CONNECTING OPERATION CONCEPTS ACROSS THE PRE-K - 8 GRADE BANDS

I. PERSISTENT MISCONCEPTIONS

Evidence abounds that secondary school students and adults, most of whom have "mastered" school algorithms for operating with fractions, decimals, and signed numbers, fail to apply their learning appropriately to practical situations. Baroody (1991) quotes Post (1981) commenting on results of a National assessment: only about 1/3 of 13-year-olds gave the correct answer to 3/4 + 1/2. It is likely that most of these students were well aware of the total value of 3 quarters and 1 half dollar, but apparently few could draw on this informal experience to assist them in answering the problem. This writer has asked preservice elementary teachers for a word problem to model -5 - (-2) and received many blank or confused responses. If asked how far the temperature changed if it started at -2 and went to -5, these same students could respond easily, but they saw no connection between that situation and the expression

-5 - (-2).

Other studies indicate that students have incorrect informal concepts of operations, particularly of multiplication and division involving rational numbers. Greer (1988) cited a study by Ekenstam and Greger (1983) in which 12 - and 13 - year- olds were given two problems: 1) A piece of cheese weighs 5 kg. One kg costs 28 kr. [a Dutch unit of money]. Find the price of the cheese. 2) A piece of cheese weighs 0.923 kg. One kg costs 27.50 kr. Find the price of the cheese. Eighty-three percent of the participants correctly chose multiplication for the first problem, but only 29% believed the second problem should be solved by multiplication. Their reasoning: the piece of cheese must cost less than 27.50, so division must be used to obtain a smaller number.

This same misconception was found to be prevalent among preservice teachers in studies conducted by Graeber (1988) and by Thipkong (1991). In the latter study, 32 of 65 participants believed one should divide to obtain the number of minutes in .25 hours. The study by Graeber included a two-part problem similar to the first cited above: "If a motorcycle travels 15.5 miles on 1 liter of gas, how far will it travel on a) 3 liters? b) .75 liters?" Even with this arrangement, some students chose to multiply for a) but to divide for b), so strong is the belief that "multiplication makes bigger, division makes smaller."

II. POSSIBLE CAUSES

Brown and Burton (1978) proposed a theory of "procedural bugs" to describe students' errors in carrying out mathematical computations. They emphasized that student errors tend not to be random, but rather due to a consistent "bug" in the student's procedure. A second paper by Brown and VanLehn (1980) attributed the origin of "bugs" to incomplete learning and forgetting: a student leaves out a step altogether, or reaches an impasse and invents a way out.

While helpful in emphasizing the tendency of students to act from a consistent cognitive base, Brown's theory might not go deep enough. Giyoo Hatano (1996) studied elementary students' use of buggy algorithms and found the same student was prone to use both the correct procedure and a buggy procedure on different problems on the same page. Hatano suggests that students believe the buggy procedures to be effort-saving, equally valid variants. Interestingly, Hatano notes a different relevant concept: students who understood the "trade principle" tended not to use buggy algorithms on multi-digit subtraction. This result suggests that children could resist the temptation to save some effort because they understood why the correct procedure was necessary. Thus, the most profitable direction for helping students use procedures correctly might be to focus on helping them understand why the operations work as they do. If the principles are understood, it seems possible the errors and confusion might be avoided.

Efraim Fischbein, et. al. (1985) published a landmark study giving insight into the incomplete concepts students often bring to their understanding of operations, particularly multiplication and division. This group of researchers postulated the following: "Each fundamental operation of arithmetic generally remains linked to an implicit, unconscious, and primitive intuitive model." For addition, the intuitive model seems to be "putting together," for subtraction, either "take away" or "building up." Multiplication is viewed as "repeated addition,"and division is either "partitive," (i.e., sharing) or "measurement," (i.e., repeated subtraction).

If this hypothesis is correct, students will have difficulty with decimal or fractional multipliers: how do you add a number .75 times? In division, both of the primitive models require the dividend to be larger than the divisor, and the partitive model requires a whole number divisor (how do you share cookies among .4 friends?). Note in particular that the primitive models will follow the rule cited earlier, "multiplication makes bigger, division makes smaller."

