse math problem solving. I’d discovered this amazing question of the day program where students had one lengthy problem scenario, full of numbers and information and five questions, one for each day of the week. I thought this was brilliant! I set aside about 15 minutes each day for the task and my students got to problem solve every day. It’s a win-win! I used this structure for a couple of years before I finally realized that my students were not problem solving; they were solving problems.

Problem Solving vs. Solving Problems

It’s funny that as a math specialist I never considered the difference before. However, when my students were completing the question of the day task, they were solving routine word problems. Now, don’t get me wrong, there is a place for this in our day-to-day curriculum; however, it was not helping my students develop the problem solving strategies that I set out to develop my very first year in the classroom. While the question of the day task was quite successful and, after using a structure to help my students attend to the fundamentals of analyzing word problems, they became strong word problem analyzers. This excited me, but when it came to problem solving, they were not as efficient.

So, what does this mean? There is a difference between problem solving and solving problems. While the difference may seem minute, it makes a huge difference for instruction.

Solving Problems

Many of the word problems our students routinely solve are procedural in nature. Once students determine the “formula” to use, they can apply the formula with ease, such as by using keywords. (Remember my blog post earlier this year about the dangers of keywords? If not, read it here !) While many teachers consider solving problems to be the same as problem solving, these tasks lack the cognitive demand necessary to develop our students’ critical thinking skills.

As I mentioned earlier, there is a place for these kinds of problems in our everyday curriculum; however, it’s important to consider how we utilize them. Use the strategies below to get the most from these tasks.

• Include more details than what is needed to solve the problem
• Have students represent their solutions in multiple ways, i.e. pictures, symbols, and/or words
• Require students to justify their answers
• Ask students to explain how they know their answer is reasonable

Problem Solving

On the other hand, problem solving tasks are non-routine problems that can not be solved with basic procedures and “require complex and non-algorithmic thinking” where a “predictable, well-rehearsed approach or pathway is not explicitly suggested by the task” (p. 18). Because these tasks are more complex, they do take additional time to complete; however, my students have loved the challenge, especially once they had a set of problem solving strategies on which they could rely. Use the strategies below to get the most from these tasks:

• Infuse problem solving tasks to extend and build on content and skills
• Utilize tasks that have multiple entry points to allow all students an opportunity for success
• Allow productive struggle (Read more here !)
• Provide opportunities to solve these types of tasks on a regular basis
• Encourage students to use a variety of strategies to solve the problems
• Employ math talk strategies to discuss student work

Developing a good balance between reinforcing strategies for solving word problems and implementing effective math problem solving routines takes time. To get you started, I will be blogging about how to start a problem solving routine and how to convert word problems into problem solving tasks over the next few weeks. Be sure to stay tuned!

In the meantime, be sure to grab a copy of my Solve It! Back to School Math and Problem Solving Activities Pack to get some great examples of non-routine tasks to start your year! Click here  or on the image below to grab your copy!

Reference: National Council of Teachers of Mathematics (2014). Principles to actions: Ensuring mathematical success for all . Reston, VA: National Council of Teachers of Mathematics.

Shametria Routt Banks

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Daily Routines to Jump-Start Problem Solving, Grades K-8

• John J. SanGiovanni - Howard Public School System
• Description

Finally! A book that helps solve the problem of teaching problem solving!

Learning to be a problem solver is hard. Teaching students how to be problem solvers themselves can be even harder. Some students may learn to mimic procedures to come up with correct answers, but are they really learning to solve problems? To become independent problem solvers, students need to practice exploring, tinkering, and, most important, thinking!!

The bite-size routines in this guide are perfect for teachers looking for the interesting, engaging, and doable practice students need to become problem-solving masters. These flexible, modifiable bursts of quality practice are designed to get students to look at problems at different ways, sparking discussion, making connections, and boosting mathematics achievement. This collection addresses the common challenges students and teachers face when learning to problem solve by offering guidance on

• Developing students’ mathematical reasoning and conceptual understanding
• Building students’ skills with various problem-solving strategies
• Nurturing mathematical confidence and improving identity and agency

Fortified with standards for math practices and processes, the ideas in this guide develop the reasoning and critical-thinking skills needed to become independent problem-solvers for life!

See what’s new to this edition by selecting the Features tab on this page. Should you need additional information or have questions regarding the HEOA information provided for this title, including what is new to this edition, please email [email protected] . Please include your name, contact information, and the name of the title for which you would like more information. For information on the HEOA, please go to http://ed.gov/policy/highered/leg/hea08/index.html .

For assistance with your order: Please email us at [email protected] or connect with your SAGE representative.

SAGE 2455 Teller Road Thousand Oaks, CA 91320 www.sagepub.com

Problem solving has long been a challenge for students—until now! This book provides teachers with a bank of routines to engage students in thinking, reasoning, and sense making. Everything you need to teach students to be effective problem solvers is in your hands right now!

John J. SanGiovanni has written yet another masterful book on routines focused on problem solving. My mentor once told me the way you get better at solving math problems is making sense and solving more math problems. Many students typically shy away from problem solving, yet in this book SanGiovanni has developed culturally relevant problems students can engage in to help build their problem solving skills. He has brilliantly put together a book of problem-solving routines to help all students be more successful. If we want equitable mathematics classrooms, ALL students must be engaged in daily problem solving and Daily Routines to Jump-Start Problem Solving is a resource to ensure that!

I have been using resources from John J. SanGiovanni since 2018! His strategies have helped propel thinking and student engagement in my classroom as well as classrooms across my district. His activities and routines have increased students’ enjoyment of mathematics concepts, which has led to higher scores across the district.

Teachers often say, “My students can solve math equations, but not word problems.” With SanGiovanni’s Daily Routines to Jump-Start Problem Solving , teachers can ensure students develop the confidence to solve word problems without using hazardous tricks and gimmicks. Students will be encouraged to think about the mathematics behind the problem. The ideas in this book should be part of every mathematics class.

Instead of “cookbook” problem-solving prompts, SanGiovanni provides teachers with step-by-step guidance on implementing problem solving through the use of classroom-ready instructional routines that allow students to engage in rich problem-solving experiences and promote reasoning. Additionally, this book encourages productive discourse, facilitates “just in time” scaffolds, and supports existing high-quality curriculum materials.

The routines in this book are easy to follow and help set the stage to spark student engagement and discourse in any math class. As a math coach, I am always looking for ways to support teachers in helping students to be better thinkers and to reason with math, and these quick bursts of quality practice are an amazing resource. The graphic organizers provided to support the routines help ensure all students have an entry point to problem solving.

Daily Routines to Jump-Start for Problem Solving offers a plethora of problem-solving routines to promote reasoning and critical thinking skills. A resource for administrators, instructional coaches, and math teachers, each protocol has a “Something to Think About” section, images and examples that encourage productive discussions, and variations of each routine to meet all students’ needs.

Daily Routines to Jump-Start Problem Solving is truly a game changer for K–8 educators. Every problemsolving situation embodies critical thinking and comprehension, thus building student self-efficacy. SanGiovanni provides practical tools and support to breathe new life into math language routines, promoting equitable problem solving for all students. Routines showcased in this book allow students to take the driver’s seat as they engage in real-world problem solving.

As math educators and leaders, we have all been waiting for a book like this to aid in facilitating rich discussions centered on problem-solving, all in the first few minutes of daily instruction! SanGiovanni provides 20 research-based strategies to help students become problem-posers and problem-solvers while building their confidence, agency, and identity. This book is truly a gift to the math education community.

