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Reflecting on the students' weaknesses and

strengths in solving subtraction problems

MSTE 504

Assignment 3

Fatimah Alsaleh

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Introduction

Many students, especially children, consider mathematics to be a difficult subject and face

difficulty in learning mathematics, whether the students do or do not have learning

disabilities. It is suggested that much more than 6% of students are ranked as low achievers

and experience severe difficulty in learning mathematics, as they display poor results (Peter,

2004, 2008). Van Kraayenoord and Elkins (2004) indicate that more than half of the 377

schools that participated in a national survey of schools reported that between 10% and 30%

of their students experienced difficulty in mathematics.

There are many different reasons for students’ difficulties in numeracy. One reason could be

the contexts in which the students learn. Difficulties could also result from unusual patterns

of individual characteristics, such as brain development, poor teaching strategies and

insufficient preschool home experiences. Students’ preferred ways of learning, along with

inappropriate choice of strategy used and lack of mental flexibility, are referred to as sources

of students' numeracy difficulties. Children's difficulties could also be due to the mismatch

(gap) between the students' learning characteristics and the instructional materials and

practices presented (Dowker, 2004; Van Kraayenoord & Elkins, 2004; Peter, 2004).

The aspects of students’ numeracy misconceptions also vary. Some students have difficulty

in recalling basic facts, understanding the concept of place value, regrouping numbers and

moving groups of numbers. Other students may experience problems with thinking mentally

and using appropriate strategies. Many students struggle with solving multiplication and

division problems, and some struggle with addition and subtraction problems. Although

addition and subtraction are usually presented at the same level of difficulty, many teachers

observe that students find addition easier than subtraction, and children often make errors in

solving subtraction problems but not in solving addition problems (Kamii, Lewis & Kirkland,

2001). According to Cebulski and Bucher (1986), some studies conducted on the aspect of

students’ mathematics performance reported that students face difficulty in subtraction more

than in addition, especially in the primary grades. In particular, solving subtraction problems

by working backwards is usually hard for most children and adults (Fosnot & Dolk, 2001).

One aspect most commonly found as difficult by many students is memorising basic facts

(Dowker, 2004). Remembering the basic combinations is a big stumbling block for many

children experiencing mathematical difficulties (Baroody, Bajwa & Eiland, 2009). Peter

(2008) presented similar findings: many children who have learning difficulties have

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problems recalling number basic facts and waste much time working out simple basic facts

using their fingers or a calculator. According to the Tertiary Education Commission (2008),

students are more likely to have difficulty with recalling subtraction facts than with addition

facts. The difficulty in recalling basic facts, for some students with more severe mathematical

difficulties, may result from their reliance on counting strategies (Dowker, 2004).

Another aspect of difficulty for students is understanding place value. Many students have

difficulty in understanding place value until they reach the middle grades (Cawley, Parmar,

Lucas-Fusco, Kilian & Foley, 2007). Research studies have revealed that many young

children have problems in learning the concept of place value (Nagel & Swingen, 1998).

However, Van Kraayenoord and Elkins (2004) state that a failure to understand the concept

of place value is common in not only young students but also in upper primary students and

secondary students. Teachers in primary schools observe a wide range of students’

understanding and misunderstanding of place value when they work on solving multi-digit

addition and subtraction problems. Furthermore, some students face difficulty applying and

representing place value because they struggle to solve two-digit subtraction problems

mentally and with regrouping. Some children can solve single-digit problems, but they lack

understanding of tens, ones and place value. Other children can carry out two-digit

addition/subtraction problems with regrouping or mentally without regrouping (Dowker,

2004). Consequently, Flores (2009) point out that students’ insufficient knowledge of the

regrouping process results from their lack of a conceptual understanding of place value.

Students can understand place value through using a variety of models such as ten-frame and

coloured counters (Van Kraayenoord & Elkins, 2004). Ten-frames are useful for developing

students’ part–whole thinking, especially in addition and subtraction (Isaacs & Carroll, 1999).

Although ten-frames help students to be able to visualise number relationships and how to

partition numbers (Anthony & Walshaw, 2009), “some students have difficulties in replacing

physical counting strategies using objects to count with verbal counting and, later, with

retrieval from memory” (Van Kraayenoord & Elkins, 2004, p. 36).

