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Early years Subtraction

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by on 1 November 2014

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Transcript of Early years Subtraction

Yarden Alon 25106643
Kirsten Tulloch 25140183
Yuqing Wang 24583979
Brianna Pellow 25171771
What is subtraction in the early years?
Subtraction is just taking away?
Let's take a look...
Key words for subtraction
There are three main types of subtraction situations:
Separation problems
Comparison problems
Part-whole problems
Early years Subtraction
Thank you!
Separation Problems
"Separation, or take away, involves having one quantity, removing a specified quantity from it and noting what is left" (Reys et al., 2012, p. 200).

Comparison Problems
This involves comparing or finding the difference between 2 objects without taking away (Reys et al., 2012). This involves having 2 quantities, matching them 1 to 1 and noting the quantity that is the difference between them. Problems of this type can be solved by subtraction, even though nothing is being taken away.
Part-Whole Problems
"Nothing is being added or taken away- you simply have a static situation involving parts and a whole" (Reys et al., 2012, p. 200). You know how many are in the entire set and you know how many are in one of the parts. You need to find out how many must be in the remaining part.
So subtraction is far more than just take away. Take away is just
one
of the
three
types of subtraction situations.
Why is subtraction important in early mathematics?
The importance of subtraction:
Subtraction is basic knowledge and a skill for a great number of mathematical concepts throughout different levels. Including:
- division (repeated subtraction)
- fractions
- plus many more
Subtraction is a mathematical tool for supporting cross-curricular subjects such as, literacy, music, sports and arts.
Subtraction is an underlying mathematical skill for problem-solving.
Key Concepts
Basic facts and place value
"Subtraction of multidigit numbers requires knowledge of basic facts and place value" (Reys et al, 2012, p. 255). Children need to be explicitly taught about trading and regrouping. This can be a common problem among children, because without explicit teaching, children may struggle to understand the columns, instead subtracting the smallest number from the largest number, without regrouping, therefore getting the answer incorrect. Using concrete materials such as MAB blocks can assist a child’s understanding of this.

Introducing vocabulary and symbols
The inverse relationships between addition and subtraction
Reys et al. (2012) emphasise that students should be encouraged to use different subtraction sentences, for example, "5 minus 3 equals 2" instead of always using 5 take away 3 equals 2," otherwise it results in a misunderstanding from the other two situations.
For each addition equation, there will be two subtraction equations using the same numbers. For example, 8 + 2 =10, 10 – 2 = 8 and 10 – 8 = 2. It is important that children make these connections, which is why tasks such as ‘Number of the Day’ are important, where children can describe what equations make for the number of the day. This assists children’s understandings that subtraction is the inverse of addition.

Problem-solving knowledge
Place value:
“A key idea underlying the typical approach is that, in the range one to 100, concepts of tens and ones, that is place value concepts, should be taught prior to and separate from addition and subtraction" (Wright, 1994, p. 25).
To build on these skills, children benefit from problem-solving questions based on subtraction. This allows the child to interpret the question and lay out how they would find out the answer, instead of having the layout already given to them. As Reys et al. (2012) suggest, problem-solving can be a difficult skill to learn, and so it is essential that educators teach it at the correct development level.

“Problems are the source of the meaning of mathematical knowledge" (Stylianides & Stylianides， 2007， p. 107). Learning mathematics with understanding can be promoted through students’ engagement in problem solving activities. Several researchers emphasise that curriculum activities that engage students in problem solving reflect an emphasis on learning mathematics with understanding (Fennema et al., 1999; Romberg & Kaput, 1999; Schoenfeld, 1992; Stylianides & Stylianides， 2007，p. 106).

Algorithms
Algorithms are beneficial to students learning, particularly with larger digits and with multiple digits, because this is difficult to do using fingers or other concrete models.

