Self-Powered Math

I sat across from Grace for an assessment conference, with just a small whiteboard and expo marker between us. “Can you come up with a function for which you would need to use the product rule in order to find the derivative?” She wrote a function on the whiteboard. “How do you know that you need to use the product rule here?” I asked. She explained, “because the function I came up with is the product of two other functions.” I then prompted her to apply the power rule and take the derivative of the function she had written.

In an alternate universe, I could have simply administered a pencil and paper test with 5 functions and the prompt “Use the power rule to take the derivative of each function.” Each student would do so silently and I’d check their answers.

By having Grace generate the problem herself, I accomplished a few different things. Firstly, I was able to tell that she understood the context and purpose of the power rule and when it was necessary to use it. The ease with which she came up with an example showed me that she had worked out plenty of other examples to prepare for this assessment. To generate her example she needed to make a function that was a product of two other functions. I was also able to determine her comfort level with taking derivatives of other functions based on her choices of which two to include. Students with a high comfort level might use two complicated functions for each component while students with less comfort might choose two simpler functions.

This example highlights the beautiful benefits of asking students to generate their own math (I’ll explain what I mean by this in more detail later). When they do so, they make connections, think critically, learn more deeply and are more readily able to transfer their knowledge. Every student has an entry point, providing natural differentiation, and it is nearly impossible to cheat.

Student-Powered math can take a variety of different forms. We’ll talk about each one in this post. Students can generate their own:

problems to solve.
mistakes to analyze.
driving questions for projects.
conjectures about mathematical truths.

In each case, it is the students applying what they have learned in order to create their own “math”.

Student Generated Questions

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One of my most memorable experiences in high school was preparing for a math exam. We had learned to find the area of sectors of a circle (a pizza slice shape). To study, I repeated problem after problem to practice. The night before the exam, I was in a half-sleep when I wondered: how could I find the area of a shape like the shaded region below?

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I got out of bed and worked on the problem. To my surprise - a very similar problem was on the exam! By generating my own example and self-testing, I was better prepared than my classmates. I still remember how to solve this problem today.

Student generated questions can refer to a variety of things but for the purpose of today’s post, we’ll focus on content-related questions. (If you’re interested in the wide-ranging types of questions students ask and, in particular, those in the “wonderment” category, you can check out my book, “Creating Curious Classrooms”).

My memory of the circle test exemplifies what a collection of research has been able to prove about student generated questions. Compared with reviewing notes, writing a list of questions at the end of class or before an exam helps students recall facts more readily and successfully complete transfer tasks (King, 1989 & Ebersbach et al., 2020).

Furthermore, when students generate their own questions they must think about the material in a deeper way. They are typically asked to learn a procedure and run it from start-to-finish routinely. Coming up with a question in which the procedure is needed ensures that they understand the context of it and can apply it appropriately.

There are many ways to prompt math students to generate their own questions about the content. Here are a few of my favorite:

Student led conferences: As demonstrated in the opening vignette, sit down with students and assess them orally. Ask them to demonstrate their knowledge of the material by coming up with their own example questions.
Self-testing: Ask students to write their own test questions and use them to quiz their classmates. You can assign this task right before a big assessment or as a regular reflection at the end of each class period.
Reverse multiple choice: Similar to self-testing: give students the answer to a multiple choice question and ask them to generate the question resulting in that answer as well as the other options. This forces them to also think about common mistakes and misconceptions (which we’ll discuss in the next section).

Student Generated Mistakes

Math is a field where students are particularly fearful of making mistakes. Bad handwriting resulting in the loss of a minus sign somewhere can have “catastrophic” results by the end of a long calculation. At least it might feel catastrophic to a student because they didn’t end up with the “right” answer. One of my favorite ways to empower students to call out mistakes and recognize that they are part of learning is to have them generate errors on purpose.

I use a protocol in which small groups of students reflect on a common error that they have seen or made themselves.

They write a problem and solve it incorrectly, intentionally making the mistake.
Everyone rotates and the next group locates the error and solves the problem correctly.
Finally, a third group looks at both the incorrect and correct work and explains the logic behind each solution.

I go into detail on this procedure in this Edutopia post. I love this activity because every student has an entry point. They’ve all made mistakes and can each bring something to the board for discussion.

Student Powered PBL

One of the most powerful ways to hand students the reins in the classroom is by giving them the opportunity to design their own Driving Question for a Project Based Learning Unit. Not every unit of study is conducive to allowing students to design the project from start to finish. In fact, not every unit is conducive to project based learning at all! However, I encourage you to look for opportunities where students can take the lead here.

In my experience, statistics is a natural place for students to design their own research questions. Here are a few examples of PBL units I facilitated with my statistics class and the questions they generated.

Observational Studies: I organized for the students to do an observational study at an elementary school. The driving questions they designed for the study included: “Is there a relationship between a student’s race or gender and their participation in class?” “Are students more likely to participate in class with a male or female teacher?” “How does student participation fluctuate over the course of the day?” After designing these questions, my students organized themselves. They created a data collection method and talked authentically about the best practices for observing a sample of the population rather than every student.
Representations of Data: I curated about a dozen datasets in csv format for my students and gave them two days to simply explore each data set to see which sparked their interest. Once they chose one to study, they generated five questions about it that they could answer using the data. For each question, they created both visual and numeric representations of the data to support their conclusions.
Hypothesis Testing: I generated an overarching driving question for the class, “Can statistics be used to spot injustice?” Within that, students identified different issues and their own, more specific, guiding questions to explore. Some examples included: “Is there a statistically significant relationship between race and college admission?”, “Is there a statistically significant difference between the drug overdose rate in DC compared to the entire US?”, “Is there a statistically significant relationship between police killings and income level?” and “Is there a statistically significant correlation between the type of government and malnutrition rates?” To answer each question, students needed to find an appropriate data set and perform a hypothesis test like a two sample t-test, an ANOVA test, a Chi-Squared test, or a t-test on the slope of a linear regression.

In all of these projects, students were meeting the content standards within each unit but were doing so by exploring their own curiosities.

Student Generated Conjectures

When my husband taught Discrete Math he designed a mini unit in which students debated the question: “Is mathematics discovered or invented?” Regardless of how you answer this philosophical quandary, one thing is certain: mathematical concepts align with beautiful and predictable patterns. With the right scaffolding, students are capable of both inventing their own conjectures and discovering mathematical truths on their own by careful observation of these patterns.

A lesson in which students “discover their own math” follows the design of “inductive learning.” The idea here is that you prepare a variety of examples for students to study. Then, students look for a pattern and make a conjecture about the mathematical truth behind it. Next, they test their conjectures on a new set of examples and refine their hypotheses.

Here is an example lesson that I designed to guide precalculus students to identify the rules for determining vertical and horizontal asymptotes.

Here are a few other examples of math lessons that naturally lend themselves to inductive learning.