How to Learn Math Part 4: Teaching for a Growth Mindset

I’m halfway through Jo Boaler’s online course “How to Learn Math”. Normally I would devour a course like this as quickly as I can, but as a foster parent to a 3-month old baby, I find that I don’t have as much free time as I used to. (What a surprise.) In this session in particular, I could tell I didn’t have as much energy as I wrote my reflections, so I’m not quite as proud of my work. Nonetheless, I’m determined to finish the second half of the course by the end of September. It’s been such a valuable learning experience, and I don’t want to miss anything.

So in session 4, we returned to the idea of a growth mindset. Whereas in the earlier session we compared and contrasted fixed and growth mindsets, this session focused specifically on activities and actions we can take to help develop a growth mindset in our students/children. Here are my reflections from this session:

The lesson opened with a clip from a classroom. A teacher poses the problem 1 ÷ 2/3. There’s no context or story, just a bare naked math problem. She asks the students to solve the problem in a way that makes sense to them. How long does she spend on this computation problem? 15 minutes. That’s right, she keeps her class engaged and thinking for 15 minutes. It was enjoyable to watch.

After the video was over, we were asked to discuss what the teacher did to support her students’ learning:

She held off confirming if anyone was correct or incorrect. Every answer was considered valid, the students just had to explain why it made sense to them. Her classroom discussion showed that it is okay to show your work in a variety of ways, as long as it makes sense to you. In this problem we saw a circle, a number line, and an equation as three different representations of the solution. She never made anyone feel like they had done something bad because they made a mistake. Even though three people had come up and shown that 1 1/2 is the answer, she still had someone who thought it was 6 come up to the board to show their work. This actually helped some students solidify their belief that the answer was 1 1/2 because they realized that 6 was too large of an answer.

I liked this video clip especially because it acknowledged that there is an algorithm that can be used to solve the problem, but the teacher showed that what she valued more was understanding why you were doing what you were doing, whether it was using the algorithm or some other method. What she has done is create a classroom culture that values sense making, as messy and dirty as that may be. I’m sure it’s scary for many teachers because they feel the pressure to teach all the content before the test, but as Jo Boaler points out, students who have been taught to problem solve rather than answer endless test questions actually performed very well on the dreaded standardized assessments.

Next we reviewed an activity from Fawn Nguyen’s blog. I’ve been following Fawn on Twitter (@fawnpnguyen) for the past year, so I couldn’t help but smile when I saw that I was being directed to her blog. I admire her work, and I’m thrilled that everyone in the course got a little taste of it. You can (and should!) check out Fawn’s activity by clicking the link above. We were asked what ideas this task gave us about what goes into a quality math task.

Quality tasks are open-ended. There were some basic constraints that everyone had to work within, but the students were able to personalize the problem. It also wasn’t clear what would be needed in order to solve the problem. Just like in real-life problems, the students needed to analyze it and figure out what tools, strategies, and skills were going to be needed.

After analyzing a quality task, we had to actually do a task and analyze it. We were shown the following image:

Image of 3 different groups of stacked blocks. The first group has, from left to right, 1 block, 2 blocks, and 1 block. The second group has, from left to right, 1 block, 2 blocks, 3 blocks, 2 blocks, and 1 block. The third group has, from left to right, 1 block, 2 blocks, 3 blocks, 4 blocks, 3 blocks, 2 blocks, 1 block.

and asked these questions:

How do you see this shape growing?

How many cubes are in case 100?

In every case, the outside towers are kept. The inside tower is duplicated and a new inside tower is added whose height is one greater than the previous inside tower. In case 100 there is going to be a series of towers on the left that are 1 + 2 + 3 + 4 + 5… + 98 + 99. This same sum will appear on the right. The middle tower will be 100. So the total number of cubes is double the sum of all numbers 1-99 and then add 100 to that amount.

Then we had to analyze the task using a Growth Mindset Task Framework that Jo Boaler presented.

1. Openness

The question asking how this shape is growing is a very open task. There isn’t just one correct answer. The question about the 100th case is more of a closed task because there is one correct answer.

2. Different ways of seeing

In the growth question we saw two different descriptions and both were different from the description I gave, but they all described the growth in a way that was happening. In the 100th case question, there are different ways of seeing. Some students may use/see a quadratic equation, but it is not required. I saw it as the sum of the numbers from 1-99 doubled plus 100. I think this is the problem some teachers have, especially in middle and high school, where they get what skill a problem is meant to utilize so they automatically jump to that when solving it. As an elementary school teacher, I didn’t think about quadratics. I talked about it in terms of addition.

3. Multiple entry points

In the growth problem, someone could start by drawing more cases or they might start by getting out some blocks and making models of the cases. Others may not even make any more. They might just analyze the ones they are given. In the 100th case question, there is less variety in entry points. You need to get from case 3 to case 100. Chances are students will need to start solving more cases. Some might continue solving until they get to case 100 while others might stop to look for a pattern to save them the work of solving to case 100.

