BOOK REVIEW

Thomas P. Carpenter, Megan Loef Franke and Linda Levi (2003).

Thinking mathematically: Integrating arithmetic and algebra in the ele-

mentary school. Portsmouth, NH: Heinemann. ISBN 0-325-00565-6.

INTRODUCTION

‘Thinking Mathematically’ is both one of the most exciting books for

primary school teachers that I have read recently, and in some ways

one of the most frustrating. As the title suggests, the book addresses a

way in which it is possible to work with children in elementary school

on the inter-related nature of arithmetic and its generalisation into

algebra.

The book is a development of the ideas from the authors’ (Carpen-

ter et al., 1999) previous book Children’s Mathematics written in con-

junction with Elizabeth Fennema and Susan B. Empson. In that first

text the reader was introduced to Cognitively Guided Instruction as a

method of working with elementary children on problems solving,

mathematical communication, and teaching for understanding. ‘Think-

ing Mathematically’ takes these ideas, and the children, further, into

the world of generalised arithmetic and algebra. For readers of ‘Chil-

dren Mathematics’ the format will be familiar. The text is clearly set

out, with shaded areas containing examples and quotations from the

teachers who were involved in the project, and a CD is included con-

taining video clips of class teaching and individual interviews with chil-

dren explaining their mathematical thinking.

It starts from the premise that ‘for many students and adults, arith-

metic represents a collection of unrelated and arbitrary manipulations

of numbers and equations and algebra is perceived as a separate col-

lection of meaningless procedures that are only tangentially related to

arithmetic’ (p. 133). Carpenter, Franke and Levi draw on their work

over five years with a group of elementary school teachers and chil-

dren in the United States. The study has focused on encouraging chil-

dren to think more fluently about the arithmetic they encounter in

elementary school. Their intention is that through reading this book,

and using the related CD, teachers will be enabled to develop the

Journal of Mathematics Teacher Education 7: 383–391, 2004.

� 2004 Kluwer Academic Publishers. Printed in the Netherlands.

understanding and skills to carry out such teaching in their own class-

rooms.

THE CONTENT OF THE BOOK

The content is exciting for those of us who believe that even very

young children are able to think logically and mathematically when

given the right contexts. Following an introduction in Chapter 1 which

discussed the purpose and structure of the book, Chapter 2 offers ideas

for the introduction and development of the concept of equality. This

emphasises the need to teach children that the equals sign indicates

equality on either side rather than being an instruction to carry out a

calculation and write an answer. A set of benchmarks describe the

stages children may pass through in their development of this concept,

which could be used by teachers to evaluate their children’s responses.

The causes of misconceptions about the equals sign are discussed and

ways of teaching that avoid or remediate such misconceptions are sug-

gested. For example, they suggest that not all equations are written in

the form 3 + 5 =h, since children then interpret the equals sign as

an instruction to add, but to consider equations such as 3 + ? ¼ 8,

3 + 5 ¼ 6 + 4, etc. They also suggest that teachers avoid the use of

the equals sign ‘as a shorthand for a variety of purposes that do not

represent a relation between numbers’, such as J J J J J ¼ 5

(p. 20).

This discussion of children’s understanding of equality moves, in

Chapter 3, into the development and use of relational understanding

as children are asked to solve equations such as 43 + 28 ¼ h + 42

without the need to calculate. One teacher, Ms K., is shown teaching a

single pupil, Emma: together they work on a series of calculations

which move Emma from using calculation to using relational thinking

to solve such problems. The authors then discuss the use of true and

false to label equations, starting with simple statements such as 12 ) 9 ¼3, then moving on to statements which encourage the use of relational

thinking such as 27 + 48 ) 48 ¼ 27, and 54 + 17 ¼ 17 + 54.

Chapter 4 focuses on the development of conjectures as children are

asked to make their implicit knowledge explicit. At this stage the con-

jectures are articulated in everyday language e.g. ‘when you add a

number to another number and then subtract the number you added,

you get the number you started with’ (p. 54), or ‘when you add two

numbers, you can change the order of the numbers you add and you

will still get the same number’ (p. 55). A summary of conjectures

384 BOOK REVIEW

about basic properties of number operations follows, including addi-

tion and subtraction involving one, multiplication and division involv-

ing zero and one, and commutativity.

