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Reading: Learning to See the Wind, unpublished paper by Sid Rachlin, Pg. 1–9.

Please attempt all problems. 2 1. a. Find two fractions in lowest terms with unequal denominators whose difference is . 13

3 1 1 b. What added to the sum of 5 4 , 1 3 , and 4 4 equals 99?

ALGEBRAIC CONCEPTS AND RELATIONSHIPS Problem #9—Page #1 2. Two ways to use tiles to represent the expression 3x–2 are:

Each diagram could be described by equations. For example, for the first diagram [2x + (–1)] + [x + (–1)] = 3x – 2; for the second diagram 4x + (–x) + 2 + (–4) = 3x – 2 or for the second diagram [2x + 1] + [2x + 1] + [(–x) + (–4)] = 3x – 2. a. Show two ways to represent the expression 2x + 2

b. Write two equations for your representations in part a.

c. Show two ways to represent the expression 3 – x.

d. Write two equations for your representations in part c.

e. If you were to use a tile to represent x2, what would it look like? Why?

Problem Set #9—Page #2 Name...... 3. a. There is a red tag sale at a store. Which deal is better for you? Explain your answer. (1) The store first marks down an item 30% and then marks it down an additional 25%. (2) The store first marks down an item 25% and then marks it down an additional 30%.

b. Which deal is better for you? Explain your answer. (1) The store first marks down an item 30% and then marks it down an additional 25%. (2) The store has a half price sale.

ALGEBRAIC CONCEPTS AND RELATIONSHIPS Problem Set #9—Page #3 4. a. As a part of the Click-it or Ticket campaign the signs throughout Greenville report seat belt use. Assuming that have been rounded to the nearest per cent, what is the smallest number of cars that could have been sampled to get the record seat belt use of 93%? b. What is the smallest number of cars that could have been sampled to get “last month’s” seat belt use of 88%? c. Suppose the same number of cars are sampled each month. What is the smallest number of cars that could have been sampled to get the percentages indicated on the sign?

Problem Set #9—Page #4 Name......

5. Draw a diagram representing how you could use base five materials to demonstrate the following operations. After you perform the given , represent your answer using the smallest possible number of objects.

a. The sum of 412 Base five and 234 Base five.

b. The difference of 412 Base five and 234 Base five. Cube Flat Long Unit

ALGEBRAIC CONCEPTS AND RELATIONSHIPS Problem Set #9—Page #5 6. Consider the numbers 97, 167, and 391. (Problem is adapted from a task used in a seminar given by Rina Zazkis.) a. List all factors of 97. How did you make sure you found all factors?

b. List all factors of 167.

c. List all factors of 391.

d. Multiply 97 and 167. Call this number J.

e. Will 7½J? (This reads, will 7 divide J, meaning will 7 divide J without a remainder?)

f. Will 13½J? What method did you use to answer this question?

Writing Assignment #9: Authors of mathematics textbooks are sometimes criticized for writing a narrow set of problems that make students appear to be successful, but do not give them a well-rounded conception of a particular idea or operation. For example, students may only think about as “take-away” and may even use take-away instead of the word subtract. Below is one way to think about categorizing and subtraction tasks. Write tasks for the missing sections.

Result Unknown (m) Change Unknown (r) Start Unknown (£) £ Æ r = m Deena found 8 books on Write a join task with Norman had some change in JOIN dolphins for her report. Serena change unknown that his desk. He took 83¢ out of gave her 6 more. How many involves fractions. his pocket and put it in the books does Deena have now? desk. He now has $3.52 in the desk. How much money was in the desk to start with? Result Unknown (m) Change Unknown (r) Start Unknown (£) £ – r = m Jorge has a deck plank 8 and Reliable Motors had 127 vehicles Write a separate task SEPARATE 1/2 feet long. If he cuts off a on the lot before their annual with start unknown piece 5 and 3/4 feet long, how inventory reduction sale. They only that involves decimals. long is the remaining piece? had 78 cars at the end of the sale. How many cars did they sell? Difference Unknown (m) Compare Quantity Reference Quantity £ – r = m Unknown (r) Unknown (£) Write a compare task COMPARE with difference unknown Stan is 2 and 1/2” taller than his Marie’s time for 1600m was that involves whole brother. If his brother is 5'10”, how 3.7 seconds better than numbers. tall is Stan? Jonell’s time. If Jonell’s time was 5:02.6, what was Marie’s time?

