IMPACT OF CHANGES IN TEACHING STRATEGIES ON HOW TEACHERS WORK WITH A TEXTBOOK Jarmila Novotná and Petr Eisenmann Charles University in Prague,...

IMPACT OF CHANGES IN TEACHING STRATEGIES ON HOW TEACHERS

WORK WITH A TEXTBOOK

Jarmila Novotná and Petr Eisenmann

Charles University in Prague, [email protected] Evangelista Purkyně University, [email protected]

ICMT-2014, 29-31 July 2014, University of Southampton, UK

Acknowledgement:Acknowledgement: The research was supported by the project GAČR P407/12/1939

ProgrammeProgramme

IntroductionIntroduction Theoretical BackgroundTheoretical Background

– TextbooksTextbooks– Heuristic strategiesHeuristic strategies

Our researchOur research Discussion and resultsDiscussion and results Concluding remarksConcluding remarks




IntroductionIntroduction

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.


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.


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.


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.




Research backgroundResearch backgroundTextbooksTextbooks

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 backgroundResearch backgroundHeuristic strategiesHeuristic strategies

Strategies that pupils use to solve problems in another way than using school algorithms (Polya, 1973), (Schoenfeld, 1985).

12 heuristic solving strategies:•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.




Our researchOur researchResearch questionsResearch 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 researchOur researchResearch methodologyResearch 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 researchOur researchData collectionData 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.




Discussion and resultsDiscussion 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.


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.


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


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.


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 self-study 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.”



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.”



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.


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


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.”


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.


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


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.”


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.).”




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 attentionThank you for your attention

Acknowledgement: The research was supported by the project GAČR P407/12/1939

Our researchOur 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.



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.



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.



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.



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.



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


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.”

IMPACT OF CHANGES IN TEACHING STRATEGIES ON HOW TEACHERS WORK WITH A TEXTBOOK Jarmila Novotná and Petr Eisenmann Charles University in Prague,...

Documents

introduction textbooks

research discussion

use of mathematics textbooks

mathematics textbooks

textbooks karp

nordic network of research

results discussion

teachers work