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Teaching Math for Understanding

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Wendy Ward Hoffer

793 X 10 = 7,930

793 x 100 = 79,300

793 x 1000 = 793,000

  • What do you notice about the above expressions?
  • What is happening to each of the digits of 793?
  • Write the zero rule for multiplication as you understand it?
  • Explain why the zero rule for multiplication works using your understanding of our base ten place value system.

These questions served as the heart of a math workshop last month when I visited lab hosts Andrea Overton and Rachel Gardener’s fourth-grade classroom at STEM Launch school in Thornton, CO. They dispensed with the typical worksheet featuring rows and rows of multiplication problems and instead sought to highlight conceptual understanding by inviting learners to look deeply at just a few expressions.

Students dove independently into their “thinking sheets,” responding in writing to the questions above.  Then their teacher brought a small group together for conversation. A student listening to a peer exclaimed, “Oooh!” and Rachel turned to him with an invitation to share his brain’s growth.

“My brain growth is that when you add a zero, it’s growing by ten. When you add a zero, you are basically making your number bigger by ten.”

Another student chimed in, “That makes a lot of sense to me now. I didn’t get it at first, but now it made a lot of sense.”

Rachel affirmed, “You didn’t give up when it didn’t make sense. That shows me you have a lot of perseverance.

Another student inquires, “So, you just add the zeroes?”

Rachel asks, “I heard you say, ‘just add those zeros.’ Why would you do that?”

A third student explained, “ It is because of the place value chart when I am making a jump…” as she began to explain, Rachel realized an illustration would help the group understand and invited this learner to the board. She drew:

She showed how each column goes up by ten and explained, “To get to ten from one, you have to make one jump…”

Rachel revoiced, “What I heard you say is if you move that seven over one jump, and everything moves over one jump, you are multiplying by ten.” Then she pointed to the one’s place and asked, “So, what comes over here?”

“Zero!” several students replied.

“What is the zero representing?”


“We are starting to understand.” Rachel affirmed these learners, then asked them to self-assess using thumb signals, their own sense of the zeros rule.  “Show me where you are.”

To learn more about teaching math for understanding and to see a lab classroom like Rachel and Andrea’s in action, join us for the PEBC Minds on Math Institute Dec. 12 – 13, and sign up for the Dec. 14 lab.



Excerpt from
Hoffer, Wendy Ward. Developing Literate Mathematicians: A Guide for Integrating Language and Literacy Instruction into Secondary Mathematics. Reston: The National Council of Teachers of Mathematics, Inc., 2016. (15-16)

Vygotsky, L. “Interaction between Learning and Development.” In Mind and Society, translated by M. Cole, pp 79-91. Cambridge: Harvard University Press, 1978

Boaler, Jo, and David Foster. “Raising Expectations and Achievement: The Impact of Wide Scale Mathematics Reform Giving All Students Access to High Quality Mathematics.” 2014.

Learn More at an Upcoming Institute


Enjoy Denver, the Mile High City, when you attend the Minds on Math institute

Minds on Math Institute
In this institute, you will learn how to address all eight of the common core standards for mathematical practice within workshop model instruction. We offer this institute twice a year, we hope you can join us at one!

November 14-15, 2017
March 6-7, 2018
Aurora, CO

Explore how explicit thinking strategy instruction can promote sutdents mathematical understanding while preparing them with problem-solving skills they can apply on standardized assessments and real world situations that require them to think as mathematicians.
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See Wendy Ward Hoffer in person at a PEBC event

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