Decomposition Algorithm:
renaming the sum

Standard Algorithm
:

Research evidence
What are recommended approaches to the teaching and learning of early years subtraction?
Pedagogical approaches:
Examples and Manipulatives
EDF2304 Assignment two:

Topic: Early years subtraction
Subtraction in the early years is important because children experience subtraction in their everyday living even from a very early age (Mannigel, 1992). At a young age, children are already "building up experience and thinking about aspects of subtraction in their own lives, in many practical situations" (Manningel, 1992, p. 141)
“Subtracting 1 and 2: Once they have learned strategies for adding 1 and adding 2, most children find it rather easy to learn the related subtraction facts involving 1 and 2. They can profit from work with materials and from observing patterns similar to those used for addition facts" (Reys, et al., 2012, p. 212 ).
“And sometimes, we are not just subtracting by 1, but subtracting by 2 with each verse, as in ‘Ten Fat Sausages’. We should begin children’s journeys into the world of mathematics by consolidating their knowledge of number names and their values and developing a facility with these concepts before we begin more advanced mathematical processes like subtraction.” (Goddard, 2010, p. 5).
-
Doubles:
"The strategy for doubles may need to be taught more explicitly for subtraction facts than addition facts. It rests on the assumption that children know the doubles for addition. like 8 +_ = 16, 8 + 8 = 16, so 16 - 8 = 8" (Reys et al. 2012, p. 212)

-
Equivalent statements:
making connections between these is important for children to further their understandings of subtraction. For example, 6 + 2 is the same as 4 + 4, because they both equal 8.

-
Counting back: "
The strategy of counting back is related to counting on in addition. It is most efficient when the number to be subtracted is 1 or 2: Think 9…8, 7, so 9 - 2 = 7" (Reys et al. 2012, p. 212)

-
Counting on:
"The starter of counting on is used most easily and efficiently when the difference is 1 or 2. Think addition by counting on: Think 6...7, 8, so 8 - 6 = 2" (Reys et al. 2012, p. 212)

A specific error that children often make is called "smaller-from-larger (e.g. 216 - 109 =113)." (Olteanu & Olteanu, 2012, p.807). In the case of the example, the student subtracted 6 from 9 because 6 was the smaller number/ digit. Olteanu and Olteanu (2012) go on to say that the assumption with this is because subtraction is cumulative. They have identified some strategies to help children with this, these use addition and subtraction together, so before using the strategies, the students would need to know the basic facts of both addition and subtraction. The strategies are called jump, split and compensation. Example for jump: 47+18:47+10=57+3=60+5=65, split: 47+18:40+10=50, 7+8=15,50+15=65, and compensation: 47+18: 47+20=67−2=65 (Olteanu & Olteanu, 2012, p.807)
Reys et al. (2012) suggests that a good way to teach children subtraction is to encourage them recognise, think about and use the relationships between addition and subtraction facts. It is possible to find answers by considering the missing addends.

Problem-solving based questions are important when teaching subtraction, because problem solving in the classroom allows for a more ‘real’ maths experience, whereby students are able to feel the necessity and importance of maths in everyday life, not just in the classroom to ‘pass tests’ (McClure, 2013).
Davenport and Howe (1999) suggest that allowing for peer collaboration can lead to an increased awareness of the cognitive processes and strategies of solving number problems, such as subtraction problems. When children explain number problems to other children, it can really be beneficial to them (Webb, 1992, as cited in Davenport & Howe, 1999). Davenport and Howe (1999) found that working in groups on problem solving tasks results in cognitive gain. It was also stated that children who rate as below average in ability level to solve mathematical problem solving tasks, appear to greatly benefit from working in groups as it allows them to have the opportunity to listen to the other children in their group discuss problem solving in language that they can more easily understand.
This means that as teachers, it is important to include group work when subtraction problem solving tasks are involved, as this allows students to gain a greater understanding of the concept as they can learn from one another.

There is no precise time frame for how long it takes to teach students the concept of subtraction (Page, 1994). This varies depending on the needs of the children. It is paramount to take your time when teaching subtraction and to link children’s own experiences to your teaching (Page, 1994). Page (1994) further states that children come to school with an experience of subtraction, such as eating cookies or losing crayons, they just do not have the name for these experiences. As a teacher we must make the link between the child’s everyday experiences and the mathematical concept of subtraction. By linking these to the real world, children are able to see the relevance of this mathematical concept (J. Cheeseman, personal communication, October 16, 2014).

Children usually find it easy to discuss their subtraction experiences such as eating cookies or losing crayons, but the struggle commences when we move from discussing their experiences, to numerical representation (Page, 1994). As teachers we must not bypass the mathematical language, as this is an essential bridge between the experiences and the numerical representation (Page, 1994). It is crucial to link the children’s informal mathematical ideas with mathematical symbols (Page, 1994).

