Principles and Methods of Teaching Mathematics
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A book on mathematics education that I read and summarized around 2002. It criticizes lessons that deductively present finished definitions and proofs, and discusses principles and methods for helping students rediscover mathematics through intuition, experience, conjecture, and trial and error. It brings together the historical-genetic approach, intuitive models, discovery learning, metacognition, problem solving, and theories of mathematization, with examples from education.
Chapter 1. Socrates’ Method of Midwifery
Socrates’ view of education:
In fact, Socrates did not believe that learners discovered or invented knowledge. According to Socrates, the soul is immortal and possessed all true knowledge while dwelling in the realm of Forms, but forgot it upon being joined to the body. Anyone who perseveres in inquiry can recall all knowledge. Knowledge is already within us, but it is not clear: it exists dimly and vaguely, as in a dream, and can be brought into clarity through dialogue. Learning is the process of recalling such forgotten knowledge; helping with this is the task of education and the teacher. —p14
The anti-didactical method:
The opposite of this Socratic method is to explain objectified mathematics according to its deductive system. Mathematicians have a tradition of objectifying the results of thought and organizing them rigorously. Mathematics textbooks therefore remain faithful to an objectifying attitude that publishes polished results rather than the thinking that led to them. Hans Freudenthal calls this attitude among teachers and textbook authors ‘anti-didactical.’ Ready-made, objectified mathematics may satisfy mathematicians who seek its complete aesthetic beauty and are preoccupied with mathematical systems and rigor, and may allow them a clear understanding. But it is difficult for students who know nothing of its constructive relationships to find meaning in it, and difficult for it to influence the development of their mathematical thinking. —p15
Chapter 2. Deduction and the Analytic–Synthetic Method
The educational problems of deductive presentation:
Asking students to state the basic properties of operations at every step of a calculation is like asking adults to justify everything they do after getting up in the morning. … Yet just as such actions should be habitual, it is far more desirable for students to be so familiar with the basic properties of operations that they use them without conscious thought. This recalls the fable of the centipede and the toad. A centipede was crawling leisurely along when it met a toad. The toad said, ‘Even with a hundred legs, you know when to use each one.’ Hearing this, the centipede began to think about which leg to use next and could no longer move. A tragic situation must soon have followed. —p29
Demanding rigorous presentation does not help students learn.
Taking logical rigor to be the spirit of mathematics, people attempt rigorous deductive presentations, treating trivial matters exhaustively and emphasizing logical approaches throughout school mathematics. This is hypocritical. Does rigorous presentation actually contribute to education—rigorous Euclidean geometry, for example, or rigorous algebra supplemented with axioms such as the closure of operations and the reflexivity, symmetry, and transitivity of equality? To begin with, it is unreasonable to demand of young students what the greatest mathematicians failed to recognize for more than two thousand years. … Moreover, proving many unimportant properties prevents us from emphasizing the main points and takes away time for important theorems. —p29-30
Mathematical discovery comes through imagination, intuition, experimentation, thoughtful conjecture, trial and error, analogy, aesthetic sense, mistakes, and failures. This also appeared in What Is Mathematics?. Without these aspects, mathematics would be a truly uninteresting discipline.
People claim to teach students how to discover for themselves while emphasizing rigorous deductive presentation. Yet mathematical discovery comes through imagination, intuition, experimentation, thoughtful conjecture, trial and error, analogy, aesthetic sense, mistakes, and failures; it does not proceed on a firm logical foundation. —p30
The point is to put deductive proof in its proper place, not to exclude it:
Deductive presentation is useful for systematically communicating what is already known, but does not promote genuine understanding and has the harmful effect of draining the life from school mathematics. This does not mean that deductive proof should be completely excluded from school mathematics; the argument is that it should be put in its proper place. —p30-31
On the overuse of symbols. How should students become familiar with them?