The study carried out by Fischbein, et. al. involved 628 pupils from grades 5, 7, and 9 in Pisa, Italy. Their difficulties on the word problems posed tended to support the hypothesis. In addition to demonstrating the conjectured difficulties, the 5th grade students showed additional difficulty with measurement division in general, leading the researchers to conjecture that initially students only recognize the partitive model for division.

Studies by Graeber (1988) and Thipkong (1991) have results which tend to support this hypothesis, specifically illustrating the persistent belief that multiplication makes bigger and division makes smaller. Greer (1988) and Bell, Swan, & Taylor (1981, cited in Greer, 1988) have documented the same misbelief. In addition, Graeber (1989) found that most incorrect answers to division problems posed were on problems where the whole-number divisor was larger than the dividend. In interviews, 22 of 33 participants reversed the roles and claimed that "in division the larger number should be divided by the smaller number."

Fischbein's view is also consistent with Sleeman's misgeneralization theory, as described by Blando (1989). This theory suggests that some errors result when a student infers too many rules consistent with an example given, rather than just the correct rule(s). Basing their beliefs on typical whole-number examples, students infer that all multiplication involves repeated groups, resulting in a larger product, and all division follows the pattern "larger divided by smaller," resulting in a quotient which is smaller than the dividend.

Mulligan and Mitchelmore (1997) take issue with Fischbein's models. They propose that all multiplication and division solutions involve equal-sized groups, and the distinguishing characteristics in solution relate only to student's methods of counting: direct counting, repeated addition, or multiplicative operation. They believe the intuitive model doesn't reflect any particular feature of the problem, but rather the student's imposed structure. They do note, however, that problem difficulty varies with the semantic structure of the problem--in particular, students found the comparison problem ("4 times as many books" to be moderately difficult and the Cartesian product problem extremely difficult.

However, all of the problems in Mulligan and Mitchelmore's study involved whole numbers, and all of the division problems had divisors smaller than the dividend; thus, all of the problems except the two noted as "difficult" fit the primitive models. It is this writer's opinion that Mulligan and Mitchelmore's results do not contradict Fischbein's models--they simply measure other variables. The varying difficulty they note could be interpreted as how well the problem statement fits or fails to fit the intuitive structure the student would like to impose, i.e., the primitive model the student would like to match up to the problem. The fact that students' choice of counting method seems independent of the semantic structure of the problem may simply demonstrate that in the familiar setting of a problem which follows the constraints of the primitive model students feel confident enough to vary their methods.

Thus, this writer believes that Fischbein's theory of primitive models has a great deal to commend it--it offers a potentially useful way to explain the difficulties students encounter when extending their knowledge of operations to signed numbers and rational numbers. As Greer notes, these misconceptualizations are overgeneralized from the whole numbers, persistent, resistent to instruction, protected by distortions, and coexistent with later knowledge even when contradictions are produced. Fischbein sumamrizes a fundamental educational dilemma: in order to introduce operations in a meaningful way, students must be taught from concrete models which mirror their intuition. However, there is a danger that these models then become so entrenched in the student's thinking that they exert influence on their actions in domains beyond the whole numbers, where these first models are no longer adequate.

With such dire effects, consideration must be given to whether these consequences could be mitigated. Is there any way to introduce operations with signed numbers and fractions so that students extend their primitive models to include the new types of results obtained?

III. POSSIBLE SOLUTION

Semadeni, (1984), recommends an alternative approach to presenting definitions of operations which merely preserve formal rules. In his view, although dividing fractions via the invert-and-multiply method can be formally justified, there are different, more intuitive approaches which should be pedagogically preferred. Semadeni suggests a "principle of concretization permanence," in which a well-chosen concrete setting is used to illustrate operations in first the familiar domain of whole numbers and then the unfamiliar domain. For example, multiplication of fractions can be introduced in the following manner:

Consider boxes of cakes:

3 boxes of 4 cakes is 3 x 4 or 12 cakes

1/2 box of 4 cakes is 1/2 x 4 = 2 cakes

If a small box only contains 1/2 cake, how much is 1/2 of a small box?

