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ED 373 960 

SE 054 456 




Masingila, Joanna 0.; And Others 

Mathematics Learning and Practice In and Out of 

School: A Framework for Making These Experiences 


Syracuse Univ., N.Y. 


19p.; Paper presented at the Annual Meeting of the 
National Council pf Teachers of Mathematics (72nd, 
Indianapolis, IN, April. 1994) . 
Speeches/Conference Papers (150) ~ Reports - 
Research/Technical (143) 

EDRS PRICE MF01/PC01 Plus Postage. 

DESCI, PTORS -'Educational Experience; Elementary Secondary 

Education; ^Experiential Learning; Mathematics 
* • Curriculum; ^Mathematics Instruction; Models; Prior 
Learning; *Student Experience; ^Teaching Methods 


Mathematics learning and practice in school and out 
of school differ in some significant ways which are explained by the 
fact that: (1) problem in everyday situations are embedded in real 
contexts that are meaningful to the problem solver; and (2) the 
mathematics used outside school is a tool in the service of some 
broader goal. This paper discusses research which examined 
mathematics practice in everyday work situations by comparing 
in-school and out-of-school practice. It presents a framework for 
gaining insight into the interplay between socio~cultural and 
cognitive developmental processes through the analysis of practice. 
Discussion of the research illustrations includes goals of the 
activity, conceptual understanding, and flexibility in dealing with 
constraints. Suggestions for teachers in connecting in-school with 
out-of-school experiences are given. Contains 25 references. (MKR) 


* Reproductions supplied by EDRS are the best that can be made 

* from the original document. 

****************** **************^ 




Mathematics Learning and Practice In and 
Out of School: A Framework for Making 
These Experiences Complementary 

Joanna O. Masingila 
Susana Davidenko 
Ewa Prus-Wisniowska 
Nkechi Agwu 

Syracuse University 
Department of Mathematics 
215 Carnegie Hall 
Syracuse, NY 13244-1150 

(315) 443-1471 



^ This document hit been reproduced ts 

received from the person or organization 

C Minor changes have been mede to improve 

originating it 

J* 0. Masinqi 1 a 

reproduction quality 

• Poirta ot view or opinions stated in this docu- 
ment do not necessarily represent official 
OfeRI position or pohcy 


Some of the research in this paper was supported by a grant to the first author from the 
Fund for Innovation, Office of the Vice Ciancellor of Academic Affairs, Syracuse 



Mathematics In and Out of School 


A variety of researchers in the last fifteen years have described how people use mathematics in 
out-of-school situations to solve problems and achieve goals (e.g., Lave, 1988; Masingila, 1992a; 
Millroy, 1992; Scribner, 1985). Furthermore, it is generally accepted that mathematics learning "is 
not limited to acquisition of the formal algorithmic procedures passed down by mathematicians to 
individuals via school. Mathematics learning occurs as well during participation in cultural 
practices as children and adults attempt to accomplish pragmatic goals 1 ' (Saxe, 1988, pp. 14-15). 

However, there are differences between mathematics practice in and out of school, as well as 
mathematics learning in and out of school. Lave (1988) has found evidence that mathematics 
practice in everyday settings differs from school mathematics practice in a number of ways. In 
everyday settings: (a) people look efficacious as they deal with complex tasks, (b) mathematics 
practice is structured in relation to ongoing activity and setting, (c) people have more than sufficient 
mathematical knowledge to deal with problems, (d) mathematics practice is nearly always correct, 
(e) problems can be changed, transformed, abandoned and/or solved since the problem has been 
generated by the problem solver, and (f) procedures are invented on the spot as needed. 

Researchers who have investigated how persons solve problems in school-like situations and 
solve mathematically-similar problems in everyday contexts found that in the former situation 
people "tended to produce, without question, algorithmic, place-holding, school-learned 
techniques for solving problems, even when they could not remember them well enough to solve 
problems successfully" (Lave, 1985, p. 173). When the same people solved problems in 
situations that appeared different from school, they used a variety of techniques and invented units 
with which to compute (Lave, 1985). 

These differences in mathematics practice appear to be explained by the fact that: (a) problems 
in everyday situations are embedded in real contexts that are me aningful to the problem solver and 
this motivates .and sustains problem-solving activity (Lester, 1989), and (b) "the mathematics used 
outside school is a tool in the service of some broader goal, and not an aim in itself as it is in 
school" (Nunes, 1993, p. 30) 

Just as mathematics practice in and out of school differs, so does mathematics learning. 
Whereas school learning emphasizes individual cognition, pure mentation, symbol manipulation 
and generalized learning, everyday practice relies on shared cognition, tool manipulation, 
contextualized reasoning and situation-specific competencies (Resnick, 1987). 

Knowledge constructed in out-of-school situations often develops out of activities which: (a) 
occur in a familiar setting, (b) are dilemma driven, (c) are goal directed, (d) use the learner's own 
natural language, and (e) often occur in an apprenticeship situation allowing for observation of the 
skill and thinking involved in expert performance (Lester, 1989). Knowledge gained in school all 
too often grows out of a transmission paradigm of instruction and is largely devoid of meaning 
(lack of context, relevance, specific goal). Furthermore, Resnick (1987) has argued that "[t]he 
process of schooling seems to encourage the idea that . . . there is not supposed to be much 
continuity between what one knows outside school and what one learns in school" (p. 15). 

