IMPACT OF CHANGES IN TEACHING
STRATEGIES ON HOW TEACHERS WORK
WITH A TEXTBOOK
Jarmila Novotná and Petr Eisenmann
Charles University in Prague, jarmila.novotna@pedf.cuni.cz
Jan Evangelista Purkyně University, Petr.Eisenmann@ujep.cz
ICMT-2014, 29-31 July 2014, University of Southampton, UK
Acknowledgement: The research was supported by the project GAČR P407/12/1939
Programme
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
Introduction
Theoretical Background
– Textbooks
– Heuristic strategies
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

Our research
Discussion and results
Concluding remarks
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
Introduction
Theoretical Background
– Textbooks
– Heuristic strategies



Our research
Discussion and results
Concluding remarks
Introduction
Textbooks - one of the basic teachers’ instruments in
planning and conducting their lessons.
• Baldwin and Baldwin (1992): In Canada, teachers
were using textbooks 70 to 90% of the teaching
time.
• Askew et al. (2010): According to TIMSS 2007
data, 65% of 5th grade math teachers and 60% of
9th grade teachers work with the textbook most of
the teaching time.
Introduction
Textbooks in the center of attention of teachers,
educators and researchers for a long time
• e.g. (Triantafillou, Spiliotopoulou, Potari, 2013),
(Veilande, 2014), Nordic Network of Research on
Mathematics Textbooks.
Three areas of the research on mathematics
textbooks (Rezat and Sträßer, 2014):
• research that focuses on the mathematics textbook
itself,
• research on the use of mathematics textbooks,
• research on the impact of mathematics textbooks.
Introduction
Textbooks in the center of attention of teachers,
educators and researchers for a long time
• e.g. (Triantafillou, Spiliotopoulou, Potari, 2013),
(Veilande, 2014), Nordic Network of Research on
Mathematics Textbooks.
Three areas of the research on mathematics
textbooks (Rezat and Sträßer, 2014):
• research that focuses on the mathematics textbook
itself,
• research on the use of mathematics textbooks,
• research on the impact of mathematics textbooks.
Introduction
Longitudinal research project Development of culture
of problem solving in mathematics in Czech schools:
focuses on improving culture of problem solving
by pupils through the use of various heuristic
solving strategies
This contribution:
• classifies how teachers work with textbooks in
mathematics lessons
• follows the changes of approaches in the use of
mathematics textbooks in the classroom by
teachers participating in the research.


Introduction
Theoretical Background
– Textbooks
– Heuristic strategies



Our research
Discussion and results
Concluding remarks
Research background
Textbooks
Mathematics lessons should develop pupils’ creativity
and independence in their search for suitable solving
strategies.
Little or no attention paid to whether or how this is
supported by textbooks (Karp, 2013)
Research background
Heuristic strategies
Strategies that pupils use to solve problems in another
way than using school algorithms (Polya, 1973),
(Schoenfeld, 1985).
12 heuristic solving strategies:
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Guess – Check – Revise,
Systematic experimentation,
Use of false assumption,
Graphical representation – Solution drawing,
Introduction of auxiliary element,
Working backwards,
Generalization and specification,
Specification and generalization,
Problem reformulation,
Decomposition into simpler cases,
Omitting a condition,
Analogy.


Introduction
Theoretical Background
– Textbooks
– Heuristic strategies



Our research
Discussion and results
Concluding remarks
Our research
Research questions
• Do the selected textbooks use heuristic strategies
in problem solving? If so, to which extent?
• How do teachers use textbooks in mathematics
lessons and how was a teacher’s approach to
textbooks influenced by long-term experimental
inclusion of heuristic strategies into their lessons?
Our research
Research methodology
Questionnaire and in-depth structured interviews
used for investigating the original and the changed
teachers’ attitude after their participation in the
experiment.
Respondents: 11 teachers participating in the
research experiment
• Throughout the period of the experiment,
teachers used problems designed for the
experiment and encouraged their pupils to solve
them using a heuristic strategy.
Our research
Data collection
342 pupils aged 12-19 from 11 lower and upper
secondary school teachers in the Czech Republic.
• Classes selected with the intention of having a
variety of classrooms as far as geographical
position, specialization and pupils’ intellectual
levels are concerned.
• No special training of teachers
Questionnaire survey: sent out and collected by
email
Interviews conducted by the researchers.