Problem Set #9—Page #6 When I was at the University of Hawaii, a colleague of mine told me a story of an experience he had as a first grader growing up in Hawaii nearly forty years earlier. It was a simpler time, one in which a trip from one side of the island to the other was no small feat, since they traveled over a small mountain road called the Pali that stretched through a pass in the mountain range. At the top of the mountain, his family stopped to enjoy the view. As he watched from the windy edge of the mountain's lookout, he made a discovery he deemed worthy of sharing at show-and-tell the next Monday. Contrary to what he had been taught in class, he could see the wind!

On Monday, he anxiously awaited his turn to share his discovery. But to his disappointment, when he told the class of his adventure, rather than share in his excitement the teacher corrected him. “Now David,” she said. “You know that you can not see the wind. What you saw was the dust and dirt being blown around by the wind—the wind is invisible.”

There are two things that struck me as important in my colleague’s tale. First, how meaningful an experience this must have been for him to have held it in memory for nearly forty years. What made this a memorable experience in his life? And second, how mathematics is like the wind.

As my colleague spoke, my mind wandered to thoughts of the old Claude Raines movies of the invisible man. How did one catch an invisible man? In one Abbott and Costello version, the invisible man was captured in a kitchen by tossing flour in the air. Amidst the flour the invisible man became visible and was caught. But what was it that was seen? After all, the invisible man was by definition invisible, so all that we saw was flour.

And what of mathematics? Are the symbols of mathematics really mathematics or merely the bits of flour that we throw about to help us find mathematics. Too often mathematics is taught as if the flour is what is important. Yet all of the symbols, and even the manipulatives, , and the computer software, are only flour. The pages of dittoed ink do not make mathematics. The base ten sets and counting aids do not make mathematics. The texts and technology, manipulatives and worksheets provide opportunities for our students to see mathematics — to see the invisible man. But only if our students are looking for mathematics.

Mathematics is like the wind. It is seen, when some would argue it cannot be seen. It is touched, when some would argue it cannot be touched. It is taught, when some would argue it cannot be taught. It is understood, when some would argue it cannot be understood.

Learning to See the Wind —1 The challenge of teaching mathematics is to provide an environment in which students can see, touch, understand, and learn. The learning environment we provide, will set the probability of students understanding, re-creating, applying, and retaining the patterns of mathematics. While manipulatives, computer software, cooperative learning and portfolios may be used by students to facilitate their learning, it is the role of instruction to coordinate the alternatives and maximize learning. To this end, instruction and curriculum are inseparable. The curriculum must be organized to enhance the opportunities for students to learn. The instructor must present the curriculum in a manner that increases the learning potential.

Mathematics is learned through communication — “students need opportunities not just to listen, but to speak mathematics themselves (Silver, Kilpatrick, & Schlesinger, 1990).” Although it is not uncommon for students to be asked to put homework problems on the chalkboard or the overhead projector so that other students can see their work, students are seldom asked to put into words not only what they did but also how and why they did it. This is particularly true when the tasks they are asked to explain are repetitions of trained algorithms. I am reminded of a third grade classroom that I had an opportunity to visit. The teacher was explaining how to do three-digit subtraction with regrouping.

“Suppose I have 568 – 279.

15 1 You can't subtract eight from nine, so you'll have to regroup. Eighteen subtract nine is 4 568 nine. You can't subtract five from seven, so you'll have to regroup. Fifteen subtract seven is –279 eight. And four subtract two is two. The answer is 289.” 289

Although the teacher's was correct, her words were not. You can subtract eight from nine and five from seven. After she presented another example to the class, she sent five students to the board—each with a different subtraction problem. After all of the students completed this problem and the class had an opportunity to finish all five problems, she asked the students to explain what they had done. Each student in turn parroted the same incorrect justification. For example, if the problem was 253 – 168, the child would begin by saying “You can't subtract three from eight.”