SanGiovanni shares a creative collection of quick, high-quality practice opportunities that will help students sharpen their problem-solving skills. Teachers have been longing for purposeful activities like these to engage their students in thinking and sense-making as they strive to develop confident problem solvers in mathematics. This book will become one of teachers’ best-loved instructional resources!

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Pentathlon Institute

Active Problem-Solving

Learning to resolve problems that are continually undergoing change.

The Mathematics Pentathlon ® Program provides experiences in thought processes necessary for Active Problem Solving. The series of 20 Mathematics Pentathlon games provide K-7 students with experiences in deductive and inductive reasoning through the repeated use of sequential thought as well as nonlinear, intuitive thinking. Exposure to such forms of thought helps students relate to real-life problem-solving situations and learn to “think on their feet.”

The Four Sections below explain the following: Active Problem Solving Defined, What is Mathematics, Three Types of Mathematical Thought, and Conceptual Understanding Using Concrete and Pictorial Models.

Active-Problem Solving Defined

Problem solving can be divided into two categories:, routine & non-routine.

Routine Problem Solving , stresses the use of sets of known or prescribed procedures (algorithms) to solve problems. The strength of this approach is that it is easily assessed by paper-pencil tests. Since today’s computers and calculators can quickly and accurately perform the most complex arrangements of algorithms for multi-step routine problems, the typical workplace does not require a high level of proficiency in Routine Problem Solving. However, today’s workplace does require many employees to be proficient in Nonroutine Problem Solving.

Nonroutine Problem Solving , stresses the use of heuristics and often requires little to no use of algorithms. Unlike algorithms, heuristics are procedures or strategies that do not guarantee a solution to a problem but provide a more highly probable method for discovering a solution. Building a model and drawing a picture of a problem are two basic problem-solving heuristics. Studying end-of-game situations provides students with experiences in using the heuristics of reducing the problem to a similar but simpler problem and working a problem backwards, i.e. from its resolution to its initial state. Other heuristics include describing the problem situation, classifying information, and finding irrelevant information.

There Are Two Categories of Nonroutine Problem Solving:

Static and active.

Static-Nonroutine problems have a fixed, known goal and fixed, known elements that are used to resolve the problem. Solving a jigsaw puzzle is an example of a Static- Nonroutine problem. Given all pieces to a puzzle and a picture of the goal, learners are challenged to arrange the pieces to complete the picture. Various heuristics such as classifying the pieces by color, connecting the pieces that form the border, or connecting the pieces that form a salient feature to the puzzle, such as a flag pole, are typical ways in which people attempt to resolve such problems.

Active-Nonroutine problems may have a fixed goal with changing elements, a changing goal or alternative goals with fixed elements, or changing or alternative goals with changing elements. The heuristics used in this form of problem-solving are known as strategies. People who study such problems must learn to change or adapt their strategies as the problem unfolds.

What is Mathematics?

There is a growing belief in the mathematics community, as well as society in general, that the study of mathematics must develop in all students an understanding of mathematics that continues throughout one’s lifetime and evolves to meet ever-changing situations and variables. From our perspective, mathematics is an area of investigation that develops the ability to critically observe, classify, describe, and analyze data in a logical manner using both inductive and deductive methods. In contrast to the sterilized and unrelated manner in which school mathematics has often been taught, mathematics is a creative and aesthetic study of patterns and geometric and numerical relationships. It is dynamic rather than passive in nature and should involve students in strategic thinking by exploring multiple possibilities and variables that continually change, much like life.

The Mathematics Pentathlon® Program, which integrates Adventures in Problem Solving, Activity Books I & II, the Mathematics Pentathlon® Games and Investigation Exercises, Books I & II was designed to implement the definition of mathematics described above. The games are organized into four division levels by grade, K-1, 2-3, 4- 5, and 6-7 with five games at each level. The name of the Program, Mathematics Pentathlon®, was coined to liken it to a worldwide series of athletic events, the Decathlon component of the Olympics. In the world of athletics the Decathlon is appreciated for valuing and rewarding individuals who have developed a diverse range of athletic abilities. In contrast, the mathematics community as a whole has rarely valued or rewarded individuals with a diverse range of mathematics abilities. The Mathematics Pentathlon® Games promote diversity in mathematical thinking by integrating spatial/ geometric, arithmetic/computational, and logical/scientific reasoning at each division level. Since each of the 5 games requires students to broaden their thought processes, it attracts students from a wide range of ability levels, from those considered “gifted and talented” to “average” to “at-risk.”

The format of games was chosen for two reasons. First, games that are of a strategic nature require students to consider multiple options and formulate strategies based on expected countermoves from the other player. The Mathematics Pentathlon® further promotes this type of thought by organizing students into groups of four and teams of two. Teams alternate taking turns and team partners alternate making decisions about particular plays by discussing aloud the various options and possibilities. In this manner, all group members grow in their understanding of multiple options and strategies. As students play these games over the course of time, they learn to make a plan based on better available options as well as to reassess and adjust this plan based on what the other team acted upon to change their prior ideas. Through this interactive process of sharing ideas and possibilities, students learn to think many steps ahead, blending both inductive and deductive thinking. Second, games were chosen as a format since they are a powerful motivational tool that attracts students from a diverse range of abilities and interest levels to spend more time on task developing basic skills as well as problem-solving skills. While race-type games based on chance are commonly used in classrooms, they do not typically capture students’ curiosity for long periods of time. Students may play such games once or twice, but then lose interest since they are not seriously challenged. The Mathematics Pentathlon® Games have seriously challenged students to mature in their ability to think strategically and resolve problems that are continually undergoing change. As a result, we view active-problem solving and strategic thinking as described above as a critical focal point of the mathematics curriculum.

3 Types of Mathematical Thought

Integration of spatial/geometric, computational, and logical/scientific reasoning.

Most mathematics instruction stresses students’ knowledge of basic arithmetic facts. While the Mathematics Pentathlon Program provides a great deal of practice with mastery of the basic facts, it goes far beyond learning arithmetic skills. The Mathematics Content and Standards Chart for the 20 Mathematics Pentathlon Games shows how each game addresses several mathematical content and process objectives (see inside back cover of manual). These objectives have been clustered into logical/scientific reasoning, computational reasoning, and spatial/geometric reasoning. Each of these categories is described below.

Spatial/Geometric Reasoning

Spatial visualization involves the ability to image objects and pictures in the mind’s eye and to be able to mentally transform the positions and examine the properties of these objects/pictures. A large body of mathematics research concludes that spatial reasoning ability is highly related to higher-level mathematical problem-solving and geometric skills as well as students’ overall achievement in mathematics. Many of the Mathematics Pentathlon Games stress spatial reasoning and several integrate this form of thinking with logical and computational reasoning.

Computational Reasoning

Many of the Mathematics Pentathlon Games incorporate computation into the game structure. More time-on-task practicing arithmetic skills does indeed result in students’ increased performance in the classroom as well as on standardized tests. But in the Mathematics Pentathlon Games that stress computation, it is not sufficient to rely on arithmetic skills alone. To be successful in these games, students must also use their logical reasoning abilities to consider several options and to decide which ones will maximize their ability to reach the game’s goal(s).

