We Ask, We Listen, We Learn

I’ve been a math educator for more than fifty years, and I thought that by now I’d probably heard every possible answer to problems I’d given students. But “120 and 30/5” for the answer to 12.6 × 10 was new to me. This answer was given by a fifth-grade boy at Malcolm X Elementary School in Berkeley, California. My colleague Ruth Cossey and I were visiting his class along with the students from Ruth’s math methods course at Mills College in Oakland, California. Ruth and I were part of the Math Solutions team that developed the Math Reasoning Inventory (you can read more about the MRI below). Ruth had prepared her Mills students to use the tool, and we blitzed the class, breaking up into pairs and interviewing all of the fifth graders. Then, after school, we met together with the teacher and the principal to talk about what we had learned. It was a terrific professional learning experience.

From developing the MRI, Ruth and I knew that the two most common incorrect answers students give for 12.6 × 10 are 120.6 and 12.60. (Of 7,881 students interviewed nationwide, 51 percent answered incorrectly, and these two answers represent 39% of the wrong answers given.) To hear a sampling of how students reasoned, watch these four video clips of students who answered incorrectly.

In contrast, the following video clip of a student who got the correct answer of 126 shows an explanation that revealed the student’s understanding of the distributive property.

We didn’t videotape the boy who gave the answer of 120 and 30/5, but the students from Mills College who interviewed him wrote down his explanation:

First I multiplied 12 x 10 and got 120.
Then I changed .6 to the fraction 3/5.
I multiplied 10 times 3/5, and that’s 30/5.
So the answer is 120 and 30/5.

In our after-school discussion, we first talked about whether “120 and 30/5” is a correct answer. Some were sure it was wrong, others felt it was right, and others weren’t sure. (Yes, the answer is correct, but it’s certainly not conventional. One way to think about it is that 30/5 is equal to 6, and 120 + 6 is 126.) Trying to make sense of the boy’s explanation reminded me that understanding the math we teach is essential for understanding how students reason. (On reflection, I think it would have been a good discussion to present the fifth graders with two possible correct answers—120 and 30/5, and 126—and talk about why they were equivalent.)

About the Math Reasoning Inventory (MRI): It’s a free online assessment tool for learning about students’ numerical reasoning, appropriate for students in grade 5 and up. You can learn about the tool and how to use it with your students at https://mathreasoninginventory.com/. There are three individual interview assessments—whole numbers, fractions, and decimals. After students answer each question, whether the answer is right or wrong, we ask them to explain how they reasoned. I’ve found MRI to provide insights I never had access before about students’ numerical understanding and reasoning ability. The video clips I posted here are part of the MRI Video Library of almost 100 clips that you can search by students or by problem.

Classroom Suggestions
Try the following with students who are studying about decimals. Write 12.6 × 10 on the board and ask students to figure out the answer in their heads. After a few moments, have students share their answers and how they reasoned with a partner, and then lead a class discussion. Here are questions that may be useful for the discussion:

  • How would you explain to someone why 12.60 can’t be the correct answer?
    [Since 12.6 and 12.60 are equal, 12.60 can’t be 10 times 12.6.]
  • How could you use addition to figure out the answer?
    [Add 12.6 ten times: 12.6 + 12.6 + 12.6 …]
  • Why doesn’t the rule for adding a zero when you multiply a number by 10 work here?
  • Why is moving the decimal point one space to the right the same as multiplying by 10?

Then write on the board the fifth grader’s answer: 120 and 30/5. Tell the students that this is an unusual way to express the answer but that it’s mathematically correct. Use the think–pair–share instructional strategy to engage students in thinking about this possible solution.

25 thoughts on “We Ask, We Listen, We Learn

  1. ps: It appears that your computer’s clock is off by 10 minutes, +/-, and that it has you located in the Mountain Time Zone; i.e., the time here in Florida is 10:51 pm EST, and your time stamp indicates about 9:01 pm. Just for your information….

    D

    • Yes, I noticed the the WordPress time is off. Learning WordPress has been a very steep learning curve (harder than learning long division). I’m trying to fix. Bear with me.

  2. I am retired, when I was a teaching math to primary students in 1st, 2nd & 3rd grades, I always believed that analyzing errors was an important tool in planning next steps. Thanks for this. I hope you will put me on your list to receive blogposts and updates.

  3. This was great Marilyn! Thank you so much for starting this blog! Look forward to learning more from you directly in my inbox! 🙂

  4. I used to believe that one of the most important skills I had as a teacher was the skill of “talk”. On my journey of continuous improvement, I have learned that the most important skill I could possess is the skill of “listen”. Students teach me so many things if I would just take the time to listen! Thank you again for sharing! 🙂

  5. I am very old and not a math major but I solved the problem by moving the decimal point one position to the right, and got 126. However I then read in your classroom suggestions “Why doesn’t the rule for adding a zero when you multiply a number by 10 work here?” And, I don’t know the answer to that because, it did seem to work for me.

    I think your deciding to ‘blog’ is a great idea!

    • For 12.6 x 10, the students who applied the whole number rule of “adding a zero to multiply by 10” arrived at the answer of 12.60. Though 12.60 is equivalent to 12.6, not 10 times greater, students don’t typically notice. These students didn’t apply the rule that you did of “moving the decimal point one place to the right,” which gives the correct answer of 126. What’s important with applying any rule is to check that the answer you get makes sense.

  6. Thank you for the sharing. I am very interested to do research regarding children’s reasoning in solving mathematics problem. Hope to hear from you soon. Thanks again..