Appropriate use of symbols can aid communication, but excessive use burdens semantic memory and makes the content difficult to understand. Symbols can conceal ideas and, psychologically, they frighten students. Symbols have vitality, meaning, beauty, and implications for thought only when the ideas they represent have already been learned and understood. The issue is to use language with clear meaning, rather than rigorous language employing symbols. —p32
A problem with deductive presentation: its connection to the real world grows weaker.
Just as numbers are formally extended by raising questions about the solutions of new types of equations within an existing number system and extending existing laws of operation to a new domain through the ‘principle of permanence of forms,’ deductively presented mathematics introduces new concepts by posing mathematical questions about concepts already learned. It is presented as something that reproduces itself. Mathematics presented in this way becomes mathematics for mathematics’ sake, isolated from reality and other fields of knowledge. The deductive structure thus established is then thought to happen to fit some real situation and to be applicable to practical problems. In fact, however, mathematics introduced in this way goes no further than solving artificial textbook exercises: it is unrelated to the real world and cannot be applied to it. —p32
One major reason teachers favor the axiomatic/deductive approach to mathematics education is that it is the mode of presentation used in higher mathematics and, above all, is easy to teach. … But formal presentations that rigorously deduce trivial matters; draw pedantic, meaningless distinctions such as those between numbers and numerals or between angles and their measures; emphasize trivial things with inflated terminology or symbols; conceal rich substance while rigorously presenting abstract forms; offer dull, dry general principles isolated from other knowledge; emphasize sophisticated forms for simple ideas; or treat deep ideas formally and superficially corrode mathematics’ healthy vitality and are far less useful than traditional teaching materials. They present form at the expense of substance, disregarding mathematics’ relationships with the real world and its applications, as well as educational principles. They emphasize theory at the expense of education. —p32-33
Further criticism: the separation of mathematics from physics.
We cannot teach the spirit of mathematics by introducing new concepts without sufficient discussion of their connection to concrete facts, introducing general unifying concepts without the experience of unification, or repeating concepts introduced without concrete, genuine applications. Premature formalization leads to barren thought. One major defect of school mathematics today is that by severing mathematics from other fields of knowledge and inquiry, particularly physics, it has lost its origins, purposes, usefulness, and motivation for learning.
From the seventeenth century to the end of the nineteenth century, the main concern of great mathematicians was to understand natural phenomena. Many achieved more in astronomy and physics than in mathematics. … Mathematics was considered a kind of science, and there was little distinction between it and theoretical physics. … Mathematicians still invoke the achievements of great mathematicians of the past to argue for mathematics’ potential scientific value and speak of creating scientific models, but without knowledge of science they cannot actually do so. … Most mathematicians concentrate on rapidly producing many conspicuous research results in pure mathematics. Classical mathematical research with scientific purposes requires a broad background explored over hundreds of years and is pursued by only a few applied mathematicians. —p33-p34
Emphasizing the formal character of mathematics is educationally inappropriate:
Modern abstract mathematicians sometimes put theoretical beauty ahead of mathematics’ applied value. But the idea of mathematics as a game of formal rules is educationally inappropriate. What matters is the outlook of Archimedes and Isaac Newton, who regarded mathematics as a tool for understanding the world and as the language of nature. Mathematics is the language of science and an indispensable tool for it; separating mathematics from science means losing its most important source. —p34
Presentation according to the historical-genetic principle is more appropriate than logical presentation:
Science is most readily assimilated in its formative state. Reading original texts, explaining the sources of its genesis, and retracing its historical formation are therefore natural methods of learning and teaching. Presentation according to the historical-genetic principle is educationally more appropriate than logical deductive presentation. —p35
Motivating learning through interest is important:
We cannot expect young students to take material seriously merely because they might need it in the future. Motivation arises when it engages their interest at the time of study. Persuading students to study mathematics in order to enter university, without giving any indication of how school mathematics is useful, is even inhumane. —p36
Failing to give school mathematics meaning is like teaching students to read musical scores without allowing them to play music. Motivation is more than a psychological stimulus: genuine motivation comes from insight into the meaning of mathematics. For example, the quadratic expression
acquires meaning when it is understood as relating to free fall, and an ellipse becomes more than a simple curve when it is understood as the orbit of a planet around the sun. Problems involving quadratic expressions or ellipses then become meaningful through their connection to physical situations. Physical meaning gives students the power to think about mathematical problems. —p37-38
The importance of heuristic presentation.
