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Understanding Students Understanding Mathematics
Milan Hejný
Introduction
The rule shall therefore run as follows: The scholar should be trained to express everything that he sees in words, and should be taught the meaning of all the words that he uses . For he who cannot express the thoughts of his mind resembles a statue, and he who chatters, without understanding what he says, resembles a parrot.
Comenius, 1649 (translated 1896, p. 329)
A lot has changed in schools in the four centuries since Comenius wrote these words in his Didactica Magna, but the basic problem he describes still remains: Understanding, not chattering, must be the goal of the teaching/learning process. Anna Sierpinska, one of the most accomplished world experts on the teaching of mathematics, has written a whole book on the issue of understanding. In the introduction, she asks questions that are relevant to anyone teaching mathematics today:
How can I teach so that students understand? Why, in spite of all my efforts to explain things clearly, do they still not understand, and still make all these nonsensical errors? What exactly don't they understand? What do they understand and how?
A. Sierpinska (1994, p. xi)
The aim of this article is to discuss these questions and to suggest some solutions. I would also like to encourage teachers to experiment with new approaches and will offer them specific strategies to accomplish this.
Knowledge with or without understanding
Knowledge without understanding might be termed parrot knowledge. Parrot knowledge is not true knowledge, and the phrase itself is something of an oxymoron. Yet we can understand and use this phrase, just as we use the term "soy milk" to denote a milk-free beverage.
In fact, the distinction between parrot knowledge and actual knowledge (with understanding) is only a theoretical tool that enables us to talk about this phenomenon. We would rarely find these two extremes in real life.
Parrot knowledge may be thought of as a cognitive "disease" that is spread by rote learning. In schools where students' independent thinking is suppressed, parroting becomes an epidemic.
An advanced case of parrot knowledge
The following story happened 20 years ago, but it could just as well have happened yesterday. (All eight examples cited in this article are from the author's archive. For additional commentary on stories 1 [Iva], 3 [Fanka], 4 [Filip], and 8 [Ela] see Hejný and Kurina, 2001.)
Story 1: Iva (grade 5) has a good memory, is conscientious, and does all her assignments very well. Her previous teacher regarded her as a top student. However, her new teacher wants the students to understand mathematics.
Teacher 1: All right, so we want to draw an obtuse triangle and divide it into two right-angled triangles. Who can tell me what an obtuse triangle is? Iva.
Iva 1 (squints her eyes and spits out): A triangle that has all three angles obtuse is called an obtuse triangle.
T 2: Now, Iva, you have kind of rushed into it. Come up to the board and show us which of these triangles here are obtuse (there are 7 triangles on the board, two of which are obtuse).
I 2 (coming to the board and announcing): I don't know how to show it. I only know how to say it.
T 3 (surprised by the girl's admission): All right, so tell me what an obtuse angle is.
I 3 (quite confidently): An angle that is bigger than a right angle and smaller than a straight angle is called obtuse.
T 4: And could you draw those for us the right, straight, and obtuse angles?
I 4 (taken aback): No, I only know how to explain it. I don't know how to draw it. (confidently rather than fearfully asks) Did I say it wrong?
T 5: The point about the angles was correct, but the point about the obtuse triangle was wrong.
I 5 (feeling hard done by, she has briskly found a boxed text in her textbook): Here it is:
A triangle whose three angles are all acute is an acute triangle.
T 6: That's right, but that applies to an acute triangle, not to an obtuse triangle.
I 6 (realizing her mistake and blushing): Yeah, I got it mixed up. It's here (quickly finds a different box with the definition of "an obtuse triangle").
Iva repeats memorized words, but the words do not correspond to concepts. Terms such as obtuse triangle or obtuse angle are meaningless for her. This story illustrates two cognitive characteristics of formal knowledge (1 and 2 below); and two basic attitudes that go hand in hand with this knowledge (3 and 4):
- Knowledge is stored in the memory as isolated data (in the example above, as a verbal definition).
