Leveraging Digital Tools for Problem Posing

I have blogged a few times about problem posing using print materials and lately I’ve become really interested and excited about the potential for digital tools in this work!

If you are new to problem posing, below are a few slides from Jinfa and my NCSM presentation for background – each image is linked to an associated research paper.

What is problem posing?

Many activities can easily be adapted to provide opportunities for problem posing by removing task questions (left) and replacing it with different prompt options (right).

original curriculum activity

problem-posing prompt options

How can digital tools enhance problem-posing experiences?

Being relatively new to both problem posing and digital lessons, I have learned so much trying things out in math classes this year. As always, the more I learn, the more questions/ideas I have. Below are two digital lessons that involve different flavors of problem posing.

Lesson 1: Our Curious Classroom

You can click through the lesson screens to see the full flow, but in a nutshell, students answer questions about themselves and explore different data displays.

After answering the first survey question, we asked students for problems they could answer about their class data and recorded their responses (sorry for the blurry image, I had to screenshot from a video clip:).

The students then worked at their table to answer the questions based on their choice of display.

The lesson continues with more survey questions, data display exploration, and ends with students personalizing their own curioso character (see bottom of post for unrelated, cute idea).

Things I learned:

1. Student responses can be collected and displayed so quickly with digital which saved us more instructional time for posing and solving problems.
2. The capability to see data displays dynamically change from one to another enhanced the discussion about which display was most helpful to answer the problems and why.
3. Students were so motivated to answer questions about themselves, learn about their classmates (audio clip below), and ask and answer questions about their own class, not a fictitious one.

“What did you like about the lesson?”

Things I wonder:

1. While having the teacher record the questions on the board worked perfectly, I wonder if or how younger students might digitally input their own questions w/o wearing headphones for voice to text or having spelling errors that are challenging for others to interpret? Maybe something like a bank of refrigerator magnets to choose from?
2. During the lesson, could the teacher input student questions onto cards in the Card Sort in Desmos so they could then sort the problems based on structure before solving?

Lesson 2: Puppy Pile

In this lesson, students generate a class collection of animals, are introduced to scaled bar graphs, and create scaled bar graphs. This one has a different problem-posing structure than the the first lesson which was interesting!

In this lesson, students use the Challenge Creator feature. In order to pose their problem to the class, students create their own set of animals (left) and then select a scale and create a bar graph (right).

After submitting their challenge, students then pick up one another’s problems and solve them.

Things I learned:

1. Students were extremely motivated to create their own problems and solve the problems of others.
2. This version of problem posing allowed students to have more control over the situation around which they were formulating problems, which they really enjoyed.
3. Challenge Creator is an amazing tool for repeated practice that is MUCH more engaging than a worksheet of problems.

Things I wonder:

1. How could this activity structure support or extend the problem posing experience in Lesson 1?
2. What other K-5 math concepts would be great candidates for a Challenge Creator problem-posing activity?

Final thoughts

I think problem posing is such an important instructional structure whether done in print, digital, or a hybrid of the two. It is important, however, to also consider the math, student motivation, and amount of time students spend engaging in the problem-posing process when choosing the format we use.

I would love to hear about what you try, learn, and wonder whether you try these lessons or adapt other lessons for problem posing!

Unrelated by Adorable Idea…

After Lesson 1, Katie printed out their personalized Curiosos for the wall;)

Gallery Walks: Engaging Students in Other’s Ideas

One instructional strategy that I love for collaboration and public sharing of student ideas is a gallery walk. In a gallery walk, students create displays of their thinking on chart paper or white boards and then the small groups walk around the room and visit each other’s posters. And even though students create such beautiful displays of their ideas, it is always challenging for me to structure the walk in a way that actively engages them in one another’s ideas. Like any problem of practice, it takes trying out new ideas to see what works, when, and for whom.