1/2 box of 1/2 cake is 1/2 x 1/2 = 1/4 cake

Semadeni's recommendations seem to fit particularly well the need described by Fischbein. If indeed the problem is that the primitive models do not adequately describe many problem situations yet students persist in believing their models, it seems clear that models are very important in students' minds. Thus, teaching procedures or formal justifications or even offering calculators will not address the fundamental issue: students need to extend their models to accomodate new types of numbers and the new types of results obtained.

In fact, Graeber (1989) offers evidence that students may interpret procedural knowledge in ways that serve to strengthen their incorrect concepts. She reports that some students believed the rule for moving the decimal point before carrying out a long division problem implied a conceptual rule, "you must divide by a whole number." Similarly, changing fraction division by the invert-and-multiply procedure was interpreted by some to mean "you can't divide by a fraction."

Greer (1988) also recommends that fraction multiplication and division be introduced in ways which help students extend the concept, not just the algorithm. Harrison (1989) reports on a study in which an experimental group of students was given this opportunity; the experimental group had significantly higher achievement on a fractions problems test and similar scores on a fractions computation test when compared to the control group. As Harrison notes, this goal of extending the concept (in this case, of multiplication to include fractions) fits Skemp's theory, as well. Relational thinking has the goal of relating new concepts to an existing schema or modifying an existing schema to accomodate the new idea.

Once a new or extended model is introduced, Graeber (1988) points out that students must be encouraged to reflect thoroughly on its implications. She suggests that students solve word problems which violate primitive constraints, compare and contrast multiplication and division in rational and whole number domains, and use estimation and intuition for problems like .5 ¸ .25 or .5 x .2 before carrying out a procedure. Students can also be asked to write their own word problems for expressions like these (Graeber, 1989).

IV. POTENTIAL MODELS FOR FRACTIONS AND INTEGERS

Semadeni makes a number of specific recommendations for models which can be used to introduce fraction and integer operations. Other educators have proposed models, as well. Regarding fraction multiplication, Graeber (1990) recommends extending student's early experiences with multiplication as a rectangular arry to the notion of rectangular area, and then introducing fractions as dimensions of the rectangle. Greer (1988) recommends a fuel pump which displays cost and number of gallons as a familiar example. Mulligan (1997) emphasizes the connection of equal-sized groups, and recommends that students be taught that .63 times something means to partition it into 100 equal sized groups and take 63 of them. Kouba (1989) noted that the children in her study seemed to view multiplication as a two-step process: make equivalent sets and then put them together. She recommended students be encouraged to make one referent set and operate on it in one of three ways: take a part of it, take several of it, or take several and a part.

Semadeni relates dividing fractions to filling sugar bags of various sizes and determining how many bags can be filled by 6 kg of sugar. Beginning again with whole numbers, he would ask a series of questions such as, "How many 3 kg bags can be filled from 6 kg? How many 1/2 kg bags? How many 1/3 kg bags? How many 2/3 kg bags?" Baroody (1991) offers a partitive interpretation of a division problem like 6 ¸ 1/2: If six crayons fill 1/2 of a box, how many crayons does it take to fill the whole box? He also recommends exploiting unit price and rate examples which involve fractional amounts, e.g. if roast beef costs $8 for 3/4 pound, how much is 1 pound?

The need for common denominators when adding fractions can be motivated, Semadeni suggests, by noting that 20 min + 1/2 hour must first be changed to like units before the addition can proceed. Alternatively, lengths may be used, as in 2 feet + 3 inches + 1 yard.

Semadeni suggests that subtraction of negative numbers can be motivated by two-color counters: -2 - (-5) can be modeled as a take-away problem if -2 is represented with 3 positive and 5 negative counters. An alternate method would be to consider the comparison model of subtraction as modeled on a number line. However, this writer encountered some resistance from preservice elementary students when presenting both of these concepts. In particular, they had a great deal of difficulty accepting -2 - (-5) as the directed distance from -2 to -5. Another widely used model is that of assets and debts, where removing a debt has the effect of increasing one's net worth. A similar linear example involves gains and losses in a football game: subtraction could be modeled as removing a 5-yard penalty.

