# Development of Multiplication and Division

Understanding and Strategies

*Overview*

- Big ideas
- Development of multiplication and division
- Models related to multiplication and division
- Multiplication, division, and related properties concepts

Critiques often target teachers and schools when they feel there is a lack of memorization of multiplication tables and proficiency of use of multiplication and division algorithms.

Using emotional words like:

New math, fuzzy math, soft math, and claim it is dumbing down students learning of mathematics.

However, the mindless use of algorithms is the real dumbing down.

## Big ideas (generalization) for multiplication and division

Multiplication and division are ways to describe how two (or more) values can be operated on to create another value that is equivalent to the initial values in the following ways:

*Multiplication*

- As repeated addition of equal groups, unitizing.
- Measurement of like units, area, volume. Results in a product that is something different than the factors (square feet is the product of linear feet, cubic cm...). Arrays, Matrices
- Combinations or Cartesian products

*Division*

- Measurement (repeated subtraction),

Example: Old lady lives in shoe gives candy bars to her children. If she has 144 bars and gives one a day to each of her 22 kids, how many days will the bars last? - Distribution or partition or fair share.

Example: Old lady lives in shoe has 144 candy bars and wants her children to share them. If she has 22 kids, how many bars will each child get?

## Development of multiplication and division strategies

### Initial use of multiplication related ideas

Young chldren will form concrete groups and their solution focuses on *counting* the objects one by one
to find the total value of collections of equal groups.

Young children will find how many objects are needed if they share a given number of objects among a given number of groups by measuring with equal groups or units. For example: If everyone in a class gets 5 M&M's how many M&M's will we need? Will make groups of M&Ms for each person in the class. Will use concrete objects, tally marks, or numerals on paper.

Young children will find how many objects each person would get if a given amount were shared with a given number of people by distributing equal groups or units. For example: We have 55 cookies and 18 students, how many will each get? May start by dealing out 1 for each student and repeat until run out of cookies. Or may start with a larger number and run out. Needing to start again. The process of distributing cookies for young students is trial and error not systematic.

Young children can skip count by 10. Will use it to compose and decompose numbers related to the operations of addition and subtraction, not with an understanding of multiplication, because they are not able to unitize. Similarly may also work with 100 and 1000.

### Unitize as requirement for multiplication

Children in the middle elmentary grades begin to *unitize*. They must be able to simultaneously understand two groups of four represents two single groups of two, and two equal groups of four, eight. They are able to switch from one to the other easily recognizing the equality of the values and the interchangeability of factors as single units or equal groups with those units. In other words they can represent five nickels as five nickels and 25 cents as being equa simultaneously. They don't have a problem, as young children do, understanding the place value of the two and four in 24. Young children when asked to show what the 2 and 4 represent with counters will put two counters by the 2 and four counter by the 4. When asked what about the other 18. The will slide them aside or look befuddled. Older children will easily unitize, know, and explain the 2 is 2 tens while saying it can also be, at the same time (simultaneously), 20 tiles, or 20 units/ ones.

Being able to unitze, middle grade students understand the equality of the product to the relationship
of the factors. This relationship can then be used to explain the *reciprocal* relationship
between multiplication and division. If we know a part and the whole
we can find the other part and if we know the parts we can figure
the whole. This relationship must be understood efficiently or quickly to be able use the basic facts
for solving problems requiring the operations of multiplication and division.

### Multiplication of larger numbers.

Students first understand addition and subtraction by counting on and counting back. It takes a lot of experience before the are able to understand on operationalize addition and subtraction with as two or more digits with different place values. Eventually they learn that to add 23 and 34, they can more efficiently than counting on, add the tens and the ones and then combine the tens and the ones. When they get to multiplication they discover that multiplication of two digit numbers requires a bit more complex procedure. Multiplication requires the number in each place operate on each number in all the other places. This is one way that multiplication is unlike addition and many adults who conceputalize multiplication only as repeated addition never fully realize the potential of multiplication. This presents problems for students as they move further into higher branches of mathematics, algebra and beyond.

While the following models help provide concrete models for multiplication students will need to struggle with how multiplication is and isn't related to addition. Likewise they will need to struggle with division.

*Models related to multiplication and division*

- Grid or square model and one connection to algebra
- Rectangle sample 1
- Rectangle sample 2
- Uses array and number-line models as tools
- Uses money as model in calculating
- Uses open array to represent strategies
- Models multiplicative situation as branching
- Models volume as 3-D array
- Models multiplicative situation as area
- Models multiplicative situation as repeated addition on a number line

*Arrays* - view as one directional (count all squares
in a one directional path). View as a set of rows or columns, but
not both. View as both a set of rows and columns but not both simultaneously,
can't see how corners can be both a row and column. Can view corner
squares, and others, as being both a row and column

Once students have constructed unitizing they can use multiplication to solve division problems

Multiplication by .1, .01, .001

Multiplication and Division of Whole Numbers Concepts for Computation/Algorithms

Concepts, misconception, and outcomes for multiplication, division, and their properties