Solving 100 ÷ 3 Mentally: A Surprise

I love interviewing students to find out how they reason numerically―no paper or pencil, problems that are accessible mentally, with my focus entirely on how they explain their thinking. Here I was listening to learn what Jonah knew about division, and this was one of twelve questions I asked him. Take a look.

 

About the video: Jonah used almost half of the time in this 46-second clip to sit quietly and think about how to solve the problem of dividing 100 by 3. That felt to me like a long, long time to wait. But I didn’t interrupt Jonah because I noticed he was working to figure out the answer. Jonah then took about the same amount of time to explain how he arrived at the incorrect answer of 33¼.

Notice that I neither corrected Jonah nor let on that his answer wasn’t correct. When I’m interviewing, I focus on listening to understand how students reason. Then, what I’ve learned informs my teaching decisions, where I plan instruction to build on what I’ve learned about what they know to help them think about ideas new to them.

About the stretch during which Jonah was thinking: I describe this as an example of “wait time” in action, an idea that has had a huge impact on my teaching. If you’re interested in learning more about wait time, which was coined by Mary Budd Rowe, check out her 1986 article from the Journal of Teacher Education, “Wait Time: Slowing Down May Be A Way of Speeding Up!”

Thinking Back to a Classroom Experience with 100 ÷ 3
Because of my experience over many years interacting with students about their ideas about 100, I wasn’t surprised by Jonah’s response. Two Ways to Count to Ten is a children’s book I’ve read to many classes across the elementary grades. It’s a retelling of a Liberian folktale written by Ruby Dee and illustrated by Susan Meddaugh.

I remember reading the book to a class of fifth graders, the same grade as Jonah’s. This was quite a while ago but I remember the lesson. After talking about the ways we can count to 10―by 1s, 2s, and 5s―we talked about how to count to other numbers. We agreed on several ideas: You could count to any number by 1s, you could count to any even number by 2s, you could count to any number that ends in a zero by 10s, you could count to any number that ends in zero or 5 by 5s. This was low-hanging mathematical fruit.

I then said, “I’m thinking about the number 100. Do you agree that we can count by 1s, 2s, 5s, and 10s and land on 100?”

There was agreement about this.

Then I asked, “What about if we counted by 3s? Would we land on 100?”

What I expected was that they’d either know without having to figure, since 100 is a common benchmark number, or would easily figure out that you couldn’t. What I wasn’t prepared for was that almost all of the students didn’t know without having to figure, most hadn’t ever thought about this, and it was tough for them to reason. Since then, in many other classes, I’ve found that it’s not immediately obvious to students that you won’t land on 100 when counting by 3s.

Using the Video with Another Class
I often “borrow” classes at John Muir Elementary School in San Francisco to try out teaching ideas. After I had interviewed Jonah, I made this teaching plan to try with the Muir fifth graders:

  1. Teach a problem-solving lesson based on solving 100 ÷ 3 but in a context, presenting the problem of dividing 100 sheets of Origami paper equally among three classes. (This was to prepare them to listen to Jonah’s explanation.)
  2. Show them the video clip of Jonah solving 100 ÷ (They were familiar with being interviewed and were interested in watching me interview Jonah.)
  3. Give them the writing assignment of writing a letter to Jonah about whether they agreed or disagreed and to explain their thinking. (As committed as I am to conducting one-on-one interviews, I’m also committed to making writing an integral part of math instruction.)

The students were interested and curious about how Jonah thought .

 

Here are my takeaways from the interview with Jonah, also from the experience with the fifth graders years ago when I read the book, and more recently from my experience with the fifth graders who I asked to write to Jonah:

  • A general reminder to self: Be cautious about the assumptions I make about what students understand and remember that students’ understanding is often fragile.
  • Another reminder: Partial understanding and confusion are part of the process of learning.
  • And one more reminder: Be sure to use wait time in classroom instruction as well as during interviews.
  • An observation about Jonah: He knew, with confidence, that 4 fourths is equal to 1 and that 33 + 33 + 33 equals 99.
  • A takeaway about showing video clips like these: They can give students experience with creating viable arguments and critiquing the reasoning of others, an important Mathematics Practice Standard.
  • A takeaway from the letter writing: Writing is another way for me to gain valuable insights into students’ reasoning. (If you’re interested, I wrote Writing in Math Class a while ago.)

There are other takeaways, but I’m more interested in hearing about yours. Please post any feedback. As much as I’ve embraced listening to learn from students, I know that it’s also important for me to listen and learn from teachers.

Letters the Students Wrote to Jonah
Below are some of the students’ letters. I loved reading how they explained their thinking and also how kind they were in their comments.

Erick showed how he split the numbers and also included a drawing.

 

Gio’s letter is clear and succinct.

 

Melody gave Jonah encouragement at the end of her letter.

 

Santos used multiplication to solve the problem. He also gave Jonah encouragement at the end.

 

Lamberto explained how he interpreted Jonah’s error.