Logical/Scientific Reasoning

One of the most important life skills, not to mention mathematical skills, is the ability to think logically. The process of observation, classification, hypothesizing, experimentation, and inductive and deductive thought are required for logical reasoning. Yet where do children learn these fundamental life skills? Strategic games provide students the opportunity to develop this form of thinking. Each of the Mathematics Pentathlon Games is a strategy game that develops students’ logical reasoning skills through the process of investigating a variety of options and choosing better options.  At the same time students develop scientific reasoning skills by learning how to be better observers of game-playing variables and options. Playing the games over the course of time allows for hypothetical reasoning to evolve since students analyze sequences of “if-then” situations and make choices based on linking inductive and deductive thought.

While each of the 20 games may stress one form of the mathematical thinking over another, each game integrates at least two categories of mathematical thinking. Furthermore, the five games at each Divisional level balance the three types of reasoning.

Conceptual Understanding Using Concrete and Pictorial Models Understanding

Since its inception, the National Council of Teachers of Mathematics (NCTM) has called for a conceptually-based curriculum in schools throughout the country. The most recent psychological and educational research has shown that conceptual understanding is a key attribute of individuals who are proficient in mathematics. Furthermore, a large body of research over the last four decades suggests that effective use of physical and pictorial models of mathematics concepts improves students’ conceptual understanding, problem-solving skills, and overall achievement in mathematics. Research also indicates that the use of concrete and pictorial models improves spatial visualization and geometric thinking.

The  Mathematics Pentathlon ®   Program  incorporates a variety of concrete and pictorial models to develop students’ conceptual understanding of many important mathematics concepts that involve computational, spatial, and logical reasoning. In addition, by playing these games in cooperative groups, as suggested in this publication, students also improve their oral and written communication skills through their discussion of mathematical ideas and relationships.

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Daily Routines to Jump-Start Problem Solving, Grades K-8

Cultivate reasoning, critical thinking, and mathematical agency with quick bursts of quality practice, or routines, outlined in this indispensable guide designed to produce independent, problem-solving students.

Full description

• Grade Level: PreK-12
• ISBN: 9781071888261
• Published By: Corwin
• Series: Corwin Mathematics Series
• Page Count: 232
• Publication date: April 12, 2023

Price: $38.95

For Instructors

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Description

Finally! A book that helps solve the problem of teaching problem solving!

Learning to be a problem solver is hard. Teaching students how to be problem solvers themselves can be even harder. Some students may learn to mimic procedures to come up with correct answers, but are they really learning to solve problems? To become independent problem solvers, students need to practice exploring, tinkering, and, most important, thinking!!

The bite-size routines in this guide are perfect for teachers looking for the interesting, engaging, and doable practice students need to become problem-solving masters. These flexible, modifiable bursts of quality practice are designed to get students to look at problems at different ways, sparking discussion, making connections, and boosting mathematics achievement. This collection addresses the common challenges students and teachers face when learning to problem solve by offering guidance on

• Developing students’ mathematical reasoning and conceptual understanding
• Building students’ skills with various problem-solving strategies
• Nurturing mathematical confidence and improving identity and agency

Fortified with standards for math practices and processes, the ideas in this guide develop the reasoning and critical-thinking skills needed to become independent problem-solvers for life!

John J. SanGiovanni

John J. SanGiovanni is a mathematics coordinator in Howard County, Maryland. There, he leads mathematics curriculum development, digital learning, assessment, and professional development. John is an adjunct professor and coordinator of the Elementary Mathematics Instructional Leadership graduate program at McDaniel College. In addition to this Figuring Out Fluency series, some of his many Corwin books include Daily Routines to Jump-Start Problem Solving, Grades K-8 , Answers to Your Biggest Questions about Teaching Elementary Math , the Daily Routines to Jump-Start Math series, and Productive Math Struggle: A 6-Point Action Plan for Fostering Perseverance . John is a national mathematics curriculum and professional learning consultant who also speaks frequently at national conferences and institutes. He is active in state and national professional organizations, recently serving on the board of directors for the National Council of Teachers of Mathematics (NCTM) and currently on the board of directors for the National Council of Supervisors of Mathematics (NCSM).

Table of Contents

Part I: Why Jump-Start Routines for Problem Solving?

Part II: Jump-Start Routines for Problem Solving

I Would Say

Asked and Answered

What’s the Question?

Same Data, Different Question

How Do You…?

Same and Different

What’s the Sitch?

They Did What?

This Number, That Number

Imagine That

Math the Equation

Another Question

It Can’t Be

Is It Reasonable?

Sticky Patterns

Lots and Lots

Part III: Where to Go Next

Problem solving has long been a challenge for students—until now! This book provides teachers with a bank of routines to engage students in thinking, reasoning, and sense making. Everything you need to teach students to be effective problem solvers is in your hands right now!

John J. SanGiovanni has written yet another masterful book on routines focused on problem solving. My mentor once told me the way you get better at solving math problems is making sense and solving more math problems. Many students typically shy away from problem solving, yet in this book SanGiovanni has developed culturally relevant problems students can engage in to help build their problem solving skills. He has brilliantly put together a book of problem-solving routines to help all students be more successful. If we want equitable mathematics classrooms, ALL students must be engaged in daily problem solving and Daily Routines to Jump-Start Problem Solving is a resource to ensure that!

I have been using resources from John J. SanGiovanni since 2018! His strategies have helped propel thinking and student engagement in my classroom as well as classrooms across my district. His activities and routines have increased students’ enjoyment of mathematics concepts, which has led to higher scores across the district.

Teachers often say, “My students can solve math equations, but not word problems.” With SanGiovanni’s Daily Routines to Jump-Start Problem Solving , teachers can ensure students develop the confidence to solve word problems without using hazardous tricks and gimmicks. Students will be encouraged to think about the mathematics behind the problem. The ideas in this book should be part of every mathematics class.

Instead of “cookbook” problem-solving prompts, SanGiovanni provides teachers with step-by-step guidance on implementing problem solving through the use of classroom-ready instructional routines that allow students to engage in rich problem-solving experiences and promote reasoning. Additionally, this book encourages productive discourse, facilitates “just in time” scaffolds, and supports existing high-quality curriculum materials.

The routines in this book are easy to follow and help set the stage to spark student engagement and discourse in any math class. As a math coach, I am always looking for ways to support teachers in helping students to be better thinkers and to reason with math, and these quick bursts of quality practice are an amazing resource. The graphic organizers provided to support the routines help ensure all students have an entry point to problem solving.

Daily Routines to Jump-Start for Problem Solving offers a plethora of problem-solving routines to promote reasoning and critical thinking skills. A resource for administrators, instructional coaches, and math teachers, each protocol has a “Something to Think About” section, images and examples that encourage productive discussions, and variations of each routine to meet all students’ needs.

Daily Routines to Jump-Start Problem Solving is truly a game changer for K–8 educators. Every problemsolving situation embodies critical thinking and comprehension, thus building student self-efficacy. SanGiovanni provides practical tools and support to breathe new life into math language routines, promoting equitable problem solving for all students. Routines showcased in this book allow students to take the driver’s seat as they engage in real-world problem solving.

As math educators and leaders, we have all been waiting for a book like this to aid in facilitating rich discussions centered on problem-solving, all in the first few minutes of daily instruction! SanGiovanni provides 20 research-based strategies to help students become problem-posers and problem-solvers while building their confidence, agency, and identity. This book is truly a gift to the math education community.

