“Don’t Kids Need the Fundamentals?”
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.”
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.
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).
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
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
See Yarieliz’s work
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.
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.
Boaler, Jo. 2016. Mathematical
Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring
Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass.
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.