Teachers try to move students beyond counting strategies and push them towards using part–

whole strategies because counting strategies are not the best way for children to develop more

advanced computational skills. In addition, over-reliance on counting strategies may delay

children’s development of more complex mathematical skills and may lead to them being

incapable of solving large number addition/subtraction problems (Cheng, 2012). There are

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different strategies that students can use to solve addition and subtraction problems, including

basic strategies (doubles, commutativity, adding 10, tens facts), derived strategies (near

doubles, adding 9, build to next 10) and reasoning strategies (doubles, near doubles, adding

10, adding 9, commutativity, combinations for 10, part–whole strategies and retrieving

answers from memory). When children develop a variety of strategies, it is important that

from the various strategies, these children can choose which one is appropriate for solving a

problem. However, not all children are able to develop each strategy (Gervasoni, Hadden &

Turkenburg, 2007). Some children may focus on using doubles, as doubles-based strategies

are commonly used by children. Children also use derived fact strategies or bridging up or

down through ten. Other students may use known addition facts to derive unknown

subtraction facts, such as 15 – 8 = 7, since 7 + 8 = 15 (Isaacs & Carroll, 1999).

It is important that students have the capacity to develop knowledge and strategies because

both of them are needed to improve each other. More advanced and complex strategies are

related to a richer knowledge base. Research has found that there is a strong relationship

between number knowledge and number strategies, and the children who possess less

sophisticated strategies are more likely to have less sophisticated number knowledge. In

addition, children are less likely to possess number knowledge when they tend to use

counting all strategies (Biddlecomb & Carr, 2011). The strategies that students have are

derived from breaking the numbers into parts and thinking about them first, looking at the

number relationships and then playing with those relationships (Fosnot & Dolk, 2001). For

example, remembering basic facts is really important in order to be able to use strategies for

solving different problems. Knowing basic facts helps students to extend their repertoire of

more efficient strategies for harder facts. Consequently, increasing students’ repertoire of

strategies enables students to recall more basic facts (Isaacs & Carroll, 1999). If students have

difficulty with remembering basic number facts, they will work out a problem using more

time-consuming strategies (Dowker, 2004). In addition, students must learn the concept of

place value before they start learning the basic operations of multi-digit addition, subtraction,

multiplication and division because without understanding place value, the procedures of

solving these problems will be more difficult (Hunting, 2003; Flores, 2009). Similarly,

Baroody (2004) found that children who do not have a meaningful understanding of grouping

and place value usually face difficulty working out multi-digit numbers. Part–whole thinking

provides students with a conceptual understanding of the place value system and enables

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them to be more flexible in mentally moving different units such as hundreds, tens and ones

(Hunting, 2003).

Observation is commonly used to conduct educational research and it is an important tool in

developing a complete picture of children and their learning. “Observation is the act of

looking at something—without influencing it—and recording the scene or action for later

analysis” (Yount, 2006, p. 3). There are different types of procedures used to observe

children and they can be grouped into three types of observational methods: informal

observations, indirect observations and formal observations (Cross, 2007). Indirect

observation does not require the presence of the researchers on site and is achieved through

the use of video cameras, which can be reviewed or watched by others in addition to the

original observer (Slack & Rowley, 2001). Teachers can learn new knowledge from watching

other teachers give a lesson either live or on video (Star & Strickland, 2008). Video footage

helps the researchers to record the participants’ interactions and allows the researchers to

reflect on the events that took place during the recorded lesson in more detail after the actual

observation (Menter, Elliot, Hulme, Lewin & Lowden, 2011). In addition, observation helps

the researcher to collect several kinds of information; however, this means that the researcher

has to analyse more data, which can be time-consuming. Another weakness of using indirect

observation is that it creates a distance between the researcher and the participants, which

may lead the researcher to miss some of the participants’ interactions (Slack & Rowley,

2001).

The purpose of this study was to investigate the aspects of the strengths and weaknesses or

misconceptions that students have in solving subtraction problems by determining the

students’ ability to use strategies and number knowledge to solve these problems.