SanGiovanni shares a creative collection of quick, high-quality practice opportunities that will help students sharpen their problem-solving skills. Teachers have been longing for purposeful activities like these to engage their students in thinking and sense-making as they strive to develop confident problem solvers in mathematics. This book will become one of teachers’ best-loved instructional resources!

Daily Routines to Jump-Start Problem Solving should be on every elementary mathematics teacher’s desk! These routines include practical approaches that will support all students in developing problem solving strategies and skills. There is also substantial background information for teachers to deepen their own pedagogical understanding of how to further student success in approaching and solving problems.

SanGiovanni does an excellent job empowering teachers to bridge the gap between some of their favorite math routines and problem solving. He provides a roadmap that will walk alongside teachers as they look to engage students and improve thinking beyond the first 15 minutes of math class.

Other Titles in: Mathematics | Elementary Maths | Secondary Mathematics

Related Professional Learning

• Routines to Jump-Start Problem Solving

Related Resources

• Daily Routines to Jump-Start Problem Solving Webinar  [Study guide]
• Develop an Approach to Analyzing Problems  [Lessons and strategies]
• Know Your Students & Learn From Your Students  [Podcast]
• Routine to Build Students' Problem-Solving Skills  [Lessons and strategies]

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Problem Solving in Mathematics

• Math Tutorials
• Pre Algebra & Algebra
• Exponential Decay
• Worksheets By Grade

The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition  problems:

Common clue words for  subtraction  problems:

• How much more

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:

• Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
• What did you need to do in that instance?
• What facts are you given about this problem?
• What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:

• Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
• If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you’ve solved the problem, ask yourself the following:

• Does your solution seem probable?
• Does it answer the initial question?
• Did you answer using the language in the question?
• Did you answer using the same units?

If you feel confident that the answer is “yes” to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

• What are the keywords in the problem?
• Do I need a data visual, such as a diagram, list, table, chart, or graph?
• Is there a formula or equation that I'll need? If so, which one?
• Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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• Study Tips for Math Homework and Math Tests
• How to Do Algebra Word Problems

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Math Problem-Solving: Combining Cognitive & Metacognitive Strategies

• Reading the problem. The student reads the problem carefully, noting and attempting to clear up any areas of uncertainly or confusion (e.g., unknown vocabulary terms).
• Paraphrasing the problem. The student restates the problem in his or her own words.
• ‘Drawing’ the problem. The student creates a drawing of the problem, creating a visual representation of the word problem.
• Creating a plan to solve the problem. The student decides on the best way to solve the problem and develops a plan to do so.
• Predicting/Estimating the answer. The student estimates or predicts what the answer to the problem will be.  The student may compute a quick approximation of the answer, using rounding or other shortcuts.
• C omputing the answer. The student follows the plan developed earlier to compute the answer to the problem.
• Checking the answer. The student methodically checks the calculations for each step of the problem. The student also compares the actual answer to the estimated answer calculated in a previous step to ensure that there is general agreement between the two values.
• The student first self-instructs by stating, or ‘saying’, the purpose of the step (‘ Say ’).
• The student next self-questions by ‘asking’ what he or she intends to do to complete the step (‘ Ask ’).
• The student concludes the step by self-monitoring, or ‘checking’, the successful completion of the step (‘ Check ’).
• Verifying that the student has the necessary foundation skills to solve math word problems
• Using explicit instruction techniques to teach the cognitive and metacognitive strategies
• Ensuring that all instructional tasks allow the student to experience an adequate rate of success
• Providing regular opportunities for the student to be engaged in active accurate academic responding
• Offering frequent performance feedback to motivate the student and shape his or her learning.

Attachments

• Say-Ask-Check Student Self-Coaching (Metacognitive) Prompts
• Burns, M. K., VanDerHeyden, A. M., & Boice, C. H. (2008). Best practices in intensive academic interventions. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp.1151-1162). Bethesda, MD: National Association of School Psychologists.
• Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.
• Montague, M., & Dietz, S. (2009). Evaluating the evidence base for cognitive strategy instruction and mathematical problem solving. Exceptional Children, 75, 285-302.

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6 Strategies for Increasing Critical Thinking with Problem Solving

By Mary Montero

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For many teachers, problem-solving feels synonymous with word problems, but it is so much more. That’s why I’m sharing my absolute favorite lessons and strategies for increasing critical thinking through problem solving below. You’ll learn six strategies for increasing critical thinking through mathematical word problems, the importance of incorporating error analysis into your weekly routines,  and several resources I use for improving critical thinking – almost all of which are free! I’ll also briefly touch on teaching students to dissect word problems in a way that enables them to truly understand what steps to take to solve the problem.

This post is based on my short and sweet (and FREE!) Increasing Critical Thinking with problem Solving math mini-course . When you enroll in the free course you’ll get access to everything you need to get started:

• Problem Solving Essentials
• Six lessons to implement into your classroom
• How to Implement Error Analysis
• FREE Error Analysis Starter Kit
• FREE Mathematician Posters
• FREE Multi-Step Problem Solving Starter Kit
• FREE Task Card Starter Kit

Introduction to Critical Thinking and Problem Solving

According to the National Council of Teachers of Mathematics, “The term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development .)”

That’s a lot of words, but I’d like to focus in on the word POTENTIAL. I’m going to share with you strategies that move these tasks from having the potential to provide a challenge to actually providing that challenge that will enrich their mathematical understanding and development. 

If you’re looking for an introduction to multi-step problem solving, I have a free multi-step problem solving starter kit for that! 

I also highly encourage you to download and use my free Mathematician posters that help students see what their “jobs” are as mathematicians. Giving students this title of mathematician not only holds them accountable, but it gives them greater confidence and gives me very specific verbiage to use when discussing math with my students. 

The impacts of Incorporating Problem Solving

When I made the shift to incorporate problem solving into my everyday instruction intentionally, I saw a distinct increase in student understanding and application of mathematical concepts, more authentic connections to real-world mathematics scenarios, greater student achievement, and notably increased engagement. There are also ripple effects observed in other areas, as students learn grit and a growth mindset after tackling some more challenging problem-solving situations. I hope that by implementing some of these ideas, you see the very same shift.

Here’s an overview of some problem solving essentials I use to teach students to solve problems.

Routine vs. Non-Routine Problem Solving

Routine problems comprise the vast majority of the word problems we pose to students. They require using an algorithm through one or more of the four major operations, have relevance to real-world situations, and often have a distinct answer. They are solvable, and students can use several concrete strategies for solving, like “make a table” or “draw a picture” to solve.

Conversely, non-routine problem-solving focuses on mathematical reasoning. These are often more open-ended and allow students to make generalizations about math and numbers. There isn’t usually a straight path leading to the answer, there isn’t an algorithm readily available for finding the solution (or students are going to have to come up with the algorithm), and it IS going to require some level of experimentation and manipulation of numbers in order to solve it. In non-routine problems, students learn to look for patterns, work backwards, build models, etc. 

Incorporating both routine and non-routine problems into your instruction for EVERY student is critical. When solving non-routine problems, students can use some of the strategies they’ve learned for solving routine problems, and when solving routine problems, students benefit from a deeper understanding of the complexity of numbers that they gained from non-routine problems. For this training, we will focus heavily on routine problems, though the impacts of these practices will transition into non-routine problem solving.