School mathematics should be presented heuristically rather than deductively. A heuristic approach means letting students rediscover and recreate mathematics. This is possible by guiding and encouraging them to think intuitively; it is difficult to expect within the framework of a logical presentation that artificially reconstructs the highly polished results of original creative work. —p39
The historical-genetic principle for heuristic presentation.
The historical-genetic principle is a great help in guiding students to rediscover mathematics. According to this principle, the historical order of mathematics’ development is a natural order for organizing teaching material, and the difficulties mathematicians experienced are the difficulties students face. —p39
Logical presentation must not precede discovery and intuition.
Understanding is achieved intuitively, and logical presentation can, if mishandled, decisively obstruct it. Logical presentation must not precede discovery and intuition. Logic merely authorizes what intuition has conquered; it is a health regimen mathematicians follow to keep intuited ideas healthy and strong. —p39
Analysis and synthesis. In a proof problem, synthesis is the proof itself.
In analysis, we assume that what we are asked to do has already been done, that what we seek has already been found, or that what we must prove is true. We then ask from what antecedent the desired result could be derived, and continue asking for that antecedent’s antecedent until we reach something already known or assumed to be true. Alternatively, we assume that what the problem asks for has already been obtained, derive a proposition from it, and continue deriving propositions until we reach one whose answer we already know. This procedure is called analysis, or working backward. Synthesis reverses the analytical process: starting from its final point—a proposition already known or assumed to be true—it retraces the steps through deduction until it reaches the proposition originally required. Analysis discovers a plan of solution, and synthesis carries it out. In a proof problem, synthesis, the deductive process that retraces analysis in reverse, is the proof itself. —p43
Since Euclid’s Elements, synthesis has been excessively emphasized over analysis.
Greek mathematics regarded the dialectical unity of analysis and synthesis as the essence of mathematical thinking. Yet the Elements takes fundamental principles—axioms, postulates, and definitions—as mathematics’ logical and ontological foundation, and regards synthesis, the deduction of theorems from them, as the method of constructing mathematical truth and guaranteeing its understanding. The Elements, which conceals analysis, the logic of discovery, was used as a textbook. To this day, this is treated as an inviolable tradition and serves as the model for organizing mathematics textbooks and teaching mathematics. As noted earlier, this seems to stem from Socrates’ philosophy discussed in Chapter 1. —p43
Chapter 3. Intuitive Methods
Intuitive teaching influenced by Francis Bacon’s empiricist philosophy. See p57.
