My wife Anne-Marie isn’t always impressed when I talk about mathematics, especially when I ask her to try something out for me, but on occasion I can get her to really think mathematically without her realizing how much math she is actually doing. Here’s a quick story about one of those times, along with some considerations:

A while back Anne-Marie and I were preparing lunch for our three children. It was a cold wintery day, so they asked for Lipton Chicken Noodle Soup. If you’ve ever made Lipton Soup before you would know that you add a package of soup mix into 4 cups of water.

Typically, my wife would grab the largest of our nesting measuring cups (the one marked 1 cup), filling it four times to get the total required 4 cups, however, on this particular day, the largest cup available was the 3/4 cup.

Here is how the conversation went:

**Anne-Marie:** How many of these (3/4 cups) do I need to make 4 cups?

**Me:** I don’t know. How many do you think? (attempting to give her time to think)

**Anne-Marie:** Well… I know two would make a cup and a half… so… 4 of these would make 3 cups…

**Me:** OK…

**Anne-Marie:** So, 5 would make 3 and 3/4 cups.

**Me:** Mmhmm….

**Anne-Marie:** So, I’d need a quarter cup more?

**Me:** So, how much of that should you fill? (pointing to the 3/4 cup in her hand)

**Anne-Marie:** A quarter of it? No, wait… I want a quarter of a cup, not a quarter of this…

**Me: ** Ok…

**Anne-Marie:** Should I fill it 1/3 of the way?

**Me:** Why do you think 1/3?

**Anne-Marie:** Because this is 3/4s, and I only need 1 of the quarters.

The example I shared above illustrates sense making of a difficult concept – division of fractions – a topic that to many is far from our ability of sense making. My wife, however, quite easily made sense of the situation using her reasoning instead of a formula or an algorithm. To many students, however, division of fractions is learned first through a set of procedures.

I have wondered for quite some time why so many classrooms start with procedures and algorithms unill I came across Liping Ma’s book *Knowing and Teaching Mathematics*. In her book she shares what happened when she asked American and Chinese teachers these 2 problems:

Here were the results:

Now, keep in mind that the sample sizes for each group were relatively small (23 US teachers and 72 Chinese teachers were asked to complete two tasks), however, it does bring bring about a number of important questions:

- How does the training of American and Chinese teachers differ?
- Did both groups of teachers rely on the learning they had received as students, or learning they had received as teachers?
- What does it mean to “Understand” division of fractions? Computing correctly? Beging able to visually represent what is gonig on when fractions are divided? Being able to know when we are being asked to divide? Being able to create our own division of fraction problems?
- What experiences do we need as teachers to understand this concept? What experiences should we be providing our students?

#### Visual Representations

In order to understand division of fractions, I believe we need to understand what is actually going on. To do this, visuals are a necessity! A few examples of visual representations could include:

A number line:

A volume model:

An area model:

#### Starting with a Context

Starting with a context is about allowing our students to access a concept using what they already know (it is not about trying to make the math practical or show students when a concept might be used someday). Starting with a context should be about inviting sense-making and thinking into the conversation before any algorithm or set of procedures are introduced. I’ve already shared an example of a context (preparing soup) that could be used to launch a discussion about division of fractions, but now it’s your turn:

**Design your own problem that others could use to launch a discussion of division of fractions. Share your problem! **

#### A few things to reflect on:

- How do you use contexts and visuals to help your students make sense of concepts?
- What does day 1 look like for any new concept in your classroom?
- How do you know if your students “understand” a topic? How would you define “understanding“?
- How do you assess your students’ understanding? How does this assessment help you know where to go next?
- How do we get better as teachers to understand the mathematics we teach?
- If you know that a specific concept is difficult for you to visually represent, where do you turn to continue your own learning?

As always, I’d love to hear your thoughts. Leave a reply here on Twitter (@MarkChubb3)

Great article! I found the American vs. Chinese teacher responses to be especially telling. Sensemaking is always the way to go!! Thanks, Mark.

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Informative and engaging post. People should stop thinking that Math deals with an unfeeling set of numbers and are exclusively tackled in the classroom. With the story you mentioned above, it is clear that Math is a necessary incorporation in everyday lives. In fact, people do Math in many ways without being aware of it!

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