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

I also talked about this article in an earlier blog post:

Give students opportunities to make sense of varying increases

Give students opportunities to make sense of varying increases

2018 began with news articles about varying increases:

In our recent research article, we report on a study identifying learning conditions that help early secondary students to “discern variation in unidirectional change.” Or put another way, make sense of accelerating growth.

We identify two conditions:

Provide students opportunities to conceive of attributes as capable of varying and possible to measure.

Possible to Measure

How students think about the attributes matters. Attributes such as length can be easier for students to conceive of measuring, than attributes such as volume or time.

capable of varying

Give students opportunities to make sense of change as happening. Too often, students only have opportunities to think about how much change has accrued.

Want more? Click the link to read our article.

Johnson, Heather Lynn, and Evan McClintock. “A Link between Students’ Discernment of Variation in Unidirectional Change and Their Use of Quantitative Variational Reasoning.” Educational Studies in Mathematics, Springer Netherlands, Jan. 2018, pp. 1–18.

Want to try this with students?

Use one of the Techtivities we developed in collaboration with Dan Meyer and the team at Desmos.

Make Graphs about Relationships with Cannon Man

In math classes, students work with graphs. A LOT.

Yet, what do students think graphs are? Why might students sketch or use graphs?

A powerful way for students to think about graphs: As relationships between “things” that can change

Together with Dan Meyer and the team at Desmos, I developed activities, “Techtivities” to provide students opportunities to think about graphs as representing relationships.

In this audio clip from a recent presentation, I talk through one of the Techtivities, the Cannon Man:

Want to find out more about how we’re using the Techtivities? See our ITSCoRe project website.

Tried one of the Techtivities? Have questions about when or how the Techtivities? I would enjoy hearing and responding to your comments.

Are transformations of functions giving your students trouble? Try a covariation approach.

Consider this problem:

Sketch a graph of a function y=f(x).

Now sketch a graph of y=f(ax) for some constant value a>1.

Reflect.

What kind of graph did you sketch for y=f(x)? How did you decide what to sketch? Did you choose a particular value for a? How does your graph of y=f(ax) compare to your graph of y=f(x)? What is similar? What is different?

It is no secret that students have difficulty making sense of transformations of functions. A covariation approach can help. Here’s how.

A graph of y=f(x) represents a relationship between two “things” that can change: “y” and “x.”

Think about a graph of y=f(x) as representing a relationship between two “things” that can change: “y” and “x

The notation y=f(x) means that y is a function of x. Functions are special kinds of relationships between two “things” that can change. When y is a function of x, as the values of x change, the values of y change in predictable ways.

A graph of y=f(ax) represents a relationship between two “things” that can change: “y” and “ax

The notation y=f(ax) means that y is a function of ax. When y is a function of ax, as the values of ax change, the values of y change in predictable ways. Because a is some constant value greater than one, does not change. Therefore, the changing values of ax depend on changes in the values of x.

Working from a relationship between “y” and “x,” students can make sense of a relationship between”y” and “ax.”

For a function y=f(x), as the values of x change, the values of y change along with them.

For a function y=f(ax), the values of x change by a factor of “a.” This means that “1/ath of the change in x will yield the same amount of change in y as for y=f(x).

Using relationships between changing “things” can help students to make sense of graphs of transformations of functions.

For example, sketch a graph y=f(x). 

Now sketch a new graph with this constraint: “1/2 the change in x will produce the same change in y as for the original function.”

Reflect.

What kind of graph did you sketch for y=f(x)? How did you decide what to sketch? How does your new graph compare to your graph of y=f(x)? What is similar? What is different?

A covariation approach helps students to focus on mathematical relationships. Using relationships is more powerful than trying to remember rules.

Graph Makeover: It’s about time

Think back to a time when you encountered a “real world” graph in a math class. What was on the horizontal axis?

Probably TIME.

Graphs represent relationships between two things. When working with graphs, it is important for students to form and interpret relationships between TWO things that can change. Yet, if one of those things is passing time, students might be thinking about only ONE thing changing.

Want to provide students opportunities to think about TWO things that can change?

Have students interpret graphs that represent two changing lengths.

Wonder how to get started?

Try these Ferris Wheel Interactives on NCTM Illuminations.

Screen Shot 2017-03-06 at 11.22.45 AM

It’s about time that students have opportunities to encounter “real world” graphs representing things other than time.

 

 

How would you use mental math to solve 36×25?

Last week I asked #MTBoS followers how they would use mental math to solve 36×25.

Below are a few responses that I received:

 

Having students use mental math to solve multiplication problems such as 36×25 can provide them with opportunities to informally use mathematical properties that are fundamental to algebraic reasoning.

For instance,

 

When students talk about their mathematical thinking, they have opportunities to clarify their ideas and to make meaning from symbol sentences. Tina, Annie, and Max all mentioned how they interpreted the symbol sentences they wrote. I think it is especially interesting how Tina said that the first two sentences were how she “thought it through.”

 

When I investigate students’ reasoning in my own research, I have learned that students may use the same written representation, but think about it very differently. The selection of Tweets that I included here is only a sampling of the interesting ways that #MTBoS followers thought about 36×25.

 

I wonder what might have changed if I had asked #MTBoS followers how they would use mental math to solve 25×36.

 

This semester, I am teaching a fully online class – Expanding conceptions of algebra— #MTED5622. In #MTED5622 we are investigating how students in grades K-12 engage in Algebraic Reasoning. One of the resources we are using is from NCTM’s Essential Understanding Series –Developing Essential Understanding of Algebraic Thinking Grades 3-5. This fall, I’ll be working to include blog posts focused on my work in teaching this course.

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