Increasing Critical Thinking in Problem Solving

When tackling a problem, students need to be able to determine WHAT to do and HOW to do it.  Knowing the HOW is what you likely teach every day – your students know how to add, subtract, multiply, and divide. But knowing WHAT to do is arguably the most essential part of solving problems – once students know what needs to be done, then they can apply the conceptual skills – the algorithms and strategies – they’ve learned and will know how to solve. While dissecting word problems is an excellent starting point, exposing students to various ways to examine problems can help them figure out the WHAT. 

Being faced with a lengthy, complex word problem can be intimidating to even your most adept students. Having a toolbox of strategies to use when you tackle problems and seeing problems in various ways can enable students to get to the point where they feel comfortable knowing where to begin.

Shifting away from keywords

While it isn’t best practice to rely solely on operation “keywords” to determine what operation needs to occur when solving a problem, I’m not ready to fully ditch keyword-based instruction in math. I think there’s a huge difference between teaching students to blindly rely on keywords to determine which operation to use for a solution and using words found in the text to guide students in figuring out what to do. For that reason, I place heavy emphasis on using precise mathematical vocabulary , including specific operation keywords, and when students become accustomed to using that precise mathematical vocabulary every day, it really helps them to identify that language in word problems as well.

I also allow my students to dissect math word problems using strategies like CUBES , but in a way that is more aligned with best practice. 

Six Lessons for Easy Implementation

Here are six super quick “outside the box” word problem, problem solving lessons to begin implementing into your classroom. These lessons shouldn’t replace your everyday problem solving, but are instead extensions that will help students tackle those tricky problems they encounter everyday. As a reminder, we look at all of these lessons in the FREE Increasing Critical Thinking with problem Solving math mini-course .

Lesson #1: What’s the Question?

In this lesson, we’ll encourage students to see. just how many different questions can be asked about the same statements or information. We start with a typical, one-step, one-operation problem. Then we cross out or cover up the answer and ask students to generate possible questions.

After students have come up with a variety of questions, ask them to determine HOW they would solve for each one.

Reveal the question and ask students how they would solve this one and see if any of the questions they came up with match.

This activity is important because it demonstrates to students just how many different questions can be asked about the same statement or information. It’s perfect for your students who automatically pick out numbers and start “operating” on them blindly. I’ve had students come up with 5-8 questions with a single statement!

I like to do this throughout the year using different word problems based on the skill we’re focused on at the time AND skills we’ve previously mastered, but be careful not to only use examples based on the skill you’re teaching right then so their brains don’t automatically go to the same place.

These 32 What’s My Operation? task cards will help your student learn and review which operations to use for different types of word problems! They’re perfect to use as a quick assessment, game of SCOOT, math center activity, or homework.

Lesson 2: Similar Scenarios

In this lesson, students will evaluate similar scenarios to determine the appropriate operations. Start with three similar scenarios requiring different operations and identify what situation is happening in each scenario (finding total, determining an amount, splitting or combining, etc.).

Read all three-word problems on a similar topic. Determine the similarity of all of them and determine which operation would be used to solve them. How does the situation/action of the problem help you determine what step to take?

I also created these differentiated word problem task cards after noticing my students struggling with which operation to choose, especially when given multiple problems from a similar scenario. They encourage students to select the appropriate operation for each word problem.

Lesson 3: Opposing Operations

In this lesson, students will determine relevant information from a set of facts, which requires a great deal of critical thinking to determine which operation to use. Give students a scenario and a variety of facts/information relating to the scenario as well as several questions to answer based on the facts . Students will focus on determining HOW they will solve each question using only the relevant information. 

These Operation Fascination task c ards engage students in critical thinking about operations. Each card has a scenario, multiple clues and facts to support the scenario, and four questions to accompany each scenario. The questions are a variety of operations so that students can see how using the same information can solve multiple problems.

Lesson 4: Next Level Numberless

In this lesson, we’ll take numberless word problems to the next level by developing a strong conceptual understanding of word problems. Give students scenarios without numbers and have them write a question and/or insert numbers using a specific operation and purpose . This requires a great deal of thinking to not only determine the situation, but to also figure out numbers that fit into the situation in a way that makes sense.

By integrating these types of math problems into your daily lessons, you can significantly enhance your students’ comprehension of word problems and problem-solving. These numberless word problem task cards are the ideal to improve your students’ critical thinking and problem-solving skills. They offer a variety of numbered and numberless word problems.

Lesson 5: Story Situations

In this lesson, we’ll discuss the importance of students generating their own word problems with a given set of information. This requires a great deal of quantitative reasoning as students determine how they would use a given set of numbers to create a realistic situation. Present students with two predetermined numbers and a theme. Then have students write a word problem, including a question, using the given information. 

Engage your students in additional practice with these differentiated division task cards that require your students to write their OWN word problems (and create real-world relevance in their learning!). Each task card has numbers and a theme that students use to guide their thinking and creation of a word problem.

Lesson 6: No Scenario Solving

In this lesson, we’ll decontextualize problem solving and require students to create the situation, represent it numerically, and solve. It’s a cognitively demanding task! Give students an operation and a purpose (joining, separating, comparing, etc.) with no other context, numbers, numbers, or theme. Then have students generate a word problem.

For additional practice, have students swap problems to identify the operation, purpose, and solution.

Implementing Error Analysis

Error analysis is an exceptional way to promote thinking and learning, but how do we teach students to figure out which type of math error they’ve made? This error analysis starter kit can help!

First, it is very rare that I will tell my students what error they have made in their work. I want to challenge them to figure it out on their own. So, when I see that they have a wrong answer, I ask them to go back and figure out where something went wrong. Because I resist the urge to tell them right away where their error is, my students tend to get a lot more practice identifying them!

Second, when I introduce a concept, I always, always, always create anchor charts with students and complete interactive notebook activities with them so that they have step-by-step procedures for completing tasks right at their fingertips. I have them go back and reference their notebooks while they are looking at their errors.  Usually, they can follow the anchor chart step-by-step to make sure they haven’t made a conceptual error, and if they have, they can identify it.

Third, I let them use a calculator. When worst comes to worst, and they are fairly certain they haven’t made a conceptual mistake to identify, I let them get out a calculator and start computing, step-by-step to see where they’ve made a mistake.

IF, after taking these steps, a student can’t figure out their mistake (especially if I find that it’s a conceptual mistake), I know I need to go back and do some individual reteaching with them because they don’t have a solid understanding of the concept.

This FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

Digging Deeper into Error Analysis

Once students show proficiency in the standard algorithm (or strategies), I take it a step further and have them dive into error analysis where they can show a “reverse” understanding as they evaluate mistakes made and fix them. Being able to identify an error in someone else’s work requires higher order thinking not found in most other projects or activities and certainly not found in basic math fact completion.

First, teach students the difference between a computational error and a conceptual error. 

• Computational is when they make a mistake in basic math facts. This might look as simple as  64/8 does not equal 7. Oops!
• A Conceptual or Procedural Error is when they make a mistake in the procedure or concept. 
• I can’t tell you how many times students show as not proficient on a topic when the mistakes they are making are COMPUTATIONAL and not conceptual or procedural. They don’t need more review in how to use a strategy… they need to slow down and pay closer attention to their math facts!

Once we’ve introduced the types of errors they should be looking out for, we move on to actually analyzing these errors in someone else’s work and fixing the mistake.