Olteanu and Olteanu (2012) have found that "in the field of research on subtraction, two main lines that are general in nature can be distinguished: the first is about what it means conceptually (i.e. the inverse of addition, difference, etc.) and the second is about learning traditional subtraction algorithms (i.e. smaller from larger, decomposition, etc.)" (p. 805). They also stated that "basically, there are four addition and subtraction situations that represent the real world: compare, combine, change add to and change take from." (p. 807).

Manipulatives are objects that can be moved around to assist with student’s learning in mathematics. This can be physical (e.g. counters) or virtual (e.g using a Smartboard). Some examples of manipulatives to be used in subtraction are shown in the next slides:

Post it Note Maths
All we need is just the post it notes with numbers on them, place them in a random order, and then ask student to put numbers in numerical order. This is adaptable for all ages and levels and is hands on learning.
This teaching activity can be linked to ausVELS (2014) level 1: Recognise, model, read, write and order numbers to at least 100 as students are ordering numbers. This lesson can be adapted to putting these numbers on a number line.

Activity: 100 hunt minus 10
Materials:
IWB

Method:
Using this website http://www.ictgames.com/100huntminus10.html

A number will come up on the side of the screen, students must use the 100 chart to count back by 10 from that number, this encourages the simple problem solving strategy of counting back.

This is linked to AusVELS (2014) level 1: Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts, as students are using counting back to work out the answer.

Number House
: This resource allows children to understand the link between addition and subtraction.
This activity can be used for up to grade 4.

Steps:
Write the two numbers to be added in the lower two boxes.
Write the sum in the top box.
Use these three numbers to write four true math sentences in the lower section.

This activity links to AusVELS (2014) level 3, as students are learning to recognise the connection between addition and subtraction.

Ten in the Bed

This is good for K-1, and allows the students to be able to count backwards, which is an early area of subtraction.
Peg Equations:
Using icy pole sticks with pegs allows students to physically create and solve simple subtraction problems, allowing an engaging way to solve subtraction, other than verbally or on a worksheet.

This can be used for early years when students are beginning to learn about subtraction. This can then be adapted further on with two digit numbers.
Subtraction Activity
Materials: Egg carton, cloths, icy-pole sticks.
Fact Families: Instead of making the sum facts shown in the picture, many possibilities of equations can be made into different numbers, using subtraction.
This activity links to the level 3 of AusVELS (2014), as students are learning to recall subtraction facts for single digit numbers, as well as moving on to double digit numbers.

Number Line
Materials: stickers and a long blue line

This activity allows students to be physically and visually counting on and counting back as you are asking students to step forwards and backwards with the given equations.

This activity facilitates students’ understandings of counting on and counting back which is also linked to AusVELS level 1 and 2, solving simple addition and subtraction problems using a range of efficient mental strategies.

Missing Digits
Pose this problem to the class:
I solved a subtraction task but I can only remember the answer.
It looked like this:
_ _ _ - _ _ _= 3 7

What might the missing numbers be?

Students determine some of the possible numbers and describe how they worked them out.

This activity relates to level 2 of AusVELS mathematics (2014), as students are solving a simple subtraction problem and they must use a range of efficient strategies to solve this
problem.

This topic is important because it is the inverse of addition, as well as being a necessary skill that will be used throughout life.

Addition and subtraction are the basic building blocks in mathematics. Buying and selling objects is just one example of where you need to know addition and subtraction in life.

Take away:
having a set of objects and removing a number of objects (e.g. have 10 counters and removing 3, leaves you with 7 counters).

Decomposition:
mentally calculating smaller equations within the whole equation, for example 54 – 2, the students can simply subtract 2 from the 4 and keep the 5. This is a basic component of subtraction algorithms.

Strategies and teaching
It is important when teaching mathematics (or any subject), to never tell the child that their answer is wrong. Instead, allow the child to look back over their answers, gently proposing to them how they can make their answer ‘better’ (Cheeseman, 2014).