- Knowledge is not connected with a student's life experience. (Iva cannot picture the words she uses.)
- The student is convinced that this superficial knowledge is fully acceptable. (Iva claims I can explain it.)
- The student is convinced that s/he is not capable of understanding the underlying concept. (Iva expresses this belief in entries I 2 and I 4.)
It is this last attitude that marks the most advanced stage of rote learning. In such cases any extra classes or coaching are worthless. First, it is necessary to eliminate the student's negative attitude. For a geometry class, for instance, the teacher might give the student an easy assignment, such as teaching her four-year-old brother how to build a block castle; then discuss this activity.
Story 1 points out an important feature of formal knowledge: When there is a mistake in formal knowledge, the student is not able to spot it, let alone eliminate it.
How do we understand the word understanding!
Story 2. I was eating bread and butter with a tomato and a hot pepper, and feeding my two-year-old son porridge. The boy wanted some pepper. I moved it out of his reach and warned him: A pepper is hot, hot! Yet, in an unguarded moment he got hold of the pepper, and right into his mouth it went. Surprise, agony, and tears followed, as well as treatment to soothe his burned little mouth. About a week later when we were shopping, my son pointed to a pepper and said: 'ot! [hot].
A child is eager to learn, wanting to know not only words but also things. The best way to learn is through personal experience. That's how my son came to understand his father's words: A pepper is hot, hot! It was a painful lesson, but as a result it made a profound impression. Now the boy knows not only the simple word pepper, but also the more complicated concept hot, and he knows that his father's intonation conveys important information as well.
Understanding of the concepts conveyed by language about the surrounding world and the world of abstract ideas (notions, relationships, phenomena, situations, conditions, and processes) is gained through experience. It does not always have to be personal experience, though. By reading the directions for use of a washing machine, I acquire explicit knowledge that I can later employ to operate the washing machine correctly. At the same time, having previously read instructions for other appliances, I use my experience to help me understand the directions I am reading now. If I encounter something I cannot understand because I lack the necessary experience, I have a choice: I can go ahead and acquire the needed experience hands-on, or I can ask advice from somebody more experienced.
Definition: The extent of our understanding of a concept is determined by the extent of its connections to our personal experience and explicit knowledge, to other concepts, and to ideas in our minds. If the connections are few, we are dealing with formal knowledge.
How do we come to understand mathematical concepts?
Story 3. Five-year-old Fanka liked to count things with her grandmother. She had counted 2 chairs and 3 chairs, 2 candies and 3 candies, 2 dolls and 3 dolls, several times. So her grandmother decided to ask her a question: There are 3 strawberries under my hand and 2 strawberries under the napkin. How many strawberries altogether? The girl could not see the strawberries. She looked from the napkin to her grandmother's hand, obviously thinking very hard. Then she put two of her left-hand fingers on the napkin and three of her right-hand fingers on the grandmother's hand and counted up her fingers. She called out joyfully, Five, five! The grandmother praised her enthusiastically. Fanka's eyes lighted up as she said, It's always five. Two and three together is five.
The great joy that accompanied the girl's performance came from the magic of the discovery. Fanka discovered that when counting strawberries, she could represent unseen objects with her fingers. Her last sentence, particularly the word always, indicates that her discovery transcended this specific situation. Fanka clearly understood that fingers could be used to represent not only strawberries but any objects.
This story illustrates how knowledge with understanding is formed: Knowledge that results from our own intellectual processes is knowledge with understanding.
Let us take a closer look at the process that led to Fanka's discovery. First, Fanka had clearly had some experience with counting objects and finding the sums, such as 2 dolls and 3 dolls how many altogether? or 2 fingers and 3 fingers how many altogether? She would simply count the objects, not yet knowing the relationship between these sums. Every such sum that the girl counted up was a discrete model of the abstract concept: 2 + 3 = 5. Then a challenge cropped up, a clever question asked in a climate of warmth and encouragement. The challenge involved an obstacle Fanka could not use her usual procedure of straight counting. She had to somehow overcome the obstacle, and she was strongly motivated to do so. (Motivation can be defined as tension that results from the dichotomy between I don't know and I want to know.)