I have created error analysis tasks for you to use with you students so they can identify the errors, types of errors, rework the problem, and create their own version of the problem and solve it. I have seen great success with incorporating these tasks into ALL of my math units. I even have kids beg to take their error analysis tasks out to recess to finish! These are great resources to start:

• Error Analysis Bundle
• 3rd Grade Word Problem of the Day
• 4th Grade Word Problem of the Day
• 5th Grade Word Problem of the Day

The final step in using error analysis is actually having students correct their OWN mistakes. Once I have instructed on types of errors, I will start by simply telling them, Oops! You’ve made a computational error here! That way they aren’t furiously looking through the procedure for a mistake, instead they are looking to see where they computed wrong. Conversely, I’ll tell them if they’ve made a procedural mistake, and that can guide them in figuring out what they need to look for.

Looking at the different types of errors students are making is essential to guiding my instruction as well, so even though it takes a bit longer to grade things like this, it is immensely helpful to me as I make adjustments to my instruction.

Resources and Ideas for Critical Thinking

I’ve compiled a collection of websites for complex tasks with multiple, open-ended answers and scenarios. The majority of these tasks are non-routine and so easy to implement. I often post these tasks and allow students short bursts of time to strategize and plan for a solution. Consider using the tasks and problems from these sites as warm-ups, extensions of your morning meeting, during enrichment groups, or on a Problem of the Week board. I also highly encourage you to incorporate these non-routine problems into your core instruction time for all students at least once or twice a month.

• NRICH provides thousands of FREE online mathematics resources for ages 3 to 18. The tasks focus on developing problem-solving skills, perseverance, mathematical reasoning, the ability to apply knowledge creatively in unfamiliar contexts, and confidence in tackling new challenges..
• Open Middle offers challenging math word problems that require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. They support the Common Core State Standards and provide students with opportunities for discussing their thinking. All problems have a “closed beginning,” meaning that they all start with the same initial problem, a “closed-end” meaning that they all end with the same answer, and an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.
• Mathcurious offers interactive digital puzzles. Each adventure is dedicated to exploring the world of math and sharing experiences, knowledge, and ideas.
• Robert Kaplinsky shares math strategies, lessons, and resources designed to create problem solvers. The lessons are detailed and challenging!
• Mathigon “The mathematical playground” offers free manipulatives, activities, and lessons to make online learning interactive and engaging. The digital manipulates are a must-use!
• Fractal Foundation uses fractals to inspire interest in science, math and art. It has numerous fractal activities, software to help your students create their own fractals, and more.
• Greg Fletcher 3 Act Tasks contain engaging math videos with guiding questions. You can also download recording sheets to go with each video.

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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Improving Students’ Problem-Solving Flexibility in Non-routine Mathematics

• Conference paper
• First Online: 30 June 2020
• Cite this conference paper

• Huy A. Nguyen   ORCID: orcid.org/0000-0002-1227-6173 13 ,
• Yuqing Guo 13 ,
• John Stamper 13 &
• Bruce M. McLaren 13  

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12164))

Included in the following conference series:

• International Conference on Artificial Intelligence in Education

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1 Citations

A key issue in mathematics education is supporting students in developing general problem-solving skills that can be applied to novel, non-routine situations. However, typical mathematics instruction in the U.S. too often is dominated by rote learning, without exposing students to the underlying reasoning or alternate ways to solve problems. As a first step in addressing this problem, we present a cognitive task analysis study that investigates how students without a mathematics-related background solve novel non-routine problems. We found that most students were able to identify the underlying pattern that yields the final solution in each problem. Furthermore, they tended to use various forms of visualization in their draft work, but occasionally made computational mistakes. Based on these results, we propose our plan for developing an instructional platform that leverages learning science principles to train students in problem-solving abilities.

• Problem-solving flexibility
• Non-routine mathematics

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Opportunity to learn problem solving in dutch primary school mathematics textbooks.

Marc van Zanten & Marja van den Heuvel-Panhuizen

Reflections on Problem-Solving

Designing opportunities to learn mathematics theory-building practices

1 introduction.

The ability to tackle non-routine problems – those that cannot be solved with a known method or formula and require analysis and synthesis as well as creativity [ 9 ] – is becoming increasingly important in the 21st century [ 5 ]. However, when faced with a non-routine problem, U.S. students tend to apply memorized procedures incorrectly rather than modify them or develop new solutions [ 8 ]. One possible source for this difficulty is the typical instructional focus in U.S. schools on memorization and application of routine procedures [ 2 , 6 , 7 ]. Such an approach makes students proficient at executing rote procedures, but it does little to help them understand the conceptual basis for the procedures or to think creatively about novel problems - both of which are essential for developing problem-solving flexibility.

An important first step in addressing this issue is to assess how students currently approach non-routine problem solving, so that we can design the appropriate learning interventions. In this work, we present an empirical cognitive task analysis where participants were asked to think aloud while solving a series of non-routine problems from discrete mathematics. We chose this domain because discrete math problems can often be tackled from multiple perspectives while not requiring any advanced background beyond the high school curriculum [ 3 ]. Based on the findings from this study, we propose our plan for developing a tutoring system for non-routine problem-solving ability. Then, we discuss the system’s broader implications and the challenges we need to address in deploying this system at scale.

2 Assessing Students’ Problem-Solving Skills

We conducted interview sessions with three students at a private university in a midwest US city. None of the students had a mathematics-related background. The participants were asked to solve three non-routine mathematics problems on paper in one hour. They were also encouraged to think aloud and write down their draft work. The three problems in our study, taken from [ 3 ], and a brief summary of their sample solutions, are as follows.

In an air show there are twenty rows. The first row contains one seat, the second three seats, the third five seats, the fourth seventh seats, and so on. How many seats are there in total ?

Sample solution: In the first row there is 1 seat. In the first two rows there are 1 + 3 =  4 seats. In the first three rows there are 1 + 3 + 5 =  9 seats. In the first four rows there are 1 + 3 + 5 + 7 =  16 seats. In the first five rows there are 1 + 3 + 5 + 7 + 9 =  25 seats. Based on this pattern, in the first k rows there are k 2 seats. In our case, there are 20 rows and therefore 400 seats in total.

Find all integers between 1 and 99 (inclusive) with all distinct digits.

Sample solution: there are 99 integers between 1 and 99 in total, and 9 of them have non-distinct digits, namely 11, 22, 33, …, 88, 99. Hence, the remaining 90 integers have distinct digits.

What is the digit in the ones place of 2 57 ?

Sample solution: Looking at the sequence of powers of 2–2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, … – we see that the corresponding sequence of digits in the ones places is 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, … In other words, this sequence is a cycle of length 4. Therefore the last digit of 2 57 is that of 2 53 , which is that of 2 49 , …, which is that of 2 1 , which is 2.

We then analyzed recordings of the participants’ think-aloud and their draftwork, from which we derived the following insights:

Pattern Identification.

Participants were aware that they had to find a pattern or formula to solve the problems, because it was not feasible to directly compute the final answer. All participants were able to identify the expected pattern for each problem as outlined above, except for one student who failed to do so for Problem 1 . While this participant realized that the number of seats on row k is the k-th positive odd number, this pattern alone was insufficient to solve the problem.

Visualization.

Participants tended to visualize the problem by drawing examples and making lists or tables (Fig.  1 ). They expressed that these visualizations were crucial in helping them identify the correct pattern and solve the problem.

Participants’ attempts at visualizing the problem in their draftworks.

