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Philosophy of the Course
General philosophy of the course EMTISM - Empowering Mathematics Teachers for the Improvement of School MathematicsMilan Hejný
1. What is the purpose and goal of our course?We perceive the teaching of mathematics as an opening of the world of mathematics to students. First it is important to create an enjoyable and stimulating climate which motivates students towards solving problems, discussions, creating and verifying hypotheses, looking for arguments and counter-arguments, etc. Through these activities, each student constructs his/her own mathematical world. A teacher who leads students in this constructivistic way makes a far greater effort than a teacher who tries to transmit ready-made mathematical results (theorems, formulas, definition, schemes,.) and who lets students practise algorithms and solve standard problems. A teacher's constructivistic approach towards students is more demanding for him/her, but far more effective, and brings him/her greater joy from work and better relationship with his/her students. In view of this, the goal of our course is to create a stimulating work climate and inspire practising teachers to independently gain knowledge through experiments. We have had rich experience of co-operating with practising teachers. We have often co-operated with them in our research, in preparations, realisations, analyses and evaluations of experiments. Each time, at least some of the practising teachers experienced the same surprises, joy from discoveries and disappointments from failures. Many of them stated that the experience gained during our co-operation helped them to find a different, more enjoyable and effective approach to their teaching strategies. We would hope that our course participants have a similar experience. The idea of co-operation will be elaborated in three dimensions.
1.1 Dimension of relationship (dimension of attitude)This level concerns a teacher's relationship to mathematics and students. We are convinced that the quality of a teacher's work depends on this level. A teacher who thinks about his/her work, who thinks about his/her lessons deeply, who looks forward to his/her lessons, not only has good pedagogical results, but also gains a personal satisfaction. A teacher who works in a routine way, who does not anticipate surprise which the next lesson might bring to him/her, cannot address students by the strength of his/her personality and cannot experience his/her own pedagogical work. In our pedagogical work, we have noticed many times that many teachers, after five or seven years of work at school, live through something which we could call "greying of the school reality". With nostalgia, they remember their "first students", the first years of their teaching, when they made mistakes but on the other hand, they were able to experience lessons as adventures, as unrepeatable events which they wrote about in their pedagogical diary together with their feelings. One of them aptly compared his first four years at school to love which subsided and he perceived it as a fact obtained by destiny. However, when he joined our research and began to ask himself more difficult and deeper questions about his work, when he began to discuss his opinions, something happened which he did not expect - he re-discovered his desire to discover something new and began to work on the development of his personality. The first and most important goal of our course is to address the teacher, encourage his/her both pedagogical and mathematical confidence and enhance his/her resolution to start working with more vigour, expectation and joy.
1.2 Dimension of methodologyThis level concerns ways in which a teacher can lead his/her students to gain knowledge. A poet Khalil Gibran described the basis of a teacher's work in a poetic and apt way in The Prophet: "If he is really prudent, he does not exhort you to come to the home of his wisdom, but rather leads you to the threshold of your own thinking". Forty years ago, the community of methodists believed that the main task of methodology was to find an ideal way of introducing fractions, areas, combinatorics, word problems, etc. Today, we know that this conception left out an important reality of a student's individuality. It does not take into account that what is clear and intelligible for one, can be unclear and confusing for another. With the coming of constructivism, the idea that the basis of teaching is the transmittance of knowledge, i.e. a teacher presents students with knowledge and students acquire it, was dropped. We are convinced that real mathematical knowledge must be constructed by each student him/herself with the teacher's help by leading them along a well thought out sequence of problems, questions, challenges and provocation. On the level of methodology, we will aim our attention at the process of gaining knowledge for both the teacher and the student, on its characteristics, possible deformations and re-educational procedures. We have experience in this area which we will present as input theses in discussions and as challenges to participants' own experimental activities.
1.3 Dimension of contentThis level concerns four different topics which we have chosen for the four offered modules. They are elaborated in detail in the following chapters. Here, we will emphasise one aspect only. Each topic enables work on various levels of mathematical knowledge. It means that both an elementary teacher and a teacher at a mathematical secondary school will find an adequate level of difficulty.
2. The process of gaining knowledgeThe process by which a student gains knowledge is described in detail in literature. For a teacher, such a description is beneficial provided on one hand it is comprehensible enough and on the other hand it would enable him/her to diagnose the quality of a student's mathematical knowledge and help him/her effectively. In this sense, it is important to distinguish between formal and real knowledge.
