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The books always start from a large number of definitions, postulates, axioms and some preliminary explanations that appear to the reader as nothing but dry stuff.
As a result the beginners are bored and rejected before they have got only the slightest idea of what they are expected to learn. In order to avoid this dullness attached to geometry some authors included applications in such a way that right after the theoretical treatment of the theorems their practical use is illustrated.
However, in this way only the applicability of geometry is shown without facilitating the learning of it. As any theorem precedes its applications the mind is brought into contact with meaningful situations only after having taken great pains in learning the abstract concepts. It occurred to me that geometry as well as other fields of study must have grown gradually; that the first steps were suggested by certain needs, and that these could hardly have been too high as it were beginners who made them for the first time.
Fascinated by this idea I decided to go back to the possible places where geometric ideas might have been born and to try to develop the principles of geometry by means of a method natural enough to be accepted as possibly used by the first inventors.
My only addendum was to avoid the erroneous attempts these people necessarily had to make. What arguments are put forward in favor of problem-oriented teaching?
The problem chosen by Clairaut for introducing the Pythagorean theorem was this:. Determine the side c of a square whose area is the sum of the areas of two given squares with sides a and b.
How to construct a square whose area is the sum of the areas of two different given squares? Figure 19 is well known to us: It is nothing but Fig. While this figure came out of the blue in Sect. We have in this example a good illustration for the difference between a proof embedded solely into a net of logical relationships and a proof embedded into a meaningful context.
Special case: First draw figure 14 a. You get a combination of Figures 14 a and 14 b. You get Fig. How long is the diagonal of a rectangle with sides a and b? Is it possible to recombine the four halves of two congruent rectangles to make a square whose side is the diagonal of the rectangle? A first attempt leads to Fig. We arrive at three equal sides, two right angles, and an isolated right triangle.
The question is:. Again, Fig. In marked contrast to that presentation, the figure appears here within the solution of a problem.
So we have another illustration of the difference between a formal proof within a deductive structure and an informal proof arising from a meaningful context. Can these five pieces be recombined to form a shape composed of a square with side a and a square with side b? You have to arrange the five pieces such that they cover the union of a square with side a and of a square with side b.
Hint: Fig. Reexamine the logical line in approaches 1 and 2: At what places is the assumption of right angles crucial? Concepts are the backbone of our cognitive structures. But in everyday matters concepts are not considered as a teaching subject.
Though children learn what is a chair, what is food, what is health, they are not taught the concepts of chair, food, health. Mathematics is no different. Children learn what is number, what are circles, what is adding, what is plotting a graph. They grasp them as mental objects and carry them out as mental activities.
It is a fact that the concepts of number and circle, of adding and graphing are susceptible to more precision and clarity than those of chair, food, and health. Is this the reason why the protagonists of concept attainment prefer to teach the number concept rather than number, and, in general, concepts rather than mental objects and activities?
Whatever the reason may be, it is an example of what I called the anti-didactical inversion. Our mathematical analyses have shown that the Pythagorean theorem is fundamentally related to the concepts of area and similarity. Therefore it is necessary to provide data on the psychological development of these concepts. We cannot give a systematic and coherent review of research here. Instead we concentrate on a few interesting studies that give a first orientation and—what is even more important—also provide a basis for doing similar studies.
The basic message of this chapter is this: Mathematical concepts are neither innate nor readily acquired through experience and teaching. Instead the learner has to reconstruct them in a continued social process where primitive and only partly effective cognitive structures that are checkered with misconceptions and errors gradually develop into more differentiated, articulated and coordinated structures that are better and better adapted to solving problems.
For teachers this message is of paramount importance: Concepts must not be presupposed as trivially available in students nor as readily transferable from teacher to student. On the contrary, the teacher must be prepared that students often will misunderstand or not understand what he or she is talking about.
The Greek philosopher Plato ca. Relevant for teaching and learning and therefore frequently referred to is his dialogue Meno that centers around the fundamental questions if virtue can be taught and where knowledge does come from. One part of this dialogue is particularly interesting as perhaps the oldest recorded lesson in mathematics: Socrates teaches, or better interviews, a boy on how to double a square Plato Although the boy predicts the area of this new square as 8 square feet, nevertheless his first suggestion is to double the sides.
Again Socrates arouses a cognitive conflict by having the boy calculate its area: to his own surprise the boy finds 9 square feet instead of the expected 8! Clinical Interviews on Area and on Doubling a Square. These Greek philosophers believed that knowledge was already innate in human beings. In sharp contrast with this view Piaget sees knowledge not as something pre-fabricated inside or outside the learner but as the continued personal construction and reconstruction of the learner while interacting with and trying to adapt to the natural and social environment.
He or she must not be content with just listening to students, but has to stimulate them to express their mental processes with words or actions, always following their fugitive thoughts. For this reason it was called clinical. The book contains a chapter on doubling area and volume Chap.