Émile:
The true founder of intuitive teaching was Johann Heinrich Pestalozzi, who was influenced by Jean-Jacques Rousseau’s naturalistic educational thought. … Jean-Jacques Rousseau grounded education in developing people’s inherent nature, educating the natural person according to natural laws of development, as nature itself educates. His naturalistic educational thought, which sought to align educational activity with the natural laws of human development, is embodied in the educational novel Émile. So much of his thinking went into it that he confessed, ‘I spent twenty years thinking and three years writing.’ … Émile may be called the most important educational work in all human history. —p57-58
Knowing something formally differs from accepting it intuitively:
Formally knowing a rule of logic does not imply that it has been intuitively assimilated. Knowing De Morgan’s laws does not mean regarding them as self-evident. Transitivity, by contrast, is understood as self-evident at a certain age. The whole being greater than its parts is an intuitive logical rule, so we are surprised to learn that the set of natural numbers has the same cardinality as the set of even numbers. —p75
People generally tend to overestimate the accuracy of their knowledge, interpretations, and judgments:
If we did not trust what we saw when crossing the street, our responses would lack continuity and be poorly regulated. Similarly, in reasoning we must trust our representations, interpretations, and momentary solutions, at least for the moment; otherwise, the flow of thought would be paralyzed. Mental images and conceptions of reality, and hypothetical solutions, can be biased, distorted, incomplete, or entirely wrong. Nevertheless, a degree of overconfidence is necessary to trust these mental products, at least temporarily. Intuition is a cognitive process in which overconfidence plays an essential role. This means that intuitive feelings lead us to accept propositions unsupported objectively by empirical data or logical proof, and to overlook gaps in information. People generally tend to overestimate the accuracy of their knowledge, interpretations, and judgments. This is thought to be fundamentally important to the mechanism of intuition. —p76-77
People trust incorrect but intuitively acceptable answers they have never been taught more than correct answers they have learned:
Asked for the sum of
, some students who remembered learning that the answer was 2 did not actually trust this very much. Students who gave a different but intuitively acceptable answer, by contrast, had relatively high confidence in it. The important point is that they trusted an incorrect but intuitively acceptable answer they had never learned more than a correct answer they had been taught. Our memory and reasoning procedures tend to select and retain propositions that seem intrinsically trustworthy over propositions that must appeal to external support, such as explicit instruction. —p77
Visualization, diverse models, and active personal participation:
The higher the level of intrinsic confidence in an idea, the greater its chances of survival and active participation in problem solving and decision-making. What, then, are the means of creating intrinsic confidence? They include visualization, the use of various forms of models, and active personal participation. —p77
The problem of intuitive obstacles:
Imagine trying to convince a child that they can be wrong even when they strongly believe they are right. This could disrupt the child’s mathematical reasoning. Experts and intellectually educated adults know how to play a double game: they know that in principle they can be wrong, yet continue reasoning at every step as though certain they are right. Once the entire thinking process is complete, a stage of monitoring can begin. Children, however, do not yet possess this intellectual duality. If a child loses confidence in their mathematical intuitions and their attempts at a problem, how can they continue reasoning? They may lose all interest in mathematics along with their confidence. —p78
To avoid the problem of intuitive obstacles:
We must emphasize that the fact that we make mistakes and are often prone to overconfidence should not discourage us. … We must therefore learn to analyze and regulate our own mental activity. Reasoning, however, is a process guided broadly by the thinker’s goals, with details guided automatically through automatic feedback systems. Trying to control every step of thought consciously destroys its fluency and continuity and makes productive thought difficult.
After solving a difficult mathematical problem, a good student reflects deeply, analyzing and turning the problem over in their mind. Retrospective analysis develops the metacognitive ability to regulate thinking. Trying to convince students that their thinking may be wrong without providing metacognitive techniques may have the counterproductive effect of destroying their confidence. Developing metacognitive abilities improves students’ retrospective monitoring. …
One desirable strategy for developing metacognitive abilities is to develop self-regulatory schemas that operate automatically at the operational level. One method is to develop an ‘alarm’ that warns students whenever their reasoning approaches a potential trap. For example, an alarm could activate automatically when solving a potentially error-prone problem such as, ‘If 0.65 L of juice costs 2,000 won, how much does 1 L cost?’ The alarm prevents a hasty answer. —p78-79
A difficulty in geometry:
We cannot think about geometrical points or straight lines without visualizing them, yet these intuitive representations lead us into traps. —p82
Intuition about multiplication:
A formally presented and justified solution may or may not be intuitively accepted. When asked the price of 3 L of juice costing 900 won per liter, students intuitively feel that multiplication is needed, justify it numerically, and perform it. But asked the price of 0.9 L of wine costing 3,000 won per liter, they struggle; even when they know that multiplication is the correct solution, it often fails to resonate intuitively. The intuitive model of multiplication is repeated addition of equal quantities, and multiplication with a decimal multiplier has no intuitive meaning. —p87
Improving intuitive understanding through empirical and productive mental activity:
It is important to deepen students’ intuitive understanding of various concepts and propositions. This can be achieved only by creating teaching situations in which students personally and empirically engage in productive mental activity in the relevant domain. For example, successful teaching of statistics and probability cannot be achieved by teaching theorems and solution procedures alone. Students must first experience physically manipulating dice, coins, and marbles, and observing, recording, and summarizing different outcomes. They must experience situations where individually unpredictable outcomes tend to produce structure and reveal regularities when considered in large numbers. Students without this prior experience will never understand statistical and probabilistic propositions or manipulate them intuitively. Experience plays a fundamental role in forming programs of action and systems of expectation that will be organized into intuitions. —p97
Experts, too, show psychological resistance to things that conflict with intuition.
