Understanding Multiplication Concepts: Lesson Study, a Technique from Japan Where Teachers Study Ways of Teaching One Lesson for Several Hours, Is Described by Judy Mousley as She Considers Concepts of Multiplication and Division

By Mousley, Judy | Australian Primary Mathematics Classroom, Fall 2000 | Go to article overview
Save to active project

Understanding Multiplication Concepts: Lesson Study, a Technique from Japan Where Teachers Study Ways of Teaching One Lesson for Several Hours, Is Described by Judy Mousley as She Considers Concepts of Multiplication and Division


Mousley, Judy, Australian Primary Mathematics Classroom


Does 3 x 6 mean three groups of six or a group of three multiplied by six? Both? What about 30 x 6? Should we teach both ideas and aim make sure children understand that multiplication is commutative, or should we aim for consistent understandings?

Division concepts

Each year in teacher education courses Australia-wide, student teachers meet the idea of partition and quotition division. The difference is not hard to grasp, but the question of why both concepts need to be taught is always asked--or is at least preempted by lecturers.

In partition division, the action is sharing. The number of groups that need to be made is known from the start, and the aim is to find the size of the groups. It is like dealing out tickets, where the number of people is known but not how many tickets each person will receive. (To facilitate memory, I tell student teachers that the number of parts is known: 'parts-partition'.)

In quotition division, the action is partitioning. The size of groups that need to be made is known from the start, and the aim is to find the number of the groups resulting. It is like giving away bundles of tickets, where the number of tickets in each bundle is known but not how many people will receive some. (That is, the quota is known: 'quota-quotition'.) Quotition division can be likened to repeated subtraction in that one group is taken from the original amount, then another, and another, etc., until the original amount is exhausted.

This all seems rather theoretical, and not too important. We merely need to realise the need to teach the two division actions, and to give children problems of both types (see Figure 1).

Note that for both examples presented in Figure 1 the symbolic representation is

30 / 10 = []

Figure 1: Two types of division problems

Typical partition problem:

   I have one large packet of Jaffas, containing 30
   sweets. Ten children will be at the party. How many
   could each child get?

Typical quotition problem:

   I have one large packet of Jaffas, containing 30
   sweets. If each child is to get 10 Jaffas, how many
   children can be served from one packet?

Also, in both cases the answer is 3. However, different mental and/or physical actions are required to solve the two problems, so different language is likely to be used in discussing the problems. Essentially, two different division concepts need to be taught. Pupils do not have to be introduced to the terms 'partition' and 'quotition', but it would be interesting to have children of any age draw pictorial representations of the two problems in Figure 1 (or a similar set) and then discuss whether their drawings, and the thinking that led to them, displayed any differences.

Enough said about division. The difference between partition division and quotition division is quite straightforward. All that is needed is awareness by teachers and other people who prepare classroom resources, so that children get a range of experiences that give them a good understanding of each. We are left, however, with a question of whether to teach both ideas at once or to structure learning so that both ideas receive their share of attention by being taught separately or in a logical sequence.

Multiplication concepts

Given the fact that most teacher education courses deal with the above, and primary teachers generally attend to both sharing and partitioning in teaching division, it is surprising that very few courses and classroom lessons take the equivalent forms of multiplication into account.

Consider, for example, 4 x 6. Does one of the numbers represent the size of the group and the other the number of groups? What would the children in your class say? How would they draw this problem?

In junior primary years, 4 x 6 is usually interpreted as four groups, with six in each group. Words like 'four lots of six', 'four groups of six' and, later 'four times six' are used.

The rest of this article is only available to active members of Questia

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • Ad-free environment

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
Loading One moment ...
Project items
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited article

Understanding Multiplication Concepts: Lesson Study, a Technique from Japan Where Teachers Study Ways of Teaching One Lesson for Several Hours, Is Described by Judy Mousley as She Considers Concepts of Multiplication and Division
Settings

Settings

Typeface
Text size Smaller Larger
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

While we understand printed pages are helpful to our users, this limitation is necessary to help protect our publishers' copyrighted material and prevent its unlawful distribution. We are sorry for any inconvenience.
Full screen

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.

Are you sure you want to delete this highlight?