MATHEMATICAL CONTENT KNOWLEDGE -
There are three main operations to addition and subtraction, these include concrete, pictorial and abstract. Concrete includes modelling with materials, pictorial means representing with pictures, and abstract is representation with symbols. Thinking strategies for addition include commutativity, adding 1 and 2, adding doubles, counting on, combinations to 10, and adding to 10 and beyond. Subtraction thinking strategies include subtracting 1 and 2, doubles counting back, and counting on.
Addition and subtraction structures include joining, part part whole, and comparison.
Joining
- result is unknown e.g. ___+___ = ?
- change is unknown e.g. 3 + ___ = 5
- start is unknown e.g. ___ + 2 = 5
Part part whole
- whole is unknown e.g. 3 + 2 = ?
- part is unknown e.g. ___ + 2 = 5
Comparison
- Difference is known e.g. There were 5 marbles in the box, I took 3 of them out. How many marbles were left?
- Larger is unknown
- Smaller is unknown
Common misconceptions in addition and subtraction include understanding addition and subtractions being inverse operations, counting on and counting back accurately, using the base 10 structure of numbers to solve multi digit problems, the concept of zero in operations, and the use of commutative property for subtraction.
PEDAGOGICAL CONTENT KNOWLEDGE -
To introduce the concept of adding or subtracting, start with hands-on materials such as counters, blocks or plastic toys. Extend this by using informal written methods such as diagrams or drawings. End with writing in numerical expressions and sentences with the correct symbols.
Reference: (Van et al., 2015)
--------------------------------------------------------------------------------------------------------------------------------
Activity - Adding Cookies
Year Level - Foundation
Classroom Context -
For those who are visual learners, working with hands-on manipulatives they are familiar with (such as
counters, blocks, everyday objects) will allow them to physically move the items around, or have other
students and teachers move the items for them so they can visually see the mathematical process.
counters, blocks, everyday objects) will allow them to physically move the items around, or have other
students and teachers move the items for them so they can visually see the mathematical process.
Elaboration - using a range of practical strategies for adding small groups of numbers, such as visual
displays or concrete materials
displays or concrete materials
Learning Objectives -
The purpose of this activity is to help children develop knowledge in adding physical objects together
to represent addition. They will be incorporating mathematical terms used in addition such as “adding,”
“altogether,” “more,” or “how many.”
The purpose of this activity is to help children develop knowledge in adding physical objects together
to represent addition. They will be incorporating mathematical terms used in addition such as “adding,”
“altogether,” “more,” or “how many.”
Resources -
Key Mathematical Language:
How many, Add on, Altogether, More, Plus, Equals
Symbols: Using the addition symbol ‘+’ and equals symbol ‘=’ correctly when writing numerical
expressions.
expressions.
Hands-on Manipulatives/materials: counting mat (cookie jar), manipulative objects (print-out cookies)
Hands-on manipulatives are useful for students to visualize real-world objects being added together.
Prior Knowledge -
Students should have the ability to combine two groups of objects and attempt to find the total
Students should be able to compare two quantities of up to 10 and state which has more
Instructions -
- The teacher will begin with an example - e.g. ‘I had 2 chocolate-chip cookies in my jar, but Bob put 1 more jam cookie in my jar. How many cookies are in my jar altogether?’
- The children may then use the mat and the manipulatives to work this out.
- The teacher will demonstrate by breaking down the problem. They will place 2 chocolate-chip cookies in the jar and say ‘I had 2 chocolate-chip cookies in my jar,’ and then place 1 more jam cookie in and say ‘but Bob put 1 jam cookie in my jar.’ (Part part whole problem)
- The teacher will count all 3 cookies and announce the answer.
- Students will repeat these exercises with different problems.
- Example of a ‘Joining’ problems: ‘Bob and Sue shared a cookie jar with 5 cookies in it. If Bob had put in 3, how many did Sue put in to make 5 altogether?’
- Example of a ‘Comparison’ problem: ‘Bob had 6 cookies. If Sue had 2 more cookies than Bob, how many did she have?’
- The teacher can simply make any adaptations to this activity.
Questions to ask -
How many are there altogether?
Are there more cookies inside the jar or out?
How many did he add?
How many are there in the jar?