Then Fanka realized that unseen objects (the strawberries) could be represented by fingers. She mentally connected the two separate models the strawberries and the fingers and realized that one could replace the other. This discovery goes significantly beyond the situation from which it originated. It applies not only to the numbers 2 and 3, but to other numbers in the same way. Fanka recognized that in mathematical situations, fingers can serve as universal substitutes. Such knowledge significantly increased Fanka's level of arithmetical knowledge and skills. The girl perceived this change as a delightful feeling, a source of great joy.
How is formal mathematical knowledge created?
Story 4. Five-year-old Filip counted 2 apples and 3 apples to find that there were 5 apples altogether. Then he wanted to add together 2 candies and 3 candies. He made a pile of 2 candies and was just about to make a pile of 3 candies when his father interrupted him. The father showed his son that there was no point in counting again: With the apples, they had found out that 2 + 3 = 5, so the same must apply to the candies. And not only to the candies, but also to fingers, chairs, cars, any object. The boy probably understood his father's explanation, but he felt no enthusiasm for the new knowledge.
The stories of Fanka and Filip show two different approaches to education. In both of them, we are dealing with the concept of representation: The learner comes to recognize that it is not necessary to count actual objects; they can be represented by other objects, e.g., fingers.
However, in these two cases this concept was formed in the children's minds in two very different ways. Fanka worked her way up to it by generalizing from her previous experience; she created the concept by herself. With Filip, the knowledge was given to him, even forced upon him. Both Fanka and Filip will probably use this knowledge equally correctly, just as other children do. However, it is highly probable that Fanka, who managed to discover the idea of representation for herself, will be more successful later on in understanding other processes of representation, such as substituting letters for numbers in algebraic expressions.
Abstraction
Story 5: In grade 5, as part of the explanation of percentages, we did several problems such as:
In Form 5A there are 8 girls; and in Form 5B there are only 6 girls. In Form 5A 4 girls have a pet at home; in Form 5B 3 girls have a pet at home. Which form has a larger proportion of pet-owners? (This problem can be represented as: 5A: 8g, 4p; 5B: 6g, 3p, where g=girls and p=pet-owners.)
The children found the solution rather quickly: The situation is the same in both forms because there is one pet for every two girls in both forms.
Then we moved on to more difficult questions, for example:
A: 10g, 5p; B: 11g, 6p, or
A: 13g, 4p; B: 9g, 3p, or
A: 12g, 7p; B: 11g, 6p, and so on.
Then there was a very difficult question: A: 13g, 4p; B: 10g, 3p.
In this case the first solution proposed by one of the students, although it appeared logical, was inaccurate: Both forms are in the same situation because if we take away one girl from each form, there will be one pet for three girls in each form.
Some students were not happy with this solution and voiced strong objections to it. Still, for a long time we failed to resolve the problem. Martin claimed that Form B is more "pet-full," but none of his schoolmates could understand his strange argumentation. About a month later, when most students had forgotten about our unsolved question, Martin came up with a wonderful idea: In Mathland (a kind of Never-Never land, where incredible things happen) there is a school with huge classrooms, and each class has 130 girls. In Form C there are 40 pet-owners out of 130 girls; in Form D 39 pet-owners out of 130 girls. Clearly, Form C is more "pet-full." Since form C is 10 times bigger than our Form A, and Form D is 13 times bigger than our Form B, this means that our Form B is more "pet-full" than our Form A.
Martin's solution was based on his notion of a "Mathland classroom," something we can imagine even though it does not correspond to our everyday experience. By means of this abstract notion he was able to grasp the comparison of the ratios represented in the question, 13:4 and 10:3, and find the solution.