The Lesson

Last week, it was the first 3rd grade lesson about division. We decided to launch by mathematizing Dozens of Doughnuts to set the stage for the subsequent activities. If you haven’t read the book before, it is about a bear named LouAnn who keeps baking 12 doughnuts to share with a different number of guests who arrive at her door. We read the book and did a notice and wonder, anticipating we would hear something about LouAnn sharing doughnuts and the number of doughnuts, friends, or plates, which we did.

Student Displays

We then asked small groups to record all they ways that LouAnn shared her doughnuts. We purposefully didn’t specify the representation so they could look for different ways during the gallery walk.

As we walked around it was great to see the various ways students were representing the situations, but some small groups seemed to have settled on only one way. We had planned for them to look for similar and different ways during the gallery walk, but that can be so passive, with no opportunity for them to connect those new ideas to their work. So, instead of waiting for the gallery walk at the end, we decided to engage them mid-activity with each other’s ideas and allow time for them to use those ideas.

Taking a page from Tracy’s book, Becoming the Math Teacher You Wish You Had, we opted for a Walk-Around to cross pollinate ideas. We asked students to walk around and look for ideas they wanted to add to their poster. These could be new ideas or just a different way of representing an idea they already had.

You would have thought we gave them a chance to ‘cheat’ as they walked around with such intention to other’s posters. I wish I had captured the before and afters of all of their posters, but here are just a few where you can see the new addition of ideas.

After they finished adding to their posters, we paused to discuss the ideas they found from others – both new ideas they hadn’t thought about and ideas they had, but were represented in different ways.

Next Activity

Students then independently solved a few problems. It was great to see the variation we saw on the posters in their work. So many great representations to share and connect in future lessons!

More Ideas and Resources

Want to learn more about mathematizing? Check out Allison and Tony’s book, Mathematizing Children’s Literature.

Want to read more mathematizing blog posts? I have written about some of the books I used when coaching K–5.

Want to share your children’s book ideas for math class? Join me on IG!

Making Connections for Deeper Learning

In 3rd grade, students come to understand fractions as numbers. They count by them and locate them on a number line just like whole numbers. However, once they start operating with fractions in 4th and 5th grade, they tend to set aside everything they understand about whole number operations and treat fractions as numbers with their own set of ‘rules.’ I can think of many reasons why this happens, but my current wondering is how we can create more opportunities for students to make connections between their understanding of whole number and fraction operations.

Why Connections?

So many times I see students not realize all of the wonderful things they know that would be helpful in their new learning. I believe this is because we just don’t spend enough time making connections explicit to support this transfer.

One of my favorite papers on the importance of practicing connections (linked in the citation) describes the ‘why’ so nicely…

“Although there may have been a time when rote learning of facts and procedures was sufficient as an outcome for education, that is certainly not the case today. Anyone with a phone can Google to find facts that they have forgotten. But gaps in thinking and understanding are not easily filled in by Internet searches. Increasingly, we value citizens who can think critically, coordinate different ideas together, solve novel problems, and apply their knowledge in all kinds of situations that do not look like ones they have previously encountered. In short, we want to produce students with deep understanding of the complex domains that constitute the modern knowledge landscape (National Academies of Sciences, Engineering, and Medicine 2018).”

“Studies show that expert knowledge in a domain is generally organized around a small set of core concepts (e.g., Lachner and Nückles 2015) that imbue coherence to even wicked domains. Because they are highly abstract and interconnected with other concepts, core concepts must be learned gradually, over extended periods of time and through extensive practice. As students practice connecting concepts with other concepts, contexts, and representations, these core concepts become more powerful and students’ knowledge becomes more transferable (e.g., Baroody et al. 2007; National Council for Teachers of Mathematics 2000; Rittle-Johnson and Schneider 2015; Rittle-Johnson et al. 2001).”

Fries, L., Son, J.Y., Givvin, K.B. et al. Practicing Connections: A Framework to Guide Instructional Design for Developing Understanding in Complex Domains. Educ Psychol Rev 33, 739–762 (2021).