A subtraction model which has been widely used in the Netherlands is the "Witch's Cauldron." (Streefland, 1996). The cauldron contains hot cubes and cold cubes. If the witch removes hot cubes, the cauldron cools off; if she removes cold cubes, what happens to the cauldron? Streefland, however, considers this "false concretization" because it doesn't explicitly admit that -5 is really the net effect of a pair of numbers like +12 and -17. This characteristic also makes it less likely the model could be linked to an earlier addition and subtraction model familiar to students from their whole number learning.

Streefland used a model of people entering and exit a bus as it made various stops to model addition and subtraction. He found students accepted the notion of exiting passengers as negative numbers easily. He was able to model subtraction by asking a question like, "At the last stop 2 people entered the bus and 5 exited. There are now 15 passengers on the bus. How many were there before the last stop?" Several students immediately stated, "I'll make the bus drive backwards." They then wrote 15 + 5 - 2, instinctively using opposites.

Multiplying negative numbers is a notion which seems resistant to models. Certainly the fact that historically this multiplication was not even widely accepted until it was proved formally by Hankel in 1830 should forewarn educators that intuitive models are not easy to construct. Some educators feel that this is an example of a point in time where it is necessary to abandon the model approach and resort to the formalism that, frankly, does sometimes describe the discipline (Harel, p. 74). Complicated recommendations abound: a rabbit faces left and makes three hops backwards of four units each, landing at +12; starting with 12 positive and 12 negative chips, remove three groups of four negatives to obtain +12. Streefland (1996) described a "small trains" example used in the Netherlands, but stated that this setting had a "complicated logic all its own" which was probably more difficult than helpful to students. Semadani proposes a model which seems to make more sense, however: view a decrease happening over time, and ask what the situation was in the past. For example, a girl spends $3 per day on lunch. Today she has $0. How much money did she have four days ago? Transition Mathematics (Usiskin) is a pre-algebra text which recommends a similar type of thinking: If a person has been losing 2 pounds per month, then 5 months ago he was 10 pounds heavier, i.e., -2 pounds/month x -5 months = +10 pounds.

V. LINKING TO STUDENTS’ PREVIOUS MODELS

This task is more difficult than the suggestions above indicate, however. In order for students to truly believe they are not learning a new operation, but extending a previously used operation, the new model must be linked to their prior knowledge--it cannot merely replace prior models. Attempting to simply "teach the correct" way via a model without regard to informal student's notions seems no more likely to succeed than teaching the correct way via a procedure. In fact, it may be less likely to succeed, since a procedure at least can be memorized. Thus, what initially looks like a task directed at upper elementary and middle school teachers and curricula, when operations with fractions and signed numbers are introduced, is actually a challenge with implications for lower elementary school teachers and those who write textbooks for these grades.

For example, when Semadeni recommends modeling " 6 ¸ 1/2" by asking "How many 1/2 kg bags of sugar can be filled from 6 kg?" he is assuming students have access to a measurement model of division in their mental conception. This is by no means certain, however. Graeber (1988, quoting Tirosh 1986) found that when asked to provide a word problem for a division expression, 63 - 78% of preservice teachers provided a partitioning problem. This writer has experienced similar blank stares when attempting to "remind" preservice elementary teachers that 12 ¸ 3 means "how many 3's are in 12." Interestingly, Kouba (1989) reported that young children were comfortable solving both measurement and partitive types problems. Did these college students once have a stronger intuitive concept of measurement division which failed to thrive?

