Tuesday, October 16, 2018

Bar Modelling For Two Types of Division

Many people aren’t aware that there are two different types of division. Even if you search the web, only one type mostly comes up. It’s the type we learn first: that when we share one quantity, we split it into equal parts. This is ​partitive​ division, or dividing a quantity into parts. When we divide a whole amount, partitive division tells us how many items there are in each group. Let’s look at a couple of examples.

Jeanine bought 24 buns for a party. She wanted to put an equal amount on each of 6 tables. 
How many buns should she put on each table?

Jeanine should put 4 buns on each table.

Armando earned $56 for a day’s work. 
If he worked 7 hours, how much was his hourly wage?

Armando’s hourly wage was $8.

The other type is ​quotative​ division, and it’s less familiar to most people. In quotative division, we find the number of groups, not how many in each group. It is also known as measurement division, or finding how many of a certain unit it will take to measure something. Let’s look at a couple of examples.

When the party planner was setting up, he found he had 32 place cards. He would like to put four on each table. How many tables would he need to set up?

The party planner would need to set up 8 tables.

For a pizza party, a parent has a maximum budget of 6 pizzas. He plans on each child being allowed three slices. This pizza parlour slices each pizza into 8 slices. How many children can be invited to the party?

16 children can be invited to the party.

The above example is as interesting way to model dividing a whole number by a fraction (an upper primary standard in most countries). Although bar modelling may not be as useful as a tool to calculate quotative division problems, it can be very helpful to help students visualize the problem.

Often when teaching such problems, many teachers resort to multiplying by the reciprocal, or 6 × 8/3 (We need to avoid teaching Keep-Change-Flip though, as this 3 mnemonic can lead to lots to misconceptions.).

Did you know that you can also divide using common denominators? You can
change 6 to 48/8  and solve 48/8 ÷ 3/8 . See if you can figure out why this works.

Leave a comment with your thoughts!

Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

Monday, October 8, 2018

What is more important than the right answer in primary mathematics?

“There is no right or wrong in mathematics:” a quote from a memoir by the daughter of Fischer Black, a famed US mathematician, that she remembers him saying often. What does that mean?

When we learn how numbers work, we can do so in one of several ways. The way many of us have been brought up to learn mathematics is through memorization and learning procedures, such as algorithms. One example is the long division algorithm, which is opaque to most students and is arguably one of the most difficult to learn. Teachers and parents alike can relate to frustration as students confuse which number to put above the little house, which below, and more. 

Another issue with this algorithm is that the way most people learn it. The digits are independent of value. For example, when dividing 125 by 5, first you would see how many times 5 goes into 1, then 12, then… but 1 what? 12 what? If it were truly 1 or 12, wouldn’t the digits be written in different places?

Disconnecting the procedure from the meaning behind it leads to magical thinking about numbers: that it doesn’t matter how or why it works, it just does. 

Another place this shows up is in multiplying or dividing by powers of ten. How many of us think of “moving the decimal” or “adding a zero”? Both of these are not only how mathematics doesn’twork, they lead to significant mathematical errors and lack of understanding in students. 

To promote numeracy, or paying the same amount of attention to all students being as successful and capable in mathematics as they are in reading and writing, the focus needs to shift away from whether or not the answer is correct and more towards the way people think about the mathematics. This involves developing the concept so the student can apply whatever problem solving procedure works for him/her. This also means exposure to a variety of representations of the mathematics, to improve flexibility in thinking. The model of teaching mathematics in Singapore, and the US Common Core standards, both push for that type of learning. This way, students learn how mathematics works, not just how to get the right answer.

For example, ten years ago, when working with parents or teachers, I would ask the question, “What is Pi?” The answers would range from something to do with a circle, to 3.14, to “I don’t know.”

These days, after years of celebrating Pi Day in schools in March, the teachers I meet are much more likely to say, “It’s a ratio,” or “It’s the ratio between the circumference of a circle and its diameter.” This is a much more conceptual definition that allows flexibility in thinking and the ability to recreate the formulas for the different measurements of a circle. 

If teachers or parents are working with students on mathematics, why not try shifting focus from the answer to how the students are thinking? Start by giving feedback, rather than “right” or “wrong,” that focuses a student on their thinking and allows the student to identify if they made a mistake in understanding how the numbers work. This should lead to greater competence and confidence in their own ability to learn.

Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

Related article:
Singapore abolishes school exam rankings, says learning is not competition