Subtracting Whole Numbers

In 4th grade, students have been decomposing fractions into sums of fractions with the same denominator and justifying their decompositions. They naturally leveraged their understanding of whole number decomposition, but when we gave them a problem to add or subtract, they quickly looked for a ‘rule’ to find the sum or difference. And, while we want them to generalize these operations, as the numbers get more complex –mixed numbers and unlike denominators – a memorized rule absent understanding doesn’t help students reason about the problem.

The lesson last week had students representing their fraction addition and subtraction on the number line, but that representation was causing some students more angst than support so we decided to use the problems from the lesson, but focus on the connection to whole number operations instead of forcing only the number line on them.

On their whiteboards, we asked them to record all the ways they think about and can represent 13–6=?. We saw a nice mix of ideas like removing items (base 10 blocks), hopping back on a number line, decomposing 6 and subtracting in parts, adding up (relationship to addition), and the algorithm – which made them all chuckle because after ‘borrowing’ it ended up being the same problem.

Connection to Subtracting Fractions

We shared these ideas out, recorded them on the board for reference, and then asked them to erase their whiteboards and do the same thing for 13/5 – 6/5=?. Students shared the methods and representations they used and we discussed how they were like the ones they used for whole numbers. It took no time for someone to say ‘It is exactly the same, just fifths.’ One student wrote it out in words so the discussion of the change in units was perfect.

Applying the Strategies

Next we wanted students to practice a couple of problems, including mixed number subtraction where the first fraction numerator was less than the second. We were excited to see many of the same methods being used and some students really got into showing it multiple ways.

Try it out!

Any 4th and 5th grade teachers out there who want to try this out, I would love to hear what you learn about student thinking and what students learn about important connections in math class!

Small Change, Big Thinking

Adapting math activities is one of my favorite parts of lesson planning. I love it so much because of the thoughtfulness, curiosity, and creativity involved in even the smallest of changes. In making any change, I have to think about what students know, the math of the activity, how the activity addresses the learning goal, ways students might engage in the activity, and questions to ask students along the way.

Fraction Activity

In this 4th grade activity, students were writing equivalent multiplication equations for a fraction multiplied by a whole number and then discussing the relationship between the different equations. The curriculum activity was good and definitely addressed the learning goal, but there was definitely an opportunity to open it up for more student reasoning and ownership. For example, in its current form, students don’t have the chance to think about which whole numbers would work in their equations or play around with the properties.

Small Change

Adapting doesn’t always require huge lifts. For this activity, all we decided to do was change the prompt to 12/5 = ____ x _____ and ask them to find as many ways as they could to make the equation true. I got so wrapped up in their work and discussions, that I didn’t snag any pictures of that part of the lesson, but after they finished we pulled up polypad and asked them how we could show why they are are all equivalent using the fraction bars. We wanted to be sure they just weren’t proceduralizing it at this point of the unit, so pulling up the fraction bars felt like a nice grounding of the concept. The board looked like this before we erased to make space to circle the other expressions.

Making Connections

At this point, they couldn’t get enough and asked for another fraction to try, so we gave them 16/3. We saw so much great thinking and use of the commutative property when finding the whole number and numerator.

Their excitement alone was the first indicator that allowing more space for their choices was a great idea! And then, as I was walked around, a couple students asked if they could write division equations. Of course I said yes and walked away.

I came back to #7 and #8 on this board:

When I asked how she came up with those equations, she said she used her multiplication equations because multiplication and division are related. I left her with the question of how she might show that division on the fraction bars and class wrapped up. I can’t wait to check back in with her tomorrow to see what she came up with!

Next time you plan a math lesson, I encourage you to think about small tweaks you can make to open it up for more student voice, ownership, and opportunities to think big! And I don’t know if anyone is even talking much about math planning on Twitter (X) anymore, but if you are, I would love to think together about tweaking math activities. So, send some activity pics my way @MathMinds and we can flex our curiosity and creativity muscles in planning together.

