What's Wrong with this Fraction Visual?

I want to explain my reason for updating this (old) visual. First, the original:

Not bad, you say? Why change it? Since creating that visual I’ve read the incredible book Extending Children’s Mathematics and witnessed teachers’ fraction instruction that has blown my mind. 

First of all, the meaning of fractions isn’t dependent on how many parts a whole is partitioned into:

Instead of asking how many pieces something is cut into, Extending Children’s Mathematics suggests questions like “How many of these parts fit into the whole?” (p. 24). 

A choice about how to model and partition encourages students to consider the multiplicative relationship between the numerator and denominator:

I updated my original visual to leave more room for students to reason about the relationships (and the meaning of the numerator and denominator). You might ask students what they notice and wonder, or pause and ask a more specific question like “How can we be sure that is 6/4? What other fractions do you see? Could you draw it in a different way?”

But wait, there’s more: I am unlikely to show this visual to students until I’ve done a lot of contextual problem solving that focuses on student generated fraction visuals. Here’s why: students come to school with a lot of experience sharing quantities fairly and don’t need instruction in fractions to get started.

For example, here is a problem Kristin Welch did with her students where the meaning of 1/4 is very clear (1 mini-pie being shared equally between 4 people). Also, this allows fractions to be a natural extension of whole number operations (division in particular).

In addition to problems like that, student benefit from open-ended discussions of photographs like this:

You might ask: “What do you notice? Wonder?” and leave it at that. Or, you might ask: “How many/much?” They might count fourths, or bites, or servings, or piles. A picture like this leaves more space (than my visual) for students to participate in surprising and brilliant ways.

Speaking of surprising, did you know that kindergartners (5 and 6 year olds!) can solve 10 ÷ 1/2 if it comes up in a meaningful context and they are allowed to solve in any way that makes sense to them? Here’s proof from Meggan Akin‘s classroom, where a student used a digital whiteboard to create stamps (the blue circles) and partitions to solve these problems. No “invert-and-multiply” or “keep-change-flip” necessary. Just sense-making: one-half cup of food each day, with 10 cups of food in total is 20 days she can feed her dog! (This is a KINDERGARTENER!)

(Here’s the original tweet if you’d like to share with your colleagues)

To summarize: fraction visuals are great, but student generated fraction visuals (rooted in context and whole number division) are better.

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