How might we use open-ended questions to provide access for all students? How might these questions draw out a broader perspective of mathematics than what students typically associate with math?

My team and I have been integrating **Math Talks** into our work over the past two years. We were recently developing a workshop that was based on the following Texas Essential Knowledge & Skills (TEKS) standards:

Texas Mathematics Grade 6.5B: The student is expected to solve real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models.

Texas Mathematics Grade 7.4D: The student is expected to solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.

My wonderful colleague Amy suggested this prompt for a Math Talk:

## What do you know about 35%?

This prompt comes from the book Math for All: Differentiating Instruction, Grades 6-8 by Linda Dacey and Karen Gartland (2009). One of the sample excerpts from the book (see here) provides a really nice write-up of a teacher’s use of this prompt with her students. (Access this and other excerpts from the book by navigating to this listing for the book from Math Solutions and then clicking on the **Additional Information **tab.)

Amy and I worried that providing the prompt as written (*What do you know about 35%?*) might not produce as broad of an array of responses as we were hoping. How might we prompt learners to think more broadly about mathematics? To consider multiple representations, visuals, real-world scenarios, words, models, etc., as well as the more “typical” algebraic notations? Inspired by the work of Jill Gough in her #ShowYourWork learning progression, we decided to add a second sentence to our prompt:

Before reading any further, pause here. (No, really.) Set a timer for two minutes and see where your brain takes you. Record all of your thinking on paper.

..

How did it go? Were some representations easier for you to *show what you know* with than others? Does your paper reflect a broad understanding of 35%? What could be added to it?

What would the collective work of a room of middle school teachers look like for this Math Talk?

As you did, teachers recorded their individual thoughts on paper for 2 minutes, then turned and shared ideas with their tablemates, noting similarities and differences and brainstorming additional ways to show what they knew about 35%. Learners shared for a minute or two in their small groups, while the facilitator moved around the room, listening in on conversations, asking about learners’ work, and mentally sequencing some ideas to pull out in the follow-up whole group discussion.

Here’s an overview of the whole group debrief from one of these sessions…

T had the following set of equivalencies written on her paper.

I asked T to share these equivalencies *first* in order to get these algebraic notations on the table—not because they weren’t valuable (of course they were!)—but because they were a place that was *comfortable* for most of the learners in the room (middle school math teachers). We wanted to go further and/or to make more connections, but this was a great place to start.

We had a brief discussion about notation—*is it okay to have multiple equals signs on the same line? *We decided that *yes*, this was okay, because we were representing that all of these mathematical representations were equivalent.

Is this row of equivalences where I would have started a similar Math Talk debrief with students? Probably not, but this was the “comfort zone” for many teacher learners in the room, so it was a good place to start in this session.

Next, R shared some written statements with the group, including:

My cursive does not do justice to R’s beautiful script, but you get the idea. 🙂 R had some really nice *sense-making* sentences like the one above that related 35% to other benchmark percents.

J then shared the following models with us, including a sketch of a pie chart, a 100s grid, and a number line, each showing 35%:

This was interesting, as we considered and discussed how there is more than one model for representing percents—and how these models add value to students’ understanding of percents.

E continued the conversation by sharing about the real-world percent applications she had considered, including a sale for 35% off:

This (and a nomination from her tablemate) prompted J2 to share some of the real-life applications she had considered and drawn during her independent think time, including:

These real-life applications (shared by E and J2) helped other learners think about 35% in ways they hadn’t previously noted on their papers. J2 bravely shared with the group how she felt sheepish about voicing some of her ideas for 35% as she felt like they weren’t mathematical enough. This prompted a wonderful discussion about *what mathematics is*—what it *really *is—not just what the “school math pseudo-equivalent” that we were raised in is. Some of the biggest advocates for the value of J2’s contributions were those who did not have *any* real-world applications on their papers originally—and they realized now that those were missing and important.

Think of the value these real-world representations offer: The gas gauge gives me a *feel* for 35%. I’m not on empty (I have a bit more than that), but I’m less than half-full. I don’t *need* to fill up (well, *I* don’t, but my mother would!), but I’m not too far away from getting close to empty either. That’s what 35% *feels* like. And numerical representations alone (without the context) don’t always give us this same *feeling* for the value of the number.

I was grateful that J2 shared her thoughts—and her vulnerability. Both moved our learning forward as a community.

Lastly, B shared the following representation she had drawn:

*Wait*—does this represent 35%? An interesting discussion ensued where B defended that this *does* represent 35%, but that students often struggle with seeing 35% in this fashion (35% of a discrete set where the number of items in the set is not 100).

As a group we wondered: How might we show more explicitly and/or convincingly that this representation *does* represent 35%?

A few ideas surfaced:

*Maybe we could partition each dot into 5 equal pieces. Then there would be 5 x 7 = 35 shaded pieces out of 5 x 20 = 100 total pieces.*

*Maybe we could iterate this set of 7 shaded dots out of 20 total dots five times to create this picture:*

*Now there are 35 shaded dots out of 100 total dots.*

Then, J2 bravely piped up again: *What if each dot is a nickel?* **Mic drop.** How does J2’s comment help us *see* the 35% represented here in a new way?

Thank you, J2, again, for your real-world context—it’s still moving our learning forward!

Our final board after our whole-group debrief looked like this:

Consider where we started:

*What do you know about 35%? Explain your thinking using words, pictures, graphs, numbers, symbols, equations, etc…*

and where we ended.

How might we use open-ended questions to provide access for all students? How might these questions draw out a broader perspective of mathematics than what students typically associate with math?

Van de Walle, Karp, & Bay-Williams (2013) share the following graphic that displays five different representations of mathematical ideas:

Consider the collective work that came out of this Math Talk: Which representations did we develop as a group?

I love the authors’ second sentence in the caption for this figure: **Translations between and within each [representation] can help develop new concepts.** B offered the 7 shaded dots out of 20 as a manipulative model (based on two-colour counters):

But when J2 *connected* it to the real-life situation of each dot representing a nickel we made powerful mathematical connections about how we might *justify* that this visual represents 35%. *Translating between and within representations helps develop new concepts.*

Lastly, we stepped back from the task as learners and considered what middle school students might do with this task. Teachers read different student responses from this excerpt of Dacey & Gartland’s Math for All: Differentiating Instruction, Grades 6-8 (mentioned earlier) and discussed what each student work sample revealed about the student’s thinking. We all then read the **Teacher Reflection** section at the end of the excerpt. This commentary was especially powerful, as it helped us reflect on *why* a teacher would use a task like this. It articulated many of the feelings, learning, and wonderings we had experienced as a group of learners during the task and provided us with a common language to talk about *why* these moves were important.

What other benefits might open-ended Math Talks like this offer?

What other topics might work nicely with a

What do you know about ___?Prompt? Why?

**Sources:**

Dacey, L., & Gartland, K. (2009). Math for All: Differentiating Instruction, Grades 6-8. Sausalito, CA: Math Solutions.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary & Middle School Mathematics: Teaching Developmentally. New York, NY: Pearson Education, Inc.

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