-Kristin

Supporting Mathematical Habits of Mind

 “The widespread utility and effectiveness of mathematics come not just from mastering specific skills, topics, and techniques, but more importantly, from developing the ways of thinking—the habits of mind—used to create the results.“

Cuoco, Al & Goldenberg, Paul & Mark, June. (2010).

Math curriculum lessons are often aligned to the Standards of Mathematical Practice. These practices can provide opportunities for students to develop the mathematical habits of mind described by Al Cuoco, Paul Goldenburg, and June Mark.

Mathematical Habits of Mind

1. Students Should Be Pattern Sniffers
2. Students Should Be Experimenters
3. Students Should Be Describers
4. Students Should Be Tinkerers
5. Students Should Be Inventors
6. Students Should Be Visualizers
7. Students Should Be Conjecturers
8. Students Should Be Guessers

The thing I love most about these habits of mind is the fact that as I read them, I can picture the math content and activity structures that could provide opportunities for students to develop these habits. I also really like the connectedness of them, where I can easily imagine how one habit leads students to engage in another. And because my favorite Math Practice is SMP7, look for and make use of structure, I am particularly drawn to the habit of conjecturing in math class. Excitingly, last week 5th graders were engaging in a topic that provided a perfect opportunity to conjecture.

Fraction Division

This past week, 5th grade students were dividing unit fractions by whole numbers and whole numbers by unit fractions. If you have ever taught this, you probably immediately picture students overgeneralizing these two different situations. In the vein of answer-getting, they often think the quotient will either always be a whole number OR always be a unit fraction – both including the product of the denominator and whole number in some way. And even though students have engaged in a lot of the habits within this work, it was with the two situation types separately.

To address the overgeneralization, we wanted them to engage in mix of the situation types in order to compare them. We launched with the following 2 problems, purposefully choosing the same numbers to elicit the difference in what is happening in the situation and the resulting quotients.

Student Thinking

As anticipated, we saw wonderful diagrams that generally matched each situation, but we could tell by the shading and erased work on Situation B that students were thinking that because they were working with fractions, their answer had to be a fraction.

We focused our discussion on the questions, “Where is 1 cake in your diagram?”, “Where are the people in your diagram?”, “Where are the servings in your diagram?”, and “Where is your answer in the diagram?”. Through those questions we saw a lot of labeling revisions to their work to make it clearer.

Mathematicians Talk Small and Think Big

“The simplest problems and situations often turn into applications for deep mathematical theories; conversely, elaborate branches of mathematics often develop in attempts to solve problems that are quite simple to state.”

Cuoco, Al & Goldenberg, Paul & Mark, June. (2010)

While the discussion was productive and we saw a ton of sense-making, visualizing, describing, and revision, I was left wondering how this moment transfers to the next time a student engages in one of these division situations.

I love this idea of tinkering around with smaller ideas to conjecture about larger ideas as a great way for students to deeply understand a concept and be able to transfer their understanding to the next time they engage in that concept.

So, for the tables done their discussions early, I asked them to write things they think are true about the division and lingering questions they might have. Here are a couple examples:

Next Steps

The question I am always left with after students have such amazing insights and questions is, ‘How do I keep this math conversation alive?’ With the pacing of curriculum, it can be challenging to dig into each of these moments for an extended period, so we need ways to let this thinking extend across the year.

One thing we could do is ask students if we can launch the next class period with their ideas. For example, I might ask the first student if I could post, “The order matters in division.’ at the start of class the next day and have the class discuss if they think that will always be true and why. This would be a great way to elicit the difference in quotients when we divide a whole number by a fraction and vice versa.

Another option that I used in my classroom, was posting the ideas on what I called a Class Claim wall. When students make a claim or conjecture, we posted them on the wall and then anyone could revisit them at any point and time.

I think both of these options are a wonderful way for students to continually think small and big about concepts while allowing us the opportunity to communicate to them that just because a curriculum unit of study wraps up, the learning about that concept continues.