Begin to relate subtraction to ‘taking away’ and counting how many are left (5 lollies take away 3 equals 2)

Algorithms for column subtraction using concrete objects

“The errors in the application of the algorithms to numbers up to 100 can be traced back to a lack of understanding of the relationship between tens and ones. Students tend to treat the tens and the ones columns as if they were completely independent of each other. This leads to errors in the application of the rules for regrouping (Fernández & Estrella, 2011, p. 543).”

This activity allows students to place the particular number of concrete objects, i.e. cookies, blocks, into ones and tens columns, and then move the actual items as symbolic representation. This activity also connects to AusVELS level 2 as it visually helps children’s understanding of three-digit numbers as comprised of hundreds, tens and ones/units using concrete objects.

Subtraction requires knowledge of basic facts and place value (Reys et al., 2012, p. 255). Children need to be explicitly taught about trading and regrouping. This can be a common problem among children, because without explicit teaching, children may struggle to understand the columns, instead subtracting the smallest number from the largest number, without regrouping, therefore getting the answer incorrect. Using concrete materials such as MAB blocks can assist a child’s understanding of this.

As with many maths topics, special attention needs to be placed on subtracting with ‘0’.

References
Australian Curriculum Assessment and Reporting Authority. (2014). AusVELS Curriculum. Retrieved from https://ausvels.vcaa.vic.edu.au

Cheeseman, J. (2014). EDF2304 Notes [Power Point slides]. Unpublished manuscript, EDF2304, Monash University, Melbourne, Victoria, Australia.

Davenport, P., & Howe, C. (1999). Conceptual gain and successful problem-solving in primary school mathematics.
Education Studies
, 25(1), 55-78.

Goddard, J. (2010). Focus on...Maths through music, rhythm and rhyme. Early Years Magazine. Retrieved from https://www.ncetm.org.uk/resources/23975

Haylock, D. and Cockburn, A. (1989). Understanding Early Years Mathematics, London: Paul Chapman Publishing. Retrieved from: http://www.annery-kiln.eu/gaps-misconceptions/subtraction/why-subtract-difficult.html

Klein, J. S., & Bisanz, J. (2000). Preschoolers doing arithmetic: The concepts are willing but the working memory is weak. Canadian Journal of Experimental Psychology, 54(2), 105-16. Retrieved from http://search.proquest.com/docview/200317263?accountid=12528

López Fernández, J. M., & Estrella, A. V. (2011). Everyday settings, like enjoying freshly baked cookies, can foster proficiency and enhance students’ conceptual understanding. The National Council of Teachers of Mathematics, 541-548. Retrieved from http://info.eb.com/wp-content/uploads/2013/05/NCTM_Article.pdf

Mannigel, D. (1992). Young Children As Mathematicians. Wentworth, NSW: Social Science Press.

McClure, L. (2013). Problem Solving and the New Curriculum. Stage: 1 and 2. Retrieved from http://nrich.maths.org/10367/index

Olteanu, C., & Olteanu, L. (2012). Improvement of effective communication- the case of subtraction. International journal of science and mathematics education, 10, 803-826.

Page, A. (1994). Helping students understand subtraction.
Teaching Children Mathematics
, 1(3), 140-143.

Reys, R. E., Lindquist, M. M., Lanbdin, D. V., Smith, N. L., Rogers, A., Falle, J., Frid, S., & Bennett, S. (2012).
Helping children learn mathematics
. Milton, Queensland: John Wiley & Sons Australia.

Sarama, J., & Clements, D. (2013). Helping children use problem-solving strategies in the classroom. Grades: Early Childhood, Infant, PreK–K, 1–2.
http://www.scholastic.com/teachers/article/early-math-how-children-problem-solve

Selter, C., Prediget, S., Nührenbörger, M., & Huβman, S. (2012). Taking away and determining the difference- A longitudinal perspective on two models of subtraction and the inverse relation to addition.
Educational Studies in Mathematics
, 79(3), 389-408. doi: 10.1007/s10649-011-9305-6.

Stylianides，A. J., & Stylianides, G. J. (2007). Learning Mathematics with Understanding: A Critical consideration of the learning principle in the principles and standards for school mathematics.
The Montana Mathematics Enthusiast

Wright, B. (1994). Mathematics in the lower primary years: A research-based perspective on curricular and teaching practice.
Mathematics Education Research Journal,