In some instances, the difference between mathematics practice in and out of school may be 
inherent Sometimes a mathematical concept is understood and used differently in everyday 
situations than the way it is taught in school (e.g., de Abreu & Carraher, 1989). For example, 
percentage of change is a common concept in ^tailing and in school mathematics. In school, 
percentage of change is understood to be the amount of change from the original amount A typical 
textbook exercise involving this concept might be the following: 

Find tht percent of change for a video game system that costs $29 in 1980 and $99 in 
199C. (Davison, Landau, McCracken & Thompson, 1992, p. 262) 

A student finding the answer to this exercise would subtract $29 from $99 to get a $70 ' 
increase, then divide $70 by $29 to get an increase of approximately 241%. Percentage of change 
in retailing, however, is understood to be the amount of change from the retail price. Thus, for the 
situation in the textbook exercise above, a retailer would divide $70 by $99 to get an increase of 
approximately 71%. Since the final result in retailing is sales, all percentages of change are based 

Mathematics In and Out of School 

on retail prices. In this case, the solution process in the everyday context is different because of 
the different conceptual understanding of percentage of change (Masingila, 1993b). 

We believe that while some differences may be inherent in mathematics learning and practice in 
and out of school, the differences can be narrowed so that instead of being disjoint activities that do 
not influence each other, mathematics learning and practice in and out of school can build on and 
complement each other. In this wiy, students can bring to bear their mathematical knowledge 
gained in out-of-school experiences on their school mathematics. Likewise, students can use their 
school mathematics in solving problems that occur in everyday situations. Acioly and Schliemann 
(1986), in their study of lottery game bookies in Brazil, found that the bookies with school 
mathematics experience were able to understand and solve novel problems while bookies who had 
not attended school were unable to do this. In this case, the schooled bookies seemed able to draw 
on their school mathematics to use in an out-of-school situation. 

Individuals need both in-school and out-of-school mathematical experiences in our society. 
Without everyday mathematical experiences, in-school learning is solely for the sake of learning. 
Students need in-school mathematical experiences to build on and formalize their mathematical 
knowledge gained in out-of-school situations. This occurs in the classroom through the activity of 
constructing knowledge that is subject to "explanation and justification as students participate in the 
intellectual practices of the classroom community" (Cobb, Yackel & Wood, 1992, p. 7). 

The question, then, is how can teachers use students' in-school and out-of-school experiences 
so that mathematics learning and practice in these contexts can be complementary? Before 
discussing and elaborating on a framework that we think sheds some light on this, we will present 
some research from several studies (Agwu, 1993; Davidenko, 1994; Masingila, 1992a, 1993b; 
Prus-Wisniowska, 1993) that illustrates some of the points made above and lays the groundwork 
for discussion of the framework. 

Illustrations from Research 

Structure of the Studies 

In each of these studies, mathematics practice was examined in an everyday work situation. 
Workers in each context were observed and informally questioned as they worked. Data were 
collected from carpet laying estimators and installers, a dietitian, an interior designer, a retailer, a 
restaurant manager, and a textile designer. We analyzed the data through a process of inductive 
» data analysis, looking for the concepts and processes that were involved in the mathematics 
practice in these contexts. 

Selected problems that occurred in these contexts were then given to pairs of secondary 
students. We observed and informally questioned the pairs of students as they solved these 
problems. We analyzed the data by examining how each pair solved the problems, and then 
compared these with how the persons in the work context had solved the same problems. 
Comparing In-School and Out-of-School Practice 

The follow discussion focuses on three key differences that we found in our work comparing 
in-school and out-of-school mathematics practice. The differences involved the goals of the 
activity, the conceptual understanding of persons in each context, and flexibility in dealing with 

Goals of the activity. In each context (carpet laying, dietetics, interior design, retailing, 
restaurant managing, school, textile design), the distinction between the goals of the individuals in 
the out-of-school contexts were in sharp contrast to the goals of the students in the school context 
as long as the students viewed the problems as school problems. However, when the students 
were able to place themselves in the everyday situation, they appeared to view the problems 

For the individuals in work situations, the goal was to make a decision. In the course of that 
decision-making process, problems needed to be solved and mathematics was a tool to be used to 
solve the problems. The students had as their goal to solve the problems we gave them; nothing 
beyond that was required of them. We offer two examples of this distinction. 

In the restaurant management context, the restaurant manager was faced with the problem of 
changing a recipe obtained from a newspaper (Prus-Wisniowska, 1993). The recipe gave the 

i) 2 4 

Mathematics In and Out of School 

ingredients for serving six persons (see Figure 1). However, the restaurant manager needed to use 
the recipe for a dinner party for 20 people. Her goal appeared to be to decide the amount of each 
ingredient needed and give instructions for the cooks while being efficient To this end, she 
decided to make enough fruit salad for 24 portions and divide the remaining four portions among 
the 20 fruit cups. 

When asked to change the recipe for exactly 20 portions, she computed 'he factor 20 + 6 = 3.3 
on her calculator and used it to increase the amount of each ingredient. To make the recipe feasible 
for the cooks, she changed each decimal into a proper fraction. She noted that "we only work with 
halves, thirds, and fourths." For example, since the 2 cups of apples became 6.6 cups when 
multiplied by the factor of 3.3, she indicated that "Six and a half cups is fine for that." The 
restaurant manager used 1 1/2+ for the 1.65 cups of sugar in the increased recipe: "The cooks 
know to put just a little bit more in when they see the plus sign." 

However, in the case of the cream, 1.65 cups of cream she noted as 1 1/2 cups and 1 Tbls. 
She indicated that it was important to be more specific for the cream. The restaurant manager 
figured that .8 cups of vinegar were needed for 20 portions and she reasoned as follows: "The .8 
is a result of multiplying 3.3 by 1/4 so I won't be far off if I skip the .3 and multiply -3 by 1/4. So 
Til just need 3/4 cup for that." She also decided that 7 eggs were needed. 