A similar "lack of linking model" may confound efforts to model integers. Early experiences with addition and subtraction emphasize the combine model of addition and the take away model of subtraction. Fuson (1988) remarks, "All too often, school mathematics in operation opts for simplicity at the cost of richness and flexibility. Subtraction . . . initially, and in many texts for a long time, is given only one interpretation, that of take away. . ." Reviews of American elementary textbooks have found that only four of twenty possible types of addition and subtraction word problems account for 75 to 91% of the problems in the text (Stigler et. al., 1986). Stigler claims, in fact, that children entering first grade can already solve the simple word problems which are heavily represented in American texts in grades 1 - 3. In contrast, Soviet textbooks offer far more variety in the types and complexity of problems students are asked to solve. Thus, it may come as no surprise that when this writer asked students what operation would be used to determine the distance between two points on a number line, no students offered a suggestion. The attempt to link subtraction to directed distance and extend to include distances from points below zero was doomed because the students did not have the prerequisite knowledge of subtraction modeling distance or difference.

Francis Thompson (1993) encountered a similar frustration in attempting to teach fraction division when she discovered that students had never expressed a remainder as a fraction. Graeber's study (1990) found that 4th and 5th grade students had no concept that 5 ¸ 15 could be written 5/15 to give the result 1/3. The misconceptions Graeber recounted (1988, 1989) about division always being "larger number ¸ smaller number" have no opportunity to be questioned if students only work through the stereotyped problems encountered in many textbooks. Apparently this situation has persisted for many years, as indicated by this quote in Greer (1988) of Stevenson (1925): A 12-year-old boy explained his approach to word problems by stating, "If there are two lots of numbers, I adds. If there are only two numbers with lots of part (digit), I subtracts. But if there are just two numbers and one littler than the other, it is hard. I divides if they come out even, but if they don't, I multiplies."

Clearly, student's earliest experiences with operations must be far richer than they presently are in order to prepare them to extend these operations to the domains of integers and frctions.

VI. IMPLICATIONS FOR TEACHING

As has been emphasized already, new concepts cannot be simply deposited, ready-made, into children's heads. The effects of this assumption are the present state of affairs--where students fail to see any connection between their intuitive notion of what an answer should be and the operation or result given by a school algorithm. Students must connect their informal knowledge to the extensions and clarifications that will enable them to become efficient and effective users of mathematics. This by no means implies instruction is of less importance; in fact, this requires greater skill of the teacher--the teacher must first determine what informal concepts the child brings to the lesson, and then lead the child through a process which will enable the child to add to or modify his present construction in the direction the teacher knows will be most helpful.

Instruction should include well-chosen physical models. Thomas O'Shea points out that mathematics was closely related to physical devices until the 18th century, when formal schooling in the spirit of the Latin grammar schools became the prominent educational institution. Unfortunately, he notes, "The educational structures developed by the mathematical practitioners were not adopted, and instruction in arithmetic became formalized, theoretical, and divorced from practice" (O'Shea, 1993).

The specific types of manipulatives offered can have a profound impact on students' understanding. Beishuizen (1993) reports that the types of physical manipulatives and models used seem to influence the thinking that children do. Beishuizen found differences in Dutch children's addition procedures depending on the materials provided to them--some used blocks, and others used a hundredsquare. When taught to model a problem like 37 + 25 with blocks, most children combined the tens first to get 50, then the ones, 12, and then combined to get 62. This strategy was used despite specific instruction to begin with the entire first number, 37, add 20, and then add 5. In contrast, students taught to model the instruction procedure on the hundred square tended to begin with 37, move down two rows to add 20, then across 5 boxes to add the 5.

Additionally, some models should be introduced specifically because they have the potential to be extended to the integers and rations. On this basis Thipkong (1991) recommends students be introduced to linear models. Although all of the whole number addition and subtraction problems in the early grades can be done with discrete counters, a model like the number line has added value because it can be extended down to negative numbers and because it permits modeling of numbers between two integers. In fact, this writer recalls independently deciding as a second-grader that 3 - 5 was meaningful: numbers simply needed to be inserted to the left of zero on the number line that decorated the front of the classroom. Even though the teacher said, "you can't take a larger number from a smaller number," the number line model was more convincing. When negative numbers were formally introduced about five years later, it was no big surprise.

