I have been thinking hard about how students make sense of graphs.
In my April 17 Global Math Department webinar, we’ll explore ways to help students see #HowGraphsWork
I hope many are able to join us. In case you aren’t able to make it, or if you would like to access resources after the webinar, I included links in this space.
Slides from the webinar
Open Access Online activities
Desmos Activities: Cannon Man, Toy Car, and Ferris Wheel
NCTM Illuminations Ferris Wheel Interactive
a Blog post and an Article
Steve Phelps’ (@giohio) Desmos Sketches
Isosceles Triangle v4
(Graphs selected in partnership with Sharon Hessney, the Statistics Content Director at Mass Insight Education)
Sea levels aren’t just rising. They’re rising FASTER.
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
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.
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.
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.
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, a 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/a“th 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.”
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.