-Kristin

If you want to read a bit more about claims and conjectures, I was kind of obsessed with it when I was teaching and blogged a lot:

Developing Claims – Rectangles With Equal Perimeters

Geometry Is Worth The Extra Time…

The Meaning of Subtraction

Extensions: Leave Students With More Questions

I appreciate the coherent connectedness of good curriculum when I see students making connections as they move through a unit concept. At the same time we know that no one learns the same thing, in the same way, at the same moment in time, so the need for additional time with a concept and extension opportunities are always a necessity. In our school, we typically address the additional time element during small group time, but extensions seem to be more of that ‘back pocket’ in-the-moment teaching move that is challenging and fun at the same time. We have to think quickly about the purpose of the extension problem/prompt and the quick question(s) to support students in moving forward in their thinking without a lot of back and forth because the lesson must keep moving for the rest of the class.

‘I’m finished, what do I do?’

Our 5th grade class is finishing up their unit on fraction multiplication and are at the point of explaining the generalization of multiplying numerators and denominators. As I was monitoring the other day, I saw two students who quickly, and correctly, finished up and were waiting for the whole class discussion – I am sure we are all all too familiar with this scenario! I asked them to grab their math journals (highly recommend everyone has a journal) and quickly needed to come up with a problem. I had just come from our 5th grade PLC, during which we were planning for the upcoming fraction division unit by working through some 5th and 6th grade fraction problems and digging into the math as learners. Selfishly, I was curious to see how students approached one of the division problems we worked on. So, I ask them to try and solve 4 ÷ 2/3 .

Class ended and when I walked in the next day they couldn’t wait to talk to me about it. I mean, what is better than that? This is what they showed me at the start of class:

This is where I love curriculum coherence, but see the need for dropping novel problems in students’ laps to get them out of the mindset that all problems they do in this unit will be about that unit concept. While there is so much interesting thinking here, they automatically approached it like they would a fraction multiplication problem, which if they leveraged their understanding of division might have worked, but in this case they both were having the feeling that it wasn’t quite right.

‘Can you give me a hint?’

The lesson was starting but they wanted a hint. I needed a quick question I could write down and leave for them to revisit when they had time during the lesson. I tried, ‘How do you think about 4 ÷2?’ and got a questioning look, so I tried again with different wording. ‘What does 4 ÷ 2 mean?’. I saw a lightbulb as she wrote her answer so I went to the other student and dropped the same question.

‘Can that be right?’

They used repeated addition and skip counting to arrive at 6. They were so excited, but then one of them asked me how it can be 6 when that is more than 4 and they were dividing. I literally couldn’t have asked for a better question! As we ended class, I wrote the following string of problems on their papers for them to think about:

4 ÷ 4

4 ÷ 2

4 ÷ 1

4 ÷ 2/3

4 ÷ 1/2

The coolest thing was that as I was giving them these problems, other students wanted to get in on the action too!

To be continued when I go back into school this week…

Focusing Teacher Learning Around Students

When I was teaching, I often felt overwhelmed by my own learning. The list of things I needed to know and be able to do felt never ending. And then, as I chipped away at my list, it seemed like the more I learned about teaching math, the more I didn’t know.

I think David Cohen describes the root cause of my feeling perfectly:

‘To teach responsibly, teachers must cultivate a kind of mental double vision: distancing themselves from their own knowledge to understand students’ thinking, yet using their knowledge to guide their teaching. Another predicament is that although attention to students’ thinking improves chances of learning, it also increases the uncertainty and complexity of the job.’

Cohen, D. K. (2011). Teaching and Its Predicaments. Harvard University Press.

As a coach, it feels much the same way when trying to decide on areas of focus for our lesson planning and PLC sessions. With a finite amount of time for teacher learning, prioritizing is so hard when everything feels equally important. So, this year instead of the principal, the teachers, and me planning the year-long instructional focus solely based on what we think it should be, we wanted our decision to also be informed by students’ math experiences. Then, to determine if the things we are learning and trying improve student learning as evidenced by assessments (formative and summative), we also wanted to know if they impacted students’ math identity, feelings about math, and ways in which they viewed learning math.