For the students’ communication to help them make mathematics meaningful it must go beyond recitation. Instead of a repetition of the demonstrated algorithm, students might be asked to find a three-digit number that subtracted from 253 has a difference that's greater than 60 and less than 100. By opening the problem up, you provide greater opportunities for discussion. How did you solve the problem? Why did you choose to do it that way? Did anyone solve the problem another way? Is the answer unique? Notice that each correct solution provides an opportunity to discuss subtraction with regrouping. Create another open-ended problem such that all of the solutions provide opportunities to discuss subtraction with regrouping.

The learning of mathematics, beyond the level of rote memorization of formulas and algorithms, can be regarded as a kind of problem-solving process. That is, even the application of formulas to routine textbook exercises involves some degree of problem-solving activity on the part of most students, at least for a while. Accordingly, the National Council of Teachers of Mathematics (1989) proposes that problem solving not be treated as a distinct topic, but rather be viewed as a process that should permeate all mathematics instruction by providing “the context in which concepts and skills can be learned.” The successful study of mathematics requires a set of problem-solving processes that help students to reason mathematically, to provide a context in which to communicate their reasoning, and to provide opportunities for students to link their learning to prior mathematical contexts. Three of the basic processes identified as starting points are included in Krutetskii's (1976) model of mathematical abilities — reversibility, flexibility, and an ability to generalize.

Reversibility is an ability to restructure the direction of a mental process from a direct to a reverse train of thought.

For example, in the expression O + D = £ we might be given values for O and D, and be asked to find a value for £ . The reversibility of this addition incorporates three variations: where the values of O and £ are given and the value of D is to be found, where the values of D and are given and the value of O is to be found, and where the value of £ is known and both O and D are to be found. 2— Rachlin For children to possess complete reversibility of addition of whole numbers, they should be able to solve problems involving all three variations: 5 + D = 7, O + 2 = 7, and O + D = 7.

To create a non-routine routine task we need only switch the given information of the standard textbook problem. For a binary operation *, we are given that O * D R £ where * is an operation, such as addition, subtraction, , , raise to a power, find the root of; R is relation such as equal to or greater than; and O, D, and £ are numbers or algebraic expressions. Here are some examples:

Add 7 + 5 = becomes: Seven added to what number equals 12 7 + D = 12 What number added to 5 equals 12 O + 5 = 12 Find two numbers whose sum is 12 O + D = 12

2 2 Find the sum of 2x – 5xy + (2xy + y ) becomes: 2 2 2 What polynomial added to 2x – 5xy equls 2x – 3xy + y ? 2 2 2 (2x 5xy) + D = x – 3xy + y 2 2 2 The polynomial 2xy + y added to what polynomial equals 2x – 3xy + y ? 2 2 2 O + (2xy + y ) = 2x – 3xy + y 2 2 Find two polynomials whose sum is 2x – 3xy + y . 2 2 O + D = 2x – 3xy + y

- 2 3 Simplify 64 becomes: - 2 2 1 3 1 What number raised to the power - 3 equals 16 ? O = 16 1 D 1 64 raised to what power equals 16 ? 64 = 16 1 D 1 Find an that raised to a negative power equals 16 . O = 16

The effect of the position of an unknown quantity can greatly effect a student's solution processes and the level of difficulty of the problem presents. Wagner, Rachlin, and Jensen (1984) interviewed ninth grade algebra students in Athens, Georgia and Calgary, Alberta with a series of problems based on variations in the missing terms of the form £ * b = c and a * £ = c where * represents an algebraic operation and a, b, and c are whole numbers, fractions, polynomials, algebraic fractions, or radical expressions. They found wide differences in the ease with which students were able to solve these missing term tasks depending on the operation substituted for *. The difficulty of the problems varied depending on the operation and the placement of the missing term. For example, the following missing term task was a problem for most ninth grade algebra students. What number multiplied by 2 equals 3 ? 3 2

2 3 3 2 2 3 Many students simply multiplied ´ , while others were unsure whether to represent the problem as ¸ or ¸ . Students 3 2 2 3 3 2 tended to be rule-oriented and liked to state generalizations about the solution process before beginning the problem. In many cases students falsely generalized about how to do the problem, either over-generalizing about something that occurs in the problem or something they heard their teacher say (Jensen, Rachlin, & Wagner, 1982). How did you solve the problem? Why did you solve it that way? Is there another way to solve the problem? The ninth graders who solved the problem used a variety techniques.

