**By Adam Sarli, posted September 11, 2017 —**

The New York
City Department of Education just finished its second Algebra for All summer
workshop. The reading list—NCTM’s *Principles to Actions: Ensuring
Mathematical Success for All* and Smith and Stein’s *5 Practices for
Orchestrating Productive Mathematics Discussions*—and facilitators made it
clear that the workshops were meant to shift the city’s math teachers away from
an “I do, we do, you do” approach. Teachers engaged in Number Talks, engaged collaboratively
in novel math problems, and shared multiple strategies with the whole class. For
some teachers, this was a huge shift in teaching style.

It was not surprising that many expressed their doubts. In doing so, they revealed some misconceptions about problem-based math instruction. These misconceptions were familiar to me because I have struggled to overcome them myself for many years. In each of my four blog posts, I will present a misconception and attempt to challenge each misconception using my own experiences in the classroom.

“Kids need the fundamentals before they can do the math.”

The implicit belief in this statement is that there are certain rules in mathematics that students need to know before they can approach a problem. Students need the addition algorithm before they can solve an addition word problem. Or, more relevant to the middle grades, students need to know ratio strategies before they can approach a ratio problem. Let’s explore this latter assertion.

In *Developing
Essential Understanding of Ratios, Proportions, and Proportional Reasoning*, the authors put forth the big idea
that a ratio is a “composed unit” that can be iterated or partitioned while
still maintaining equivalence (Lobato and Ellis 2010, p. 12).

To help students make sense of this big idea, I designed a problem-based unit. I did not teach any “fundamentals” first; instead, I gave students a series of word problems and let them develop their own strategies over time.

Yarieliz was a student who initially struggled with proportional reasoning. After several days of word problems, she intuitively discovered that you can scale up by addition. If 2 apples cost $4, then 4 apples must cost $8, and 6 apples must cost $12. This reasoning was the beginning of Yarieliz conceiving of ratio as a “composed unit.” At this point, I helped Yarieliz organize her strategy on a double number line.

Yarieliz grew comfortable with scaling up by addition but wasn’t shifting to iteration by multiplication. So instead of modeling how to multiply both quantities by the same factor, I gave students a word problem where scaling by addition would be annoyingly inefficient.

See Yarieliz’s work at https://flic.kr/p/WnNBw5.

At first, Yarieliz still scaled all the way to 50 by multiples of 2. But then something clicked, and she could barely contain her excitement. She called me over to share her discovery: Scaling up 2:4 twenty-five times and multiplying by both quantities by 25 gave the same answer. She began drawing lines between different ratios on the table, discovering more and more scale factors relating equivalent ratios. This was powerful. Yarieliz had arrived at the big idea that a composed unit can be scaled up by multiplication. But before she could have accessed that idea, she had to enrich her understanding of the connection between multiplication and addition.

Did Yarieliz need the “fundamentals” before she could “do the math?” She developed the idea of a ratio as a composed unit by reasoning through ratio scenarios; she discovered iteration by multiplication in response to the demands of a word problem. She did this without my modeling of any strategies.

What if I had taught the fundamentals first? Many curricula on ratios propose exactly this, modeling on the very first day the multiplication of a ratio by a scale factor. The teacher puts 2:4 on the board, multiplies both by a scale factor of 25, and says you can multiply and divide ratios to find equivalent ones. Yarieliz would then have had to mimic a strategy she didn’t understand. Instead of reasoning that if 2 apples cost $4, then 4 apples would cost $8, she would simply have followed a procedure. And she certainly wouldn’t have had the chance to connect scaling up a ratio by multiplication with addition.

Jo Boaler argues in*Mathematical Mindsets*that when students are introduced to algorithms and rules at an early age, they begin to think that mathematics is a subject of memorization and rules (Boaler 2016, pp. 33–34). The beautiful thing about mathematics is that most of its truths can be arrived at intuitively. By putting problems first, we give students opportunities to make sense, rather than follow prescribed rules. In my experience, this results in students gaining a deeper and more lasting understanding of the big ideas inherent in mathematics.

**REFERENCES**

Boaler, Jo. 2016. *Mathematical
Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring
Messages and Innovative Teaching*. San Francisco, CA: Jossey-Bass.

*Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning, Grades 6–8*. Reston, VA: National Council of Teachers of Mathematics.

Adam Sarli is a middle school math teacher and math coach at MS331: The Bronx School of Young Leaders in the Bronx, New York. He has been teaching middle school mathematics for over ten years and is a Math for America Master Teacher. He is passionate about student-to-student discourse, student-created strategies, and problem-based math instruction. He tweets at @adam_sarli. His math music videos can be found on YouTube at youtube.com/user/Remikarli.

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