# Draw a Picture: Check out the webinar

In March, I did a session for Global Math Department on drawing diagrams. If you don’t know about Global Math Department, you should check it out. They present free video conferences and webinars every Tuesday night and you can join in and participate that night or you can watch the recording later if you can’t make it.  Here’s the recording of my talk.

It was an interesting experience for me. I find that whenever I have to write a blog post or prepare a presentation, I learn more about the topic. The process of synthesizing and preparing the message I want to deliver forces me to think about the content in different ways.

It was also a unique experience because it was a different type of interaction with the audience than I’m used to. When I do workshops, I can read the body language of the members and I can interact with them differently. My workshops are also much more interactive and I do far less talking than the participants. With the webinar I did, I could read the comments and interact that way but it was a different form of communication for me.

I’ve been thinking a lot about how to foster learning with teachers. What activities help them develop new ideas? When I look at my own learning, the experiences that help me are sometimes listening to talks or reading books or articles. But I think experiencing new ways of learning math and then trying these methods out with students help the most.

I’m really interested in how to create different learning experiences for teachers.  I’d love to hear any ideas you have or more about what learning experiences have been helpful for you.

# Multiplying Fractions with Meaning

When I do workshops with parents, I often get complaints about why students don’t just memorize things.  It’s often followed by, “I learned math that way and I’m fine!”

I then ask them to solve a multiplication of fractions problems, say 4/5 x 2/3.  Inevitably, one person will tell me to draw “butterfly wings” and cross multiply. Another will tell me to find a common denominator, multiply the numerators and leave the denominators the same. Eventually, someone will say that you multiply the top and then multiply the bottom.  Don’t even ask about what happens when I ask them to explain why that procedure works.

I tell this story because most of us learned multiplication of fractions without any meaning. As a result, if we forget the memorized procedure, we don’t know how to reason about it.

I’ve been using the following type of problem to help learners begin to develop meaning for what it means to multiply a fraction by a fraction.

Ms. Placa made a tray of brownies.  She put icing on two-thirds of the pan.  She then put sprinkles on four-fifths of the brownies that had icing on them.  What fraction of the pan of brownies have sprinkles and icing on them?

Before students begin, I have them estimate whether the answer is going to be bigger or smaller than two-thirds. This leads to an interesting discussion and will help addresses a major misconception later on that students have about multiplication always making things bigger.

Then students draw pictures.  Here’s one possible sequence of drawings and student thinking:

1.  I’ll draw a pan and shade two-thirds with icing.

2.  Now I’ll cut the iced brownies into fifths and put sprinkles on four-fifths of those.

3. Hmmmm, I know I have 8 brownies with sprinkles and icing on them, but what size are they?  I can’t tell because all the brownies in the pan aren’t the same size. Oh, I have to make some more cuts to have equal sized brownies.

Now I know that the brownies are fifteenths and eight of them have sprinkles and icing on them so eight-fifteenths of the pan are brownies with sprinkles and icing.

Eventually we can get to writing number sentences and to imagining what would happen with larger numbers. We can then start to generalize what rule would work for multiplying any two fractions. But I think starting with a picture and context provides a nice foundation for starting to think about multiplying fractions.

What are your thoughts?  How do you usually teach students to multiply fractions?

Want to know more?  The study below gives a more detailed progression of how this type of thinking was fostered and some of the background knowledge it requires.

Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267.

# Using pictures as a tool to justify

One of the things that is particularly interesting to me right now is how we help young students develop the ability to justify why things work in math. Often, kids know the correct rule but have no idea why the rule works. When I ask them to explain why a rule works, they wind up just listing the steps of the procedure.

Part of developing a conceptual understanding in math is being able to anticipate what procedure to use and why that procedure works.  For example, when I am trying to convert a mixed number into an improper fraction, I don’t simply need to know that I multiply the denominator by the number of wholes and add the numerator.  I also need to know why that rule works and why it will give me the correct number of parts in the mixed number.

I’ve found that children (and even adults) can have a really hard time writing or even explaining out loud their justification for why something works.  They often say “I know why but I can’t explain it.”

Recently, I have noticed that drawing diagrams or pictures is one way to help them begin to justify.  It’s as if the diagram allows them a way to make explicit what they are doing when they perform a calculation.

For example, the other day I was working with a group of elementary school teachers on fractions.  They worked in groups to draw the pictures like the one below to show why the calculations they performed worked.

It was an interesting experience for them because they had to think deeply about what it really means to say, multiply fractions, as opposed to just remembering the formula.  It also gave them something to refer to when explaining to the rest of the class.

I’m currently digging through the research on this, but I’m curious to hear what your experience has been. What do you do to help student justify in math?