2011 key understandings in mathematics learning no films

3rd National ConferenceDyscalculia & Maths LD

Key Understandings in Mathematics Learning

Professor Terezinha Nunes

© Professor Terezinha Nuneswww. learning–works.org.uk T. +44 (0)1672 512914 info@learning‐works.org.uk

Key understandings in mathematics learning

Terezinha NunesDepartment of Education

University of Oxford

SupportThe Nuffield Foundation

ESRC – Teaching and Learning Research ProgrammeRNID and NDCS

The research team

• Peter Bryant• Deborah Evans• Daniel Bell• Rossana Barros• Darcy Hallett

www oxford ac ukwww.oxford.ac.uk

Department of EducationResearchChild Learning





• Paper 1: Overview• Paper 2: Understanding extensive quantities and whole

numbers

Key understandings in mathematics learning

numbers• Paper 3: Understanding rational numbers and intensive

quantities• Paper 4: Understanding relations and their graphical

representation• Paper 5: Understanding space and its representation in

mathematicsmathematics• Paper 6: Algebraic reasoning• Paper 7: Modelling, problem-solving and integrating concepts• Paper 8: Methodological appendixDownload from: http://www.nuffieldfoundation.org/key-

understandings-mathematics-learning

Numbers, quantities and relations

• Quantities and number are not the same thing: we can think of quantities without having towe can think of quantities without having to represent them numerically

• Numbers do not have to refer to quantities to have meaning: by definition, 5 means 4+1 and 3+2

• Children learn to count and to reason about• Children learn to count and to reason about quantities quite independently and need to learn to coordinate number representation with what they know about quantities





Numbers, quantities and relations

• Numbers are also used to represent relations– Sam has eight books (quantity)g (q y)– Theo has five books (quantity)– Sam has three books more than Theo or Theo has three books

fewer than Sam (relation)

• Relations can be additive or multiplicative

• Additive relations are based on part-whole schemes andAdditive relations are based on part whole schemes and have their origin in joining and separating actions

• Multiplicative relations are based on correspondences

• Additive relations– Rob and Anne have together 15 books (quantity)– Rob has 3 more books than Anne (or Anne has 3

books less than Rob) (relation)) ( )– How may books does each one have?

• Multiplicative relations– Rob and Anne have together 15 books (quantity)– Rob has twice as many books as Anne (or Anne has

A +3 =R 15

Rob has twice as many books as Anne (or Anne has half the books that Rob has) (relation)

– How may books does each one have? A R





Modelling and mathematics

• Children understand how to use numbers to represent and think about (extensive) quantities relatively earlyand think about (extensive) quantities relatively early

• In order to use mathematical models, students need to be able to use numbers to represent relations

• One of the problems reported in the literature is that students may know how to carry out a calculation but not when to use itwhen to use it

• Arithmetic is about carrying out calculations; modelling is about how to use numbers to represent relations and make predictions using these representations

School mathematics

• One of the models that students have considerable difficulty with is proportionalconsiderable difficulty with is proportional reasoning

• Many studies have documented this difficulty





Karplus & Karplus

Mr Short has a friend Mr Tall. When we measure their heights with matchsticks:

Mr Short’s height is 4 matchsticks

This is Mr Short’s measure in paper-clips (6). How many paper-clips are needed for Mr Tall’s height?

g

Mr Tall’s height is 6 matchsticks

How many paper-clips are needed for Mr Tall’s height?

Hart et al. (N=2257) ( )

Ages 12/13 13/14 14/15)

% correct: 28 30 42

% answered 8: 51 51 39

Street mathematics

• Many Brazilian children and adults have yto use reasoning about proportions in their everyday lives

• They did not attend school long enough to be taught about proportions

• They do not make mistakes in choosing the correct operation when solving proportions problems in practice





10

20

MD (separates two lemons at a time as she counts out loud):

20

30

40

MD uses the correspondence between two lemons and 10 cruzeiros to find the total

50

60

Researcher: Here how much shrimp to you have?

Fisherman: 7 kilos.

7 kg raw 5 kg toasted

21 15R: Toasted, how much would that be?

F: 5 kilos.

R: If I wanted 15 kilos of

21 15

Scalar reasoning: relations within quantities are represented (5x3) and those between quantities remain implicit

roasted shrimp, how much would you have to fish?

F: 21, isn’t it?Functional reasoning represents relations between quantities (5x1.4); scalar relations remain implicit





• Problem: There is a type of oyster in the south that gives 3 kilos of shelled oyster for every 12 kilos that you catch; how many kilos do you have to catch for a customer who wants 10have to catch for a customer who wants 10 kilos of shelled oyster?

• Fisherman: Forty. (Int: How did you get it so quickly?) It’s because we make it simpler than using pencil. … It’s because 12 kilos give 3; 36 give 9. Then I add 4 to give the 10 you want.give 9. Then I add 4 to give the 10 you want.

• Note the reasoning by correspondences

The nature of informal reasoning

• An analysis of what people do and say when they solve proportions problemswhen they solve proportions problems outside school shows that they represent quantities explicitly

• They use correspondence reasoning to solve the problemssolve the problems





One-to-many correspondences

• Piaget started the investigations on dcorrespondences

Take one red flower for each vase

Take one yellow flower for each vase





In each straw you can only fit one flower. Pick up the exact number of straws you need to put all the flowers into a straw.