2.1 Formal and real knowledgeIllustration 1. A student, grade 7, had to find the area of a triangle drawn on paper. He did not know what to do. The teacher asked him if he knew the formula for the area of a triangle. The student wrote immediately S = h×a/2 but he did not know how to use it. The teacher advised him to choose one side, construct the corresponding height and measure the lengths of both. It did not help either. The student did not know what a height was. This illustration shows what we mean by formal knowledge. The student has a formula (definition, theorem) in his/her mind but he/she does not understand what it means of how to use it. The opposite of formal knowledge is real knowledge when a student understands the individual parts of a piece of knowledge, their mutual connections and their place in different contexts. Terms formal versus real represent the endpoints of a scale. The reality is usually somewhere between these poles. For instance, the student in the illustration knows what a side of a triangle is, therefore, he has some knowledge but at a low level.
2.2 The process of gaining knowledge as a sequence of five stagesOur model of the process of gaining knowledge is based on stages. It starts with motivation and its core is two mental lifts: The first leads from concrete knowledge to generic knowledge and the second to abstract knowledge. The permanent part of gaining knowledge is crystallisation, i.e. inserting new knowledge into the structure. The whole process can be described by a scheme.
Illustration 2. The story of five-year-old Anka illustrates five stages of the process of gaining knowledge.
This illustration suggests the main ideas within the process of gaining knowledge. A person first understands several concrete examples and then finds a more general and abstract piece of knowledge.
2.3 MotivationWe see motivation as a tension which appears in a person's mind as a result of the discrepancy between the existing and desired states. The motivation for gaining knowledge stems from the discrepancy between "I do not know" and "I need to know", or "I cannot do that" and "I want to be able to do that", sometimes from other needs and discrepancies, too. A child has a strong need to get to know things around him/her. A child is curious. He/she asks questions about everything which is within his/her perception. A stimulating environment enhances a child' curiosity. However, school is often not a stimulating environment and it can inhibit curiosity rather than encourage it. A child's motivation to get to know the world and the level of language is different from that of an adult. It often leads to misunderstanding between an adult and a child or a student. A child's motivation changes quickly. A child is interested in a dog, then plays with cubes, does not finish the building and goes to solve puzzles, etc. Parents try to keep the child in one activity until the end because they see the goal of the game in the resulting object. They consider that "such chaotic and non-systematic work" is undesirable. They are mistaken because the goal of the game is not the resulting object, but the activity itself which leads to the development of a child's abilities. A child remains at one activity for as long as his/her inner motivation keeps him/her. As soon as this inner tension disappears, motivation for a different activity comes along. A child's motivation is urgent. For instance, a child wants paper and scissors and requires them now, immediately. If he/she does not get what he/she needs, he/she directs his/her attention elsewhere and the original need is left unfulfilled. Adults can wait for the realisation of their motivational impulses for years. Therefore, they perceive the child's urgency as obstinacy. They think that if a child's request for the toy is fulfilled later, nothing bad can happen. Again, they are mistaken. As soon as the pertinent motivation subsides, a child loses interest in the activity. An adult then reproaches the child saying "A while ago you wanted the pram urgently and now you do not even look at it". Similar motivation of urgency often concerns students. A student who managed to solve a difficult problem is delighted and asks a teacher for other problems. The teacher is pleased and promises to bring them the next day. The next day, however, the student is not interested in them any more. The quality of the class climate in mathematical lessons can be evaluated by the measure of the students' inner motivation to gain knowledge. At one end of the scale, there are students who like mathematics. At the other end, there are students who fear mathematics and learn the algorithms to get a good mark or to appeal to their teacher. Sometimes, we meet different motivations, for instance a student who makes a noble effort to please his/her ill mother.
