## Increases can increase? Learn what students think

### Sea levels aren’t just rising. They’re rising FASTER.

https://www.cnn.com/2018/02/12/world/sea-level-rise-accelerating/index.html

### Yet how do students come to make sense of variation in change? How do “increasing” increases become things for students?

In a March 2018 episode of the Math Ed Podcast, I talked with Sam Otten (@ottensam) about an article I co-authored with Evan McClintock. I share results of the study and offer insights into our research process.

We found that students who discerned variation in increases also reasoned about attributes as being capable of varying and possible to measure.

Students’ willingness to share their thinking is key to our research. Learn how we position students as experts when conducting math interviews.

### Our article is available open access:

Johnson, H. L & McClintock, E. (2018). A link between students’ discernment of variation in unidirectional change and their use of quantitative variational reasoning. Educational Studies in Mathematics. 97(3), 299-316. doi: 10.1007/s10649-017-9799-7

Give students opportunities to make sense of varying increases

## Why is it so hard for students to make sense of rate?

Earlier this year, I searched for some newspaper headlines that dealt with rates. Here are a few that I came across:

• “Colorado unemployment rate among 10 lowest in the country”
• “Unintended pregnancy rate in U.S. is high, but falling”
• “Oil ends sharply higher. Logs 10% weekly gain as output draws focus”

Many of the headlines talked about rates, and not just rates, but varying rates. When I think about these headlines, I wonder how students make sense of varying rates. For example, what might a student think it means for a rate to be affected, for a rate to be low, or for a rate to be high, but falling? Furthermore, what does it mean for something to end “sharply” higher? Are there other kinds of ways to end higher? For example, what might ending “gradually” higher be like?

In my research, I investigate how students make sense of change, and more specifically, I study how students make sense of variation in change. Put another way, I want to know how—from a student’s perspective—an “increase” (or “decrease”) can be a thing that can vary. When I interact with students, I work to design learning experiences that can provide them opportunities to investigate different kinds of increases (and decreases).

When it comes to rate, I have found that it matters how students form and interpret relationships between quantities

If we want students to think about rate as something that is capable of varying, we should help students coordinate change in two different quantities, such as the height and volume of liquid in a filling bottle. Specifically, we should provide opportunities for students to think about one quantity as continuing to change while another quantity is changing along with it. For example, students could think about how the height of the liquid in a filling bottle continues to change while the volume of liquid in the filling bottle changes along with it.

I developed a framework that explains how students’ different ways of forming relationships between quantities can impact how they think about rate. I wrote about this framework in a 2015 article published in the journal Mathematical Thinking and Learning. The published version of the article is available here: http://www.tandfonline.com/doi/pdf/10.1080/10986065.2015.981946

Here is the full citation for the article (APA 6th)

Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Mathematical Thinking and Learning, 17(1), 64-90.

Here is the accepted manuscript of the article that you can download:

HLJohnson_QuantRatioRate_MTL