Enabling Prompts - For those struggling with adding on, allow them to practice counting the number of cookies already placed in the jar, then increase the complexity by ‘adding on more’.
Extending Prompts - Students may use different manipulatives of their choice and/or create their own scenarios, followed by writing the matching number sentences with the correct symbols.
Extending Prompts - Students may use different manipulatives of their choice and/or create their own scenarios, followed by writing the matching number sentences with the correct symbols.
Images -
These are the materials students will start with - a cookie jar and cookies as the manipulatives.
EXAMPLE 1) 'part part whole' problem:
I had 2 chocolate chip cookies in the jar (students will place 2 in chocolate chip cookies in)
...but Bob put 1 more jam cookie in the jar (students will pick up a jam cookie and place it in the jar)
How many cookies are in the jar altogether? Students will count all 3 cookies and the total will be 3
EXAMPLE 2) 'joining' problem:
Bob and Sue had a cookie jar with 5 cookies in it.
If Bob had put in 3 (students can circle 3 to determine how many were Bob's) how many did Sue put in to make a total of 5? Students will count the rest of the cookies and the answer would be 2
EXAMPLE 3) 'comparison' problem: Bob has 6 cookies (students will place 6 cookies in the jar).
If Sue had 2 more cookies than Bob, how many did she have? (Students will add 2 more in the jar to figure out Sue had 8 cookies altogether).
Reflection -
Although I do not necessarily remember being taught basic counting strategies in primary school, readings and
research has helped me understand the important mental processes, mathematical thinking and strategies that
children must learn in their early years to help them develop their problem solving skills later on (Van et al., 2015).
During my experiences in placement, I often see teachers use hands-on manipulatives in classrooms to teach
these basic mathematical concepts. These early number techniques of using devices or objects to assist in the
thinking process, must be covered in learner’s early years before moving on to extended mental computation
strategies.
research has helped me understand the important mental processes, mathematical thinking and strategies that
children must learn in their early years to help them develop their problem solving skills later on (Van et al., 2015).
During my experiences in placement, I often see teachers use hands-on manipulatives in classrooms to teach
these basic mathematical concepts. These early number techniques of using devices or objects to assist in the
thinking process, must be covered in learner’s early years before moving on to extended mental computation
strategies.
This activity links to the curriculum as it utilises practical strategies such as adding small groups of numbers
together (using visual displays/concrete materials) to represent addition models. It also incorporates all key
structures of addition and subtraction, such as part part whole, joining, and comparison. In the activity, models
were used to solve the contextual problems to give meaning to the number sentences. This is helpful especially
when learners are at a young age, as it enables them to better explain how they solved the problem when there
are pictures or physical materials to visually represent the problem. As discussed by Gervasoni (2002), there are
many levels used in addition and subtraction which includes count all, count on, and count back techniques. This
activity incorporated these key techniques which also allows the teacher to assess and observe which skill each
student has developed. As a future teacher of mathematics, this would enable me support the student move
towards the next level in their learning.
together (using visual displays/concrete materials) to represent addition models. It also incorporates all key
structures of addition and subtraction, such as part part whole, joining, and comparison. In the activity, models
were used to solve the contextual problems to give meaning to the number sentences. This is helpful especially
when learners are at a young age, as it enables them to better explain how they solved the problem when there
are pictures or physical materials to visually represent the problem. As discussed by Gervasoni (2002), there are
many levels used in addition and subtraction which includes count all, count on, and count back techniques. This
activity incorporated these key techniques which also allows the teacher to assess and observe which skill each
student has developed. As a future teacher of mathematics, this would enable me support the student move
towards the next level in their learning.
References -
Gervasoni, A. (2002) Growth points that describe young children’s learning in the counting, place value, addition
and subtraction, and multiplication and division domains. Paper presented at the Catholic Education Commission
of Victoria Success in Numeracy Education Strategy.
and subtraction, and multiplication and division domains. Paper presented at the Catholic Education Commission
of Victoria Success in Numeracy Education Strategy.
Van, D. W. J. A., Karp, K. S., & Bay-Williams, J. M. (2015). Elementary and middle school mathematics: teaching developmentally, global edition. Retrieved from https://ebookcentral-proquest-com.ezproxy.library.uq.edu.au