The thinking process that led Martin to his discovery is abstraction. Martin looked beyond the concrete aspects of our school, and considered the ratios 13:4 and 10:3 as objects in the number system, without any semantic restriction. The abstract knowledge contained in his discovery can be represented by algebraic symbols: If I want to compare two ratios x:u andy:v (13:4 and 10:3 in our problem about the pets), I can do this by creating equivalent ratios x:u=xy:uy (13:4=130:40 in Martin's solution) and y:v=xy:xv (10:3=130:39). Since the first number in each of these new ratios is the same, we can simply compare the second numbers, uy and vx (40 and 39).
Martin was influenced by his discovery for a long time. He tried to use the idea of proportion in different situations, connecting the idea to his previous knowledge. This process of situating new knowledge in our existing knowledge structure may be called crystallization. Sometimes the crystallization process involves a significant restructuring of our existing knowledge in light of new understanding.
Cognitive mechanism
We have described the individual parts of the cognitive mechanism involved in the formation of non-formal knowledge. We can now arrange the whole mechanism clearly in a sequence:
Motivation -> Separated models => Universal models => Abstract knowledge -> Crystallization
In this sequence, the two outer arrows (->) mark processes that occur spontaneously. Each of the two inner arrows (=>) represents a thinking step, which can be facilitated by an appropriate educational strategy. The first inner arrow is usually a process of generalization, the second of abstraction. Each of those steps is important for the quality of the resulting knowledge.
Building the concept of negative numbers: An illustration
The concept of negative numbers is very difficult for primary school students, much more difficult than learning about fractions. The history of mathematics underlines this fact. Even ancient civilizations worked with fractions. Yet mathematicians were still arguing about the legitimacy of negative numbers as recently as the middle of the 18th century, at a time when differential calculus was already well established (Kline, 1980). No wonder a sixth grader has difficulty with this concept.
When I was teaching math to students in grades 3 through 6 (1984-1986), I wanted to find a way to allow even the students who struggled with math to develop an idea of negative numbers. I developed the following strategy:
- Carefully build up various discrete models of negative numbers, starting in grade 3.
- Have students encounter negative numbers again and again. This should be done by expanding their experience of arithmetical structures.
- Make systematic use of the diversity of students' opinions to foster discussion and create a learning climate encouraging to all students, regardless of their aptitudes.
In everyday experience, I found only four discrete models for negative numbers: a thermometer (e.g., the temperature is -5 °C), an elevator (the buttons for the floors below ground level are sometimes marked -1, -2, etc.), a map of the ocean floor (the depth of the sea bed), and finances ("I owe 100 Czech crowns" is the same as "I have -100 Czech crowns." However, this last model is problematic because children find it unnatural and confusing.)
In arithmetic, we arrive at negative numbers either by counting backwards ... 4, 3, 2, 1, 0, -1, -2, or by subtracting a bigger number from a smaller one. For example, if we take away 5 from 3, we get -2. This problem can also be expressed by the apparently nonsensical question, How much do I have to add to 5 to get 3?
It seemed almost impossible to have students discover the idea of a negative number by themselves. And yet, if I wanted this concept to enter the classroom in a constructivist guise, I needed to find a challenging situation that would lead at least one student to the discovery. From him, others would come to understand through informational osmosis. I will present four strategies that proved successful in my classroom.
Story 6: (Grade 3) One day in December I introduced myself to the class as a magician. I wrote on the board
add 2
subtract 7
add 10
Then I asked the students to think of any three-digit number and carry out the three steps written on the board. For example, if I think of 156 and add 2, I get 158; then I take away 7 and get 151, add 10 and get 161. When the students had finished counting, I asked Eva what number she ended up with. Eva said 177. I told her that the number she was thinking of was 172. Then I repeated the magic with two more students. Afterwards Jarda and Katka shouted out that they could do the magic as well. They both said correctly that if the final number was 511 then the starting number was 506; and if the final number was 105 then the starting number was 100. By that time, other children had figured out the trick, and Milan said, You just have to take away 5.I asked whether Milan's rule applied to two-digit and four-digit numbers as well. The children very quickly arrived at a positive answer. Then the bell rang, and I asked in a mysterious tone whether the rule applied even to one-digit numbers. The children yelled back that it did. I let their assertion stand and left the class.
