Tag Archives: Problem-posing

Embedding Problem Posing in Curriculum Materials

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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) ↩︎

Problem Posing Fun in Fourth

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Recently, I have been learning a lot about problem posing from my friend Jinfa Cai, in particular how to infuse these opportunities within the use of curriculum materials. Because, while there are rich problem solving experiences in a good curriculum, we do not often see explicit use of problem posing, especially in K–5. The Notice/Wonder routine is probably as close as it gets.

Since I am in classrooms this year, I get to try some problem posing around curriculum activities and follow up with Jinfa. As with all learning, the more things I try, the longer my list of questions for him grows! After last week’s lessons, I was left with two questions that I can’t wait to discuss:

  1. How do we honor all of the posed problems within the timing of a lesson?
  2. How do we infuse opportunities to reason about the problems that can and cannot be answered with the information in the situation?

The curriculum task

The lesson focused on addition of fractions with like denominators. As an adaptation to the curriculum materials, I only showed students the bolded part of the task and asked them to share things they noticed and wondered. Because their ideas were all falling into the ‘wonder’ column, I quickly pivoted to the problem posing work.

Problem-posing launch

In their journals, I asked students to take a couple of minutes to write mathematical problems they could pose about this situation. They shared their problems as a whole class and I recorded.

It was so interesting, yet not surprising, that they asked the exact same problem the curriculum task was posing! I had them work as a group to solve that problem and told them that if they finished before we came back together as a group, to re-read the other problems on the board to see if they could answer them as well.

Problem solutions

Students represented their solutions in many different ways as they leveraged their understanding of fractions, addition, and multiplication. It was particularly interesting that you could see in their work how they used one expression to derive subsequent ones. Here are a couple of examples:

Additional problems

As a whole group, we compared and connected student work like in the 2 examples above. Discussing questions such as:

  • Where are the ¼ cups in the expression? 
  • Where are the ¾ cups in the expression?
  • Where is one expression in the other?
  • How did knowing this expression help you with another?

At that point, our time for the lesson was over so I quickly went around and snapped pics of the problems they posed independently at the launch.

I saw so many interesting problems that didn’t get shared during the whole group time, which made me wonder how I could have done that better next time and led to my two wonderings:

  1. How do we honor all of the posed problems within the timing of a lesson?
  2. How do we infuse opportunities to reason about the problems that can and cannot be answered with the information in the situation?

Embedding problem-posing experiences in the curriculum and answering the teaching questions that arise is such powerful learning for both students and teachers.  When students have the opportunity to pose their own mathematical problems based on a situation, they must make sense of the constraints and conditions from the given information to build connections between their existing understanding and a new understanding of related mathematical ideas (Cai, 2022). And when teachers have the opportunity to listen to connections students make, understand the math students see in situations, and make teaching decisions on how to elicit, share, and move forward with student thinking, it shares the mathematical authority in the classroom and leads to deeper, more powerful learning for all.

The other exciting part, that I wish happened more often in general, is through these experiences, Jinfa and I continually learn how research can inform practice and how practice can inform future research.