Mathematicians often used negative numbers in calculations, but only as temporary expedients; they did not accept them as conceptually meaningful entities. The psychologically significant point is that resistance to accepting negative numbers was not confined to non-experts. Mathematicians who did not object to using other ‘immaterial’ mathematical abstractions such as points and lines objected to giving negative numbers a formal mathematical status. —p98
The absence of an intuitive, familiar model:
The difficulty of accepting negative numbers as meaningful mathematical entities stems from the difficulty of identifying a good, intuitive, familiar model that consistently satisfies all their algebraic properties. In fact, no such model exists. Models can be devised, but only through artificial systems of conventions. Number-line models, future–present–past models, and left–reference-point–right models are interesting, but have serious defects. They do not make either the concept of negative numbers or the rules of signs more intuitively acceptable. —p100
What is a model?
When, on the basis of an isomorphism, descriptions or solutions in system A are consistently reflected in system B, or vice versa, system B is called a model of system A. —p103
What is a diagram?
Intuitive models using conventional pictures are generally called diagrams. —p104
What models provide:
By its nature, a model offers the problem solver a substitute better suited to human thought than the original. We think better with what can be perceived, physically manipulated, recognized as familiar, and controlled through action than with what is abstract, unrepresentable, uncertain, or infinite. The essential role of an intuitive model is to make what is intellectually inaccessible accessible and manipulable. A model translates the original material—properties, processes, and relationships—into intuitively acceptable elements. —p105
The dangers of models:
Using a model means thinking productively with it. Models stimulate solution strategies, suggest solutions, and confirm the consistency and meaningfulness of solutions obtained through the original. The chief advantage of an autonomous model is that the solver can rely on the model alone to solve various problems posed by the original. But there is a danger of becoming so captivated by the model that one draws conclusions from its properties that are inappropriate for the original. Mathematicians know well that a point is an abstract, dimensionless entity and a straight line an abstract, one-dimensional entity, yet various misconceptions in the history of mathematics arose from the uncontrolled intervention of pictorial models. —p105-106
What makes a good model?
A good model must be an autonomous entity while also serving as a trustworthy mediator between the original situation and the solver’s intellectual activity. In most cases, however, intuitive models are imperfect mediators that often lead to inaccurate and incomplete interpretations. —p106
Typical examples:
A typical example associated with a concept can influence not only how particular instances are identified, but also the strategies adopted to solve problems and the solutions themselves. One important effect of typical examples on conceptual structure is a phenomenon we might call the collapse of hierarchies. Students may be reluctant to agree that numbers such as 2 and 3 are complex numbers. Although they agree that
is the general form of a complex number and that is possible, they may think of these numbers only as natural numbers or rational numbers. —p107
A caution when giving examples: what seems obvious to a teacher is not necessarily obvious to students.
