Approaches to teaching
These general outlines for activities when used along side each other are appropriate for helping children develop meaning for subtraction.
1. Concrete – modeling with materials (Reys et al., 2012)
Use a variety of materials that students can manipulate to solve, act out and model the operation needed to solve the problem (Reys et al., 2012)
2. Pictorial – representing with pictures (Reys et al., 2012)
Provide representations of objects in the form of pictures, drawings and diagrams to help solve the problems (Reys et al., 2012). This strategy begins the process of moving away from concrete materials and closer to symbolic representation (Reys et al., 2012)
3. Abstract – representing with symbols (Reys et al., 2012)
“Use symbols (especially numeric expressions and number sentences) to illustrate the operation” (Reys et al., 2012, p. 199)
Types of problems:
· Separation problems (Reys et al., 2012)
Also known as ‘take-away’ is to remove a specific quantity from another and identifying what is left (Reys et al., 2012). There are two important ideas that must be addressed with this type of problem. The first is that research shows children find this subtraction strategy the easiest to learn. The second is the consistent use of the phrase 'take away' can leave children believing this is the only form of subtraction and can lead to further misunderstandings of the other types of subtraction. It is preferable to ask ‘10 minus 2’ rather than ‘10 take away 2’ (Reys et al., 2012).
Example problem: Have the students in your class stand in a row and use them to solve the problem.
'There are 20 students in my class, but 5 students go home (5 students sit down), how many students are left?'
· Comparison problems
"Comparison involves having 2 quantities, matching them against each other and finding the difference between them" (Reys et al., 2012, p.200).
Lesson Example:
Have 3 or more stations set up each with different comparison subtraction problems.
(These problems can be all concrete or pictorial, or a mixture of both)
Problem 1: I made 10 cupcakes for my dad’s birthday, but I ate 4 of them before everyone had arrived for the party! How many cup cakes are left?
Problem 2: I have 8 red teddies and 3 green teddies. How many teddies don’t have a partner?
Problem 3: Chloe has 7 green unifix blocks and Jack has 2 yellow unifix blocks. Who has the most unifix blocks? And how many more do they have?
· Part Whole problems
"A set of objects is separated into 2 logical parts" (Reys et al. 2012, p. 200). The total number of objects in known and so is the total of one of the parts. Therefore the total of the remaining part needs to be established.
. Counting Back
One particular mental subtraction procedure is counting down. This strategy can prove to be very difficult if students do not have a sound grasp of counting down from any given number. Before looking at, or explicitly teaching this strategy, it is essential to ensure students have the ability and confidence to count back from any given number.
Games such as ‘Blast Off’ may be used. This game involves counting down from 10 or 20 and the last person says ‘Blast off’ and sits down.
The following sheets are recourses that can be used to assess students ability to count back from any given number. Use the numbered socks underneath as starting numbers. Students pick any sock and count back 5 places from that chosen number to reach the answer. Students get a new numbered sock each time.
These general outlines for activities when used along side each other are appropriate for helping children develop meaning for subtraction.
1. Concrete – modeling with materials (Reys et al., 2012)
Use a variety of materials that students can manipulate to solve, act out and model the operation needed to solve the problem (Reys et al., 2012)
2. Pictorial – representing with pictures (Reys et al., 2012)
Provide representations of objects in the form of pictures, drawings and diagrams to help solve the problems (Reys et al., 2012). This strategy begins the process of moving away from concrete materials and closer to symbolic representation (Reys et al., 2012)
3. Abstract – representing with symbols (Reys et al., 2012)
“Use symbols (especially numeric expressions and number sentences) to illustrate the operation” (Reys et al., 2012, p. 199)
Types of problems:
· Separation problems (Reys et al., 2012)
Also known as ‘take-away’ is to remove a specific quantity from another and identifying what is left (Reys et al., 2012). There are two important ideas that must be addressed with this type of problem. The first is that research shows children find this subtraction strategy the easiest to learn. The second is the consistent use of the phrase 'take away' can leave children believing this is the only form of subtraction and can lead to further misunderstandings of the other types of subtraction. It is preferable to ask ‘10 minus 2’ rather than ‘10 take away 2’ (Reys et al., 2012).
Example problem: Have the students in your class stand in a row and use them to solve the problem.
'There are 20 students in my class, but 5 students go home (5 students sit down), how many students are left?'
· Comparison problems
"Comparison involves having 2 quantities, matching them against each other and finding the difference between them" (Reys et al., 2012, p.200).
Lesson Example:
Have 3 or more stations set up each with different comparison subtraction problems.
(These problems can be all concrete or pictorial, or a mixture of both)
Problem 1: I made 10 cupcakes for my dad’s birthday, but I ate 4 of them before everyone had arrived for the party! How many cup cakes are left?
Problem 2: I have 8 red teddies and 3 green teddies. How many teddies don’t have a partner?
Problem 3: Chloe has 7 green unifix blocks and Jack has 2 yellow unifix blocks. Who has the most unifix blocks? And how many more do they have?
· Part Whole problems
"A set of objects is separated into 2 logical parts" (Reys et al. 2012, p. 200). The total number of objects in known and so is the total of one of the parts. Therefore the total of the remaining part needs to be established.
. Counting Back
One particular mental subtraction procedure is counting down. This strategy can prove to be very difficult if students do not have a sound grasp of counting down from any given number. Before looking at, or explicitly teaching this strategy, it is essential to ensure students have the ability and confidence to count back from any given number.
Games such as ‘Blast Off’ may be used. This game involves counting down from 10 or 20 and the last person says ‘Blast off’ and sits down.
The following sheets are recourses that can be used to assess students ability to count back from any given number. Use the numbered socks underneath as starting numbers. Students pick any sock and count back 5 places from that chosen number to reach the answer. Students get a new numbered sock each time.