Two flowers per vase.

To take the same number of tubes, I will take two tubes per vase.

Children who understood one-to-one correspondence also understood one-to-two; it took a bit longer to understand other ratios.





Sharing units of different value

• Bryant and colleagues asked children to share fairly single and double sweets to two bears

• The single sweets were identifiable by colour on some trials and not so on other trials







• The majority of 5 year olds can share fairly th t h th it f diff tthe sweets when the units are of different value even if they are of the same colour

• When the units are of a different colour, some 4-year-olds succeed in the task but it is not an easy taskeasy task


• The same logical move can be made diffi lt i b h i thmore difficult or easier by changing the

objects and how the double units are presented

• Counting double units that are perceptually double is easier than counting double units that look like singles





Solving multiplication problems

• Can young children use correspondence i l t l i lreasoning also to solve numerical

problems?

In each house in this street live 3 dogs. Write down the number of dogs that live in this street.





In each hutch there are four rabbits. All the rabbits will eat together in the house where the rabbits eat. Bring to the big house the right number of food pellets to give one to each rabbit. Kornilaki (1999)

4

4





4

Percentage of children picking up the correct number of pellets (N=120)

Kornilaki, 1999





A problem with paper and pencil

• We asked children to solve this problem: in each house live four rabbits; draw the number of carrot biscuits youlive four rabbits; draw the number of carrot biscuits you need to give one to each rabbit

Nunes & Bryant, 2000





Nunes & Bryant, 2000

Percentage of children drawing the correct number of biscuits

67

Nunes & Bryant, 2000 & Watanabe et al, 2000

Individual differences in these tasks when children start school.





Do the individual differences matter for children’s learning?

• A longitudinal study– 59 children, mean age 6 years, assessed in

their first year in school– Correspondence and sharing: predictor– Outcome measure: SATs-Maths (14 months

later)

Age

Year 1 SATs Maths – Year 2

.025

Maths Achievement

BAS score

Number skills

Working Memory

.315

.210

.054

.312One-to-many

correspondences

The strength of the different measures as predictors of Key Stage 1 results





Can young children be taught to use correspondences to solve problems?

• 32 children in Year 1 (age range 4y7m to 5y7m)

• assigned randomly to intervention or control group – 2 sessions of about 45 minutes each

• intervention group solved problems by using correspondences and had representations for both variables

t l l d i l l i bl• control group solved visual analysis problems

• Pre-test, immediate post-test, delayed post-test

• Additive and multiplicative problems were included

The children improved on multiplicative reasoning problems and continued to solve additive problems appropriately.





Conclusions

• Much effort is put into teaching children ith tiarithmetic

• It would also be important to focus attention on children’s capacity to represent relations in order to model situations using numbers

Conclusions

• Addition and subtraction are modelled by joining and separating actionsp g

• Multiplication and division are modelled by correspondences

• Schools tend to spend much time helping children model additive relations through joining and separatingand separating

• Comparatively less time, if any, is spent in helping children model multiplicative relations through correspondences





Further research about representing multiplicative reasoning

• Streefland (1984)

• Start children thinking about relations that they can imagine: e.g. How many steps must a man take to keep up with a giant?

• Move on to graphical representationg p p– schematic

– mathematical

– tables

x2

x3X4

the giant’s steps

the man’s steps

1 2

4 8

x2

x3

x3X4

3 4

12 ?





Zhang, Xin, and Si (2011):

• Worked with three children with MD • Used a single-subject case study, teaching the children

about double-counting as a strategy and assessing the g gy geffect of this teaching

• Example of teaching method:T: Let me suggest the following. I’ll give you my fingers for

every tower. Okay? Every time we have a tower that’s [holds up one finger] one tower of four—how many do we now have? You can use your fingers for [counting] the four o a e ou ca use you ge s o [cou g] e ou[cubes].

C: So 4.T: What if I brought another tower [raises a second finger]?C: 8.

Zhang, Xin, and Si (2011): working with children with MD taught DC

T: What if I brought another tower [raises a third finger]?C: 12? [doesn’t look confident] No wait.T: You can use your fingers to figure it out.T: You can use your fingers to figure it out.C: [counts under her breath] 16? No!T: So we had 4—now let’s use your fingers [shows counting

on from 4 on his other hand] 5–6–7–8. And then, 9–10–11–12.

C: Yeah.T: So now we have 3 [towers], what if we added another one

[raises a fourth finger]?C: 16.T: OK another one.C: [counts on her fingers under the table] 17–18–19–20.





Example of results with one of the case study children

Conclusions

• There are many ways of teaching children about one-to-many correspondences

• It may start with insights related to understanding objects that should count as more than one unit

• This understanding can be facilitated by making the composite objects (doublemaking the composite objects (double sweets) perceptually different

• But it is important to move children on to more abstract representations (2p coins)





Conclusions

• Correspondences between variables can be represented with concrete materials (cut-out shapes, fingers)

• Double counting instead of repeated addition forgetting the other variable promotes achievementCorrespondences can also be represented• Correspondences can also be represented with schematic drawings

• Moving on to graphs and tables can add to the children’s intellectual development

www.oxford.ac.uk

Department of EducationResearch

Child Learning

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2011 key understandings in mathematics learning no films