The following study done by student teachers was inspired by both Plato and Piaget. Show the student the geometric forms and ask: Which different figures can you build with these?
Imagine that this is a pasture surrounded by a fence. It is just big enough to give grass for exactly eight cows. Now the farmer buys eight more cows and wants to fence off a pasture that is twice as big. As he likes squares the bigger pasture should be a square as well.
While playing around with the forms and laying out a variety of figures Dirk explicitly states that four triangles can be arranged to make a larger triangle see Fig.
In order to solve the pasture problem Dirk adds four squares and produces Fig. That gives me nine squares, but I need eight. Next he tries to attach a triangle to a square see Fig. He counts the squares: These are only seven squares. Must all sides be equal? Interviewer: Yes. Do you think you can do it? Dirk, after thinking for a while: I could try it with triangles. He first makes the original square see Fig.
Interviewer: Why do you think it is twice as big as the old square? Dirk: Inside you have the old square with eight triangles, and in addition eight new triangles. Interviewer: How did you hit upon the idea to arrange the triangles this way?
Stefan: These are four [ he points to the original square ] and these are Interviewer: Can you also build a pasture for exactly sixteen cows? Stefan: No. In this way I again get one more It is not possible. Interviewer: What about using these triangles? Two triangles make a square again. Either you have to take one square more or you must build a rectangle. Analyze Figures 33 c, 34 and 35 in the interview with Dirk. How far are they away from the solution? Keep in mind that the solution has to meet two requirements: it has to be a square and it has to have an area of 8 unit squares.
But compare Fig. Obviously Dirk tries to add triangles to the smaller sides of the octagon. The given triangles are too big, as Dirk recognizes: What triangles would be necessary to extend the octagon to a square in this second way? How many unit squares would this second square have? Do a similar analysis with the second interview. If they do not hit upon the square by themselves ask them to make one. The answers indicate how well he or she has attained the concept of area.
Of course the nature of operations differs from domain to domain: in arithmetic we deal with number operations, in calculus we use transformations of functions, in combinatorics we operate with combinatorial formulae, and so forth. Nevertheless in all these fields there is an operative structure of knowledge. The following analysis of operations connected to area and similarity has therefore far-reaching importance as an instructive special case.
The present subsection tries to give an idea of research that clearly follows the Piagetian paradigm and covers the age range from eight to fifteen. In clinical interviews with eight- to eleven-year-olds Wagman used the following tasks that directly reflect the basic properties of the concept of area: 1. Use of squares as units, 2. Invariance under rigid motions, 3. Unit Area Task.
The investigator presents three polygons that can be covered by unit squares and asks the child to find out how many squares are needed in each case. The necessary squares are piled besides each polygon. In the second part the child is given a large number of triangular tiles each of which is equal to one half the square tile.
The child is asked to find out how many of these triangular tiles are needed to cover the polygons. Congruence Axiom Task: The investigator presents the child with two congruent isosceles right triangles, one blue, the other one green.
The child is asked how many white triangles of half linear dimensions are necessary for tiling the blue triangle. After discovering the answer 4 the child is asked to guess without trying how many white triangles will be needed to tile the green triangle. The child is presented with two polygonal regions a. Given a set of smaller shapes the child is asked to cover each of the two polygons and to decide if they have the same or different amounts of space.
Polygons are decomposed and the pieces are rearranged to form another polygon. The child is asked to compare the areas. A machine makes holes in two equal squares of tin in two different ways see Fig. Students are asked to compare the amount of tin in A and B. A square A is cut into three pieces and the pieces are arranged to make a new shape B see Fig. Students are asked to compare the areas of A and B.
The result revealed that about 72 percent of the total population could successfully answer both questions. There were no major differences between the age groups.
However, the concept of area is by no means mastered by all secondary students. In the past decade research on the development of the similarity concept has been intensified.
A typical problem used in the International Studies of Mathematical Achievement in and in for eight graders is the following one:. On level ground, a boy 5 units tall casts a shadow 3 units long. At the same time a nearby telephone pole 45 units high casts a shadow the length of which, in the same units, is. Similar triangles appear early in the introduction to ratio in most secondary textbooks and children are expected to recognize when triangles are similar to each other and the properties they possess.
There was particular difficulty with the word when similar triangles or rectangles were under discussion. The test items dealing with similar figures not triangles or rectangles were some of the hardest on the test paper. In enlarging figures there is the danger of being so engrossed in the method to be used that the child ignores the fact that the resulting enlargement should be the same shape as the original The introduction of non-whole numbers into a problem does not make the question a little harder but a lot harder.
A satisfactory treatment of the Pythagorean theorem can only be reached within a long-term perspective of the curriculum. For coming to grips with the concepts of congruent and similar figures as well as of linear and area measure students need rich opportunities to operate with figures within meaningful contexts.
