Subtraction in early primary

This module will focus on subtraction in Years 2 and 3.

Subtraction

Introduction

Subtraction is one of four ways we work with numbers.

Generally subtraction is thought of in terms of addition. Subtraction and addition are inverse operations.

Teaching children to subtract is one of the first mathematical tasks taught in schools and constitutes a large focus of mathematics instruction in the primary grades

Why is it important?

Subtraction is a part of our everyday lives. Students need to understand how to subtract as we use subtraction when dealing with money, cooking, travel, time and other daily experiences.

Subtraction, along with other operations, provide a foundation for all later work with computation.

Q: Subtraction is

  • one of the five ways we work with numbers
  • an inverse operation of division
  • one of the first mathematical tasks students are taught in school

Language of subtraction

Conceptual knowledge

There are two types of thinking that underpin subtraction:

  • Structural thinking
  • Additive thinking

There are three models of subtraction:

  • Take away
  • Missing part
  • Comparison

Students also need to understand the connection between addition and subtraction.

The Payne and Rathmell triangle

The Payne and Rathmell triangle shows the important relationship between using concrete models   with developing language and using mathematical symbols.

There are many words that can be used to replace the word 'subtract'. Make a list of all the words you know.

In your classroom, you would normally make a word wall, or posters. You might brainstorm and then discuss the language and the different contexts they could be used in. Remember that depending on the context used, some words could represent another question.

Q: Payne and Rathmell Triangle

Label the triangle
  • Language
  • Symbol
  • Model

Structural thinking

Structural thinking

Structural thinking is built up by observing patterns and looking at how things are put together, the ability to see and represent patterns and how they're connected.

Students develop the ability to see and represent patterns and having a deep awareness of how these are put together (the relationships).

What is a mathematical pattern?

What patterns can you see in the room?

If you were to ask your students to find patterns in your classroom, what patterns would they see?

"By providing students with quality experiences in patterning, they will be more likely to see patterns in problem situations, and be able to make generalisations about their patterns, which is the basis of algebraic thinking."

Jorgensen & Dole, 2011

Jorgensen and Dole emphasize the importance of being able to see patterns in order to make generalisations as this is the basis of algebraic thinking.

Our brains are wired to look for patterns. A mathematical pattern is any predictable regularity involving number, measurement or space.

We spend a lot of our time on patterning sequences with our students, e.g. growing pattern and counting in multiples (2s counting pattern). We teach students to create and continue patterns with repeated units. This leads to generalisations and algebraic thinking later on.

What is a mathematical structure?

Mathematical structures is the way the pattern is organized.

What is the biggest mathematical structure used in number? (The base 10 system)

The place value strategy is that a number is multiplied by 10 when it moves to the place value on the left, and divided by 10 when it moves to the place on the right. 2 tenths multiplied by 10 is 2 ones, 2 tens (20), 2 hundreds (200), 2 thousands (2000) etc. Each place increases/decreases by 10.

Another mathematical structure is the hundred board. The number patterns (multiples 2, 5 and 10) are organized into a 10 x 10 grid. A difference of 10 as you go down the rows and a difference of one as you go across.

Danger of 'key words'

Students look for verbal clues when solving word problems, for example 'more' usually (but not always) suggests addition and 'less' usually (but not always) suggests subtraction.

Incorporate problems where these words suggest the opposite of what they usually do. For example:

  • Lisa had 14 books which was 2 less than Frank. How many books did Frank have?
  • Lisa had 14 books which was 2 more than Frank. How many books did Frank have?

One way to ascertain if students have developed structural thinking is to ask them to draw a hundred board from memory.

This diagnostic task was conducted with a Year 2 student.

Other similar diagnostics that you could do with your students is to give them a partially filled hundred board, and students need to fill in the missing numbers. Another diagnostic to check for structural thinking is to give students a partially filled hundred board hat has been broken up into a jigsaw. Students need to put the pieces together to complete the hundred board.

Other types of thinking that students demonstrate is additive and multiplicative thinking.

Q: Mathematics is built on

  • patterns and structure
  • problem solving

Q: Which student diagnostic shows structural thinking?

Model of subtraction

Take away

Missing part

Comparison

Problem solving

Word problems involve a lot more than just solving calculations.

Students need to be able to:

  • read the words (literacy demands)
  • figure out the mathematical operations to use
  • work through a number of steps
  • perform the calculations correctly

Think board

A think board is a useful tool to help students step through problem solving by using the Payne and Rathmell triangle.

Q: Problem solving steps

  • Understand the problem
  • Plan a solution
  • Carry out the plan
  • Check the answer
Sort Polya's problem solving steps in the correct order

Q: Solving problems is about

  • making correct calculations
  • completing a number of steps
  • reading the words
  • estimating an answer
  • selecting the correct mathematical operations to use

Q: What are the multiples of 11?

Subtraction strategies

Strategies

Automatic recall means students can quickly retrieve answers from memory without having to count. Basic number facts are an important mental computation strategy.

When students have automatic recall of facts (or fluency), they can quickly retrieve answers from memory without having to rely on counting procedures.

Lack of automatic recall is a problem as children advance because the need to rely on laborious counting procedures creates a drain on mental resources needed for learning more advanced mathematics.

The Australian Curriculum explicitly states that students should be using a range of “efficient mental and written strategies” in the primary years.

When we talk about Mental strategies, think about what strategies you use. 

When we talk about Written strategies, what strategies do you teach?

Activity

Activity: I have, who has...

This game can be played in a circle. It is important for other students to listen to the responses because if an incorrect response is made, then the loop is disrupted.

Things to note:

•Students sit down when they have read their card/s

•Make more cards than students, that way you can challenge better students by having 2 cards

•Place students who need support next to a student who can help them

•Time the game and use this time as a challenge to be bettered at each subsequent playing

Q: Match the strategy

  • Mental strategy
    compensate
  • Written strategy
    algorithm

Q: Subtraction strategies

Differentiation

Differentiation

Differentiation is what teachers do in response to the diverse learning needs of students.

The Maker model of differentiation identifies four areas where changes can be made to differentiate for students.

A major benefit of students working in mixed ability groups is that the achievement of lower achieving students increases, but not to the detriment of outcomes for high achieving students.

(Boaler, J 2009. The elephant in the classroom: Helping children learn and love maths. Souvenir Press. London.)

 

Q: Maker model of differentiation

  • Content
    What students need to learn
  • Process
    How students learn
  • Product
    How students demonstrate what they know
  • Learning environment
    How learning is structured

Mathematical literature

Mathematical literature can be a starting point for engaging and extending students' thinking. Literature builds the context for problem solving, providing motivation and purpose for approaching the mathematics. Books such as:

  • Elevator Magic - when the elevator goes down, the subtraction starts and so does the magic. Ben sees crazy things every time the door opens. Ride along as he subtracts his way down the lobby, and decide for yourself if it's elevator magic.
  • Shark Swimathon - The Ocean City Sharks have to swim 75 laps by the end of the week, and every day they figure out how many laps are left to go. Swimming and subtraction are all part of the fun!
  • Uno's Garden - Uno finds a beautiful place, decides to live there and builds a garden, as the wildlife dwindles and the buildings and inhabitants increase, their progress is charted by multiplication and square root equations along the top of the page. A page at the end explains the mathematics involved.

Q: Mathematical literature

Mathematical literature

Resources

Q: Reflection

What strategies will you take back to your classroom?

Websites