Student input

The Practical Measures work grounded our design of a quick 5–10 minute student survey that encompassed students’ feelings about math and experiences in math class. We looked at the data in our first PLC and there was so much great discussion sparked by comparing responses within and across grades. So much so that this will probably be multiple posts as we continue to aggregate the data in different ways, pair the question responses, and give the survey a few time across the year.

In our PLC, the first thing we did was notice and wonder about a pair of responses from our 3rd-5th graders:

One thing we wondered was why a student might like math but not like solving problems no one has shown them how to solve. We discussed things such as student confidence, worry about not getting the right answer right away, and the ability to transfer their understanding to a novel problem. We also hypothesized that if their prior math experiences have predominately included being shown how to solve problems with no time for revision of ideas, there could be a perception that they can’t solve problems until someone shows them what to do and that the answer they get to a problem is their ‘final’ answer.

Launch problem

Whenever we do this work together, I like to shift from hypotheses and theory to focus on an action we can take, try, and reflect on. One actionable thing we decided we could do was launch with the problem, let students try, learn from what they do, and use what we learn to adapt rest of the lesson. This often means not following the lesson plan to the letter or jumping in to rescue students by showing them what to do, but instead allowing them to use what they know, revise their ideas, and connect their ideas to others.

Here is an example of that in action from 5th grade:

The original problem: A city is designing a park on a rectangular piece of land. Two-thirds of the park will be used for different sports. One-half of the land used for different sports will be soccer fields.

In the workbook, students were given a square that represented the park and then stepped through how to think about the situation: 1) draw a diagram 2) write a multiplication expression and 3) find how much of the park will be used for soccer fields.

While that could be a great way for students to think through the problem, it is not the ‘novel problem’ experience we wanted them to have. So, we didn’t use the workbooks and instead asked student to work in their journal by themselves first. As we monitored, we noticed a variety of approaches so we asked them, as a group, to compare where they were in their thinking and finish out the problem together on a whiteboard.

As they did a gallery walk, we asked them to focus on what was similar and different in the ways groups approached the problem and then go back to their tables and make any revisions they wanted to make to their own work. For some this meant a complete revision, while others added on new connections they made.

Student thinking

Here are a few of the boards:

What I love about this when thinking about the survey prompt, ‘I like solving problems no one has shown me how to solve.’ is the multiple diagram types, the different ways students arrived at 2/6 with the same type of diagram, the multiplication and division expressions, and the equivalent answers of 1/3 and 2/6. And although the workbook problem didn’t exactly tell them what to do, it did scaffold it in a way where I could imagine their responses would have looked very similar.

For the rest of the lesson, we used their thinking to discuss their approaches, how they connected to one another, how they knew to use multiplication or division, things they noticed about their expression and product, and places where they changed or revised their thinking. We skipped Activity 2 altogether because this discussion was so interesting and important and reflected how we work through problems no one has shown you how to do or think about!

Next steps

Like all things teaching and learning, it takes time. I don’t expect this one experience to be the thing that shifts students like or dislike in solving problems w/o being told what to do nor do I expect every lesson to play out like this one. However, with repeated experiences similar to this, I hope students feel more confident in attacking a problem they haven’t been shown or scaffolded through and teachers refine their ‘double vision’ in a way that balances their own understandings and student thinking.

The best way we will be able to see if this has an impact is through students’ voice, which I look forward to digging into throughout the year in the surveys.

Students’ Brilli-ANT Connections in Math Class

This past week we planned for a 3rd grade lesson on arrays. The curriculum lesson goal was for students to build and describe arrays, in particular connecting the array structure to their understanding of multiplication as equal groups. The first activity in the lesson was written to encourage this connection, however having taught this lesson in previous years, we knew that the workbook examples could have come from students if we gave them the chance. Since we are always looking for ways to better amplify and leverage student thinking, we made some significant adaptations.