The following day Andrea announced that she had tried the magic trick at home for all one-digit numbers, and that with the numbers 1, 2, 3, and 4 it was not possible. At my request, she showed the class why it would not work for the number 3, explaining, When I add two, I get five, but now I have to take away seven and I can't do that. Then Honza said, If you take away seven, you'll get minus two! Honza's wisdom, obviously gleaned from his older brother, caught me off guard. Luckily, Andrea did not accept this minus two at all and a discussion ensued. Dan aligned himself with Honza, but most children agreed with Andrea, who insisted that Honza show her the minus two. That he could not do. No other important arguments were raised and the controversy faded away for a year. During that time, we came across negative numbers a few more times, and each time we reminded ourselves of Honza's discovery. The students probably came closest to the solution when we were working on the following problem: The temperature at noon was +7 °C. By evening the temperature had dropped by 10 °G How many degrees did the thermometer read in the evening? But again the students did not accept the negative number as "a real number." They understood the -3 on the thermometer as a code for three winter degrees.
At the beginning of grade 4, Honza came up with an idea that he believed could show when you go down to minus. He referred to the story The Searchers, which at that time was as well known as Harry Potter is today. In one episode, the searchers were forcing their way through an underground corridor, which went up or down in places. Honza drew the way his searcher was moving in the underground corridor: first two steps up, then seven steps down, and then 10 steps up. Pointing at his drawing, he explained that the exit from the underground corridor was always 5 steps higher than its entry.
Thanks to his reference to a well-known story, Honza gained another couple of recruits for his "minus" numbers theory. But none of them could answer Andrea's question, which the teacher now mentioned again: Show how much this minus two is. Despite this shortcoming, for Honza, and certainly for a few other students, the underground corridor was a discrete model we could use to descend to numbers less than zero.
Story 7: (Grades 3 and 4). In preparation for problems about age, we played a game called "Chronos" starting in grade 3. A number line with the numbers 1-20 was drawn on the floor, and the class worked together on problems such as: Adam is 3, Betka is 5 years old. In how many years will their ages add up to 20? To solve this problem, we used a dramatization:
Two students stand on the number line: Adam on number 3, Betka on number 5. The student playing the god of time, Chronos, proclaims, One year has passed now, and both students take one step forward. Adam is now standing on number 4 and Betka on 6. The student playing the watchman announces, Four and six are ten, Adam's and Betka's ages now add up to ten. Chronos issues another proclamation, the children proceed to marks 5 and 7, and the watchman announces 12. The game goes on until Adam gets to number 9 and Betka to 11. At that moment the watchman calls out, Adam's and Betka's ages now add up to twenty. We've found the solution!
We played this type of game about five times in grade 3. We rarely managed to finish the whole performance. After just the first two or three steps the students would yell out the answer, and we only had to check it.
In September, in grade 4, we were working on the following problem: The twins Petr and Pavel are 5 years old and their sister Radka is 11. How many years ago did their ages all add up to (a) 15, (b) 6, (c) 3? To solve this problem, we went backwards on the number line into the past. Two steps back and we got the solution to (a). Question (b) provoked merriment in the classroom when the children found out that the twins were 0 years old, i.e., that they had just been born. Then an argument arose about the solution to question (c). Dan, who was playing one of the twins, extended the number line backwards to include the numbers -1 and -2. Most of the children either did not understand this idea or were against it. Dan claimed that his parents had already known two years before his birth that they would name him Dan. But the class refused to contemplate the age of a person who had not been born yet. On the other hand, the children admitted that it was all right for Maruska's mother to say, with a sigh, Today, great-grandfather Arnost would have been 100 years old, because in this case the great-grandfather had once been alive. The debate was intense and went on for half an hour. The children eventually admitted and I considered this Dan's great success that the number -2 could be on the number line meaning "two years before birth."