Work has to start in the early grades with concrete materials, it has to be continued with concrete materials and drawings in the middle grades, and should gradually be extended to more symbolic settings in higher grades.
It is only in this way that students can understand the properties of area and relationships between figures based on area and exploit them with mental flexibility for solving problems as well as for proving theorems. The approach to the Pythagorean theorem via similarity is conceptually much more difficult for students than the approaches via area preserving dissections and recombinations of figures.
Therefore similarity is not appropriate for introducing the Pythagorean theorem. However, it is a good context for taking up the Pythagorean theorem at a more advanced level. Because of the fundamental importance of the Pythagorean theorem the first encounter with this theorem should take place at latest in grade 7 or 8.
In each of the subsequent years the students should meet the theorem in ever new contexts and with new proofs. The teacher is therefore confronted with the following fundamental problem: How to cope with this wide spectrum of student abilities?
The mathematical and psychological analyses and activities in the preceding sections have set the scene for attacking the central problem of this paper: the design of teaching units on the Phytagorean theorem. Students should be faced with a problem that is typical for the use of the Pythagorean theorem and rich enough to derive and explain prove the theorem.
Instead he or she has to invent them relying on his or her imagination and using the scientific basis for checks of validity, reliability and effectiveness. It is important to keep this in mind and to interpret the following units as suggestions, not as dogmatic prescriptions. Before analyzing the following teaching units resume Exploration 6 and do some brainstorming on ideas how to introduce the Pythagorean theorem.
Which mathematical or real problem situations do you think appropriate at which school level? What approaches are chosen in textbooks? Two introductory teaching units on the Pythagorean theorem are offered below. One of them is based on ideas developed in the section on heuristic approaches pages 15—24 , the other one is taken from a Japanese source and puts strong emphasis on technology.
Figure 47 is really a square. All four triangles are congruent as they coincide in the right angles and the legs a and b.
Hence it is a square. Now the square with side c is composed of the same five pieces as the two squares with sides a and b. It appears as instructive to round out the unit by comparing the measurements in the table with the values calculated by means of the formula. Also the heuristic use of the Pythagorean theorem should be derived from this special context: given the lengths of two sides of a right triangle the length of the third side can be calculated. As a result we arrive at the following plan for a teaching unit.
At the beginning of each episode the teacher has to take the initiative. His or her crucial interventions and only these! Presenting the guiding problem. Rectangles of different shapes are drawn on the blackboard Fig. The teacher explains the problem of finding the length of the diagonal.
As an example making a lath for stabilizing a rectangular frame is mentioned. It is only natural that the students will also suggest to measure the diagonals. The teacher recommends to draw a variety of rectangles and to measure the diagonals, and fixing the results in a table Fig.
At the end of this episode some data are collected in a common table on the blackboard. Redefining the problem. The teacher redefines the problem as the typically mathematical problem of finding a formula for computing the diagonal c from the sides a and b. The advantage of a formula should be plausible to students. Students are stimulated to guess what such a formula could be like. The suggested ideas are written up and tested against the values in the table. At the end of this episode the students are informed about the steps to follow: Receiving some hints from the teacher they should try to discover and prove the formula as far as possible by themselves.
Specializing the problem: Diagonal of a square. Material: Congruent paper squares. As a first hint the teacher suggests to study squares as an easier special case. Each student gets some congruent paper squares and diagonalizes them.
The task is to find an arrangement of squares such that a relationship between diagonal c and side a can be deduced. Generalizing the solution: Diagonal of a rectangle. Material: Congruent paper rectangles.
The teacher suggests to adapt the solution from squares to rectangles. Discussing the formula. The teacher informs about the history of the Pythagorean theorem and about its importance. Students check the formula by comparing the measured values episode 1 with the values obtained from the formula. The Japanese volume Mathematics Education and Personal Computers contains a case study on the Pythagorean theorem as an example for improving the traditional format of teaching Okamori , Instead of treating the whole class as one body the class was split up into small groups four or five students according to interests, academic abilities and social relationships.
The idea was to offer students different approaches to the subject matter that might better serve the individual needs and preferences.
The medium size square is dissected according to Fig. General information. The class is divided in small groups. Students are told that they are expected to do a geometric investigation by means of the computer. Then they receive some instructions how to use the system and how to interact within the groups. The three contexts for approaching the theme are explained in general terms, and the groups are asked to decide for themselves which context they would like to choose.
Introducing the task. When the students start the program three triangles appear on the screen: an obtuse one, a right one and an acute one. The sides of each triangle carry squares: the longest side a square colored red, the smaller sides squares colored green. The students are stimulated to discuss the relationship between the area of the red square and the sum of the areas of the green squares in all three cases.