Learning to See the Wind —3 Kathy solved the problem like an Curtis said that first he found Margaret solved the problem a third way. First 2 2 3 equation. First, she wrote common denominators for the and she multiplied the by to get 1. Then she 2 3 3 3 2 3 3 3 3 n = 2 , then she multiplied both and then rewrote the problem multiplied the 1 by to get . She wrote this 3 2 2 2 sides of the equation by 2 . 4 9 2 3 3 3 6 · ( ) = 6 . But 6 · 1 = 6 and as (3 · 2 ) · 2 = 2 . The fraction she was 9 3 3 9 9 4 looking for was ( 2 · 2 ) or 4 . 4 · 4 = 9 so the answer was 1 .

Lani had still another way to solve the problem. She wrote Joe solved it a fifth way. First he wrote 3 2 3 the 2 as an equivalent fraction such that 2 would divide 3 · ( ) = 2 and then he filled in the parentheses. Since into its numerator evenly and 3 would divide into its he wanted to get a 2 in the denominator, he needed a 2 in 18 the denominator of the fraction in the parentheses. But denominator evenly. She chose 12 as a fraction equivalent 3 2 18 that 2 would "cancel" with the 2 in the numerator of the to 2 that met the conditions. Then 3 · ? = 12 . Since 2 9 3 so he needed a 4 in the denominator of the fraction in 18 ÷ 2 = 9 and 12 ÷ 3 = 4, she decided her solution was 4 . the parentheses. Similarly, he said that he needed a 9 in the numerator. Why?

2 9 2 3 By changing the problem from “Multiply 3 ´ 4 .” to “What number multiplied by 3 equals 2 ?” the nature of the variety of ways that students solved the problem changed. Each student's method is rich with opportunities for discussion. Students' errors provide opportunities for new directions or to clarify misconceptions. For example, Curtis' complex fraction solution opened up a new topic for discussion.

Although the problem is not much different than the original task, the richness of the discussions is quite different. Examples of increased difficulty that results from changing the placement of an unknown can be found from the beginnings of elementary school to the university. Hiebert (1982) examined the effect of the position of the unknown term on first-grade children's representation and solution processes for verbally presented addition and subtraction problems. Forty-seven first- grade children were given three joining problems and three separating problems in an individual interview. As with the algebra students, Hiebert found that the position of the unknown had a profound effect on the solution processes and relative difficulty of the task. Fifty-five percent of the responses to the verbal problem a + b = £ and to the verbal problem a – b = £ included modeling with cubes. The percentage drops to about 40% for the a ± £ = c problems and to about 18% for the £ ± b = c problems. The latter problems were rarely modeled with cubes. Some researchers (see for example Cobb, 1986 and DeCorte, Verschaffel, & DeWin, 1984 ) have questioned the meaning of tasks such as the following example of an 8 – 3 = £ word problem used by Hiebert. Bill had 8 marbles. He gave 3 marbles to Susan. How many marbles did Bill have left? These researchers argue that the students do not truly understand the intent of such problems. For example, Cobb reported that on a similar task eight of 34 first graders would act out the problem by first counting out 8 marbles and then reach into the bag for more marbles as they heard the second sentence. Interviews revealed that these children assumed that the entire bag of marbles belonged to Bill. Although several children might play with marbles and be the temporary owner, the toys usually belong to one or more of the children who are sharing them with the others. The interpretation that Bill gives himself 8 marbles and Susan 3 marbles to play with yet Bill still has 8 marbles with which to play, is consistent with their real life experiences. In a sense, these typical story problems are somewhat artificial. Although this analysis is may suggest one source of error, a greater one occurs in the realm of content domain. As the content domain becomes less familiar errors in the applications of algorithms become more likely. In the study described earlier, Jensen, Rachlin, and Wagner (1982) designed a series tasks to examine the influence of the content domain on the students' processes of solution. Although these tasks were syntactically parallel the ways in which the students solved the problems changed as the content areas became more complex. 1. What number multiplied by 17 is 204? 2 3 2. What number multiplied by 3 equals 2 ? 4— Rachlin 3. What binomial multiplied by x + 5 equals 2x2 + 15x + 25?