Equations with multiple and repeated variables are presented, in

Chapter 5, through contextual tasks such as the distribution of a set of

mice across two cages. While the total number of mice remains the

same, the distribution across the cages can be represented by a range

of additions: 5 + 2 ¼ 7, 1 + 6 ¼ 7 etc. and the use of symbols, such

as x + y ¼ 7, to record these variables is discussed. The authors

emphasise to the reader that ‘the primary goal in giving students these

number sentences is not to teach students efficient ways to solve alge-

braic equations; it is to engage them in thinking flexibly about number

operations and relations’ (p. 73). On the CD, children are shown solv-

ing more complex equations such as 3 · p + p + 2 ) p ¼ 17. Some,

like Susan, use trial and error, ‘I first tried 4 … That was too small so

I tried a bigger number, 6. That was too big so I tried 5…’ (p. 73).

Erika ‘is more flexible in her thinking’ arguing that ‘Well, there’s a p

and you take away a p so it’s like it was never there. So it’s like 3

times p plus 2 equals 17, and 15 and 2 is 17. So 3 times p is 15, and

that makes p 5’ (p. 74). The authors assert that the children are not

being taught solutions to such problems but encouraged to explore

and discuss relationships to create their own solutions.

The use of symbols is extended in the following chapter to represent

conjectures, leading on to early ideas on justification and proof. Levels

of justification are discussed and examples show these different forms

of justification used by the children. The children are encouraged to

produce generalisations even when they are not able to prove them.

The final chapter addresses the ordering of multiple operations and

if…, then… statements. Towards the end of this chapter the text

moves from examples of work with the children to address the tea-

cher’s own level understanding of how to prove, for example, ‘why we

invert and multiply when we divide fractions’ (p. 128).

So, the book and CD clearly show some young children, beginning in

kindergarten, engaging with these abstract ideas; starting from arithme-

tic and moving through the language of generalisation towards the use

of algebraic symbolism. As such it is an exciting and stimulating read.

However, while celebrating the clarity of the text and the use of real

examples from the classroom to further illuminate this, I had several

problems with the book. One is mainly practical, relating to the CD,

while the others are more fundamental and relating to the intended

readership.

385BOOK REVIEW

ISSUES RELATED TO THE USE THE BOOK

Using the CD in Connection with the Book

The book is supported by an excellent CD, showing examples of chil-

dren working one-to-one with a researcher and in classroom situations.

However there is an assumption that the reader will have easy access to

the material on the disc as the introduction recommends that a ‘reading

of the text would not be complete without viewing the accompanying

episodes’ (p. xii). The disc requires the use of QuickTime 5.0 (or higher)

which can be downloaded free via the internet, and easily carried out at

the university. However I found that at home, without the use of Broad-

band, the downloading was difficult, expensive and time consuming.

This would be a real disadvantage for many primary school teachers

who may not have easy access to fast internet connection at school and

might, at home, find the hassle too much and give up. If the software is

free, would it not have been possible to include it on the CD?

Also, the need to watch an extract from the disc at a specific time

meant that I, as a reader, was tied to the office – not my favourite

place for reading. While the video extract was enjoyable and would be

useful for discussion in groups, for the individual reader a fuller tran-

script of the incident in the text would have allowed great freedom of

use. This raises the question of intended readership.

Who is this Book for?

This book could be used by primary/elementary teachers themselves,

or used by teacher educators in pre-service or in-service work. The

audience for the book is clearly identified by the authors as the ele-

mentary school teacher, who is addressed throughout. In the introduc-

tion (p. xi) the authors state that ‘the goal is to help you understand

your own students’ thinking so that you can help them to make sense

of the mathematics they are learning.’ But, while recognising that

many adults do not have the relational understanding of arithmetic

and algebra described here, the authors do not give evidence that

teachers will acquire this understanding through the reading of the

book.

The inclusion on page vii of the forward of the equation

½ðk� 2Þ þ 1� þ ½ðh� 2Þ þ 1� ¼ ðkþ hÞ � 2þ 2 ¼ ðkþ hþ 1Þ � 2

which aims to describe ‘when you add an odd number to another odd

number, the answer is an even number… represented compactly’

386 BOOK REVIEW

(p. vi) would, I suspect, ensure many elementary school teachers would

immediately close the book and find something better to do.

There has been considerable work carried out in the UK, and else-

where, on student teachers’ understanding of mathematical subject

knowledge, which finds that many student teachers are particularly

insecure in their own understanding of algebraic concepts and are

often afraid of algebraic notation. For example, in England, Rowland

et al. (2000) found only 43% of the postgraduate pre-service primary

teachers at their university were secure in reasoning and mathematical

argument while in Belgium, van Dooren et al. (2003) found a signifi-

cant subgroup of pre-service primary teachers were unable to use alge-

braic methods even after 3 years of teacher education.