Method

Participants

The sample consisted of a teacher and eight primary students working in two groups

presented in a video clip called ‘Lyn Peterson interaction’. Group 1 includes four girls:

Surita, Pyridnia, Kato and Summer. They all are of Pasifika heritage except for Pyridnia, who

is Indian. Two boys (Liam and Asher) and two girls (Catherine and Emma) are involved in

Group 2. The two girls are Pasifika, like most of the girls; Asher is Maori and Liam is

European.

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Procedure

The method used to assess students’ strengths and weaknesses in solving subtraction

problems was via indirect observation with video footage as a secondary source because the

video- was filmed by the original observer (the teacher). The vignette titled ‘Lyn Peterson

interaction’ was watched many times by the secondary researcher, who recorded the actions

that took place during the videotaped lesson and took notes about the participants’

interactions. Each student’s response to the tasks presented in the video and their aspects of

understanding and misconception were transcribed for later analysis.

Results

In this section, the results of this study are presented by organising them under four

subtraction problems, analysing students’ strengths, weaknesses and misconceptions

regarding their use of strategies and number knowledge, including basic facts and place

value, to solve the four problems.

13 – 7 = 6

Catherine, Summer and Kato used their knowledge of basic facts to help them solve this

problem using the deriving from doubles strategy: 6 + 6 = 12, 6 + 6 + 1 = 12 + 1, 6 + 7 = 13.

Catherine was good at using doubles to figure out this problem and she could match the four

green counters and three red counters, which were moved away from the ten-frames that were

used to represent most of the problems, to the number 7 written on the problem paper.

Summer was able to see what 6 + 6 looked like in the ten-frames and she was able to see how

6 + 6 would help to solve 13 – 7 = 6 because she said “ if you add one that makes 13”. Kato

knew that instead of 6 + 6 = 12, the story problem would be 6 + 6 + 1 = 13. Liam was able to

see that 6 + 6 + 1 = 13 is the same as 6 + 7 = 13(see appendix A).

15 – 6 = 7

At the beginning, the teacher asked the students to picture in their mind what the number 15

looked like in the ten-frames. Pyridnia was able to link the picture of 15 in the ten-frames to

the picture of 13 from the previous problem. She then added two more red counters to the 13

counters to make 15. The teacher asked the students later to imagine scooping away eight

counters from 15. Asher pictured taking away eight counters as moving away the five red

counters and the three green counters to make seven (see appendix B). Liam and Catherine

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imagined the same picture as Asher did, using their knowledge of basic facts. Liam said: “I

know that 5 and 3 was 8, so I just took away the 5 and the 3, so I know it is 7.” Kato was able

to explain what this way of solving the problem looked like in the ten-frames, “you take away

five from the reds and three from the greens and that makes eight and in the ten-frames I left

seven”. Emma used a different way (deriving from doubles) to solve this story problem. She

used her knowledge of doubles (8 + 8 = 16) and from this, she deduced that 16 – 1 = 15, 15 –

8 = 8 – 1 =7. Liam was good at seeing the relationship between Emma’s way of solving the

problem and the way that was used to solve the previous task. Summer needs a lot more

experience with manipulating the counters and moving away the groups of counters because

she was not able to see that 8 is made up of 5 and 3 or 4 and 4, or to take these groups of

numbers away, so she had difficulty with both basic facts and place value. Surita was able to

help Summer to see that 5 + 3 = 8 (see appendix B).

24 – 6 = 18

At the beginning, the teacher asked the students what 10 green counters plus 10 red counters

and four blue counters equalled. All the students were able to see those groups of counters

equalled 24 counters, which means they were able to do the combination to 10 and regroup

the numbers by 10. When the teacher asked the students to take away 6 from 24 by imagining

what they would look like in the ten-frames, Catherine tried to imagine the problem, but she

got stuck with imagining and struggled to hold on to these numbers “10 and 10 you got 20

plus 4 equal 24 and then you just take away the 6 and then you get a number far…” (see

appendix C). Liam was fairly good at imagining and his answer to his group was “I took

away 4 and that would equal 20. Then I took away one more and that would be taking away

5, which equals 19, and if I take one more, that would equal 18” (bridging down through ten).