4. Multiple paths/strategies

I think this is related to the previous item. When solving the growth problem, there are lots of different ways to describe how the pattern is growing. Students can use words, pictures, real-world objects, and/or numbers to make their explanations. In the 100th case problem there are multiple strategies as well. You could solve every case up to 100. Or you could solve some, look for a pattern, and try to generate a rule to help you find case 100 without having to solve all of the ones up to that case.

5. Clear learning goals and opportunities for feedback

The growth problem seems like it’s trying to teach me about how to analyze and describe growth patterns. The 100th case problem is trying to get me to show how to find a specific case, but it isn’t clear that it really wants me to generate a rule so that I don’t actually find all the cases from 1 to 100. Some students might just see that they are being given a big number, not that they are purposefully being given a big number to discourage them from finding every case from 1 to 100. Since there is more personal expression involved in describing the growth pattern, it seems like that problem has more opportunities for feedback.

Now that we had analyzed a task that fosters a growth mindset, we were asked to take a closed task that encourages fixed mindset thinking and revise it so that it becomes a growth-mindset task.

I’d take a problem that says so-and-so has a recipe for something. The students are given the recipe and told they need to double/triple it. How much of x ingredient will they use? To make it more open I would tell the students to each find a recipe in a cookbook or online. Then they have to determine how much of each ingredient in the recipe would be needed to make enough of the recipe so each person in class gets one serving.

Later we learned about assessments for learning. This is an interesting idea I wish I had been able to explore more while I was teaching. We had a choice of three tasks to review. I reviewed one geared towards 6th grade students. The problem has to do with optimizing the location of a security camera in a shop. We were asked to note what features we saw in the task that would support our work as teachers and how the task would support a growth mindset.

The structure of the assessment is a resource. First, students are given a chance to try out the task on their own. Then the teacher reviews each students’ work and provides guiding questions, but there is no grade. On another day, the students have time to reflect on their work and the questions the teacher asked them in the feedback before attempting a final, group solution. The assessment ends with reflection for students to think about their learning from this experience.

The sample questions for the teacher is an excellent resource. Not all teachers are going to know about the common mistakes students will make, so not only does this activity provide a list of those, but also a list of accompany questions to help students who are making one of those mistakes. This is great modeling for a teacher so they can ask similar questions in the future when they are conferring with students on other tasks. The assessment supports a growth mindset because the activity is set over three time periods. It’s not a “done in one” assessment. Students wouldn’t even think they were being assessed. Instead they are revisiting an activity and attempting to grow and improve every time they interact with it.

I like the emphasis on reflection and growth. When students first get the assignment back, they are asked to reflect on the questions they were given to think about how they could improve their response. There is no judgment that what they did is right or wrong. I also like the focus on the idea that the work they do with their group is for the purpose of creating something better than any one of them could have made individually. Finally, I liked the reflection as they compare their work to sample student work, and then their final reflections. Both of these reflections help students see how they have grown through the course of this activity.

After analyzing activities, we learned about the harmful effects of tracking/grouping students by ability level. We were asked to reflect on why we think tracking results in lower achievement for students.

First, it leads to fixed mindset thinking – “I’m dumb” or “I’m smart” – and either way that hampers achievement. Second, once students are tracked, teachers claim that students can move up to a higher group if/when they’re ready, but the trouble is that the higher groups have continued moving at a faster pace, so the students in lower groups will always have a gap between them and the higher groups. They’re stuck! I saw this quite frequently as a teacher in elementary school. Once students were identified as needing academic intervention, they always needed academic intervention from then on.

It also keeps students from encountering different points of view that can help everyone grow and achieve more. The assumption is that lower level students can’t handle the same math as higher level students. However, I’ve personally witnessed a mixed ability classroom all work on the same activity and learn a lot together. It was amazing because the “low” kids actually provided more thoughtful explanations of their work than the “high” kids because it was truly a challenge for them. They had to think and reason, and the end product for them was great learning. The “high” kids were not as challenged and so their solutions and explanations weren’t as interesting. However, they got to learn from listening to the “low” kids share their correct and thoughtful solutions to the problem.

The session ended asking us to design something we will do to foster a growth mindset in our students/children.

As a foster parent, I am taking the information about the growth mindset to heart. I have children who can come to me at a variety of ages with a variety of backgrounds. The last thing they need to feel is that anything about them is “fixed” or “stuck”. They have the ability to grow intellectually and emotionally.

One thing I can do with these children to help them develop a growth mindset is create a lifebook together. Once they arrive at our home, we can start documenting their life through words and pictures (theirs and ours). By revisiting the book together regularly, we can talk about the ways they have grown and changed since they arrived. The growth won’t be something they’ll have to “trust” me about. They’ll have the lifebook as a tangible reminder of who they have been at every step of their journey, and they can identify in exactly what ways they have grown.

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