Does 80% of 65 = 65% of 80?
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...read more
In Person or Remote: This is What Doing Math Is
Our staff has been thinking hard about how teachers are using Investigations 3 to teach math in all of the different scenarios they are faced with this year. We’ve been visiting the remote classrooms of teachers we’ve collaborated with previously, to learn from teachers and students who are teaching and learning math online, and to see how the rigor and coherence of the curriculum is supporting them in that work. This series of blogs will share some of what we are learning. (Read an...read more
Robot Fingers and Multiplicative Structure
In a blog post over a year ago, I wrote about the importance of using story contexts to support students in developing mental images of the operations. In that post, I concluded: “Along with pictures, drawings, diagrams, equations, and physical models, story contexts are critical parts of a student’s repertoire of representations. In fact, it is often the case that three kinds of representations are in play: a numerical representation, a picture or diagram, and a story. Moving among these to...read more
Multiplication in 5th Grade: What Are Some Issues?
Last year, my colleague Keith and I worked a few times with a group of 5th grade teachers. One of the questions they asked us to help them think about related to this 5th grade benchmark: “Fluently solve multidigit multiplication problems using a variety of strategies including the U.S. standard algorithm.” They told us that they had students who could multiply 2-digits by 2-digits successfully but struggled with 3-digit by 2-digit multiplication problems. They wanted to discuss the...read more
And Then, She Waited
Have you ever been teaching (or leading professional development) and asked a really good question only to be met with silence? We all have teacher moves in our back pocket for situations like this—maybe do a turn and talk, ask the student if they’d like to call on someone to help them, or ask a different question. I recently observed a third grade lesson when I saw a teacher face this exact situation. The lesson (Unit 5, Session 3.4) focused on strategies for solving division problems. The...read more
“That Seems Way Too Big”
On a recent visit to a small district in the Midwest, I got the chance to visit a third grade class that was working on division (3U5, Session 3.4). When I joined Nicole, she was in the middle of working on the following problem: Gil loves toy cars. He saved enough money to buy 32 toy cars. How many 4-packs of toy cars did he get? (SAB p. 333) Below the problem, Nicole had written I asked her to tell me about her thinking. She said she wanted to start with something she knew, which I agreed...read more
A Division Solution: Amazing or Perplexing?
At a recent professional development session on multiplication and division, my colleague and I asked participants to examine some student solutions to the problem: 1564 ÷ 36. Before looking at the student work below, think about how you would solve the problem. The following solution from a fifth grade student seemed to either amaze participants or they had a hard time making sense of it. I’ve been thinking a lot about their reactions. Take a moment to look at this work. What do you notice?...read more
A Grade 3 Q&A: Assessing the Multiplication Facts
Question: Why do the assessments of the multiplication facts in Grade 3 include a time limit?Answer: In Investigations, the overwhelming majority of students’ work with the facts is focused on making meaning of the operation of multiplication, building connections between problems and images that represent them (e.g. problems about things that come groups, arrays), and using what they know to solve what they don’t (e.g. how can knowing that 3x4=12 help with 6x4?). This work happens in...read more
What Does It Mean To Be Smart?
“Wow, you’re so smart.” These words drew my attention to a pair of 5th grade girls in a class I was visiting, who I’ll call Cassie and Sophia. They were mid-way through a turn and talk, each sharing her strategy for solving 84 x 59. I casually moved closer, curious about what prompted the comment and trying hard to see each girl’s strategy, recorded in their math journals. Upon hearing Cassie’s comment, Sophia responded in an inviting tone, “No, no. Explain to me what you did.” She...read more
A Grade 3 Q&A: Why start with multiplication and division?
Question: Why does 3rd grade start with a multiplication/division unit and not addition/subtraction?Answer: We often hear from people who wonder why Grade 3 starts with a multiplication and division unit—just like it did in the 1st edition!—rather than an addition and subtraction unit. As we decided on the sequence of the units in any grade, we considered many different things. The most important was the development of mathematical content within and across grades.In Investigations 3,...read more