When he explained his thinking for his teacher, he got stuck with imagining the coloured

counters (see appendix C). Surita was good at solving this problem by moving away six

counters from the ten-frames, but she was not able to see the relationship between the ten-

frames and the numbers that had been written because she struggled with the six counters that

were removed, which were four blue counters and two red counters. When the teacher asked

the students to see where the four blue counters and the two red counters in the written story

problem, Surita identified the four blue counters as belonging with the 4 in the 24 and when

she was asked about the two red counters, she pointed to the two tens rather than two ones.

So instead of seeing that the two and the four was coming from the six, she mixed it up

because she did not understand how the tens were a different kind of unit from the ones (see

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appendix C). By contrast, all the students disagreed with Surita’s thinking. Asher, Liam,

Catherine and Emma immediately knew that the two in the 24 were two tens and they could

see it in the ten-frames. In addition, Liam was able to connect the remaining 10 green

counters to the 10 in the 18 written on the paper, and the eight red counters to the eight ones

in 18. Surita, Summer and Pyridnia need more experience in making tens and ones with the

ten-frames and other materials to make sure they can see the tens and ones in numbers on

paper and understand the concept of place value.

83 – 5 = 78

Without using the ten-frames, the teacher asked the students not to count down but to think of

groups of numbers that could be seen in the numbers. Kato started to solve this problem using

bridging down through ten (the decade strategy) by taking away 3 from 83, which makes 80,

but when she tried to take away 2 more from 80, she struggled with that and did not know

what she needed to do (see appendix D). Surita tried to solve this problem, but she was not

sure about her thinking. She said, “I think it would equal 78 or 77; I am not sure about those

two numbers because taking away 3 and 2 from the 80 equals 98 or 97.” When the teacher

asked her if the answer could be 97, which is bigger than 80, she said, “I think it would be

78.” She seemed to know the answer, but she might get confused with her answer and could

not figure it out in her mind. Pyridnia disagreed with Surita’s thinking because she thought

that 5 was just a single number and it was not bigger than 80 and her answer was 87. So these

students might have been confused working out the answer in their mind.

All four girls (Kato, Pyridnia, Summer and Surita) needed more experience with the counting

on and counting back, because they struggled when they got to the high number (83 – 5).

They were good at visualising taking away three counters, but they got stuck on the counting

back and they did not know the combination to 10 of 10 – 2 = 8. These girls need a lot more

practice to understand subtraction from 10 and to use the bridging down through ten (decade)

strategy.

Group 2 was able to solve this large number problem because they all knew the combination

to 10 of 10 – 2 = 8. Catherine and Emma solved this problem correctly using bridging down

through ten (decade) by taking away 3 from 83, which made 80, and then taking away 2 more

from the 80 to make 78 because they knew that 3 + 2 = 5. Asher used a reversibility strategy

in solving this problem because he used known addition facts (78 + 5 = 83) to derive

unknown subtraction facts (83 – 5 = 78). Liam was able to see the similarity between this

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task and the previous one: taking away 3 from 83 is similar to taking away 4 from 24 and

taking away 2 more from 80 is just like taking away 2 more from 20 (see appendix D).

Discussion

At the beginning, most of the children had fairly good fluidity with the subtraction

combination to 10, but some of them faced difficulty when the teacher tried to push them to

use bigger numbers. Most of the students were able to connect the relationship between the

ten-frames and the numbers written in the problems, but some of them struggled with that

when the numbers were larger. Some students were able to see the link between the previous

task and the new one, and applied the same method to solving the new task (see appendix B

& D). All the students were good at using basic facts knowledge (addition and subtraction

facts, doubles facts) to solve subtraction problems and to improve a strategy that helps them

to solve the problems. This result was inconsistent with Dowker’s (2004) finding that most

students have difficulty in memorising basic facts.

The four girls in Group 1 were less able to group numbers and remove the groups of numbers

using the combination to 10. Some of them struggled with imagining what big numbers

looked like in the ten-frames (imagining how many coloured counters they would need to

solve the problem). They also had difficulty with manipulating the counters, regrouping

numbers and removing these groups of numbers. They need to work more to understand tens,

ones and place value (see appendix C), and to see the connection between the ten-frames and

the numbers written in the story problem. These results were consistent with many previous

studies (Dowker, 2004; Baroody, 2004; Hunting, 2003; Flores, 2009; Van Kraayenoord &

Elkins, 2004) where some students were found to lack understanding of tens, ones and place

value, and faced difficulty with applying and representing place value; therefore, they usually

faced difficulty in working out multi-digit numbers. All four girls struggled with counting

back when they got to the really high numbers (see appendixes C & D). Counting back for

solving subtraction problems is usually hard for most children (Fostnot & Dolk, 2001).