2.4 Isolated modelsModels of new pieces of knowledge are accumulating in the mind gradually and have a long-term perspective. For instance, the concepts of fraction, negative number, straight line, congruency or limit develop over many years on a preparatory level. For more complex knowledge, the stage of isolated models can be divided into four sub-stages. Illustration 3. A teacher shows 6-7-year-old students subtraction 6 - 2 = 4 via cubes. He places six cubes on the table, counts them and takes away two cubes from the left hand side. He counts the remaining cubes and says: "Six cubes less two cubes are four cubes." This information is perceived differently by different students. We will look at how three hypothetical students might perceive this knowledge in more detail. Ben does not listen to the teacher, he plays with his pencil. He can hear that a teacher says something but his mind is oriented elsewhere. The information does not reach his mind. Betina takes the information in but without any deep understanding. She does not understand the teacher's manipulation with the objects. She puts the statement which the teacher emphasised into her memory and a vague idea of some manipulation with objects. She elaborates it only by placing it to similar information which is already present in her mind. Boris understands everything that the teacher demonstrated. He remembers that when the teacher demonstrated the situation "six cars less two cars", he took away two cars from the right hand side. The result was the same, "four cars". The boy understood that the result of counting did not depend on the fact whether the objects were taken away from the right or from the left and he supposed that it held in other cases, too. An idea is placed in his memory which connects the previous and new experience and this discovery. There was no trace of the teacher's information in Ben's memory. Betina accepted the new information and its verbal part was placed in an appropriate part of her memory. Boris elaborated the information in such a way that he
The given illustration leads us to a more detailed decomposition of isolated models into four substages:
Determining the core of the "sameness", or perhaps it is better to say the correspondence between any two models. The models create a community. The above sub-stages can be useful for us when we investigate how a new idea gradually develops in a student's mind. It often happens that a new sub-stage, not given here, appears and that one of those presented does not appear at all. For instance, one pre-schooler understood the idea of three-digit numbers via money. She repeatedly looked at and rearranged a hundred-crown note, a twenty-crown note and a crown and created 120 crowns and 121 crowns. Each time, she placed money on the table and said the corresponding number. Via this game, she constructed the idea of a three-digit number with the help of two (not only one) models. This stage ends with the creation of the community of isolated models. In the future, other isolated models will come to a student's mind, but they will not be at the birth of the generic model. They will only differentiate it in more detail. When, for instance, Anka from illustration 2 is to find how many times the bell struck when she heard it strike first twice and then three times, it will be a new isolated model for her, but the way of counting (on fingers) will already be familiar to her.
2.5 Generic modelIn the scheme of the process of gaining knowledge, the generic model is placed over the isolated models which points to its greater universality. The generic model is created from the community of its isolated models and has two basic relationships to this community:
The first relationship denotes the construction of the generic model, the second denotes the way the model works. Illustration 4. Five-year-old Cindy wants her grandmother to buy her an ice-lolly. The grandmother agrees: "Yes, but we will buy ice-lollies for everyone. How many of them shall we buy?" Cindy: "Me, brother, mother, grandpa, grandma, father." She counts on fingers and says: "Six." In a shop, she takes the ice-lollies from the freezer, but she does not find their number by counting. She assigns ice-lollies to the members of the family. "Me, brother, ..." When asked by the grandmother how many ice-lollies she put in the shopping basket, she says "six" but then she counts them aloud. A month later, Cindy helps her mother to bake Christmas candies. On the first baking tray, which is cooling on the balcony, there are five candies which were made and counted by Cindy. Mother is placing the second tray with seven of Cindy's candies into the oven and asks her on which one there are more of her candies. After some hesitation, Cindy answers: "I will tell you when they have baked." The illustration shows that the process of the creation of the generic model is complicated and long-term. Here, the generic model for counting is the fingers. In the first story, Cindy found out via fingers that there were 6 members in the family but she did not use this knowledge when shopping. In the second story, she was to compare 5 candies and 7 candies. Even though she could model the situation on fingers, she decided to compare candies directly. It is obvious that the generic model "fingers" has not yet developed in the girl's mind. The discovery of the generic model is usually connected with the feeling of joy. We could see it in illustration 2 in which Anka discovered that "two objects and three objects are always five objects". Sometimes, a group of isolated models becomes the generic model. The community of all isolated models is divided into classes and one isolated model functions as a representative of each. The set of all representatives forms the generic model for the whole community. A nice example can be found in a textbook on solving equations by an Islamic mathematician Músá al-Chvárizmí written in the 9th century. The following three quadratic equations are solved in detail.
Negative numbers were not known at that time, therefore, the equation x2 + px + q = 0 for p and q bigger than 0, had no root. To solve this quadratic equation was meaningful only if at least one of the numbers p, q was negative. There are three such cases and they are represented by equations (*). The solution of each equation (*) is an example for the given class and the whole triad of solutions is then the generic model of the process of solution of the quadratic equation. A mathematician who learnt how to solve equations (*) was able to solve any quadratic equation. He/she proceeded on the basis of analogy, i.e. he/she used the similarity of the process of solution of the given equation and the corresponding typical equation from (*). The generic model can have the form of a direction, procedure, formula, diagram, graph, word, sign, hint,.