The authors have worked with the teachers cited in the book over a

period of time and show exciting teaching and excited children engag-

ing with issues of symbolism, justification and proof. However the

assumption that an individual primary/elementary school teacher could

engage with the ideas alone, through the reading of the book, remains

to be shown. Given the research evidence it seems likely that many

teachers, in British primary schools and perhaps in those of other

countries, would be unable, or unwilling, to engage with the mathe-

matics required to understand this book on their own.

In-service Teacher Education

Could the book therefore be used for those working with in-service

teachers? Barkai et al. (2002) studied in-service elementary teachers in

Israel – giving them statements such as ‘the sum of any five consecu-

tive integers is divisible by 5¢ – to prove or refute. They found ‘a sub-

stantial number of teachers applied inadequate methods to validate or

refute the propositions … many teachers were uncertain about the sta-

tus of the justifications they gave’ (p. 57). They conclude that ‘it seems

essential that professional development programmes attempt to

enhance elementary school teachers’ algebraic knowledge to determine

the validity of their students’ conjectures’ (p. 63–64). So, there is a

need for such in-service work: does ‘Thinking Mathematically’ provide

a resource for these programmes? The answer has to be both yes and

no!

The book provides a valuable resource for in-service teacher educa-

tion, to enable teachers to understand the powerful algebraic founda-

tions of the informal calculation methods invented by children in

their classes. With the support of teacher educators it would be pos-

sible to use the book and the CD to demonstrate what it is possible

387BOOK REVIEW

for children and teachers to do in the classroom, while at the same

time giving the teachers a range of activities to support and develop

their own algebraic confidence and competence. But for this purpose

I, as an in-service teacher educator, would require more that we are

offered here.

Continued professional development of teachers in Britain is

mostly at Master’s level and would require the teachers to under-

stand much more of the research background to the project than

we are offered here. Little is said about the research base, the num-

ber of teachers involved, the number of children, the time spent on

the project in each classroom. There is no reference at all to the

sort of research project it is. What was the teachers’ involvement?

Are they involved in the research or being researched upon (Jawor-

ski, 2003)? There is no data offered to indicate the success of the

project other than by the descriptive accounts given in the text. The

reader is given the impression that all teachers and all children were

able to benefit from this approach but there is no hard evidence of

this. Were there teachers who dropped out of the project? Were

there children who could not form conjectures or understand those

formed by their peers? In many of the CD excerpts we see one

child answering confidently, but what of the rest of the class? With-

out the answers to these questions we have to take the evidence of

the book on trust.

There is also very little reference to research literature. Only three

references are given in the entire book; one of these to the National

Council of Teachers of Mathematics (2000) ‘Principles and Stan-

dards for School Mathematics’. This is in stark contrast to the for-

mat of the authors’ previous work, ‘Children’s Mathematics’, which

contains a very useful appendix on ‘The Research Base for Cogni-

tively Guided Instruction’ supported by 22 references to published

articles on the subject. Children’s Mathematics was also available

with a Workshop Leader’s guide for Professional Development pro-

grammes. Perhaps these resources are being developed. If so they

could, I believe, be a valuable resource for mathematics education

community.

Preservice Teacher Education?

My initial response to the question of the book’s relevance to initial

teacher education was a negative one. Could the students I work with

engage with this material in a meaningful way? I doubted it. However

I also heard myself thinking – How do you know? What evidence do

388 BOOK REVIEW

you have? These questions gave direct rise to the development of a ses-

sion on one of our pre-service programmes.

The students on this programme are all teaching assistants in

schools, studying part-time to gain qualified teacher status. As such

they are generally mature students, with non-standard entry qualifica-

tions and many entered the programme with serious insecurities and

negative attitudes to mathematics. This is a four year programme and

the students with whom I was working are in their third year: 50 stu-

dents, organised into two teaching groups. They have generally strug-

gled with the mathematics at their own level, although, since they have

considerable experience of working with children in school, often sup-

porting children with learning difficulties, their understanding of chil-

dren’s learning is very good.

The session was originally planned to address early algebra in

terms of number patterns and incomplete number sentences of the

form 7 + h ¼ 12. However, following my reading of the book, and

in discussion with a colleague, we decided to try out some of the

ideas in the book. Starting from concepts of equality, then adding 0

to any number, the students developed conjectures and wrote alge-

braic equations for these. They created equations for the commuta-

tive and associative rules and recognised these as rules they had

learnt in their first year course. For example, starting from a given

addition e.g.