When teachers give an example of a theory or method, they see the generality embodied in it and may therefore fail to indicate the example’s scope or emphasize what must be emphasized for it to be recognized as an example. Students, however, have much less experience with particular instances of the situation under discussion and may not realize that other examples exist, so they focus on the example presented. They may see only the particular while taking it to be general, and consequently ‘learn the example.’ —p109
The tendency to fill gaps:
At the perceptual level, humans have a natural tendency to complete gaps in perceived figures and close them, obtaining familiar images that appear self-consistent and can be treated as single objects or wholes (see Law of closure). We have an automatic tendency to reduce uncertainty by intuitively filling gaps and closing figures. The same happens with ideas. We naturally tend to connect fragments of information and give them a unified meaning. This relates to natural tendencies toward controllability of thought, cognitive self-consistency, and internal equilibrium in mental operations. People tend not only to stop seeking evidence but to close arguments whenever possible. Closure often occurs prematurely, on incomplete information. This means the desire to decide is stronger than the desire to know, and can be explained by a basic desire to minimize hesitation and make a decision with apparently firm grounds despite incomplete information. —p119
Do not introduce new teaching methods without thorough psychological and didactical research:
Mathematics education strategies must be built on profound knowledge of the nature of the relevant concepts, their historical genesis, and their development in the individual. In particular, introducing new concepts into school mathematics without thorough psychological and didactical research is unacceptable. … To overcome educational problems successfully, we must first understand the psychological aspects of the relevant concepts. We need to know the implicit interpretations students assign to concepts, their intuitive responses, the intuitive models they use, and the effects of acquiring new concepts. We must also evaluate how different teaching methods affect the complex relationship between the intuitive aspects of mathematical concepts and their formal structures. Inappropriate strategies may destroy the productive interaction between these two elements. —p128
Chapter 4. The Historical-Genetic Principle
Chapter 4 elaborates on the arguments made in Chapter 2. Its later sections show examples of mathematics textbooks organized according to the historical-genetic principle, including the sum of the interior angles of a triangle, logarithms, definite integrals, and the fundamental theorem of calculus.
Chapter 5. Stimulus–Response Learning Principles and Programmed Learning and Teaching
No punishment:
Punishment, often used to extinguish a behavior, may have an initial temporary effect, but it is a poor method of controlling and modifying behavior and may have undesirable emotional side effects. Punishment is not reinforcement because it does not increase the probability of a response. —p184
A problem with traditional lessons: the feedback gap is too long.
From Skinner’s standpoint, traditional education has a serious problem in how reinforcement is arranged. When does a student receive reinforcement that their computational ‘behavior’ is correct? Eventually, of course, students may be able to check their own answers and receive automatic reinforcement, but in the early stages reinforcement of correctness usually comes from the teacher. The reinforcement teachers provide, however, is not optimal. Without an explicit intervening activity, even a delay of a few seconds between response and reinforcement destroys most of the effect. Yet in a typical classroom, long periods usually pass. For example, while students solve problems, the teacher walks up and down the aisles, stopping here and there to say whether an answer is right or wrong. Tens of seconds or several minutes separate the student’s response from the teacher’s reinforcement, and when assignments are collected for checking, the delay exceeds a day. It is surprising that such circumstances have any effect at all. —p185
Chapter 6. Discovery and Expository Methods
Finding interest in a boring lecture:
Robert Lee Moore arrived at the basic idea behind this rather radical method of teaching mathematics while he was a graduate student at the University of Chicago. Bright and intellectually active, he found the lecture format somewhat dull. To enliven boring lectures, instead of taking notes as the lecturer wrote a theorem on the board and proved it, he raced to see whether he could discover a proof himself. He often won. In any case, once he began trying this, he felt he was spending his time more meaningfully. —p194
Difficulties with the Moore method:
The most obvious difficulty with this teaching method is making the class homogeneous. Unless competition takes place reasonably on an equal footing, one basic motivation to learn will be weakened. … Over time, however, students’ background knowledge becomes more homogeneous, so this is not a serious obstacle.
There may also be the problem of students too timid to present proofs at the board. Try to draw them into discussion and suggest that they write its results on the board. After about six months, especially when a subtle argument grows heated, they will stand up and present their ideas without even being conscious of what they are doing.