4. What polynomial multiplied by x + y equals 2x2y + 2xy2 + 3xy + 3y2

Almost all students solved Task 1 dividing 204 by 17. Yet many students attempted to solve Task 2 by repeated "guess and test". The students’ inability to solve this task was frequently due to a fixation with one approach toward a solution. Another student was drawn by the power of her past experience with reciprocals and kept losing track of the goal of the problem. Once she was finally able to get past this barrier she incorrectly stated the resulting division problem.

In an earlier study, Rachlin (1982) included a series of missing term tasks with parallel tasks from different content domains in his study of the mathematical understanding of four successful college-level intermediate algebra students. He also found wide variability on syntactically similar tasks. In this case, students experienced difficulty with the following task: What real number added to the 3 = 6 ?

Although the students had received either an A or a B on their chapter tests on operations with real numbers, they sensed a lack-of-closure in writing their answer as 6 – 3 . Students would try anything to "finish" the problem. Some tried fractional exponents, others changed the index, and still others turned to tables in an effort to get the answer. Although these students had learned to accept fractions as both indicated operations and numbers, real number expressions still felt incomplete.

Much of the number/algebra strand of pre-college mathematics refers to actions that can be reversed. These include the actions of graphing, making a table, solving an equation or problem, simplifying an expression by performing indicated operations. Rather than merely working backwards through the steps of a algorithm, reversibility in thinking refers to an ability to find an alternative path for meeting the conditions of the problem. The development of a student’s ability to switch from one solution path to another is included in Krutetskii’s (1976) second basic problem solving process — flexibility.

Flexibility is the ability to switch from thinking of one method of solving a problem to another.

How students perceive a problem shapes the other processes that they may bring to bear on the solution of the problem. The various solution paths that a student selects establish their structure for the problem. For example, the problem “What 3 number divided by 24 equals 4 ?” has a wide variety of appropriate solution paths depending on the way in which the task is perceived; i.e., as equivalent fractions, a proportion, a division problem, an equation, etc. Students who have only one way to see a problem limit their ability to solve it and to link it to other mathematics. If a student tells you that they solved the problem like equivalent equations, what equation do you think they wrote to represent the problem? What about if they said that they solved it like a proportion? Suppose instead of using words, I had given you the problem with a blank to be filled 3 in, £ ÷ 24 = 4 , would you think of solving the task like equivalent fractions? By providing the problems in words, the students are given an opportunity to translate the problem into their own symbolic or pictorial representation. Each representation is rich with its own suggestive solution paths based on a student’s past experience.

3 ❏ 3 = 4 24 4 24)?

What is the relationship between a student's structure for problems and his or her teacher's anticipation of the student's structure. Wagner, Rachlin, and Jensen supplemented their study of students' learning difficulties in with the students' classroom teacher's analysis of the students problem-solving process in algebra. After eight interviews were conducted with each of ten students in Athens, Georgia and four students in Calgary, Alberta, the classroom teachers were asked to complete all of the interview tasks. Then they were requested to guess how each of their students would solve the Learning to See the Wind —5 problems. Finally, the teachers were able to listen to (in Georgia) or watch (in Alberta) the interviews to test the accuracy of their predictions. Rachlin (1983) reported on results of the interviews with the Calgary teacher. The teacher was very flexible with tasks such as "What number added to the sum of 17 and 6 equals 6?" and could solve the tasks in several ways. But, he was surprised to find that only one student in his advanced section solved this problem by noting the sixes in both members. Instead the others first added 17 and 6 and then subtracted 23 from 6.