When the restaurant manager was asked about buying the groceries needed for making the fruit 
and vegetable salad, she responded that she would first check to see if she already had all the 
ingredients and then order those that she needed. For instance, if she was low on sugar she would 
order a large quantity since it will be needed for otoer things as well: "Only in the case of very 
unusual meals, like scallop salad, do I order the exact amount" 

A pair of secondary students was asked to change the same recipe so that it would make 10 
portions. These students were second-year high school students in an advanced algebra course. 
They were on track to take calculus in their fourth year of high school. They immediately 
calculated 10 + 6 = 1.7 and began increasing each ingredient by this factor. When they multiplied 
the 1/2 cup of carrots by the factor 1.7 and obtained .833 cups, they were not able to interpret this 
as a proper fraction and abandoned this approach. Instead, the students set up a proportion for 
each ingredient. For example, for the apples they used the proportion 6/2 = 10/x and obtained 3 
1/3 cups of apples. Using this method, the students found the increased amount for each 
ingredient, including 5/6 cup of carrots. When asked about how they would measure 5/6 of a cup, 
the pair decided to accept only halves, thirds, fourths, and eighths. 

Hie students decided that 5/6 cup of carrots was equivalent to "2/3 cup + 1/2 of 1/3 cup." 
They said they could measure this by just filling up half of a 1/3 measuring cup. The students 
found that they needed 3 1/3 eggs and decided that they would mix together four eggs and then 
take out about 2/3 of an egg. We noticed that, unlike the restaurant manager, these students had no . 
discussion about the level of accuracy that was needed for the ingredients involved. We also noted 
that the restaurant manager was able to use mathematics as a tool in order to modify the constraints 
and make decisions that made sense in her situation. Certainly to her, using fractions instead of 
decimals allowed her to make better use of her number sense and she was able to convert the 
decimals into fractions that suited her purpose. 

The students had the mathematical knowledge to change their strategy when they ran into 
difficulties in dealing with .833 of a cup. However, their seemingly lack of number sense (e.g., to 
view .833 cup as a little more than 3/4 cup) prevented them from pursuing this strategy. But when 
they used proportions to obtain fractional representations, the students were able to make sense of 
the measurements. However, when asked to decide what groceries they would need to buy for the 
fruit and vegetable salad, the students' list contained items like 3 1/3 cups of p.pples and 1 2/3 
tablespoon of butter. It appears that these students caw the goal of the problem was simply to 
obtain measurements for the ingredients without regard to their reasonableness in an everyday 

Another pair of students, however, after thinking about the problem noted that if they really 
had to make this salad they would make the salad for 12 portions by doubling everything and then 
divide the two extra portions among the 10 people. These students were second-year high school 
in a geometry course. While these students initially saw the problem as simply using proportions 

Mathematics In and Out of School 

to find the increased amounts, they realized that they would do things less formally in an out-of- 
school context. Their list of groceries was more reasonable also, including items such as 10 
apples, 1 small can of peanuts, 1/2 dozen eggs (the smallest amount they thought they could buy), 
the smallest possible bag of sugar and flour, a pint of cream. However, like the first pair, they did 
not consider whether they would have some of these items already in stock. 

Another illustration of this difference in goals is from the interior design context The interior 
designer was deciding on the amount of materials needed for a house that was being refurbished. 
One aspect of the redecorating involved wallpapering a number of walls of the house. The 
wallpaper that was chosen was 20" wide and had a repeat length of 9" (i.e., the pattern repeated 
every 9"). The interior designer told me that what she does in considering repeat lengths is to 
divide the desired length by the length of the repeat, in this case 96" + 9 M . This gives the number 
of repeats in the desired length. She calculated that there were 10.67 repeats, "so there needs to be 
1 1 repeats in each strip of wallpaper so 1 1 times 9" means 99" needs to be considered as the length 
of the walL So, you figure the number of strips you need for the rooms and figure each strip to be 
99" long. Of course, you also have to consider that you have to place the seams for the strips at 
least two inches past the corner so it stays down better. Also, wallpaper is sold in single, double, 
and triple rolls so you must figure the best deal for what you need." The interior designer's 
solution process is aimed toward the goal of making a decision the quantity of materials needed 
while considering cost efficiency. 

W e gave a pair of secondary students a problem concerning this same situation. These 
students were from the same advanced algebra course as the first pair of students who solved the 
recipe problem,' In order to structure the situation a bit more and focus on how the students would 
deal with the repeat length constraint, we gave the following problem: 

Suppose you decide to wallpaper your bedroom If your room is 10' by 8' and the 
walls are 8 1 high, how much wallpaper do you need if the wallpaper has a repeat 
length of 9"? The wallpaper is 20" wide and a roll has 45 feet of wallpaper on it 

The concept of repeat length was explained and the students indicated that they understood. 
They began the solution process by calculating that the one wall was 120" long and the other was 
96" long. The students decided that they would need 6 strips of wallpaper on each of the 10' walls 
and 5 strips of wallpaper on each of the 8' walls. They figured that each wall was 96" from ceiling 
to floor so that 576" of wallpaper were needed for each 10' wall. When asked to explain what the 
576" inches meant, they said that there were 6 strips needed across the wall and each strip was 96" 
long "and 6 times 96 is 576." The students continued in this vein and determined that each 8' wall 
would need 480" of wallpaper. Totaling the amount needed for the walls, they indicated that 21 12 
inches of wallpaper were needed for the room. 

The students appeared to act as if their solution was complete. Finally, as one student checked 
back over the statement of the problem, she noted that they had not considered the repeat length: 
"Well, how would we do this if we were really going to wallpaper a room? Maybe we should 
draw a picture to see what this looks like." At that point, the students drew a diagram of one 10' 
wall: "The first strip doesn't matter. The pattern doesn't have to match anything, so all we need is 
96". But the next strip has to match that one." 

The students decided to divide 96 by 9 to find out how many sections of the pattern are on each 
wall. They determined that there were approximately 10.7 sections of the pattern in the first strip 
and so three inches wore going to have to be trimmed off the wallpaper before cutting the next 
strip. When asked to explain their reasoning, one student replied, "Since 99" would be a whole 
number of 9" sections, we have to cut off three more inches [after the 96"] so that the pattern will 
be starting again for the next strip." At this point, the students decided that they could treat all the 
remaining strips as 99" and come out with matching patterns. They totaled the strips and decided 
that 2175 inches of wallpaper were needed. The students made no attempt to determine the number 
of feet or number of rolls of wallpaper that would be needed but seemed satisfied to leave their 
solution in inches. 