People tend to be embarrassed by mistakes, particularly public ones, so take care not to make criticism feel like ridicule. Research mathematicians are so accustomed to making mistakes that they tend to forget the pain young students experience. —p199
The most serious difficulty with the Moore method is the teacher’s patience:
After students have investigated proofs independently for several months, it becomes very difficult to get them to read mathematics books. Rather than read and understand a book, they want to prove things themselves. But to acquire a broad mathematical education in a short time, it is essential, after a while, to set aside a period in which they work through mathematics books on their own. …
The hardest aspect of this method is the patience it requires of the teacher. That patience must come from the conviction that educating students to investigate and developing their mathematical abilities matter far more than simply transmitting knowledge. Teachers must resist the impulse to help, and must not even point out the ‘obvious.’ We all know how difficult even the obvious is before it becomes obvious. —p199
Breadth and depth of knowledge:
At every level, teaching knowledge faces a tension between the two ideals of breadth and depth. In an age when knowledge is exploding in both rate of growth and quantity, individuals can cope by grasping its relevance. What knowledge is worth knowing? Possible criteria are whether it gives pleasure, contains within it a basis for generalization that permits intellectual journeys beyond the information given, and is useful. From this standpoint, depth and continuity should take priority over breadth. An educated person should have a feel for the nature of knowledge in a field of inquiry, know the field in terms of its relevance, and have a sense of how its knowledge is acquired. This requires firsthand experience of thinking that progresses from a primitive, weak understanding of the subject to a more refined and powerful one. It can be introduced early, in very simple, limited forms without distortion, and should be revisited repeatedly. One thinking process that gives the most fundamental pleasure in human intellectual life is reducing surprise and complexity to predictability and simplicity. The most important central theme is an increase in learners’ own sense of understanding their knowledge, which produces pleasure in knowledge. —p210-211
Knowing is a process, not a product:
The body of knowledge held by university faculties and embodied in a series of authoritative books is the result of much prior intellectual activity. Teaching someone something within these disciplines does not mean having them commit the results to memory. Rather, it means teaching them to participate in the process that made the establishment of knowledge possible. We teach subjects not to create little living libraries about them, but to enable students to think mathematically for themselves, deal with events as historians do, and participate in acquiring knowledge. Knowing is a process, not a product. —p217
Chapter 7. The Principle of Operational Construction
Separating out physical properties.
His (Jean Piaget’s) interest in mathematics reportedly began with the following experience at the age of four or five. He was sitting in a garden playing with small stones. First he arranged them in a straight line and counted from left to right. There were 10. Then he counted from right to left. Again there were 10, which surprised him. Next he arranged the stones in a circle and counted them. Again, 10. Counting in the opposite direction gave the same result. He then arranged and counted them in various ways and eventually realized that the total was 10 regardless of their order. —p247
Opposing the historical-genetic principle: the psychogenetic principle.
There is another problem specifically concerning the arrangement of the mathematics curriculum. The sequence of a child’s psychological development in a subject is often closer to the subject’s own axiomatic order than to the historical order in which its concepts developed. For example, topological concepts such as one figure touching another, being separate from it, or lying inside it developed historically later than the concepts of Euclidean or projective geometry, yet form earlier in children’s development. If any special justification is needed for teaching a subject’s structure according to its proper logical—that is, axiomatic—order rather than its historical development, these facts provide it. —p252-253
Chapter 8. Lakatos’s Method of Proofs and Refutations
Imre Lakatos
Chapter 9. George Pólya’s Theory of Teaching Problem Solving
Kurt Gödel:
The idea that an algorithm could be found for every kind of mathematical problem appeared to grow more secure as mathematics developed, but was proved impossible by the publication of Gödel’s incompleteness theorems in 1931. Efforts to make problem-solving methods algorithmic seemed to pause for a while afterward. In the latter half of the twentieth century, however, research into algorithms for solving problems became active again alongside research into computers that solve problems requiring intelligence. —p312
How to solve it:
George Pólya regarded good education as systematically finding opportunities for students to make independent discoveries. Studying the question ‘How are mathematical discoveries made and problems solved?’, he attempted a systematic account of methods of discovery and invention and of teaching them. 1945, when the resulting How to solve it was published, is regarded as the boundary dividing research and education in mathematical thinking and problem solving into two eras: before and after Pólya. —p312
Artificial intelligence, another application of heuristics:
Meanwhile, as the field of AI emerged and efforts were made to make problem-solving methods algorithmic, Pólya’s heuristics attracted attention not for their educational dimension but as ‘information-processing procedures.’ —p312-313
The role of conjecture in mathematics.