The structure of the students' solutions also at times varied from classroom practice. For example, the teacher was surprised that three of the four students solved tasks such as “What trinomial subtracted from 5x2y - xy2 + 7 equals -x2y + 8?” by writing the parts vertically. With regards to the vertical form the teacher commented, “They've seen it occasionally in the textbooks, but I've never assigned the assignments that have vertical form.” At the same time that students create their own approaches, they may not realize the relationships between the approaches they do not own. For the child, there may be no reason to assume that two different ways to solve a problem should be connected. In the following account, my son Jeff was just beginning second grade.

Beside Jeff's bed hung a chalkboard. If I had been a phys. ed. teacher, I probably would have placed a trampoline there, but as a math teacher a chalkboard won out. Every so often I would come into his room and write a math problem on his board. At another time, he'd come in and solve the problem. Still later, I'd stop back and check his solution. If it was correct I'd erase it and place another problem on the board. If it was incorrect I'd call him in and we'd discuss it.

On one occasion, when Jeff had just been introduced to addition with regrouping (carrying), I wrote the problem: 24 + 16

When I later returned to the room, I found Jeff's solution: 1 24 + 16 41

After I called Jeff into the room, the following dialogue ensued. “Jeff, I think there's an error here.”

“No, there isn't Dad. Look! Four and six are ten. You put down the one and carry the one. One and two are three and one is four. The answer is 41.”

“No, Jeff, I think something is wrong here.”

“Look! [Jeff spoke a little louder to make his explanation clearly more acceptable.] Four and six are ten. You put down the one and carry the one. One and two are three and one is four. The answer is 41. You ask Miss Frame, she'll tell you how to do these.”

As a math teacher I took this as a sign that a concrete embodiment was needed. After all, using concrete objects makes math make sense. I went and got a pack of toothpicks and a box of rubber bands and returned to sit on the floor beneath the chalkboard. Jeff had grouped by tens before and had no difficulty representing 24 as two tens and four ones and 16 as one ten and six ones. He added (combined the two piles) and got three tens and ten ones or after trading in the ten ones for one ten he had an answer of 40 with the toothpicks. At this point he looked back and forth at his pile of toothpicks and the chalkboard. Finally, he said very seriously, “That's what you get when you add toothpicks, but when you work on the board you get this answer.” 6— Rachlin To my surprise, I later learned that Jeff's response is not that unusual. It has been reported by other parents and teachers (Wirtz, 1983). But nonetheless, it serves as an important reminder for teachers and parents. Why should children expect that what they get on the board should match what they get with objects? If we are using concrete objects to serve as a foundation for arithmetic operations, we must make sure that the procedures used with the concrete materials parallels the procedures used in the rote algorithms being taught. The multiple representations for problems must be linked within the students’ minds, as well as the teachers’. While it is the way that a student solves a problem that dictates whether the problem is helpful in the development of the student’s problem-solving processes, here are some examples of problems that have been designed to develop a student’s flexibility: 1a) What number subtracted from 200 equals 148? 1b) Can you solve the problem another way? 2) What number added to the sum of 48 and 64 equals 148? 3a) What number divided by 36 equals 26 with a remainder of 1. 3b) What number divided by 36 equals 25 with a remainder of 1. 3c) What number divided by 36 equals 24 with a remainder of 1. Problem 1 is used to stress multiple solutions paths for problems. The most popular way for students to solve problem 2 is by adding 48 and 64 and subtracting the sum from 148. It is only through a class discussion that these students will come to realize that there is another student way to solve the problem, that they wish they had thought of. The third problem represents a class of problems in which the solution of the first part, should change the way that the student solves the remaining parts. A further example of this later type of problem is given by the following example. 4a) Solve: 4a2 – 9 = 0 4b) Solve: 4(a + 1)2 – 9 = 0. 4c) Solve: 4(2a + 1)2 – 9 = 0.

A third basic problem-solving process identified by Krutetskii (1976) is the ability to generalize.