The original lesson

Learning goal: Build arrays with physical objects and describe them in terms of multiplication.

We decided that that the diagrams and questions in activity 1 (left image) would easily come from students’ prior understandings and experiences if we launched with a context that encouraged array thinking. Once we did that, it was then about selecting which problems in activity 2 (right image) we wanted to use. We figured we could do that on the fly depending on student work and our timing.

Adaptations

New Learning Goal: Make connections between multiplication as equal groups and arrays.

We read 100 Hungry Ants to open the lesson and asked students to mathematize the situation in a notice and wonder.

The notice and wonder elicited all the ways the ants rearranged themselves which was the perfect launchpad into the activity.

Each student had a cup of 30 ‘ants’ (beans) and a sheet of graph paper if they wanted to use it. We asked them to organize the ants into 4 groups of 6 and then captured pictures of student work to share and connect. They did not disappoint! They built the same images and made the exact connections as Activity 1, however in this version, students got to decide on the arrangement based on their understandings and experiences.

We first shared a picture of discrete groups next to an array and asked how they were the same/different and where the 4 and 6 were in each. Then, we shared arrays with 4 rows of 6 and 6 rows of 4 and discussed the same questions.

At this point we could have used Activity 2 problems, but decided that since they already have 24 counted out we could save time counting out a new set by just asking them to arrange the 24 ants in a different way. We wrapped up the lesson by asking students to write multiplication equations they used today when arranging 24 ants. It was a beautiful lesson.

Takeaways

While this is one really specific example of adapting, there are some general instructional ideas that work like this makes me think more about:

1. When using a new curriculum, teaching the lessons as is the first year is extremely helpful in making productive adaptations. Having experienced the math goal in action and understanding what students did with the lesson activities was invaluable in adapting to better center students and their ideas.
2. When we plan for lessons, we not only need to understand the content, goal, and lesson flow, we also need to look for places in the lesson where students are bringing their ideas and understandings to the table, especially when we are asking them to make new connections between concepts and representations. Side note: This is one of my favorite papers on students practicing connections.
3. Unsurprisingly, students are so much more motivated to look for similarities and differences between their own work than a workbook example. The more we can do this, the better!
4. Mathematizing children’s literature is such an incredibly engaging and powerful way to elicit and discuss math ideas. While this book is overtly mathematical, students still noticed things about the storyline and illustrations that showed wonderful sense making around the context. If you want to learn more about mathematizing, Allison and Tony wrote a beautiful book about mathematical read alouds with underpinnings, examples, and structures.

-Kristin

*If you are on Twitter (I can’t call it X yet), join me and others in sharing lesson ideas and learnings like this: https://x.com/LeahBaron03/status/1710305997472797074?s=20

Keeping Math Conversations Alive

Math routines are such a powerful tool for eliciting student ideas and making connections between them. The challenging part for me has always been ending them. Once I ask students for strategies or things they notice and wonder, the ideas are so uniquely interesting that I want to explore them all! However, when each idea can lead down a different path that may or may not be related to that day’s lesson, it is hard to know what to do in the moment. And the last thing I want to do is abandon the wonderful math ideas on the board.

Last week in 3rd grade we did a parallel choral count. Students counted by 2’s and then by 5’s as I recorded. I asked them to look for patterns they notice in either the individual counts or between the two. The lesson that followed was on multiplication, so the skip counting was helpful to lead into that lesson, but as more ideas started to emerge I found myself wondering where to go and what to do with all of these amazing ideas.