Introduction
Theoretical Background
– Textbooks
– Heuristic strategies



Our research
Discussion and results
Concluding remarks
Discussion and results
Implementation of heuristic strategies and textbook
problems supporting their use
Several mathematics textbooks for lower secondary school
widely used in the Czech Republic:
• Most of these textbooks do not work with heuristic
strategies.
• The textbooks are based on problems and tasks whose aim
is to practice and drill selected parts of mathematics
through school algorithm strategies.
• Very little attention is paid to development of pupils’
creativity.
Discussion and results
Implementation of heuristic strategies and textbook
problems supporting their use
Causes of this situation:
• Textbooks are usually designed for teachers’ practice,
therefore they are guided by the prevailing teachers’
demands; heuristic solving strategies are not commonly
used by Czech teachers of mathematics.
• Non-algorithmic character of heuristic strategies often
requires a lot of additional explanations, which exceeds the
required scope of the textbook (it would be too thick).
• There is the potential danger that misuse of heuristic
strategies will make them just another algorithmic
procedure.
Discussion and results
Use of textbooks when teaching mathematics
Classification (ignores the case when the textbook is used only
for pupils’ self-study):
The teacher
• uses the textbook only when planning the lesson but not in
the lesson,
• supplements its content with his/her own material or
modifies the content,
• works exclusively from the textbook.
Most common: use the textbook together with a collection of
problems
Discussion and results
Use of textbooks when teaching mathematics
• Often, teachers modify the problems to meet the needs of
the particular group of pupils (by changing the context,
simplifying it, reformulating the question etc.)
• They use different collections, pose their own problems, look
for problems on the internet etc.
• The textbook is a guide for making a cascade of new topics
of increasing difficulty.
Discussion and results
Use of textbooks when teaching mathematics Comments
Teacher 1: “Like my colleagues, I use Czech textbooks as a
collection of problems. I also use collections of problems. I
don’t think current Czech textbooks are good to be used with
pupils – they are very academic, austere, unsuitable for selfstudy for average and below average pupils. There aren’t
enough problems, not enough types of solutions, there aren’t
many applications. They offer the teacher no extra service –
there are no methodological teacher’s books with solutions of
problems, suitable methodology, extra materials etc. Teachers
and pupils would appreciate if there was this service offered in
the textbook.”
Discussion and results
Use of textbooks when teaching mathematics Comments
Teacher 2:
“I’m afraid textbooks available on our market don’t give much
space for use of various solving strategies. There are not many problems
supporting reasoning. I know that even the simplest reasoning is hard to
explain in writing. Textbooks most often contain problems asking for precise
mathematical reasoning. It’s up to the teacher to explain different solving
strategies. But this requires a lot of experience and ability to improvise, which
is very hard to do as a fresh graduate – beginning teachers are more likely to
insist their pupils use prescribed procedures. They are more textbook-bound.
… The aim of textbook authors is to have the texts mathematically precise.
This makes the solving procedure more difficult for pupils than they would be
in everyday life. That’s comprehensible – pupils must master certain
procedures and algorithms. For example in case of the rule of three, most
problems could be solved by reasoning. But the rule of three must be
explained as e.g. in chemistry reasoning would not do and pupils must know
the mechanical procedure.”
Discussion and results
Use of textbooks when teaching mathematics Comments
Teacher 3: “The textbooks I have contain quite a lot of real-life
and application problems but they are very artificial. My pupils
are quite good and refuse to calculate nonsense. Some of the
strategies are nice and really usable in real-life, e.g.
experimenting. But there aren’t many problems of this type.”
Most teachers do it in such a way that pupils work with the
textbook mainly when practicing.
Discussion and results
What teachers expect from a textbook:
• Large number of routine problems, short assignments,
instructive problems, sample solutions etc.
• Real-life problems (preferably not “artificial”) and problems
that can be solved using several procedures (including
heuristic strategies) that leave space for pupils’ individual
discovery.
Not many teachers who support pupils’ independent use of
heuristic strategies
Discussion and results
What teachers expect from a textbook - Comment
Teacher 2: “I do not evaluate contemporary Czech secondary
textbooks as suitable for the use by pupils - they are very
academic, unsuitable for self-study of average and weak
pupils. They do not contain enough problems, ideas for solving
procedures, applications. They do not offer the teacher any
service - there are no manuals with results, methodology,
follow-up activities etc. I would prefer textbooks with wider
services for teachers and pupils.”
Discussion and results
Changes in the approach to textbook use in case of
the teachers involved in the experiment
The success in changing students’ relationship to solving
problems requires:
• deep teachers’ involvement in realisation of the designed
activities
• active involvement in the project design.
This change of teachers’ role goes hand in hand with the
change of their pedagogical approaches and beliefs.
Discussion and results
Changes in the approach to textbook use in case of
the teachers involved in the experiment
In-depth interviews with the participating teachers:
• changes in teachers’ approaches to the use of
textbooks are independent of their approaches
before the experiment.
• shift towards creativity in the way teachers use
their textbooks (not instantaneous) 
considerable impact on pupils’ attitudes to
mathematics and problem solving
• significant increase in the teachers’ autonomy
Discussion and results
Changes in the approach to textbook use in case of
the teachers involved in the experiment - Comments
Teacher 2: “I used to insist on accurate recording of a problem,
solution usually using equations, in physics for example first a
general solution in which the unknown from a formula was
expressed, followed by substitution. Today I see that if I insist
on theoretical procedure, my pupils cannot see the general
sense in the problem. They focus on formulas and learned
algorithms and do not reason. I now appreciate if they solve
the problem anyhow. I emphasize simple reasoning. … The
goal of my lessons should be to teach the children to think not
to reiterate known algorithms.”
Discussion and results
Changes in the approach to textbook use in case of
the teachers involved in the experiment - Comments
Teacher 3: “Soon I started to select deliberately those
problems (in collections of problems) that enabled my pupils to
practice a selected strategy. I also started to pose problems
based on a good model problem from a textbook. Thus I
created sets of related problems different in parameters or
difficulty, sometimes even context. Sometimes I managed to
engage those pupils who had finished earlier than the rest of
the class in posing new problems. … I’m now more than before
annoyed by problems that ask absolutely stupid questions (for
example how many chickens and goats run somewhere if we
see 22 legs etc.).”