Mathematics In and Out of School 

The approaches to the problem-solving activity were different in the in-school and out-of- 
school contexts because the persons involved had different goals. For both the restaurant manager 
and the interior designer, solving the problems was a necessary part of their job. They used 
mathematics as a tool to help them solve problems and not as the goal of the problem. The 
students, however, seemed to view the problems as mathematical exercises and immediately started 
using algorithms that they thought would be appropriate. Although two of the pairs gained some 
insight when they tried to put themselves in the everyday problem situation, they did not stick with 
this perspective totally and did not check the reasonableness of their solution with the everyday 
context (e.g., checking for groceries already on hand, converting the number of inches of 
wallpaper needed into the unit of rolls). 

Conceptual understanding. We observed differences in conceptua 1 understanding 
between individuals in the everyday work situations and the secondary sk :>,nts. While the 
students had the procedural knowledge to solve the problems, they were not able to understand the 
concept involved and apply the procedures. On the other hand, the workers understood the 
concept, at least in this context, and had the tools necessary to solve the problems. The following 
two examples illustrate the differences between the workers and the students. 

In the carpet laying context, the concept of area is pervasive in the all work done by the 
estimators and installers. All the workers Masingila (1992a) observed converted square feet to 
square yards by dividing square feet by nine. This algorithm is essential in the carpet laying 
business since measurements are taken in feet but carpet must be ordered from a supplier in. square 
yards. In a conversation, Dean, an estimator, explained why this algorithm worked. 

Joanna: If you just know the length and width of a room, how do you find how many 

square yards of carpet you need? 
Dean: Well, if the room is 12' by 8' then you take 12x8 + 9. 
Joanna: What does the 9 mean? 

Dean: That's the way you convert square footage to square yardage. 
Joanna: Okay, but where does the 9 come from? 

Dean: I don't know. Maybe I don't understand the question. . . . Where does the 

9 come from? 
Joanna: Yeah, why isn't it 8 or 6? 

Dean: Well, when you have square footage (draws diagram with 3 by 3 grid — see 
Figure 2) . . . each of these squares is a square foot and there are three feet 
in a yard (puts x's inside the three squares in the right column of the grid) 
and then three across (puts x's inside three squares in the top row) — so that 
makes 9. (pp. 114-115) 

By using a diagram Dean was able to illustrate, although not fully articulate, that in one square yard 
there are nine square feet and to convert from square feet to square yards involves dividing by 

In contrast, several pairs of secondary general mathematics students were given a similar 
problem and did not understand that the concept of area was involved. The problem and 
conversation with two students, Jim and Matt, are given below. 

Suppose you need a piece of carpet 12 feet by 9 feet How many square yards should 
you order from the carpet supplier? 

Matt: I don't know nothin' about square yards. 

Joanna: Well, let's see. What does a piece of carpet 12' by 9' look like? 

Jim: (draws a rectangle and labels the dimensions 12' and 9') 

Joanna: Alright. Now how would you change that to yards? 

Matt Divide by 3. • 

Joanna: Why? 

Matt 'Cause it takes 3 feet to make a yard. 


Mathematics In and Out of School 

Jim: (writes "4 yds" and "3 yds" and scribbles out 12' and 9') 
Joanna: Okay, now how many square yards is that? 

Jim: Square yards? Oh . . . well, there's two 4's and two 3's — one on each side. 

So that's 4 square yards and 3 square yards. 
Joanna: What docs square yards mean? 
Matt: I don't know. (Jim shakes his head.) 

Joanna: What it means is area; finding the square yardage of this carpet is finding the 
. area. 

Jim: So that's 4 2 — 8 and 3 2 — 6 and take 8x6. 
Joanna: Where did you get the 8? 

. Jim: 4 2 — 4x4 — no, that's not 8. Area is length times width times height, (pause) 
I'm not sure. 

Joanna: Area of a rectangle is length times width. So what's the area of this carpet piece? 
Jim: You'd multiply 4 x 3 — no, 8x6 because those are square yards. - 
Joanna: So the area or square yards is what? 
Jim: 8x6. 

Joanna: Matt, do you agree with that? 

Matt Yeah. (Masingila, 1992a, pp. 235-237) 

None of the six pair of students who worked this problem, including this pair, understood that 
finding the square yardage of the piece of carpet was the same as finding the area of the carpet 
piece. However, these students had studied area with square units for several years. In fact, in the 
textbook they had used the previous year there were exercises that were similar to this problem. 
The main difference may have been that the exercises were in a chapter on area and in a lesson on 
area of rectangles so the students knew what procedure to use. 

Contrast this with Dean's explanation of how to convert from square feet to square yards. He 
knew the algorithm, dividing by nine — because he used it regularly in his job. However, Dean 
also understood that he was dealing with area, and that in one square yard there are nine square 
feet It is our conjecture that if Dean had been asked to explain this conversion when he was a 
ninth grader, his explanation would not have been much different than that of these general 
mathematics students. However, through his day-to-day experience working with rectangular 
area, Dean had come to a fuller understanding of this conversion and was able to construct a reason 
for its mathematical validity. 

In the dietetics context, the concepts of ratio, proportion, percentage, and conversion of units 
are used by dietitians in a variety of problem situations and in varying levels of difficulty. Most of 
the calculations and algorithms required to solve the problems are simple; however, we observed 
that a conceptual understanding of the concepts of ratio and proportion is necessary in order to 
properly interpret, model, and solve complex problems (Davidenko, 1994). 