If you cannot follow my recording (how have I not gotten better at this after all these years:), here are some of the great math the students brought forward:

• There are some of the same numbers in both counts, but in different locations.
• All numbers in the 2 count are even and every other number in the 5 count is even.
• The 5 count gets to a larger number faster than the 2 count.
• Every number in the 2 count is the same number being added together – doubles.
• In the 5 count, there are always 2 numbers with the same digit in the tens place.
• At the top there is 2 + 5 = 7 and that is similar to the bottom row of 20 + 50 = 70
• Even + even = even, odd + even = odd, and odd + odd = even
• Someone added on that the bottom row is the same as 2×10 = 20 and 5×10 = 50

Every time I am in this situation I think about Joan Countryman’s book Writing to Learn Math. In there she describes math journals as a way to keep math conversations alive. That is exactly what I want to do with these ideas, keep them alive for more discussion. I am also a HUGE fan of math journaling, so I don’t need much of a nudge to use them!

Since we need the dry erase board for other things, the ideas cannot live forever on that board. I wondered about giving each student a copy of this picture to tape in their math journal. Then, when students finish up something early, they could find one of these ideas to explore further. I am thinking prompts like “The pattern I am exploring is…..” and “This pattern happens because….” might help students structure their explanations a bit.

Another idea that is more collaborative could be to replace an upcoming lesson warm-up with an idea from this count. We could display the picture on the board, highlight one of the patterns and ask students to work together to figure out why that pattern is happening and decide if they think it will always be true.

I would love to hear others’ ideas for not losing all the great math there is to explore in routines like this!

Embedding Problem Posing in Curriculum Materials

In mathematics education, problem posing refers to several related types of activities that entail or support teachers and students formulating and expressing a problem based on a particular context, such as a mathematical expression, diagram, table, or real-world situation (Cai & Hwang, 2020).

Because problem posing is so dynamic, multi-faceted, and varied between classrooms, I understand why it is hard to write into published curriculum materials. However, understanding and trying out the structure of problem posing makes it a really impactful teacher tool for adapting curriculum materials.

Why adapt ?

Curriculum activities typically require students to jump right into solution mode which explains why many students pluck numbers from word problems and operate without first making sense of the context. However, when students have the opportunity to pose their own mathematical problems based on a situation, they must make sense of the constraints and parameters that can be mathematized. They then extend from that sense-making activity to build connections between their existing understanding and the new context and its related mathematical ideas.¹ This provides opportunity for increased student agency and sense making in any lesson.

How to adapt?

Last week, the third grade teachers and I planned for a lesson that involved students answering questions about data in a scaled bar graph from the prior lesson. Here is the data and graph they were working from.

Instead of asking students to jump right into answering the questions in their workbook, we removed that day’s warm-up to make time for problem posing and adapted the activities that followed.

First, we displayed the graph and asked students “What math questions can we ask about this group of students?” Below are 2 different class examples.

Having such a rich bank of questions, we could have asked students to jump into solving them, however we decided to spend some time focusing on the structure of their questions. We asked them to discuss, “Which questions are similar and why?.” The discussion ranged from similarities based on the operation they would use to solve, whether they could just look at the graph and answer the question without any operation, and the wording problems had in common or not. Such great schema for solving future word problems!

Now that students had made sense of the context and problems, we asked them to solve as many problems as they could. As they solved, we asked them to think about which problems they solved the same way and which ones they solved differently. As we wrapped up the lesson, we shared student solutions and focused on their solution strategies leading to an amazing connection about using addition or subtraction to solve the ‘how many more or less’ problems.

What was the original activity?

If we had followed the curriculum, these are the questions students would have solved. As you can see the students came up with similar, if not the same, questions and SO much more!

1. How many students are represented in the graph?
2. How many students chose spring or fall as their favorite season?
3. How many more students chose summer than winter?
4. How many fewer students chose spring than fall?

While having solid curriculum materials is extremely important, they can be made so much better by adapting lessons in ways that provide the space for students to make sense of problems and have ownership in the problems they are being asked to solve. I am so grateful for the teachers, admin, and Jinfa’s partnership in this work and look forward to sharing our work and learnings at NCSM DC!

1. (PDF) Making Mathematics Challenging Through Problem Posing in the Classroom(opens in a new tab) ↩︎