 

About Interviewing Students
The problem I asked Jonah to solve is included in one of the ten interviews from Listening to Learn, a K–5 digital resource that I’ve created with my colleague Lynne Zolli to help teachers conduct one-on-one math interviews with students. It will be ready for schools in Fall 2021 from Heinemann. Along with the interviews to help teachers learn how their students reason, Listening to Learn generates reports for guiding instruction, includes an extensive video library (of more than 200 videos), and provides What’s Next? instructional suggestions. If you’re interested in information, sign up at ListeningtoLearn.com for updates.

11 thoughts on “Solving 100 ÷ 3 Mentally: A Surprise

  1. Another superb Burns/Lesson. The letters are magnificent, complete with the graphics. You can teach me math any day you’re free.

  2. I was so excited to see a new blog post from you this morning! Despite having read your Writing in Math Class book years ago and doing tons of student interviews, I have never thought to bring a student interview into the classroom and have students write a letter to the student in the video. What a great opportunity for kids to engage in meaningful writing and math practice standards while solidifying their own understanding of division. Thanks for sharing!

  3. I love the idea of using a video of student reasoning for students to consider and then write to that student with whether they agree/disagree. I love the kind of “audience” this provides for student writing. And I really appreciate your reflection around the lesson – with all your expertise, you still model what it means to be a learner!

    I want to raise a potential complication with the use of this video for other teachers to consider. I was hoping that I might use the video of Jonah with students in our (racially and culturally diverse) school, but I am worried about potentially unintentionally reinforcing the stereotype of a black boy who is not academically successful in that moment. While I think this is context dependent, in my context it would be important to position black students as having “math status” during a lesson like this.

    • I understand your sensitivity to this issue. While I have many clips of black boys answering correctly, and the rest of the interview I did with Jonah shows him answering correctly and displaying understanding, I understand the problem of a 46-minute clip out of context. That said, the school’s focus is on students constructing viable arguments and critiquing the reasoning of others. What I loved about this experience, and the students’ letters, is the sensitivity and kindness they showed toward Jonah. And, about the video, it will be included in the video library of an interview resource that will be available in Fall 2020 titled Listening to Learn, from Heinemann. This clip is one of more than 250 that will be included. I’ll be sensitive to posting others until the full library is available. In the meantime, maybe I’ll post another clip of Jonah at the end of the blog to give a different look at his thinking. Thanks for your comment.

  4. Hi, Marilyn–I have spent the last year working with students online to bolster their SAT and/or ACT test scores in math. Since most are 11th or 12th graders, they have rarely, certainly not lately, been asked to think out-loud through their approaches to math problems.

    I found that having students do this with me (via WhatsApp or Skype), I could help them understand and accept that they had all or most of the math skills they needed; they just needed to THINK before they started “working”. Most were used to pushing forward quickly, especially in timed-test situations such as ACT’s and SAT’s, and even more so if they were in higher-level math classes. Also, it helped them to see how they jumped into an algorithm way too fast. (Personally, I am allergic to algorithms, unless you make them up yourself, and my tutees know this!) This methodology also helped them to see their “silly mistakes” and to understand that they had probably let go of their focus and concentration before they had finished whatever calculation. THINK and STICK WITH IT.

    I taught students about WAIT TIME for themselves, for their own thinking. Students practiced waiting 15 seconds before they did any “work” on a problem. Not very long, but about 14-3/4 seconds longer than they had before! And for them, it seemed like an eternity! I had them actually put their pencils down while I timed the 15 seconds.

    Thank you for your ongoing work and for sharing it with us.

    Very best to you/Ginny Woolley

    P.S. We met YEARS AGO at a NESA Teachers Conference in Athens, Greece! We rented a car together (with your partner) and drove somewhere to visits temples and take in some other sites. Lots of fun! I also worked with you on creating TERC’s Investigations curriculum. I am now retired, living in Boquete, Panama, and continuing to tutor math both locally and online. Come visit!

    • Thanks so much for your thoughts, and also for the reminder of that time in Greece at the NESA conference! I still remember that trip, which was probably about 30 years ago.

      • I do think it is important to use wait time.

        Having the students explain how they solved the problem gives you an insight into what they know and understand and what they don’t.

  5. Hi Marilyn,
    Here it is day #2 of ‘Shelter in Place’ and I had the time to read your blog and I loved it! I’m working with some others to prepare math activities for 2nd graders to do at home during this time. I’ve been thinking of you, and hope we can get together again when this crisis is over….

  6. Thank you for sharing this video with me. It has helped me learn how I need to stop and allow students to explain and this video shows in a way how important it is for students to show their work. I could use this in a classroom for students this could show the importance for me as a teacher to understand how they got their answers.

  7. Hi Marilyn! I think this post was so insightful and a great reminder to think of our approach. In our classes we are taught that the most important feedback is what you receive from your students. I my mind I always thought of self assessments and other formative forms of assessment. I never thought about the timing of replies and their explanation is a great way to process their feedback. This was great, thank you!

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