We gave a dietitian and several pairs of students the following problem: 

You buy a 12 lb roast for which you pay $3.98 per lb. The waste, when removing 
fat and bones is about 18%. Then, when you cook it, the roast will shrink about 14%. 
What is the cost of a 3 oz portion of the cooked roast? 

Judy, the dietitian, read the problem and looked a little puzzled. She read it again and said, 
"First, the goal is to find 32% of 12 pounds and find what is available." She worked a bit on some 
calculations and then realized that 32% was not right. Judy mentioned that she faces similar 
problems: "For example, we can buy two different brands of ham at different prices. One has 
13% fat, the other has 15% fat, and the fat has to be removed. Then we have to compare prices 
per unit to see which is the best buy." During the ensuing discussion, it was clear that Judy 
understood the concept involved in these problems — finding the unit price of a product after 
discarding unusable parts — and was then able to solve the problem. 

6 O 


Mathematics In and Out of School 

We gave this problem to several pairs of secondary students who were second-year students in 
a geometry course. We explained what was shrinkage and they all indicated that they understood; 
all the students approached the problem in the same way. The first step the students took was to 
multiply 12 x 3.98 to find the total price of the roast, $47.76. Next the students tackled the 
problem of accounting for the removal of fu and bones. One pair's dialogue is as follows: 

Nina: It is 18% waste, so let's take 18% of 47.75 and subtract it from the total price. 
Todd: No, it was 18% waste and 14% shrinkage, so that is 32% of $47.76. 
Nina: Let's take 18% of the $47.76 and then 14% of that. 
Todd: Wait! Shouldn't we take 18% of the 12 pounds? 
Nina: I don't know. 

Nina and Todd discussed whether to take 18% of the cost or of the weight Finally, with some 
guidance from one of the researchers, they convinced themselves of the procedure to use: "We 
have to take 18% of the weight because what we are doing is reducing the amount of roast, not 
reducing the cost Then we take 14% of the usable part." 

Whereas Judy understood the concept involved in the problem immediately, the. students 
started by looking for an algebraic solution without understanding the concept. They first 
performed the calculations suggested by the first part of the problem. Then they continued with the 
next sentence, and in doing so they sought to reduce the price and not the weight. The students 
were able to deal procedurally with taking percentages of a number, but they did not understand 
(without guidance) the concept behind the procedure (i.e., removing fat and bones reduces the 
amount of usable meat but not the price paid). 

We attribute the differences in conceptual understanding between the individuals in everyday 
work contexts and the students to a lack of experience on the part of students in dealing with these 
concepts in problematic situations where mathematics is used as a tool rather than an object We 
observed with our respondents that the workers were able to understand the problems within the 
context and had the conceptual understanding to solve the problems within that context 

Flexibility in dealing with constraints. Problems that occurred in each of the eveiyday 
situations that we examined were filled with constraints. We observed noticeable differences in the 
ways that workers in these contexts and the students were able to deal with these constraints. 

A problem that occurred in the restaurant management context is what we call the Order 
Problem (Prus-Wisniowska, 1993). One of the many responsibilities of the restaurant manager 
was to order necessary food and supplies while considering the constraints of delivery, storage 
space, and efficiency. Figure 3 shows the problem that the restaurant manager faced in ordering 
meat for each week. 

In dealing with all the constraints, the restaurant manager chose to minimize the number of 
delivery days and maximize the amount of meat in the freezer. Another priority was to schedule all 
deliveries at the beginning of the week since the end of the week is often hectic with many 
functions at the restaurant scheduled on Thursday and Friday. Furthermore, Friday is the day she 
does inventory and plans deliveries for the next week; she preferred not be bothered with additional 
things like meat deliveries. Thus, the problem of ordering and scheduling deliveries was a real 
problem for the restaurant manager and she dealt with all the constraints through optimization and 

Two pairs of secondary students worked on this problem. One pair consisted of second-year 
students from an advanced algebra course and the other students were second-year students in a 
geometry course. Both pairs produced minimal order solutions: For each delivery day they 
decided to order only the amount of meat that would cover the needs until the next delivery day. 
For example, on Monday they decided to have 50 pounds delivered (25 pounds for both Tuesday 
and Wednesday), on Wednesday, 30 pounds to cover Thursday's needs. 

When we questioned the students about if other solutions were possible, each pair reorganized 
their order schedule so that one less delivery day was needed; one pair eliminated delivery on 
Friday and the other eliminated delivery on Thursday. Neither pair seemed to consider that a 
- delivery could be for more than 55 pounds (the capacity of the freezer) since each day some meat 

O 7 ■« g 


Mathematics In and Out of School 

had to be taken out of the freezer to defrost for the next day. Both pairs struggled to keep track of 
all the constraints involved in the problem and appeared unable to consider all the constraints in 
formulating their solutions. 

The carpet laying context contains a variety of constraints: (a) floor covering materials come in 
specified sizes (e.g., most carpet is 12' wide, most tile is 1' by 1'), (b) carpet pieces are 
rectangular, (c) carpet in a room (and usually throughout a building) must have 'he nap (the dense, 
fuzzy surface on carpet formed by fibers from the underlying material) running in the same 
direction, (d) consideration of seam placement is very important because of traffic patterns and the 
type of carpet being installed, (e) some carpets have patterns that must match at the seams, (f) tile 
and wood pieces must be laid to be lengthwise and widthwise symmetrical about the center of the 
room, and (g) fill pieces for both tile and base must be six inches wide or more to stay glued in 
place. Some particular situations have more constraints, such as a post in the middle of a room that 
is being carpeted (Masingila, 1992a). 

The ninth-grade general mathematics students who worked problems from the carpet laying 
context often had difficulty dealing with the constraints involved in the problems. For example, in 
a problem involving the installation of tile, the students struggled to figure our a way to install the 
tile so that the constraints about lengthwise and widthwise symmetry and fill pieces being at least 
six inches wide were fulfilled (see Masingila, 1992a, for more discussion about the students' 
problem-solving work). 