The four children in Group 2 were different from those in Group 1 because they were able to

use the combination to 10, and to use their ability to regroup numbers and remove numbers in

groups using counters or mentally calculations. They were able to know the properties of

numbers where they removed the actual materials and they were able to imagine the numbers

even when the size of the numbers were really high (see appendix C & D). These students

verified Dowker’s (2004) opinion that some children can carry out two-digit

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addition/subtraction problems with regrouping or mental strategies without regrouping.

Generally, they had a clear understanding of using their knowledge of basic facts and place

value for improving operation strategies. One of the girls in this group (Emma) was really

focussed on using the deriving from doubles strategy and she may have just looked for

doubles, counting on or counting back in working on maths problems because she did not go

towards the combination to 10 or regrouping numbers by 5 or 10. This finding was similar to

what Gervasoni, Hadden and Turkenburg (2007) found, namely that not all children have

each strategy available, and to Isaacs and Carroll’s (1999) study that stated that doubles-

based strategies are commonly used by children. Other strategies that were often used by the

students in this study were bridging down through 10 and using known addition facts to

derive unknown subtraction facts (Isaacs & Carroll, 1999) (see appendix C & D).

Emma needs to be pushed more to develop other strategies and to get the ability to see that

using doubles will only enable her to solve certain small problems, but she cannot generalise

further from that. There will be a range of number problems that she will not be able to use

doubles to solve, whereas grouping by fives or tens would enable her to move into much

bigger numbers. To build Group 1’s understanding of place value, they need to work more to

gain experience in making tens and ones with the ten-frames and other materials to make sure

they can see the tens and ones in numbers on paper.

The observation tool used to conduct this study was very helpful for investigating students’

strengths and weaknesses. In addition, it allowed for reflection on the students’ interactions

and responses without the need for sitting with the students and teaching them (Slack &

Rowley, 2001). This was helpful to me because I am not familiar with using ‘mental

mathematical strategies’ in English and I do not have sufficient knowledge about teaching the

maths curriculum in New Zealand (NZ) schools. I have a different approach to teaching from

what they are familiar with, so I do not want to be responsible for confusing their

mathematical thinking and understanding. Another advantage of using indirect observation is

that I have learnt how subtraction problems are taught to students in NZ (Star & Strickland,

2008). However, I missed part of the lesson that was at the beginning because the teacher said

she did a quick check of the students’ knowledge about the basic facts and combinations to

10, but this was not shown in the video. Part of the teaching for the first task did not appear in

the video, which missed some of students’ interaction with that task and what omitted to

show their strengths and weaknesses were in solving those problems (Slack & Rowley, 2001).

This study raised questions about what these students’ strengths and weaknesses would be if

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the lesson taught other types of problems. It could also be questioned what these students’

strengths and weaknesses would be if I had taught them the same lesson or if the lesson was

taught by other teachers.

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Appendix

Appendix A

13 – 7 = 6

Teacher: who can see there (written problem) the number that tells about the four green counters and

three red counters?

Catherine: it is seven

Teacher: Catherin would you think of your doubles to help you, which double was helping you with

that problem?

Catherine: 6 + 6

Teacher: who know the answer for 6 + 6?

Summer: 12

Teacher: could you move six more counters over there, what would you left?

Summer: one

Teacher: who can see how this problem (6 + 6 = 12) help us to solve( 13 – 7 = 6)? Summer what do

you see?

Summer: if you add one that makes thirty

Teacher: Instead of 6 + 6 = 12 what would be the story problem?

Kato: 6 + 6 + 1 = 13

Teacher: Could anyone combine this one (6 + 6 + 1 = 13) with something else?

Liam : 6 + 7

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Appendix B

15 – 6 = 7

Teacher: I am making the number 15 in my ten-frames and I am using the red counters and the green

counters, what the number 15 looked like in the ten-frames?