Generalization is the ability to deduce the general from particular cases to form a concept and the ability subsumes a particular case under a known general concept.

Krutetskii's first type of generalization of mathematical material is closely related to Dienes' (1965) definition of abstraction as a process of class formation. The following are examples of problems that might be used to have students form their own generalizations. 1. Find two 3-digit numbers whose difference is a 2-digit number 2. Find two numbers such that when you divide them you get a remainder of 201. 2 3. Find two fractions in lowest terms with unequal denominators whose difference is 13 . 4a) Find two fractions with different denominators whose sum is less than 1. 4b) Find two fractions with different denominators whose product is less than 1. 4c) Find two fractions with different denominators whose quotient is less than 1. 5a) Find two proper fractions whose sum is not a proper fraction. 5b) Find two proper fractions whose difference is not a proper fraction. 5c) Find two proper fractions whose product is not a proper fraction. 5d) Find two proper fractions whose quotient is not a proper fraction.

Learning to See the Wind —7 One technique for doing this is to have one student predict another’s solution. Since the answers to these are not unique, you can follow this by asking the student if he or she can tell you anything about their classmates’ solution. For example in problem number 2 above, the students should come to suggest that the divisor must be larger than 201 and the dividend must be at least 201. The generalizations that the students provide will indicate their understanding of the concept of a remainder. Problems 4 and 5 lead students to search for patterns. The students’ generalization for 5c should go beyond saying that they can’t find any proper fractions to meet the conditions to an explanation of why it is impossible for anyone to find two proper fractions that meet the condition. These problems lead nicely into other what if explorations. Krutetskii's second type of generalization of mathematical material has been characterized by Dienes (1965) as an extension of an already-formed class. In the following activity, students are given multiplication facts and asked to use these known facts to find other products. The trick to the successful use of a discovery lesson such as this one, is to keep individuals from shouting out their observations. Instead they are just to tell you the products as quickly as they can. “We know that” is said before each line is read from the first column followed by the appropriate “So what about.” For example, you might say to the class, “We know that 8 ´ 8 = 64, so what about 7 ´ 9? We know that 4 ´ 4 = 16,so what about 3 ´ 5?”

We Know That: So What About: So What About: 6 ´ 6 = 36 5 ´ 7 = 4 ´ 8 = 8 ´ 8 = 64 7 ´ 9 = 6 ´ 10 = 4 ´ 4 = 16 3 ´ 5 = 2 ´ 6 = 10 ´ 10 = 100 9 ´ 11 = 8 ´ 12 = 9 ´ 9 = 81 8 ´ 10 = 7 ´ 11 = 7 ´ 7 = 49 6 ´ 8 = 5 ´ 9 = 12 ´ 12 = 144 11 ´ 13 = 10 ´ 14 = 30 ´ 30 = 900 29 ´ 31 = 28 ´ 32 = 60 ´ 60 = 3600 59 ´ 61 = 58 ´ 62 = 90 ´ 90 = 8100 89 ´ 91 = 88 ´ 92 = 15 ´ 15 = 225 14 ´ 16 = 13 ´ 17 = 25 ´ 25 = 625 24 ´ 26 = 23 ´ 27 =

´ = 2 ( – 1) ´ ( + 1) = ( – 2) ´ ( + 2) =

This notion of generalization is commonly reflected in the ordered series of exercises found in most mathematics texts in which increasingly more complicated extensions of a form are made. One alternative to giving students these ordered exercises is to have students create their own problems that meet some given conditions. The variety of solutions given in class can be used to challenge the students to give more complicated solutions. As they search for solutions, the above average student stretches to increase the range of acceptable solutions and the average and below average students follow this lead of creating new examples of the enlarged generalizations. 1) Find two numbers whose difference is 10. 2) In order to have an A average, Ferdinand must score an average of 95%. If each test is worth 100 points and he takes a total of 5 tests, find two possible combinations of test scores for Ferdinand to score exactly 95%.