The students were also not as flexible as the experienced workers in seeing more than one way 
to solve a problem. In a pentagonal-shaped room that needed carpeting, the students were able to 
see only one way (without guidance) to install carpet. The estimator, on the other hand, was able 
to visualize how the carpet would be laid if it were installed with the nap running in the direction of 
the maximum length of the room and with the nap running in the direction of the maximum width 
of the room. By having more than one solution, he was able to weigh cost efficiency against seam 
placement and make a decision while considering these constraints (Masingila, 1992a, 1992b). 

One of the interior design problems we gave to pairs of students was as follows: 

You need to purchase some materials for upholstering some chairs. If you buy the 
whole bolt, which has 60 yards of material, it will cost $5.00 per yard. If you purchase 
less tfuzn a full bolt, it will cost an additional $1.50 per yard. At what point does it 
become more economical to purchase th? whole bolt of material? 

One pair of second-year geometry students approached the problem through trial and error. 
They first found that 30 yards of material would cost $195, 40 yards would cost $260, and 50 
yards would cost $325. At this point, one of the students said, "Well, the whole bolt only costs 
$300 so it must be less than 50 yards." After some more calculations they decided that 46 yards 
was the cutoff point; "Since 46 yards costs $299, anything more than 46 yards would cost more 
than that — so it would be cheaper to buy the whole thing." 

The interior designer, when faced with this problem, found that the whole bolt cost $300 and 
then divided by $6.50 to find that 46 yards and 5 inches is approximately the amount at which it 
becomes more economical to buy the whole bolt However, she decided that if she needed an 
amount close to 46 yards, "like if I need 44 yards, I will buy the whole thing because I'm spending 
less than 5% over what I need and I can most likely use the material for something." For the 
students, this problem had only one answer; for the interior designer, the answer depended upon 
the situation. 

Our interpretation of this difference in flexibility on the part of the students and the workers is 
that the students, for the most part, have not been exposed to problems with real-life constraints 
that must be considered and addressed in order to find solutions (Masingila, 1993b). Although 
there are many exercises in school textbooks that are set in these contexts, the exercises are 
typically devoid of real-life constraints and, as a result, do not require students to engage in the 
type of problem solving required in the everyday contexts (Masingila & Lester, 1992). 


Mathematics In and Out of School 

Interplay Between Mathematics In and Out of School 

Saxe (1991) has delineated a "research framework for gaining insight into the interplay 
between sociocultural and cognitive developmental processes through the analysis of practice 
participation ,, (p. 13). The theoretical underpinnings of the framework are based on b^th Piaget 
and Vygotsky, but the framework moves beyond them in considering this interplay. Saxe 

The framework shares the underlying constructivist assumptions of the Piagetian and 
Vygotskian formulations, and, with respect to core constructivist assumptions, the model 
... is consistent with both approaches. However, the framework . . . targets a level of 
analysis that is not addressed by either of these formulations. Unlike the Piagetian approach, 
my concern is to treat cognitive development on a level of analysis in which activity-in- 
sociocultural context is a critical focus and cognitive developmental processes are analyzed 
with reference to these contexted activities. Unlike the Vygotskian writings, which do not 
develop core developmental and sociocultural theoretical constructs with reference to 
systematic analysis of core domains of knowledge, the present approach is concerned 
with a systematic analysis of mathematical cognition that integrates cognitive developmental 
and sociohistorical perspectives, (pp. 13-14) 

Although Saxe's framework is a method for studying the interplay between sociocultural and 
cognitive developmental processes, we find it helpful in thinking about working towards in-school 
and out-of-school mathematics learning and practice being complementary. Thus, we discuss his 
framework with illustrations from our own research, and then elaborate on ways to make this 
interplay between in-school and out-of-school contexts more deliberate. 

Saxe's (1991) framework consists of three analytic components: (a) goals that emerge during 
activities, (b) cognitive forms and functions constructed to accomplish those goals, and (c) 
interplay among the various cognitive forms. Goals are "emergent phenomena, shifting and taking 
new form as individuals use their knowledge and skills alone and in interaction with others to 
organize their immediate contexts" (p. 17). Forms are "historically elaborated constructions like 
number systems, currency systems, and social conventions." As these forms are "acquired and 
used by individuals to accomplish various cognitive functions" (e.g., counting, measuring), they 
become cognitive forms (p. 19). Interplay among the various cognitive forms occurs as 
individuals, "in order to accomplish goals in one setting, . . appropriate and specialize cognitive 
forms linked" to another (p. 22). 
Emerging Goals 

Saxe outlines four parameters that influence the emergence of goals: (a) the goal structure of 
activities, (b) social interactions, (c) conventions and artifacts, and (d) an individual's prior 
understandings. Figure 4 illustrates this four-parameter model. We will use examples from our 
own re search to illustrate these parameters. 

The goal structure of an activity consists of the tasks that must be accomplished in the activity. 
For example, in order to run a store a retailer must buy and reprice items for sale. A principal 
concern for the retailer is to sell an item for as much money as possible while selling as many of 
the item as possible. Thus, mathematical goals that emerge in marking items up and down are 
guided by this economic concern. 

Social interactions that occur during activities may also influence the emerging goals. In the 
carpet laying context, installers worked with helpers in a master-apprentice relationship. The 
discussion and interaction that occurred between installers and helpers often allowed helpers to 
engage in activities they would not have been able to unassisted (Masingila, 1992a). 

