Johnson, H. L., Hornbein, P., & Bryson, D. (2016). Designing online playgrounds for learning mathematics. Mathematics Teacher, 110(4), 298-303. Link to abstract. (Full text available to NCTM members)

I share highlights from our research study published on February 28, 2020. (Scroll for article link.)

What are students trying to do when sketching graphs?

What do students think graphs “should” represent?

What We Did

We conducted a set of three individual interviews with 13 high school students (8 eleventh grade, 5 ninth grade). Students interacted with Desmos “How Graphs Work” Activities, that we developed in collaboration with Dan Meyer and the Desmos team.

Each graph related different distances, given in an animation. Below is an example of a graph from the Toy Car activity.

We identified three graphing goals, and linked those goals to students’ conceptions of graphs.

How do I show what I see?

How many graphs do I need?

What are the things I am graphing?

What We Found

When working on the Desmos activities, students shifted their goals. We were most successful helping students who started with goal #2, to shift to goal #3.

An additional factor impacted students’ graphing: their notions of what graphs should do. For example, if students thought that graphs should not “squiggle,” it affected their work.

Why This Matters

Graphs represent relationships between attributes. When students wonder what those “things” are, they are well positioned to be critical consumers of graphs.

Sketching accurate graphs is not the only important thing. If students are too focused on sketching something that looks “right,” they may miss the relationships.

As researchers who interview students, our methods are not neutral. Our lenses shape what we see. We need to create space to learn with and from students, who are experts in their own reasoning.

Opportunities for reasoning matter. Students need math to be more than a race to find answers to other people’s questions.

How to create those opportunities? We worked with Dan Meyer and the team at Desmos to develop activities for students to learn how graphs work. You can check out the activities here.

Great activities are a place to start. How students experience those activities makes all the difference. We worked with instructors to leverage the activities to notice the richness of students’ reasoning, rather than to fill in gaps in students’ graph accuracy. We developed facilitation guides to help with this.

We found that opportunities for reasoning impacted students’ attitudes toward math. In particular, we found statistically significant differences in students’ perceived competence. And the opportunities impacted students’ performance on the final exam.

Want to know more? Read our conference paper for #PMENA19

To open opportunities for students’ math reasoning, change the questions.

Instead of What iS? Ask What iF?

The questions look similar. Yet they imply very different responses.

What iS? “Give an answer”

What iF? “Consider the possibilities”

Too often, students experience math as a pursuit of “What iS?” rather than an exploration of “What iF?“

Take graphs and functions for example,

What iS a graph of a function?

What iF a graph represents a function?

The first question implies that students should give an answer – a graph of a function. The second question implies that students consider possibilities for graphs that represent functions.

Questions communicate what gets privileged.

Rather than just telling students that their math reasoning is important, ask questions that privilege reasoning rather than answer finding.

Students have many opportunities to use different types of representations to show the same relationship between variables (e.g., graphs, tables, equations).

Students also benefit from opportunities to use different forms of the same type of representation (two different looking graphs) to show the same relationship between variables.

Wonder how that can be?

Check out this Desmos Toy Car Activity.

Students explore how each variable changes, then they sketch different graphs to represent the same relationship between variables.

Students might be surprised that the sliders stay on the vertical and horizontal axes. When only one variable is changing, a line is sufficient to represent that change.

Graphs can be more meaningful for students when they understand what the variables are measuring. To begin, ask students how they might measure the toy car’s total distance traveled and the toy car’s total distance from the shrub.

Students’ reasoning is more important than the accuracy of their graphs. Encourage students to question, explore, and discover!

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.

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.

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

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.

I need at least 15 minutes for a writing session. I need a break after 2 hours of writing.

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

I have more JOY in writing. I look forward to writing sessions.

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.

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.

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.

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.

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.

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.

Progress. Recording my progress helps me to chunk writing projects into smaller, more manageable portions.

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.

What kinds of opportunities help students to make sense of accelerating growth?

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.

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.