Pyridnia : you have added two to the reds

Teacher: How would you scoop away in your mind 8 and what that would leave behind? (Move in

thinking groups)

Asher: I take away the five red counters and the three green counters and that make seven

Liam : I know that 5 and 3 was 8, so I just took away the 5 and the 3, so I know it is 7

Emma: 8 + 8 = 16 , 16 – 1 = 15, 15 – 8 = 8 – 1 =7

Kato: you take away five from the reds and three from greens and that makes eight and in the ten-

frames I left seven

Teacher: summer could you move for us what Kato just describe?

Summer: ten ……take away five reds

Teacher: and what do you think we need to scoop next, how could help Summer?

Teacher: Emma what she (Kato) said she scoop next?

Emma: three

Teacher: Can you try to scoop three from the greens Summer?

Teacher: Summer why do you think Kato said she would take the five reds and the three greens

away?

Teacher: that a tricky question? Do you want someone to add to your thinking?

Summer: Surita

Surita: if you put 4 here and put these three over there that equals eight and fifteen take away eight

equal seven

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Appendix C

24 – 6 = 18

Teacher: If I use all my green and red counters, does anyone know how many I am using?

Catherine: 20

Teacher: I am going to use more four blue

All : 24

Teacher: in your thinking group I want to focus on not counting back, can you think of taking groups

of numbers away? We got 24 and in your thinking group I want you to think about what would

happen if you took six away how many would be left and can you actually say what they would look

like in the ten-frames?

Catherine: 10 and 10 you got 20 plus 4 equal 24 and then you just take away the 6 and then you get a

number far I can’t….

Teacher: you get stuck imagining

Liam : 24 I took away 4 and that would equal 20. Then I took away one more and that would be

taking away 5, which equals 19, and if I take one more, that would equal 18

Asher: 24 take 6 equal 18 and then you just like putting 6 back and that make 24 straight away

Teacher: Liam can you try explain again your thinking using the coloured counters?

Liam : I took away four of the greens and that would equal 19, no would equal 20 and then I take

away 5 greens and that would equal 19

Teacher: I think you get a bit confuse with the counters

Teacher: Who was sure they can remember what 24 looked like?

Pytidnia: it looked like ten counters of reds, ten more counters of greens and four more of blue

Teacher: The question was 24 – 6 think of groups of numbers

Surita: it was 18, I took away the 4 and then I took away 2 more of the reds which equal 18

Teacher: How can think can follow Surita’s thinking enough to do with the counters?

Catherine: she took away the 4 and then she took away the 2 to make 18

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Teacher: why do you think surita took away those blue and reds? Summer

Summer: because it is 24 take away 6

Liam : that ( reds) eight and this( greens) like ten, ten and eight is eighteen

Teacher: Can you tell us where the green counters there, Liam?

Liam : the ten

Teacher: And what about the red counters?

Liam : The eight

Teacher: How can tell me where these counters (2 reds and 4 blue) if we look at the story problem?

Emma: 6

Surita: this 2(red) would be that 2 (24) and these 4(blue) would be that 4 (24)

Teacher: Think if you agree or disagree?

All : disagree

Teacher: Asher why you disagree?

Asher: The four covers the ‘o’ on the twenty so I just added the 4

Teacher: So you said there is a twenty there and that four telling us there are four ones no ones

Teacher: Where are the four ones?

Asher: there (4 blues)

Teacher: Asher what do you think this two (24) is then? Do you think is it 2 reds?

Asher: no

Teacher: What do you think these two if we look at board?

Asher: The twenty

Teacher: So what is the two telling in the twenty?

Asher: 2 tents

Teacher: Do you see that Surita, do you agree or disagree?

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Surita: That what I was thinking, because I was thinking this four has tens in it and this two has tens

in it.

Appendix D

83 – 5 = 78

(No ten-frames)

Teacher: How many counters could I but in my frame?

