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


webinar video

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   

Isosceles Triangle v4 

New York Times Column: What’s Going on in This Graph?

(Graphs selected in partnership with Sharon Hessney, the Statistics Content Director at Mass Insight Education)

Increases can increase? Learn what students think

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

Keep track of your writing progress to grow your writing practice

Ask yourself these questions:

  1. How often do you write?

  2. How long is your typical writing session?

  3. What counts as “writing”?

Had you asked me these questions earlier in my career, I probably would have responded: (1) Not often enough, (2) A few hours, (3) Work on a paper.

Keeping track of my writing progress

A few years ago, I decided to start keeping track of my writing progress to learn more about my writing habits. What I learned surprised me.

3 things I learned from keeping track

  1. I write more frequently when I am in the midst of a project. I procrastinate the most when I am working to develop new ideas.
  2. I need at least 15 minutes for a writing session. I need a break after 2 hours of writing.
  3. I have a wide variety of activities that count as writing. Some are harder for me than others.

Growing my writing practice

Keeping track of my writing progress helped me to grow my writing practice.

3 ways my writing practice has grown

  1. I have more JOY in writing. I look forward to writing sessions.
  2. I use my writing practice to learn and grow ideas. I know that ideas that sound good in conversation need to go through the “writing fire” to develop and grow. I use free writing to develop and nurture new ideas.
  3. I can anticipate and handle challenging portions of a writing project. And plan accordingly. I prioritize more challenging writing activities to help me to make the most of my writing sessions.

How I keep track of my writing

I use a google spreadsheet. The spreadsheet has six columns: Date; Activity; Time of Day; Hours; Progress; Next Steps.

  1. Date. I aim to write every week day. Some months I do better than others. If I have a heavy meeting day, I had better write in the morning or it won’t happen.
  2. Activity. Saying yes to that new conference paper or book chapter can take more time than I realize. If I have too many projects going on, I engage in less free writing, which is one of my favorite aspects of my writing practice.
  3. Time of Day. My most favorite time of day to write is the late afternoon/early evening. And I write at all times of the day.
  4. Hours. 15 minutes is really enough time for me to make progress on small tasks. Even though a six hour writing session seems like it might be a good idea, it is too much for me all at once.
  5. Progress. Recording my progress helps me to chunk writing projects into smaller, more manageable portions.
  6. Next Steps. Plans for my next writing session helps me continue to make progress with a writing project.

“I don’t have enough time to write. Let alone keep track of my writing.”

I don’t have enough time to NOT keep track of my writing.

How do you keep track?

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.

How do you “half-work”?

Leisure is not an enemy of productivity. “Half-working” is.

How do you keep your working time productive?

Actively staving off “half working” helps me to hold myself accountable for working during my scheduled work time. And affords me real space to not work.

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.


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.”


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.

Let’s share our ideas to grow them

How do we determine when we are “ready enough” to share our voice with others?

When do we think we have an idea that is “enough” to be worth sharing with a broader community?

What might happen if we start sharing our ideas to grow them?

I think about new ideas as living organisms rather than completed products. As I share ideas, I grow them.

I enjoy having written products, because they serve as records of my thinking at particular moments in time. And written products help me to learn how ideas have grown and continue to grow.

Here is one of my very first pieces of writing that I shared with a broader community

I describe a lesson that I submitted for my PAEMST application.

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Using Data and Linguine to Discover the Triangle Inequality_PCTM_2003

Forging ahead with new ideas

In my current work, I design online tasks to provide students’ opportunities to engage in mathematical reasoning about difficult to learn concepts such as function and rate. In Why is it so hard for students to make sense of rate? I share where my ideas have come and are going.


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.

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



Use Tech to Broaden Students’ Opportunities for Math Reasoning

Subtitle: The reason I’m giving this talk at #NCTM2017.

Think of technology as “playground equipment” that teachers can use to create online “learning playgrounds” for students.

By using different kinds of equipment, we can broaden students’ opportunities to engage in mathematical reasoning.

If you subscribe to NCTM’s Mathematics Teacher journal, you can read more here.Screen Shot 2017-04-01 at 11.49.22 AM