56þ 27 ¼ 83

they talked about how they solved

56þ 28 ¼

then generalised this verbally e.g. ‘if you add two numbers together to

get a total, when one is added to the second number the total will be

one more’, and were able to write an equation such as:

xþ ðyþ 1Þ ¼ ðxþ yÞ þ 1

For some students this gave new insight into the use of brackets. For

the first time they realised why they needed the brackets, to represent

their spoken language. They then realised that this did not only apply

to +1 but to the addition of any number resulting in

xþ ðyþ zÞ ¼ ðxþ yÞ þ z

389BOOK REVIEW

which they recognised as the associative law for addition. They

expressed amazement that this law, which they had learnt but not

really understood earlier in the programme, was relevant to aspects of

the mathematics they taught in school, and also that they had been

able to generate the algebraic equation for it so easily.

We also discussed what age the children they were teaching would

be able to engage with the ideas and they were very vocal about the

relevance to the children in their classes (aged 4–11 years). In the

National Numeracy Strategy (DfEE, 1999), which they use to plan

and teach in school, the concept of using the answer to one calculation

to find the answer to a similar problem is explicitly taught. But the use

of an explicit conjecture to form a generalisation for this understand-

ing was new to them. Many students commented on how being

encouraged to express the generalisation in words increased their own

understanding and would be useful with children.

At the end of the session almost all the students said that this had

been one of the best mathematics lessons they had ever had. It had

allowed them to make some early links between arithmetic and algebra

which they had not previously understood, and they were enthusiastic

about the way in which these ideas could encourage children to suc-

ceed in mathematics at a higher level, in a way in which they them-

selves had not been to in secondary school. This session, while not

specifically using the text or video clips from the book, addressed the

basic concept of integrating arithmetic and algebra at an elementary

school mathematics level, and demonstrated that, for these students at

least, the integration was helpful to improve their own understanding

of the arithmetic ideas and the progression into algebraic ideas. Fur-

thermore, they could see the potential for this integration when work-

ing with elementary aged children. In future sessions I would wish to

use the text and the video clips more explicitly to show how this

potential could be realised.

This is, for me, the beginning of new ideas for working with all our

students. Will it work for the one year Postgraduate Certificate in

Education students? What about the more traditional BA students,

many of whom come straight from school with less experience of chil-

dren’s learning? These are things I will have to explore, and must leave

the reader to explore in their own context.

I have therefore come to the conclusion that the book, despite its

shortcomings, provides much that will be of interest to those

involved in mathematics teacher education at primary/elementary

level, and look forward to seeing how these ideas can be used in

390 BOOK REVIEW

teacher education and in schools to increase children’s mathematical

understanding.

REFERENCES

Barkai, R., Tsamir, P., Tirosh, D. & Dreyfus, T. (2002). Proving or refuting arithme-

tic claims: the case of elementary school teachers. PME 26 proceedings, Vol. 2

(pp. 57–64), Norwich, UK.

Carpenter, T., Fennema, E., Loel Franke, M., Lebi, L. & Empson, S. (1999). Chil-

dren’s Mathematics: Cognitively Guided Instruction. London: Heinemann.

DfEE (1999). The National Numeracy Strategy: Framework for teaching mathematics

from Reception to Year 6. London: DfEE/ CUP.

Jaworski, B. (2003). Research practice into/influencing mathematics teaching and

learning development: Towards a theoretical framework based on co-learning part-

nerships. Educational Studies in Mathematics, 54(2&3), 249–282.

National Council of Teachers of Mathematics (NCTM) (2000). Principles and Stan-

dards for school mathematics. Reston, VA: NCTM.

Rowland, T., Martyn, S., Barber, P. & Heal, C. (2000). Primary teacher trainees’

mathematical subject knowledge and classroom performance. In T. Rowland &

C. Morgan (Eds), Research in mathematics education, Vol. 2, London: British Soci-

ety for Research into Learning Mathematics.

van Dooren, W., Verschaffel, L. & Onghena, P. (2003). Preservice teachers’ preferred

strategies for solving arithmetic and algebra word problems. Journal of Mathemat-

ics Teacher Education 6(1), pp. 27–52.

Oxford Brookes University Alison PriceHarcourt Hill CampusOxford, OX2 9ATUK

391BOOK REVIEW