Before proving a mathematical theorem, we must first make a conjecture. Before carrying out the details, we must conjecture an idea for the proof. We must combine observations and draw analogies. We must try repeatedly. The result of a mathematician’s creative research is deductive reasoning—a proof. But that proof is discovered through plausible reasoning and conjecture. —p315
George Pólya’s heuristics:
George Pólya’s modern heuristics take the form of questions and recommendations in dialogue, divided into four stages: understanding the problem, devising a plan, carrying out the plan, and looking back. The underlying thinking strategies can be summarized as follows.
- Understanding the problem: focus on the goal; attend to the problem’s main parts; survey it with attention to its conditions; draw a figure and introduce suitable notation; break down and write out the conditions.
- Devising a plan: bring relevant knowledge to bear; look for useful patterns; identify related problems or theorems; think of problems with the same or similar unknowns or conclusions; use the results and methods of related problems and introduce auxiliary elements; restate the problem; revisit definitions; try solving related problems that are simpler, more general, more specific, or analogous; alter the unknowns, conditions, and data to formulate auxiliary problems and solve part of the problem; reassess the situation when no progress is made; check whether the data, conditions, and key concepts have been used.
- Carrying out the plan: proceed while checking every step.
- Looking back: check the result and argument; seek other solutions; survey the solution at a glance; find problems to which the result or method can be applied.
Pólya also emphasizes working backward, simplifying, drawing analogies, considering the simple, easy, and familiar first, understanding a problem as a whole before attending to details, classifying problems by type, looking for patterns, attending to symmetry, considering equivalent problems, checking extreme cases, and using dimensional analysis as important heuristic strategies. —p317
Chapter 10. Induction, Analogy, and Metaphor
What is inductive reasoning?
Inductive reasoning draws a hidden general law that might hold for an entire class of cases by attending to common features in particular observed instances. Its importance was emphasized and systematically studied by Francis Bacon in England in the seventeenth century, responding to the period’s concern with methods of investigating nature as modern science developed. Supplemented by John Stuart Mill, it has been regarded as the model of scientific thinking. …
Since David Hume questioned induction’s validity, however, the problem of induction has been a major controversy in modern philosophy. More recently, scholars such as Karl Popper and Imre Lakatos have advanced the notable argument that scientific knowledge is not discovered by inductive reasoning, but grows through conjectures based on individual observations and experiments with typical cases, refutations by counterexamples, and improved conjectures. —p337-338
Errors arising from induction:
These include overlooking cases that should be observed because of preconceptions or carelessness; making distorted observations through illusion or prejudice; generalizing hastily; attempting to generalize from a mere enumeration of facts without testing; drawing analogies from superficial similarities alone; and errors in causal inference, such as misidentifying causes, overlooking causal interactions, reversing cause and effect, or confusing causes with conditions. —p339-340
Mathematical induction is deduction, not induction.
Mathematical induction is not induction but deduction using fundamental properties of natural numbers. A law obtained through inductive reasoning is only a conjecture; to guarantee its truth, deductive reasoning—a proof—must follow. In school mathematics, however, it is often considered enough merely to check further cases to gain greater confidence in a conjectured general property. —p341
What is analogy?
Analogy conjectures the properties of an object from those of a similar object, on the basis of their resemblance. When objects of two kinds are found to resemble each other in some respects, and the first kind has another characteristic, reasoning that the second kind also has that characteristic is analogy. Its strength depends on the resemblance between the two kinds of objects in the relevant respects. —p342