  7. I would love to hear your thoughts on the CCLS for mathematics. Specifically, what are your thoughts on requiring students to use specific methods to solve their problems? The homework my daughter brings home is so confusing. The problems are simple but some of the strategies over complicate matters. Your thoughts?

    • I support the spirit and the intent of the Common Core and I’m especially appreciative of the Common Core Standards for Mathematical Practice that describe, for students in all grades, the expertise they need to be successful doing math. The third practice standard reads: “Create viable arguments and critique the reasoning of others.” To me, this translates to making discussions an integral aspect of classroom teaching so students understand not only their own way of thinking but also the strategies that others use. Understanding multiple ways to approach problems is valuable, and I encourage your daughter to make sense of strategies she’s expected to apply.

  8. Hi Marilyn,
    I like the idea of using the Math Reasoning Inventory as a lead-in for group discussions. It gives me a chance to listen to how students successfully or unsuccessfully solve a problem. The class can then analyze the various strategies and in an informal way they can develop reasoning skills that make sense to them.
    I’m retired, but I still volunteer in two primary classrooms. I’ll forward your blog post to my teachers.
    Bill

  9. The answer 120 and 30/5 reminded me of an answer I received when 5th grade students worked on the math problem about 2 pizzas and 3 people sharing it. How much would each person get? One group came up with 5 1/3 sixteenths. They said pizzas were cut into 8 slices, making 16 slices altogether. Each person would get 5 slices with one leftover. That one piece would be divided into thirds, hence 5 1/3 sixteenths. At first, because of my years of one way and one answer, I said I would have to think about that answer. Then as I thought about it, I decided it could be right, just not conventional. In the class discussion the next week I explained to the students that I wanted to return to the pizza discussion and the various answers. During the discussion a student explained that he liked that answer but did not like 5 1/3. He said you had to have the sixteenths so you would know how the pizzas were divided. The students now ask if they get a week to change my mind about answers.

    • Your comment reminds me of the importance to listening to how students are thinking, not for the answer I was hoping to hear. When I first gave this problem to students, I was hoping that students would come up with the answer of “2/3” for each person’s share, meaning “2/3 of a pizza.” But one student insisted that the answer should be “1/3” for each person’s share, and explained that each person would get “1/3 of the two pizzas.” And now your student came up with the answer of “5 1/3” for each person’s share, meaning “5 1/3 of the 16 slices.” Each answer relies on a different view of what the whole is — one pizza, the two pizzas, or the sixteen slices. It’s so important, with fractions, for students to be clear about what the “whole” is. And it’s also so important to have students explain their thinking for their answers.

  10. I’m wondering if there is an assessment tool similar to the Math Reasoning Inventory that you have or recommend for grades 3 and 4.
    Thank you.

      • I’m really glad to hear that you are working on an assessment for Grades 3 and 4! I used the tool several times last year and found it to be very informative of student thinking. Initially, I had difficulty making an account and had to contact customer support. Since then, I have passed the website along to colleagues and they are also having difficulty creating accounts. Perhaps it’s because it’s being edited? I wouldn’t want to lose this great tool! Thank you for your great blog and resources!

  11. I am having trouble creating an account with MRI. An admin from my district has invited a group of us and we get an error after filling out the form to create an account. Any words of advice?

  12. I’m teaching a grades 3,4,5 class and am also very interested in using this tool and look forward to the earlier grades being worked on.
    Thank you
    Peggy

  13. I think the strategies of adding a zero or moving the decimal when multiplying by 10 are causing the misunderstanding because they’re not accurate. What’s really happening is that the numbers are moving up the place value positions. The decimal doesn’t actually ever move.

    • I agree. When I’m working with elementary students, they notice the pattern of what happens when they multiply a number by 10, which they typically describe as “just add a zero.” It’s complicated. First of all, they’re talking about whole numbers only, which is dangerous as they later apply the pattern when they later learn about decimals. Secondly, even when I point out that “adding a zero” results in the same number; that is, when multiplying 6 x 10, adding a zero to 6 is 6 + 0, which equals 6, they push back. Then some students argue semantics, such as “I really meant tack on a zero” or “You put a zero at the end of the number.” Then we talk about how doing that shifts the value of the 6 because it’s now in the tens place. But their language is still “but what I do is just put a zero on the end.” All this is testimony to the realization that teaching by telling isn’t often effective. What’s also interesting to me is how some students, when figuring that 12.6 = 12.60, don’t make that error when it’s in a context, such as “Pencils cost $1.39. How much do ten pencils cost.” You may be interested in video clips of Natasha first incorrectly solving 12.6 x 10 () and then correctly solving the contextual problem (). Thanks so much for your comment. It’s inspriring me to write a blog about this to learn what others think.

    • The decimal point does move.

      If I want to take a number like 0.0000000000000000000000663 (ie Planck’s constant) I want to move the decimal point to something sensible, so I know what the number means.

      So I call it 6.63 x 10^-34 because the decimal point moves from where it is in Planck’s constant to where I want it to be in scientific notation.

      Human beings can’t do decimals. We only understand whole numbers. So 1.23 x 4.5 has to be done as 123 x 45 and a strategy needed to get the decimal point in the right place.

      It’s just an algorithm to get the right answer.

      2 x 6 = 12

      20 x 60 is 12 hundred.

      20×600 is 12 thousand

      0.2 x 0.6 is 12 hundredths

      They are all the same sum (2×6=12). All we need is a way of getting the decimal point in the right place. And that is just counting up powers of 10.

      So basically, just move the decimal point.

      Scientists do it all the time.

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