Saxe (1991) writes of conventions and artifacts as "cultural forms that have emerged over the 
course of social history, such as ... the Oksapmin indigenous body-part counting system and . . . 
a particular currency system" (p. 18). Sometimes individuals within a culture develop a set of 
conventions that may be unique to their particular situation. For example, the restaurant manager 
developed a notation system to keeping track of the restaurant's inventory. In counting items for 


Mathematics In and Out of School 

inventory purposes she used different units for different items (e.g., pound, 5 pound, box, case, 
pack, each). These units were usually the same as the units used for delivery purposes. 

However, she adopted a different unit for French fries. French fries were only delivered in full 
cases, where 1 case = 10 boxes and 1 box = 10 pounds of French fries. The restaurant manager 
found it difficult to operate with case as the unit for French fries since the restaurant was rather 
small and 10 pounds of French fries would often be more than was needed for a particular meal. 
So she decided to use zero to denote a case less than half full; a zero indicated to her not that there 
were no French fries, but rather that it would soon be time to order more. 

After using this convention for some time, the restaurant manager found it, too, was 
inconvenient because sometimes five boxes of French fries sufficed for one week; other times it 
did not, and so the distinction between zero and one became critical. In the end, the restaurant 
manager decided to change her notation to using box as the unit Even though the French fries 
continued to be delivered in cases, she reported each case as ten boxes and so from this time on her 
inventory indicated the amount of French fries with an accuracy of one box. Thus, the convention 
used by the restaurant manager influenced emerging mathematical goals of activities associated 
with the inventory. 

The prior understandings that "individuals bring to bear on cultural practices both constrain and 
enable the goals they construct in practices" (Saxe, 1991, p. 18). In solving the carpet problem 
that involved converting from square feet to square yards, the students' prior understandings about 
area as a formula, dependent upon the geometric shape appeared to constrain their goals. 
However, for another problem that involved a pentagonal-shaped room to be carpeted, one student 
knew from personal experience that the room had to be treated as a rectangle and this enabled him 
to construct goals that were different from students who tried to determine how to lay carpet in a 
five-sided room (Masingila, 1992a). 
Form-Function Shifts 

The second analytic component of Saxe's (1991) research framework is the dynamic in the 
"shifting relations between cultural forms and cognitive functions as they are interwoven with the 
socially textured goals linked to practice participation" (p. 19). He describes how the cultural form 
of body counting shifted in function as individuals' levels of economic participation changed. 

This phenomena also occurred in the carpet laying context as the helpers gained experience 
through participating in the practice of installing floor coverings. For example, one convention that 
was present in this context was an algorithm for laying tile. The algorithm was an agreed-upon 
proc sdure for laying tile so that the tile was lengthwise and widthwise symmetrical about the center 
of the room and that fill (partial) pieces were at least six inches wide (Masingila, 1992a). 
However, as the helper participated in the tiling process and, as was sometimes the case, became 
an installer, the procedure (form) became a cognitive tool (function) to be used for making 
decisions when complicating factors compounded the installation. 
Interplay Among Various Cognitive Forms 

In studying Oksapmin schoolchildren, Saxe (1985) found evidence that the children used out- 
of-school cognitive forms to bring to bear on in-school problems. Other researchers have 
determined that persons in out-of-school contexts may use knowledge gained in school to address 
problems they encounter (Acioly & Schliemann, 1986). Thus, there can be interplay between 
'cognitive forms that may be appropriated and specialized in one setting and their use in another. 

Saxe (1991) has specified a generalized portrayal (see Figure 5) of how cognitive 
developmental and sociocultural processes are " interwoven with one another in complex ways" (p. 
186). Saxe notes: 

As the figure shows, in our daily lives, we are engaged with multiple practices. Within 
practices, goals emerge that must be accomplished, avoided, or reckoned within the 
achieving of larger objectives. Across practices, the understandings we generate in one 
may be appropriated and transformed to structure and restructure goals in another, (p. 186) 


I 9 

Mathematics In and Out of School 

Linking the Framework to Classroom Practice 

As mentioned previously, although Saxe's intent was to outline a framework for conducting 
research to better understand th* interplay among various cognitive forms through practice, we find 
the framework useful in thinking about ways to bring about more and deliberate interplay between 
developmental processes in different settings. 

We have discussed ways in which mathematics learning and practice often differ in school and 
everyday contexts. However, individuals do make use of knowledge in one context that was 
situated in another context when they view the problem situations as being similar (Stigler & 
Baranes, 1988). We suggest that if we, as teachers: (a) can create situations where students 
experience their mathematics learning and practice in school as similar to mathematics learning and 
practice out of school, and (b) encourage students to participate in activities out of school in which 
the mathematics learning and practice may be similar to their mathematics learning and practice in 
school, then these experiences can become complementary to each other. 

Connecting in-school with out-of-school experiences. First, in order to create in- 
school experiences similar to out-of-school experiences, the goal structures of activities must be 
similar for in-school and out-of-school activities from which students may construct similar 
mathematical knowledge. This means that the curricula includes a wide variety of problem 
situations that engage students in doing mathematics in ways that are similar to doing mathematics 
in out-of-school situations. Thus, problems are embedded in situations that are real and 
meaningful to students, and mathematics practice can be structured in relation to these problematic 
situations. It also means that mathematics is a tool to be used and that procedures and processes 
are learned as they are needed in the midst of accomplishing emerging goals. 

We further suggest that in order to structure classroom experiences like this instruction should 
be via problem solving; In teaching via problem solving, "problems are valued not only as a 
purpose for learning mathematics, but also as a primary means of doing so. The teaching of a 
mathematical topic begins with a problem situation that embodies key aspects of the topic, and 
mathematical techniques are developed as reasonable responses to reasonable problems" 
(Schroeder & Lester, 1989, p. 33). Teaching via problem solving deviates from the traditional 
instructional approach of the teacher presenting information and then assigning exercises in which 
students practice and apply this information. Using a teaching via problem solving instructional 
approach means that mathematical understandings are constructed by students as they seek to 
accomplish emerging goals through problematic situations. 