The teacher suggests to draw the squares on graphic paper and to estimate the area. The discussion within the groups and with the whole class should lead to the conjecture of the Pythagorean theorem for right triangles.
Defining the task. The groups are given the following task: Try to find out from the figures and transformations offered by the computer program why the conjectured relationship must hold. Give a written account of your reasoning. Use the prepared worksheets.
The groups are handed out worksheets that present the essential figures and give some hints for the solution. Groups that have finished their task may switch over to another context. Context 1 : The group has to express the lengths of the relevant segments and the areas of the relevant figures by means of letters and to derive the Pythagorean theorem by means of algebraic formulae.
The worksheet for the context is shown in Fig. The worksheet for this context is shown in Fig. Context 3 : Students are asked to fit the four parts of the medium size square and the small square into the big square and to prove that the five parts fill the big square exactly.
The results of the groups are corroborated. The outlines of each of the units on the Pythagorean theorem given in the preceding section are restricted to a short description of a sequence of phases. It might be tempting for student teachers to fill these holes with a step-by-step script promising control and a reduction of the uncertainties of teaching.
This, however, would run counter to the conditions of effective teaching and learning as described in the introduction. Instead of a straitjacket the teacher needs a concept for his or her teaching, and this can be provided only by a professional tool-kit consisting of appropriate general principles. By their very nature these principles go beyond individual teaching units.
They secure that present learning is rooted in past learning and oriented towards future learning, and thus they provide teaching and learning with a direction, locally and globally. It is of paramount importance for appreciating new developments in the teaching of proofs to understand that the evaluation of different types of proof has been controversial in mathematics and in mathematical education over history, particularly in the twentieth century.
In mathematics education, too, it was admired, emulated as far as possible, and hardly ever questioned, apart from a few outsiders see, for example, Clairaut This use of deductive and axiomatic methods focuses attention on an extraordinary accomplishment of fundamental interest: the formulation of an exact notion of absolute rigor.
Such a notion rests on an explicit formulation of the rules of logic and their consequential and meticulous use in deriving from the axioms at issue all subsequent properties, as strictly formulated in theorems. Once the axioms and the rules are fully formulated, everything else is built up from them, without recourse to the outside world, or to intuition, or to experiment An absolutely rigorous proof is rarely given explicitly.
Most verbal or written mathematical proofs are simply sketches which give enough detail to indicate how a full rigorous proof might be constructed. Such sketches thus serve to convey conviction—either the conviction that the result is correct or the conviction that a rigorous proof could be constructed. We also related this to soccer, which is also popular among my students. The concepts they talk about in this video can actually be related to most sports.
After watching, you can ask students to explain how the Pythagorean Theorem can be applied to another sport. I use this video for an anticipatory set after students have a background with the theorem. What a great way to start a daily lesson. I think when you watch it you kinda feel like you could have saved this person from losing their money.
This is a great anticipatory set and a real life example of math. Questions you could ask students to answer during the video:. This can be an excellent way to start a class period on day 3 or 4 of your unit. What conversations will it start with your students?
How many of your students will figure it out? This Shmoop video uses a silly story about Peter Pan and his shadow to illustrate the Pythagorean Theorem. Most 7th and 8th graders seem to appreciate the ridiculousness in these videos. I love to use short math videos as anticipatory sets, and this specific one is a quick review of the Pythagorean Theorem.
Also, you can use it for students who have missed a day or two. It can help them see the Pythagorean Theorem in a different way. My two colleagues have tried this as well.
We needed to do a little more upfront research before diving in. Doodle notes are a cool concept because they let students be creative in math while taking notes.
There are a lot of ways to use task cards. Really, task cards are just a way to practice problems, but they can also be a powerful engagement tool. To practice the Pythagorean Theorem, I use this set of 27 mini task cards available in both printable and digital formats. The digital format is created in Google slides. SCOOT is a game where you put one task card on each desk. Then, students answer the question on their desk. Next, everyone moves one desk to the right and they are given time to do that problem.
Finally, you just keep repeating until students are back to their desk. If you have a small class like me you could have two task cards on each desk. Kids love this activity. One great way to review for test or to see how the class is doing is through playing whole class games. One game I love playing with my class is called knockout. It can be played with or without an interactive whiteboard.
All you need to have is a projector. A student chooses from one of the objects on this screen. After that a question is revealed. Some of the questions have bonuses. The bonuses can be good or bad.
Each question has a point value. Then, after you show the answer, you go back and choose another question. See more in this post. This particular game has solving for a missing side problems.
Also, it contains converse of the Pythagorean Theorem questions as well. That makes it work really well to prep for the unit test. I play this with all students using a whiteboard to work out each problem. Each student answers every problem, and then they keep track of their points for the problems they get right. When needed, I coach students through the problems if they seem to be struggling.
We play this type of game the day before or the day of a test. These games work as a final boost for students.
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