2.6 Abstract knowledgeThe scheme of the process of gaining knowledge has three floors. On the ground floor, there are motivation and isolated models, on the first floor, there are generic models, and on the second, top floor there are abstract knowledge and crystallisation. Climbing up the floors gives: isolated models -- > generic model(s) --> abstract knowledge. (**) Illustration 5. David, grade 7, solved the following difficult problem: Find the sum of the first 100 odd numbers. He took the calculator and added 1 + 3 + 5 + 7 + 9 = 25, 11 + 13 + 15 + 17 + 19 = 75. Then he added 25 + 75 = 100 and began to think about it. After a while he continued with the previous solving strategy. He found out that 21 + 23 + 25 + 27 + 29 = 125. He interrupted the calculations and after an hour he returned to the problem and started in a different way. He wrote 1 + 3 = 4, 1 + 3 + 5 = 9 = 3 × 3, 1 + 3 + 5 +7 = 16 = 4 × 4. Then he copied 1 + 3 + 5 +7 +9 = 25 and added = 5x5. He excitedly wrote: "The sm of the first 100 odd numbers is 100 × 100 = 10 000." (He omitted letter "u" in sum.) When asked by the experimenter, David wrote after several attempts the result in the language of letters. 1 + 3 + 5 + . + 2n - 1 = n2. (***) A month later, David solved a similar problem: Find the sum of the first 100 even numbers. He wrote 2 + 4 = 6, 2 + 4 + 6 = 12, 2 + 4 + 6 + 8 = 20, 2 + 4 + 6 + 8 + 10 = 30. Then he crossed it out and began to create a table: first what is in the first seven columns, then the next two columns and last the last column.
David's solving process shows individual stages of gaining knowledge. In the first problem: concrete partial results are separate models; the new solving strategy led to the discovery of the regularity, i.e. the construction of the generic model; the experimenter's challenge led to the abstract understanding of the knowledge. In the second problem, David begins to apply the same procedure, but then he realises that he can use the previous knowledge, because each even number can be written as an odd number plus one. The ability to use the result gained in a different context shows that the piece of knowledge (***) is on an abstract level.
2.7 CrystallisationAfter its entrance into the cognitive structure, a new piece of knowledge begins to look for relationships with the existing knowledge. When it discovers disharmonies, the need arises to remove them: To adapt new knowledge to the previous knowledge and at the same time, to change the previous knowledge according to the new knowledge. For instance, new abstract knowledge 2 + 3 = 5 affects other, already existing sum connections and begins to deprive them of dependence on the world of things: knowledge 4 objects + 2 objects = 6 objects changes into the abstract knowledge 4 + 2 = 6. There will be far more, less visible changes. The given description of crystallisation is imprecise in two aspects: First, it suggests an image that crystallisation only begins when the abstract knowledge has been constructed. Then it supposes that the only thing which was added to the cognitive structure and which takes part in the process of crystallisation is the abstract piece of knowledge. Neither is true. Each new mental step playing a role in the creating of the new abstract knowledge immediately becomes a part of the whole cognitive structure and plays a role in crystallisation. None of the pieces of knowledge which a student constructs has a final form and each is being polished, changed, broaden,... all the time. This permanent development of knowledge is a typical sign of the quality of non-formal knowledge. A mutual discussion among students helps crystallisation profoundly. The deeper the knowledge, the more such a discussion is needed. It is usually not finished and the same topic is discussed repeatedly whenever a student discovers a new idea. In our experimental teaching, for instance, the problem of Achillus and a tortoise was opened in grade 5 and the discussion was not finished until grade 8. The following illustration shows the class discussion on the concept of minus. Illustration 6. Ela, grade 1, is a bright and curious student. She heard her father ask her mother how much money she had. Mother replied: "Minus two hundred. I have a hundred, but I am going to pay three hundred for the telephone." Ela was intrigued by her mother's "minus" word. She had the heard word minus in the context of a thermometer, in a lift and maybe somewhere else as well. At this moment, all three meanings of minus connected and the girl called out: "You have not two hundred crowns." She was delighted, she jumped and kissed her mother. She understood that minus was "not". Ela's mother, a teacher, told this story in grade 5. Students liked Ela's idea and some began to develop it further: a car going at minus speed is backing; we won the match by minus two goals means that we lost by two goals; I lost minus 100 crowns means that I found 100 crowns, etc. These statements were written by students in an allocated part of a notice-board labelled "minus = not". Later a student discovered that all these statements were wrong because they did not use the word "not". After all, to reverse does not mean not to go forward, to lose is not not to win, to find is not not to lose. On students' request, the teacher added the word "conversely" on the notice-board. After several weeks of discussion, the class agreed on four ideas:
Special attention should be paid to such crystallisation which lead to the restructuring of a part of the cognitive structure. An example is the discovery of negative numbers which not only widens the structure but changes some situations as well. For instance, at the beginning stage of number the expression 3 - 5 + 4 had no meaning because the students would undertake the operations as they are written. After the discovery of negative numbers the given expression becomes legitimate. Similarly, with the introduction of complex numbers the expression Ö-1 becomes legitimate. The transfer from real to complex numbers represents a deep restructuring of the concept of number. We have finished the description of the model of the process of gaining knowledge. In conclusion, one more "stage" of gaining knowledge will be mentioned. It does not belong to the process of gaining knowledge, but it is often emphasised in the teaching process and mistakenly considered as gaining knowledge. It is the automation of a piece of knowledge.
2.8 AutomationTeachers very often see the main goal of the teaching of mathematics in grade 1 - 4 in the automation of the basic connections of addition, subtraction, multiplication and division. They want their students to learn the "reel off" count. A student who answers the question "What is five times four?" immediately with "Twenty." has already automatised the connection 5 × 4 = 20. It means that the connection is stored in a long-term memory and is easily available. Question "What is five times four?" is a signal in a student's mind which immediately brings about the correct answer "twenty". The answer is quick and does not use nearly any intellectual energy. Automatised connections ease our work because they release intellectual energy for more demanding mental operations. A driver who has automatised his/her driving can speak to his/her co-driver because he/she does not need nearly any mental energy for driving in a normal situation. In mathematics, the automation of e.g. the multiplication table enables a student to concentrate on the organisation of the calculation when multiplying multiple-digit numbers because he/she knows all basic connections. On the other hand, these connections do not denote anything about the quality of the articulated knowledge. It can be strongly formal. The fact that a student answers quickly, correctly and with confidence does not imply that his/her answer is based on the appropriate image. For instance, a student knows that 5 × 4 = 20 but he/she cannot answer how much he/she has to pay for 5 lollypops 4 crowns each. His/her knowledge is burdened with formalism. On the other hand, another student who cannot answer as quickly can solve the word problem without any difficulty. The automation can be an important agent which brings a germ of formalism into a student's mathematical structure. Next, the described model of the process of gaining knowledge will be used for the investigation of goals and methods of the teaching of mathematics.
2.9 Consequences of the above considerationsFirst of all, the model of the process of gaining knowledge clearly points to the causes of the illness of formalism: formal knowledge originates if the sequence of stages is not followed, when the stages of isolated and generic models are skipped or abridged in the teaching process and students are presented with ready-made abstract knowledge. This basic thesis shows how to diagnose and cure formalism: The diagnosis of formalism resides in the evaluation of how richly a piece of knowledge is based on separate and generic models in a student's mind. Illustration 7. This is an illustration of the case in which there is no model in a student's mind. Filip, grade 7, precisely and quickly recited the SAS theorem for the congruence of two triangles. He was asked to find out if any two of five triangles drawn on the blackboard (each was described by the lengths of three sides and the size of one angle) were congruent. He did not know what to do. The teacher asked him to draw two congruent triangles himself. Filip drew one isosceles triangle and after a while he added that it had two congruent sides. Illustration 8. A student has one generic model of the knowledge. Two examples are presented:
Illustration 9. A student has a limited set of models of the given knowledge. Three cases will be presented.
3. ConclusionIf you decide to start your own research, you will be leaving the course with quite a clear idea of the problem which you are going to solve and the conception of the experiment. We look forward to being in an e-mail contact with you and to being able to experience the surprises and joy which your research will bring you. We will gladly discuss your problems with you. We believe that this work will be enjoyable and meaningful to you, too. It is possible that the result of your work will be made available to other teachers in written form or on the internet. It is possible that you will invite your colleagues to become part of your research. Both would be considered a success for both you and our course.
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