Second, social interactions are an essential part of this classroom mathematics practice. In 
working individually and collectively to accomplish emerging goals, mathematical knowledge is . 
developed within a meaningful context and cognitive development occurs as students work 
together with peers and teacher to negotiate shared meanings. As Saxe (1991) noted, social 
interaction is a key influence on the emerging goals of an activity. 

Third, in-school activities should make use of cultural artifacts and conventions that students 
can use ' 3 interpret problems and make sense of them. Students should also be encouraged to 
generate conventions that may be helpful to them in the course of accomplishing their emerging 
goals. For example, students may invent notation to indicate when objects are the same size and 
shape, in the course Of working in a measurement context, before they have formalized the concept 
of congruence. 

Finally, teachers can build on students' prior understandings. All students bring to school 
mathematical knowledge acquired in other contexts. This knowledge is often hidden and unused 
by students in school as they learn to use the mathematical procedures that teachers demonstrate 
and evaluate (Masingila, 1993a). If teachers engage students in conversation about their everyday 
experiences, listen to them, and encourage and observe their informal methods of mathematizing, 
they can learn much about students' prior understandings. Similarly, teachers can encourage 
students to bring to bear their prior understandings by having students: (a) create their own 
problem situations, (b) solve problems in more than one way and share their solution methods with 
each other (Lester, 1989), and (c) focus on semantics rather than syntax. 

Connecting out-of-school with in-school experiences* Besides creating experiences 
in school that may complement out-of-school mathematics learning and practice, teachers can guide 

n '■" 13 

Mathematics In and Out of School 

students in reflecting on how in-school learning and practice are used out of school. In a study 
examining middle school students* ideas about their out-of-school mathematics practice, Mjsingila 
(1994) observed that with encouraged reflection students were able to note a number of ways that 
they used mathematics outside of school. Sixth- and eighth-grade students were interviewed 
before and after keeping a log for a week in which they recorded their use of mathematics. 
Although students reported ways they used mathematics they classified as "non-school math," they 
also indicated many instances where they used knowledge they categorized as "school math." 

We suggest that an important aspect of in-schbol and out-of-school mathematics experiences 
becoming more complementary is to encourage students to be aware of their mathematics learning 
and practice outside of school. This involves having students discuss their out-of-school 
experiences and what mathematics concepts and processes they used in those experiences. 
Additionally, teachers can have students reflect on how their in-school mathematical experiences 
influence this learning and practice. Teachers can also ask students to think of out-of-school 
experiences that are similar in some aspects to mathematical problem situations they have 
encountered in the classroom. Students and teachers can have a good discussion concerning 
similarities and differences between these situations that can help students to see .the value of 
mathematics practice in both contexts. 

In both in-school and out-of-school experiences, students participating in mathematics practice 
will become engaged with novel mathematics goals that require form-function shifts. Teachers 
who observe these gradual and complex shifts, gain valuable assessment information about 
students and can serve to facilitate the process of students acquiring mathematical knowledge to use 
as cognitive tools. 

Concluding Remarks 

Mathematics learning and practice in school and out of school differ in some significant ways. 
Some of these differences may be inherent because a concept is learned and used differently in 
school than out of school. However, we believe that many of the differences can be narrowed by 
creating experiences that engage students in doing mathematics in school in ways similar to 
mathematics learning and practice outside of school. The framework Saxe (1991) outlined for 
examining the interplay between sociocultural and cognitive developmental processes targets 
cultural practices as important contexts for study. Similarly, our discussion has used Saxe's 
framework to suggest how more and deliberate interplay can be encouraged between these 
developmental processes by focusing on mathematics learning and practice in everyday contexts as 
starting points. We believe that by making in-school and out-of-school mathematics experiences 
more complementary, student learning and practice in both of these situations can be enhanced. 


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Mathematics In and Out of School 

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' 15 


Mathematics In and Out of School 

Apple and Vegetable Salad 

Salad fcgedjgnis 

2 cups diced apples 
1/2 cup shredded carrots 
1/2 cup chopped peanuts 

2 eggs 

1/2 cup sugar 
lTbl flour 
1 Tbl butter 
1/4 cup vinegar 
1/2 cup cream 

Mix apples, carrots, and peanuts together. Cook dressing 
until it thickens. Add cream and allow to cool. Pour 
dressing over salad mixture and serve. Serves 6. 

Figure 1 


onverting Square Feet to Square Yards 

Figure 2 




J 6 

Mathematics In and Out of School 

Ordering Problem 

You are a manager in a small restaurant Each Friday you have to decide what will be cooked 
during the period of the next week (from Tuesday to Saturday and Monday in the following week) 
and send a suitable order to the commissary shop. Today is Friday, January 14 and by looking at 
the catering book and taking into consideration possible trends, you found out that for each day 
you will need the following amount of meat (in pounds): 

Tue Wed Thu Fri Sat Mon 

1/18 1/19 1/20 1/21 1/22 1/24 

Needed 25 25 30 20 10 20 

The commissary shop offers you good quality and very cheap meat but they deliver their goods 
only four times a week: on Monday, Wednesday, Thursday, and Friday. You can purchase things 
in advance and keep them in storage but the meat freezer capacity is only 55 pounds. So on 
Friday, January 14, you have to plan very carefully how much meat has to be delivered each 
delivery day to cover your needs. Meat comes frozen so it needs one day to be put aside and 

Plan the delivery schedule for the coming week: 

Mon Tue Wed Thu 

1/17 1/18 1/19 1/20 

Meat to 

be no 

delivered delivery 
in pounds 

Figure 3 

Fri Sat 

1/21 1/22 




Mathematics In and Out of School 

Four-Parameter Model* 





*(Saxe, 1991, p. 17) 

Figure 4 




Mathematics In and Out of School 

Expansion of Four-Parameter Model* 




Figure 5 

*(Saxe, 1991, p. 185)