Computation.

Participants occasionally made computational mistakes while calculating the initial sequence values, especially in Problem 3 . As a consequence, they could not identify any pattern based on the wrong values, and took some time to realize the mistake. All students who corrected their mistakes were able to subsequently solve the problem.

In summary, we found that participants were aware of the idea behind identifying patterns, and they all did so via some kind of visualization. On the other hand, computational mistakes, while not directly related to our learning objectives, can be detrimental to the overall problem-solving process. From these insights, we propose the following next steps.

3 Developing a Tutoring System for Flexible Problem-Solving

Moving forward, our plan is to iteratively conduct more cognitive task analysis interviews and develop a prototype of the system. Our initial conceptualization of how the system will work is as follows. A single round of exercise in the system incorporates four learning stages, all of which are built on established learning principles: 1) Reviewing a worked example of a non-routine mathematics problem, 2) Explaining the worked example to a partner, 3) Solving a new problem which is isomorphic to the worked example problem, and 4) Explaining the isomorphic solution to a partner. Between rounds, the student can review previous solutions, look at materials related to the problem space, or practice basic math skills. This design is intended to (1) formally introduce students to a complete solution through worked examples, (2) reinforce their understanding of the worked example through self-explanation, and (3) assess students’ learning through an isomorphic problem. Our hypothesis is that through the learning system, students will get a better sense of how to approach a novel non-routine problem, so that in case they have not yet found the solution – for example, like the participant in our study who did not identify the true pattern in Problem 1 – they can still adopt a different viewpoint and explore other strategies.

We have already begun mapping the problem space by developing a non-routine problem-solving flowchart and identifying sets of potential non-routine problem solutions. Once we have tested our solution space, we will develop and pilot a low fidelity paper prototype version of the system with college students to further refine the mathematical content and identify areas for revision to the design. We are also looking at which technological features could be useful for students learning in this domain. As a first step, our system will include a canvas for students to perform their draftwork on, as well as a simple calculator interface with basic arithmetic operations to help students avoid computational mistakes. An important follow-up question is whether students’ draftwork can be analyzed to infer their thinking process, which could in turn guide the design of appropriate feedback mechanics. While this task has previously been performed manually by domain experts [ 1 ], employing a machine learning technique to automate it to some extent would greatly enhance the system’s adaptive support functionality and scalability.

4 Conclusion

This research will provide concrete, generalizable evidence about the utility and implementation of worked examples, multiple solutions, and self-explanation to promote skills in non-routine problem solving. Results will inform future tutoring system design by identifying how and when the instructional features are most beneficial for developing problem-solving skills. We also intend to have a practical impact by distributing a tutoring system that is accessible to a wide range of students, including lower-performing students who would typically not be exposed to these types of problems and strategies [ 1 , 4 ]. In addition, we will provide a teacher’s guide to support educators in using the system adaptively to support their instructional goals.

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Huy A. Nguyen, Yuqing Guo, John Stamper & Bruce M. McLaren

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Carleton University, Ottawa, ON, Canada

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University of Malaga, Málaga, Spain

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Nguyen, H.A., Guo, Y., Stamper, J., McLaren, B.M. (2020). Improving Students’ Problem-Solving Flexibility in Non-routine Mathematics. In: Bittencourt, I., Cukurova, M., Muldner, K., Luckin, R., Millán, E. (eds) Artificial Intelligence in Education. AIED 2020. Lecture Notes in Computer Science(), vol 12164. Springer, Cham. https://doi.org/10.1007/978-3-030-52240-7_74

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Mathematical Mysteries

Revealing the mysteries of mathematics

How to Solve Math Problems: Non-Routine Problems

Nonroutine Problem Solving , stresses the use of heuristics [3] and often requires little to no use of algorithms. Unlike algorithms, heuristics are procedures or strategies that do not guarantee a solution to a problem but provide a more highly probable method for discovering a solution. Building a model and drawing a picture of a problem are two basic problem-solving heuristics. Studying end-of-game situations provides students with experiences in using the heuristics of reducing the problem to a similar but simpler problem and working a problem backwards, i.e. from its resolution to its initial state. Other heuristics include describing the problem situation, classifying information, and finding irrelevant information. [1]

There Are Two Categories of Nonroutine Problem Solving: Static and Active

Static-Nonroutine  problems have a fixed, known goal and fixed, known elements that are used to resolve the problem. Solving a jigsaw puzzle is an example of a Static- Nonroutine problem. Given all pieces to a puzzle and a picture of the goal, learners are challenged to arrange the pieces to complete the picture. Various heuristics such as classifying the pieces by color, connecting the pieces that form the border, or connecting the pieces that form a salient feature to the puzzle, such as a flag pole, are typical ways in which people attempt to resolve such problems. [1]

Active-Nonroutine  problems may have a fixed goal with changing elements, a changing goal or alternative goals with fixed elements, or changing or alternative goals with changing elements. The heuristics used in this form of problem-solving are known as strategies. People who study such problems must learn to change or adapt their strategies as the problem unfolds. [1]

What is non-routine problem-solving in math?

A non-routine problem is any complex problem that requires some degree of creativity or originality to solve. Non-routine problems typically do not have an immediately apparent strategy for solving them. Often times, these problems can be solved in multiple ways.

Incorporating non-routine problem solving into your math program is one of the most impactful steps you can take as an educator. By consistently allowing your students to grapple with these challenging problems, you are helping them acquire essential problem-solving skills and the confidence needed to successfully execute them. [2]

Step 1: Understand

This is a time to just think! Allow yourself some time to get to know the problem. Read and reread. No pencil or paper necessary for this step. Remember, you cannot solve a problem until you know what the problem is!

• Does the problem give me enough information (or too much information)?
• What question is being asked of me?
• What do I know and what do I need to find out?
• What should my solution look like?
• What type of mathematics might be required?
• Can I restate the problem in my own words?
• Are there any terms or words that I am unfamiliar with?

Step 2: Plan

Now it’s time to decide on a plan of action! Choose a reasonable problem-solving strategy. Several are listed below. You may only need to use one strategy or a combination of strategies.

• Draw a picture or diagram
• Make an organized list
• Make a table
• Solve a simpler related problem
• Find a pattern
• Guess and check
• Act out a problem
• Work backward
• Write an equation
• Use manipulatives
• Break it into parts
• Use logical reasoning

Step 3: Execute

Alright! You understand the problem. You have a plan to solve the problem. Now it’s time to dig in and get to work! As you work, you may need to revise your plan. That’s okay! Your plan is not set in stone and can change anytime you see fit.

• Am I checking each step of my plan as I work?
• Am I keeping an accurate record of my work?
• Am I keeping my work organized so that I could explain my thinking to others?
• Am I going in the right direction? Is my plan working?
• Do I need to go back to Step 2 and find a new plan?
• Do I think I have the correct solution? If so, it’s time to move on to the next step!

Step 4: Review

You’ve come so far, but you’re not finished just yet! A mathematician must always go back and check his/her work. Reviewing your work is just as important as the first 3 steps! Before asking yourself the questions below, reread the problem and review all your work.

• Is my answer reasonable?
• Can I use estimation to check if my answer is reasonable?
• Is there another way to solve this problem?
• Can this problem be extended? Can I make a change to this problem to create a new one?
• I didn’t get the correct answer. What went wrong? Where did I make a mistake?