Summer: 30

Teacher: I am going to do bigger than 30, I am going to make it in the 80, I do not want to

count down, think about groups of numbers, with your group talk about 83 taking away 5

Kato: take away the 3 and 2 more from the 80… mmmm I do not know

Teacher: Ok can anyone help her

Surita: I think it would equal 78 or 77; I am not sure about those two numbers because taking away 3 and 2 from the 80 equals 98 or 97

Teacher: could it be 97, which is bigger than 80

Surita: I think it could be 78

Teacher: Pyridnia do you agree or disagree with Surita

Pyridnia: I disagree because 5 is just a single number and it is not bigger than 80

Catherine: it is 78 because If we got 83 you just take away 3 and take away 2 and that makes 5

Emma: you just take away 3 and 2 and that is 78

Asher: 78 + 5 equal 83 and then you can add or take away, 83 take away 5 equal 78 and then 78 and 5 equal 83

Teacher: How is confident with the answer?

Emma: 78

Teacher: does anyone agree with Emma? Does anyone have different answer

Pyridnia:87

Teacher: Kato you herd something from your groups can you tell us what you herd?

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Kato: I take away the 3 and 2 more from the 8 and makes 78

Teacher: 83 – 3 how much would be left?

Liam: 80 as the 24 one

Teacher: Emma can you tell us what another part that Kato said she would take away?

Emma: the 2

Teacher: where does it come from?

Emma: Because OF THE 5 + 3 = 2

Teacher: see it again what plus what?

Emma: Oh 2 + 3 = 5

Teacher: So if we have 80 and we took 2 away what left behind? Now you count back to solve that or you think of facts to 10, if I have 10 – 2 who know the answer?

Asher: 8

Teacher: is that could help you with 80 – 2 , what did you get as the answer?

Asher: 78

Appendix E

Summary

Teacher: At the beginning I did a quick check on basic fact to do combination to 10 addition

and subtraction. I did that because it is important to know if we going to push the kids to part

whole thinking and to move beyond counting back and counting on with numbers because

they will need to have this combination to 10 in order to do that. At the beginning most of

them had a fairly good fluidity with the combination to, a little bite lower to subtraction. Few

like Pyridnia much slower when asking quick fire type of subtraction combination to 10

questions. She really dropped off there and she struggling to imagine the numbers and when I

try to push them to bigger numbers and she was struggling to imagine group the number by

this combination to 10. You can see that interaction between the knowledge the kids have

about and their ability to operate with them and do strategies thinking .

The four girls (Kato, Pyridnia, Summer and Surita) were less able to group those numbers

and removing the groups of numbers. Surita was fairly able to do it. Kato at times was able to

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do it, but when we to imagining she struggle a little bite more to hold on these numbers.

Pyridnia and Summer need a lot more experience with the counters, moving the groups of

counters, seeing the 8 is 5 and 3 or 4 and 4 we can put these groups of numbers away.

For all the four girls, they need more experience with the counting on and counting back

because when they go to the really high number for 83 – 5 they actually struggled. They were

able to remove the 3 counters visualise taking away 3, but they got stuck with the counting

and they do not know the combination to 10 of 10 – 2 = 8, so they were kind of left nothing

there. Pyridnia had the idea of 78, 87. She might get quite confuse with her answer 87. Those

girls need to more to get the subtraction from 10 sold.

Interaction of the concept of tens and ones when we started to talk about the number 24 we

have made and seeing the relationship between the ten-frames and the numbers have been

written. Surita struggled with the 6 we have removed which was 4 counters and 2 counters.

She started to see that as 24 and was not the 2 tens in the ten-frames was 2 tens that showed

in the 24, whereas by contrast Asher, Liam, Catherine and Emma straight away know that the

2 tens was in the 24 and they can see it in the ten-frames. Surita, Summer and Pyridnia need

more experience in making tens and ones with the ten-frames to make sure connect to the

tens and ones in the paper.

Asher, Liam and Catherine were very similar in their ability to be able to use combination to

10, use their ability to group numbers and remove numbers in groups and all the three kids

were able to think about the problems. Emma is really use doubling strategy, she need to be

pushed more towards adopted other strategies. Emma in her maths working on maths

problems would properly result on just looking for doubles or counting on or counting back

because she does not naturally go towards the combination 10 roof of grouping numbers by 5

or 10. She had the ability to do it, but she needs to work more to give her the ability to see

that using doubles only give her so far, but she cannot generalise so far from that, whereas

grouping of 5 or 10 allows her to move into much bigger numbers and arrange of number

problems she will not able to just use the doubles for.