The first problem above provides an example of how students of different ability levels will search for something to make a problem more interesting. For some, it will be the tendency to look for the easiest way to solve something; for example, using 0 in and addition or subtraction problem. Others will naturally slide into a game of can you top this. The numbers selected will be meant to wow the class. At first they will choose large numbers, but they will soon move to , fractions, and decimals. At the high school level this may extend to radical and complex numbers.

Algebra is viewed in many different ways. To some it is the generalization of arithmetic. To others it is the link between different systems of representation. And to a third it is the set of actions dealing with the use of variables— solving, simplifying, and graphing. To ask if we can teach algebra in the elementary school is to ask if we can teach mathematics in 8— Rachlin the elementary school. As we learn to view the arithmetic of the elementary school mathematics as more than a series of physical or symbolic manipulations, we are teaching algebra.

How can we help students develop the foundations of algebra? By providing an environment in which students begin to develop their ability to generalize, reversibility, and flexibility the elementary school teacher plants the seed for students to grow. Children’s ability to generalize includes their ability to see and identify patterns and their ability to recognize that a specific task falls within the class of a set of tasks. The key here is not that they see generalizations, but that they make generalizations. For a child to develop an ability to generalize they must learn to both make and apply generalizations.

Unlike the non-routine problems and applications that you may need to search for in calendars, conference presentations, professional publications, and math texts—it is possible for you to easily create non-routine routine problems. All that is required is a model and set of lenses for you to view your instruction through. One such set of lenses suggested here are the processes of reversibility, flexibility, and an ability to generalize (Krutetskii, 1976). While these processes appear to develop naturally in students gifted in mathematics, they are, according to Krutetskii, lacking in students of average ability and almost untraceable in students measured to have low mathematical abilities. The very essence of mathematical behavior is problem solving. Mathematical power comes from the development of students’ beliefs that mathematical understanding is something they are responsible for developing and further is something they can expect to achieve. The first steps to algebra may well be the creation of an instructional environment in which students begin to develop these problem-solving processes. It is in this environment that they may learn to see the wind.

References Dienes, Z.P. (1965). Current work on the problems of mathematics learning. International Study Group for Mathematics Learning. Hiebert, J. (1982). The position of the unknown set and children's solutions of verbal arithmetic problems. Journal for Research in Mathematics Education, 13, 341-349. Jensen, R., Rachlin, S.L. & Wagner S. (1982). A clinical investigation of learning difficulties in elementary algebra: An interim report. Paper presented at the Special Interest Group/ Research in Mathematics Education and the Research Advisory Council of the National Council of Teachers of Mathematics joint meeting, Toronto. Krutetskii, V.A. (1976). The Psychology of Mathematical Abilities in School Children (J. Kilpatrick & I. Wirszup, Eds.). Chicago: University of Chicago. Rachlin, S.L. (1982). The Processes Used by College Students in Understanding Basic Algebra. Columbus, Ohio: ERIC Clearinghouse for Science, Mathematics, and Environmental Education (SE 036 097). Rachlin, S.L. (1982). A Teacher's Analysis of Students' Problem-Solving Processes in Algebra. In Sigrid Wagner (Ed.), Proceedings of the Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education . Athens, Georgia: Department of Mathematics Education, 140-147. Rachlin, S.L., Matsumoto, A.N., and Wada, L.A.T. (1992). Algebra I: A Process Approach. Honolulu, Hawaii: University of Hawaii Curriculum Research & Development Group. Silver, E.A., Kilpatrick, J., & Schlesinger, B. (1990). Thinking through mathematics: Fostering inquiry and communication in mathematics classrooms. New York, New York: College Entrance Examination Board. Wagner, S., Rachlin, S.L., and Jensen, R.J. (1984). Algebra Learning Project Final Report. Athens, Georgia: University of Georgia Department of Mathematics Education. Wagner, S. and Rachlin, S.L. (1981) Investigating Learning Difficulties in Algebra." In Post, T. and Roberts, M.P. (Eds.), Proceedings of the Third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education . Minneapolis, Minnesota: University of Minnesota, 174-180. Wirtz, R.W. & Kahn, E. (1982). Another look at applications in elementary school mathematics. Arithmetic Teacher, 30 (1), 21-25.

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