Introduction
Theoretical Background
– Textbooks
– Heuristic strategies



Our research
Discussion and results
Concluding remarks
Concluding remarks
• Textbook problems very often designed to support
application of algorithmic strategies
• Use textbooks for development of heuristic
strategies requires a very active work with textbooks
• Experiment showed changes in the participating
teachers’ approaches to the use of mathematics
textbooks in mathematics classrooms (consequence
of experimental teaching):
• Shift towards creativity in teachers’ approaches to
the use of textbook.
Importance for pre- and in-service teacher training
Thank you for your attention
Acknowledgement: The research was supported by the project
GAČR P407/12/1939
Our research
Studied heuristic strategies
Guess – check – revise: This is a strategy in which we
first, drawing from our experience, make a guess about
the solution to the given problem. Then we check
whether the solution meets the conditions of the
assignment. The next guess is made with respect to the
previous result. We carry on in this way until we find a
solution.
Our research
Studied heuristic strategies
Systematic experimenting: Systematic experimenting is
a strategy in which we try to find the solution to a
problem using several experiments. First we apply some
algorithm that we hope will help us solve the problem.
Then we proceed in a systematic way and change the
input values of the algorithm until we find the correct
solution.
Our research
Studied heuristic strategies
Working backwards: This is a very common strategy in
mathematics. We assume that what we have to
find/prove/construct holds/exists. Then we try to
deduce from this assumption something we already
know or something that is easy to
prove/calculate/construct. Thus we in fact try to get
from the end to the starting situation as close as
possible. The procedure is reverted in the final
calculation/proof/construction.
Our research
Studied heuristic strategies
Introduction of an auxiliary element: When we use this
strategy we try to transform a given problem to a
problem we have already managed to solve, or we
transform it into a simpler problem we are able to solve.
An example of an auxiliary element in problems in
geometry is e.g. introduction of straight line or line
segment, but it can also be a more complex geometrical
figure. In algebra, we often introduce a new variable.
Our research
Studied heuristic strategies
Omitting a condition: Problem assignment often
involves several conditions. If we are not able to fulfil
all these conditions when solving the problem at once,
we can ask: What is it that makes the solution of this
problem so difficult? If we manage to identify which of
the initial conditions is the difficult one, we can try to
omit it. If we are then able to solve the simplified
problem, we can go back to the omitted condition and
try to finish solution of the original problem.
Our research
Studied heuristic strategies
 Guess – check – revise, Systematic experimentation,
Working backwards: strategies of algorithmic nature;
pupils can use them successfully even if they do not
have very good insight into the structure of the
problem; use of these strategies does not always ask
for very active involvement of pupils’ creativity.
 Introduction of an auxiliary element, Omitting a
condition: require creative activity from the solver
and depend on the solved problem
Discussion and results
What teachers expect from a textbook - Comment
Teacher 2: “I teach in a way that textbooks are used for assignment
of problems and it’s up to my pupils to look for the solution. If it’s
something new, I don’t follow explanations as they’re presented in
the textbook but I explain it my way. Sometimes I use several
procedures and strategies. Sometimes (very rarely) I look at the
presented solving procedure if it’s something I can’t explain easily. …
I often explain several strategies – somebody visualizes the situation,
other pupils need an illustrative drawing, somebody needs analogy.
My aim is that everybody should find a procedure they understand. I
welcome all pupils’ procedures; they explain them at the blackboard.
However, there’s a group of pupils who don’t like this variety of
several procedures. They only want one which they learn. More
possible strategies make it more difficult for them.”