Story 8: (Grade 5) All the students had accepted the system of negative numbers as an extension downwards of the natural numbers. But problems persisted with the semantic understanding of these numbers. I managed a breakthrough with the story of Ela (a first grader), which I told the class in November. Ela's father asked her mother how much money she had and the mother answered, Minus 200. Right now I've got 100, but I owe 300 for the phone. Ela cried out very happily, That means you don't have 200 crowns!, because she had understood the word minus.
Two of the students took to Ela's idea right away, and in their mutual conversation they started to use the word minus jokingly in the sense of negation. Instead of I owe you 10 crowns they would say You owe me minus 10 crowns; or instead of We lost by one goal they would say We won by minus one goal and so on. This way of speaking quickly spread among the other students. Initially some of the students made inaccurate and nonsensical sentences. For example, one girl reformulated my statement The register is not here as Here is a minus register. But within a month, the use of the word minus had become clear to almost all the students. One student who usually struggled with math told me that he had explained to his grandfather, When I said it was minus 5 °C below zero, I actually meant it was 5° above zero. The boy added that his grandfather did not grasp it anyway.
Story 9: (Grade 6) The students had accepted the concept of negative numbers. We now used a toy figure called Puk to illustrate problems on a number line. For addition and subtraction problems (e.g. 5 + 3 = ?, 5-3 = ?, 5 + (-3) = ?, and 5 - (-3)= ?, (-5) + (-3) = ?, (-5) - 3 = ?) the figure moved on the number line according to the following rules:
1. The first number indicates Puk's starting position. Initially Puk always faces to the right: If he starts on 5 he is facing toward 6; if he starts on -5 he is facing toward -4.
2. If a - sign follows the first number (subtraction), Puk makes an about-face; if a + sign follows (addition), Puk stays facing in the same direction.
3. The next number indicates how many steps Puk takes, and in which direction: The number 3 (or +3) means Puk takes three steps forward; the number -3 means Puk takes three steps backward. (My students like to call these backwards steps "pets.")
Here are some examples:
5 + 3 = ?
Puk starts on the number 5, facing toward
6. He goes 3 steps forward and stops on 8.
5 - (-3) = ?
Puk starts on 5, facing toward 6. He makes an about-face (subtraction) so he is now facing toward 4. He goes 3 steps backwards (-3) and finds himself on the number 8.
-5 - (+3) = ?
Puk starts on negative 5 (-5), facing toward -4. He makes an about-face and is now facing -6. He goes 3 steps forward and ends up at -8.
The four illustrations in Stories 6-9 above are only some of the possibilities for a constructivist approach towards negative numbers. I encourage you to try your own experiments, and I wish you good luck. You will undoubtedly encounter disappointments and disillusionment, but you will also find unique moments of pleasure in your students' unexpected performances.
Acknowledgement: This case study was funded by Project VZ J 13/98/114100002.
References
Comenius, J.A. (1896). The great didactic of John Amos Comenius. Translated into English and edited with biographical, historical, and critical introductions by M.W. Keatinge. London: Adam and Charles Black. (Original work published 1649)
Hejný, M. & Kurina, F. (2001.) Dite, skola a matematika [Child, School and Mathematics]. Prague: Portal.
Kline, M. (1980). Mathematics: The loss of certainty. New York: Oxford University Press.
Sierpinska, A. (1994). Understanding mathematics. London & Washington, DC: Falmer.
Milan Hejný is a Professor in the Faculty of Education at Charles University, Prague, Czech Republic.
E-mail milan.hejny@pedf.cuni.cz.
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Understanding Students Understanding Mathematics (Milan Hejny)