[1] “Pentathlon Institute Active Problem-Solving”. 2023.  mathpentath.org . https://www.mathpentath.org/active-problem-solving/ .

[2] Tallman, Melissa. 2015. “Problem Solving In Math • Teacher Thrive”. Teacher Thrive. https://teacherthrive.com/non-routine-problem-solving/ .

[3] A heuristic is  a mental shortcut commonly used to simplify problems and avoid cognitive overload .

Additional Reading

“101 Great Higher-Order Thinking Questions for Math”. 2023.  elementaryassessments.com . https://elementaryassessments.com/higher-order-thinking-questions-for-math/ .

⭐ “Developing Mathematics Thinking with HOTS (Higher Order Thinking Skills) Questions”. 2023. saydel.k12.ia.us . https://www.saydel.k12.ia.us/cms_files/resources/Developing%20Mathematics%20Thinking%20with%20HOTS%20Questions%20(from%20classroom%20observations)PDF.pdf .

“Higher Order Thinking Skills in Maths”. 2017.  education.gov.Scot . https://education.gov.scot/resources/higher-order-thinking-skills-in-maths/ .

“How to Increase Higher Order Thinking”. 2023.  Reading Rockets . https://www.readingrockets.org/topics/comprehension/articles/how-increase-higher-order-thinking .

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

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The math problem that took nearly a century to solve

UC San Diego mathematicians unlock the secret to Ramsey numbers

University of California - San Diego

Ramsey problems, such as r(4,5) are simple to state, but as shown in this graph, the possible solutions are nearly endless, making them very difficult to solve. 

Credit: Jacques Verstraete / UC San Diego

We’ve all been there: staring at a math test with a problem that seems impossible to solve. What if finding the solution to a problem took almost a century? For mathematicians who dabble in Ramsey theory, this is very much the case. In fact, little progress had been made in solving Ramsey problems since the 1930s.

Now, University of California San Diego researchers  Jacques Verstraete  and Sam Mattheus have found the answer to r(4,t), a longstanding Ramsey problem that has perplexed the math world for decades.

What was Ramsey’s problem, anyway?

In mathematical parlance, a graph is a series of points and the lines in between those points. Ramsey theory suggests that if the graph is large enough, you’re guaranteed to find some kind of order within it — either a set of points with no lines between them  or  a set of points with all possible lines between them (these sets are called “cliques”). This is written as r(s,t) where  s  are the points with lines and  t  are the points without lines.

To those of us who don’t deal in graph theory, the most well-known Ramsey problem, r(3,3), is sometimes called “the theorem on friends and strangers” and is explained by way of a party: in a group of six people, you will find at least three people who all know each other or three people who all don’t know each other. The answer to r(3,3) is six.

“It’s a fact of nature, an absolute truth,” Verstraete states. “It doesn't matter what the situation is or which six people you pick — you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.”

What happened after mathematicians found that r(3,3) = 6? Naturally, they wanted to know r(4,4), r(5,5), and r(4,t) where the number of points that are not connected is variable. The solution to r(4,4) is 18 and is proved using a theorem created by Paul Erdös and George Szekeres in the 1930s.

Currently r(5,5) is still unknown.

A good problem fights back

Why is something so simple to state so hard to solve? It turns out to be more complicated than it appears. Let’s say you knew the solution to r(5,5) was somewhere between 40-50. If you started with 45 points, there would be more than 10 234  graphs to consider!

“Because these numbers are so notoriously difficult to find, mathematicians look for estimations,” Verstraete explained. “This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?”

Math students learn about Ramsey problems early on, so r(4,t) has been on Verstraete’s radar for most of his professional career. In fact, he first saw the problem in print in  Erdös on Graphs: His Legacy of Unsolved Problems,  written by two UC San Diego professors, Fan Chung and the late Ron Graham. The problem is a conjecture from Erdös, who offered $250 to the first person who could solve it.

“Many people have thought about r(4,t) — it’s been an open problem for over 90 years,” Verstraete said. “But it wasn’t something that was at the forefront of my research. Everybody knows it's hard and everyone’s tried to figure it out, so unless you have a new idea, you’re not likely to get anywhere.”

Then about four years ago, Verstraete was working on a different Ramsey problem with a mathematician at the University of Illinois-Chicago, Dhruv Mubayi. Together they discovered that pseudorandom graphs could advance the current knowledge on these old problems.

In 1937, Erdös discovered that using random graphs could give good lower bounds on Ramsey problems. What Verstraete and Mubayi discovered was that sampling from  pseudo random graphs frequently gives better bounds on Ramsey numbers than random graphs. These bounds — upper and lower limits on the possible answer — tightened the range of estimations they could make. In other words, they were getting closer to the truth.

In 2019, to the delight of the math world, Verstraete and Mubayi used pseudorandom graphs to solve r(3,t). However, Verstraete struggled to build a pseudorandom graph that could help solve r(4,t).

He began pulling in different areas of math outside of combinatorics, including finite geometry, algebra and probability. Eventually he joined forces with Mattheus, a postdoctoral scholar in his group whose background was in finite geometry. 

“It turned out that the pseudorandom graph we needed could be found in finite geometry,” Verstraete stated. “Sam was the perfect person to come along and help build what we needed.”

Once they had the pseudorandom graph in place, they still had to puzzle out several pieces of math. It took almost a year, but eventually they realized they had a solution: r(4,t) is close to a cubic function of  t . If you want a party where there will always be four people who all know each other or  t  people who all don’t know each other, you will need roughly t 3  people present. There is a small asterisk (actually an o) because, remember, this is an estimate, not an exact answer. But t 3  is very close to the exact answer.

The findings are currently under review with the  Annals of Mathematics . 

“It really did take us years to solve,” Verstraete stated. “And there were many times where we were stuck and wondered if we’d be able to solve it at all. But one should never give up, no matter how long it takes.”

Verstraete emphasizes the importance of perseverance — something he reminds his students of often. “If you find that the problem is hard and you're stuck, that means it's a good problem. Fan Chung said a good problem fights back. You can't expect it just to reveal itself.”

Verstraete knows such dogged determination is well-rewarded: “I got a call from Fan saying she owes me $250.”

Annals of Mathematics

10.4007/annals.2024.199.2.8

Method of Research

Experimental study

Article Title

The asymptotics of r(4,t)

Article Publication Date

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Title: chatglm-math: improving math problem-solving in large language models with a self-critique pipeline.

Abstract: Large language models (LLMs) have shown excellent mastering of human language, but still struggle in real-world applications that require mathematical problem-solving. While many strategies and datasets to enhance LLMs' mathematics are developed, it remains a challenge to simultaneously maintain and improve both language and mathematical capabilities in deployed LLM this http URL this work, we tailor the Self-Critique pipeline, which addresses the challenge in the feedback learning stage of LLM alignment. We first train a general Math-Critique model from the LLM itself to provide feedback signals. Then, we sequentially employ rejective fine-tuning and direct preference optimization over the LLM's own generations for data collection. Based on ChatGLM3-32B, we conduct a series of experiments on both academic and our newly created challenging dataset, MathUserEval. Results show that our pipeline significantly enhances the LLM's mathematical problem-solving while still improving its language ability, outperforming LLMs that could be two times larger. Related techniques have been deployed to ChatGLM\footnote{\url{ this https URL }}, an online serving LLM. Related evaluation dataset and scripts are released at \url{ this https URL }.

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