I recently read Digging Deeper: Making Number Talks Matter Even More by Ruth Parker and Cathy Humphreys. (This book is a follow up to their book Making Number Talks Matter.)
Both books are brilliantly written and describe the authors’ process and thinking about what I consider the original number talks—a 10-15 minute daily routine where students solve computation problems mentally and discuss their strategies. One quote from Digging Deeper caught my eye about the power of number talks, and captured what I have always valued as a teacher and learner of mathematics:
“Many of us learned that mathematics consists largely of rules and procedures to be memorized and practiced. Number Talks, however, are different. They open up mathematics to discovery and help students learn different ways of interacting with mathematics and with one another. They can help students come to care about the thinking of others and learn not to be content until they understand the mathematical ideas at play. Number Talks have the power to chip away at the prevailing culture of mathematics that students have come to know and rely on.” (p. 57)
Reading this book made me think of a number talk I observed years ago in a 5th grade classroom in Las Vegas. The class had been studying percents for several weeks and were solving “80% of 65 =”. (Take a moment to mentally solve this problem yourself.)
The number talk proceeded pretty much as I anticipated. The teacher, Miss Holder, gave students enough time to mentally solve the problem—indicated by the majority of students showing a “thumbs up”—and recorded the list of students’ possible answers.
Most of the class agreed the answer was 52. As each student defended their answer, Miss Holder and students asked for further explanations or justification. The first few solutions shared were some variation of treating both numbers as whole numbers and multiplying 8 or 80 × 65, then making an adjustment for the percent. For example, one student multiplied 8 × 60 to get 480, 8 × 5 to get 40, added those numbers to get 520, and explained that because the question was what 80% of 65 was, the answer was 52 because 100% would be 65. As the class shared their thinking, the students who got incorrect answers realized they had made a multiplication error or had forgotten to adjust their answer for percents.
I was impressed by how easily and comfortably students moved back and forth between whole numbers and percents. Students seemed confident and most of the class seemed to be following the explanations. I definitely wasn’t expecting what happened next, which shouldn’t have surprised me. As Humphreys and Parker wrote in their first book:
“Number Talks, if they are to be meaningful, will be organic in nature. While we can plan for each Number Talk by choosing the problem carefully, the discussion about the problem—and where the discussion goes—depends on how our students are thinking.” (p. 6)
Miss Holder asked again if anyone solved the problem a different way.
Isabella: I got 52 too, but I did it differently. 80% of 65 seemed too hard to me, so I did 65% of 80.
Wow! Think about that for a moment. Is 65% of 80 “easier” to solve than 80% of 65? Does 80% of 65 = 65% of 80?
Even though this happened more than 20 years ago, I still vividly remember my reaction and what Isabella said. Miss Holder and I exchanged glances of surprise. In the moment, I wasn’t even sure what Isabella was suggesting would work or would work every time. I was intrigued by her strategy of switching the percent sign, and the math behind it, and was curious to hear her solution.
Isabella: 65% of 80 seemed easier to me. 50% of 80 is 40, 10% of 80 is 8, and 5% is half of that, or 4. 40 + 8 + 4 = 52.
I was so impressed with Isabella. She was so confident in her answer and she trusted that what she knew about multiplying whole numbers would work when multiplying percents. She also knew she could break 65 into 50 + 10 + 5, and that’s why 65% of 80 was easier for her to solve. Miss Holder told me afterwards that Isabella often looked for ways to break up numbers where she could take ½ of one factor, and also use the relationship between multiplying by 10 and multiplying by 5.
I still find myself thinking about Isabella’s solution. I wonder why it wasn’t immediately obvious to me that 80% of 65 = 65% of 80. If you’re multiplying, of course you can switch around the factors— 80 × 65 = 65 × 80. One way to think about the math is because 80% of 65 can be written as (80 × 1/100) × 65, then (80 × 1/00) × 65 = (65 × 1/100) × 80.
I wonder if my uncertainty was because of the way I learned to compute with percents—without much understanding—just learning rules to follow and trying to remember what formula I was supposed to use? Or is it because when we deal with percents, we’re dealing with parts of a whole, and while 80% of 65 equals 65% of 80, the “whole” isn’t the same, so it’s not evident the answers would be equivalent? Think of shopping—would you be more likely to buy a coat that was 80% of $65, or 65% of $80?
Reading Digging Deeper: Making Number Talks Matter Even More and remembering Isabella reminds me how number talks (and Investigations) focus on student thinking and how such a focus changes “the prevailing culture of mathematics that students” (and teachers) “have come to know and rely on.” (p. 57) It was obvious that number talks were an established routine in Miss Holder’s classroom. She and the students had worked hard to build a strong mathematical community where she trusted her students to understand and solve the problems, and more importantly the students exhibited mathematical agency and identity—they trusted themselves to be able to think deeply about the math and solve problems. The outcome of all this work? The mathematical power and brilliance of children. Just like Isabella.