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)

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.

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.

Screen Shot 2017-08-01 at 11.52.48 AM.png

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.


I study students’ mathematical reasoning, and I take real breaks.

This week I revisited an advice column, Workload Survival Guide for Academics, which I came across last year.

Professor Andrew Oswald identified a price that comes along with the privilege of being a university faculty member: No clearly defined leisure time.

One way I make sure that I have leisure time:

Schedule time for real breaks.

One reason why my leisure time is so important: JOY

Joy in being.

Joy in scholarship.

How do you make time for leisure time?

Reply in the comments, or Tweet me @HthrLynnJ.

Narrating students’ mathematical reasoning

Stories of “storytelling” have captured my interest this weekend:

When I narrate students’ mathematical reasoning, I engage in storytelling.

I work to analyze what students say and do to better understand their perspectives.

I have the great privilege of learning from students who are willing to talk with me about their work to solve mathematics problems.

This weekend’s stories remind me of the responsibility that comes with that privilege.

To read one of my analyses of students’ mathematical reasoning, click the link below:

Johnson, H. L. (2015, July). Task design: Fostering secondary students’ shifts from variational to covariational reasoning. In Beswick, K., Muir, T., & Wells, J. (Eds.) Proceedings of the 39th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 129-136). Hobart, Tasmania: University of Tasmania