Three cautions are in order about the use of manipulatives. Beishuizen found some students seemed to be just "reading off" the answer from their blocks, without actively engaging in thinking processes (1993). Thus, when the blocks were not available, these students were much less capable at solving problems. Second, the manipulatives need to be closely related to the symbolic notation if students are to make the connection (Mack, 1995). Expect this process to take time and to require some reteaching and overcoming misconceptions. Finally, Constance Kamii specifically cautions against the assumption that using base ten manipulatives in first grade will "deposit" the notion of place value in children's heads (Kamii, 1998). Others have used base ten blocks with second-graders, however (Fuson, 1988; Beishuizen, 1993), and reported that these somewhat older students find the blocks supportive in their computation. By this age, apparently, many students have successfully constructed the notion of a "ten" as a single entity.

Despite these cautions, however, manipulatives and practical situations present a wonderful opportunity. Students do not need to have formal procedures for subtracting negative numbers or dividing by a fraction to be successful considering simple problems which they can easily model. Teachers can challenge students to make sense of problems like the following: "The temperature started at 8 degrees and went down 10 degrees. What temperature is it now?" or "Joe has six candy bars. He plans to eat 1/2 of a candy bar each day. How many days will his candy last?" These early experiences can help prevent the deeply held but very mistaken misconceptions that so many older students retain.

Most importantly, elementary teachers need to have in mind a long-term, global view of the mathematics their students need to learn. Understanding how students will eventually need to extend their concepts of addition, subtraction, multiplication and division to negative numbers and to fractions, teachers must provide them with the models and early versions of extendable concepts which they will need later.

VII. IMPLICATIONS FOR "PRINCIPLES AND STANDARDS"

The overview of the Number and Operation Standard addresses the issues presented in the early part of this paper, stating explicitly that in grades 6 through 8, "students' intuitions about operations need to be revised." The recommendations for the earlier grades, however, do not completely address the need for young students to be introduced to the models which will most successfully extend to integers and rationals. I recommend changes to the pages indicated below.

Page 109: Under the bullet "Understand the meaning of operations and how they relate to each other"

(line 28), I suggest making separate sub-bullets for addition and subtraction. The first would specify "understand addition as both combining and increasing," and the second, "understand subtraction as both take away and as compare or equalize." Separating these gives appropriate emphasis to the centrality of these concepts in the curriculum of the early grades.

Page 113: After the sentence (lines 2 and 3) "Many varied contexts . . . such as addition and subtraction," I believe sentences should be inserted with more specific recommendations about those contexts: "Children need experience both with combining discrete objects and with increasing (or decreasing) a continuous amount, such as pouring liquid into a container. They should also relate addition and subtraction to changes in location, e.g. by an elevator model at first and later on a number line."

Grades 3 - 5:

Page 156: Under the bullet "Understand the meaning of operations and how they relate to each other" (line 23), I recommend separating multiplication and division into two separate sub-bullets. The first could state, "understand the meaning of multiplication as repeated addition, as an array, and as an area." The second may read, "understand the meaning of division as both repeated subtraction and as sharing."

Page 159: I believe one or two sentences like the following should be added to the end of the first paragraph elaborating on multiplication (line 10): "Multiplication should also be extended from an array to the continuous model of an area, leaving the 'square units' intact in the diagram. This continuous model will be an important link when students need to model multiplication with fractions in middle school."

The discussion of division highlights both the partitive and repeated subtraction model, which is very important for students' later understanding of division statements like "6 ¸ 1/2." However, I believe that the discussion of division needs to mention some additional concepts that students will need in order to successfully extend division to the rational number domain. Wording like the following is suggested:

"Students should also consider various ways to express the remainder and how it differs depending on the context. For instance, packaging 58 cookies in packages of eight yields seven packages with two leftover cookies. Dividing 58 pounds of mulch into eight piles of equal weight, however, yields 7 2/8 or 7 1/4 pounds per pile.

"Finally, students should consider problems where the divisor is larger than the dividend--for instance, six friends sharing three candy bars. They should learn that 3 ¸ 6 can also be written 3/6, and should use a model or diagram to illustrate that the answer is 1/2."

I believe it is important to make these changes to highlight the importance of students' early encounters with rich, extendable models. Just as a language-rich environment promotes a child's verbal development, I believe an environment rich in models which embody many facets of mathematical operations promotes a child's ability to understand mathematical concepts.

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