Belief About Mathematical problem Solving student beliefs about problem solving may also influence the way they plan a method for reaching a solution. Perhaps the most destructive belief concerning the planning process is the idea that math problems must be solved by applying meaningless procedures. Schoenfeld (1992) summarizes this belief as follows: “Ordinary students cannot expect to understand mathematics, the expect simply to memorize it and apply what they have learned mechanically and without understanding” (p. 359). For example, lester, Garofalo, and kroll (1989) reported that many third graders believed that “all story problems could be solved by applying the operations suggested by the key words present in the story (e.g., in all suggests addition, left suggests subtraction, share suggests division)”. (lester et al., 1989, p.84). where did such abizarre belief come from? According to lester and colleagues, such a belief was well founded because “most of the story problems to wich these children had been exposed could be answered correctly by applying their key-word method” and in many cases “teachers had taught them to look for key words” (p. 84).

Another common belief that prevents students from using productive planning processes the idea that “students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less” (schoenfeld, 1992, p. 359). The effect of this belief is that students will give up on a problem if they are unable to solve it within a few minutes. For example, when schoenfeld (1998) asked high school students how long it should take them to solve a typical homework problem, the average estimated time was 2 minutes. When he asked them how long they would work on a problem before giving up, the average estimated time was 12 minutes. The belief that all math problems can be solved quickly is based on students’ experience in mathematics clasess: “students who have finished a full twelve years of mathematics have worked thousands upon thousands of ‘problem’-virtually none of which were expected to take the students more than a few minutes to complete” (schoenfeld, 1988, pp. 159-160). In monitoring their problem solving, students are likely to quit when they reach a major obstacle even though they might have solved the problem if they had persevered.

One of basic elements of mathematical proficiency is productive disposition, which is the “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy” (Kilpatrick et al., 2001, p. 5). One source of information about productive disposition comes from surveys of student attitudes. For example, the National Assessment of Educational Progress (NAEP)---a nationwide assessment given to U.S. students---reported that 54% of the fourth graders and 40% of the eighth graders thought that mathematics was mostly a set pf rules and that mathematics learning means memorizing the rules (silver & Kenney, 2000). In general, girls have more negative attitudes about mathematics than do for boys (Leder, 1992; silver & Kenney, 200).

Observations of student problem solving also provide information about productive disposition. Students often solve mhatematics problem by manipulating symbols without understanding; that is, by combining numbers in the problem to produce an answer. For example, consider this problem: “john’s best time to run 100 meters is 17 seconds. How long will it take him to run 1 kilometer?” When this problem was given to middle school students, almost all of them (i.e., 97%) proceeded to carry out computations such as 10 X 17 = 170 seconds (Verschaffel, greer, & De Corte, 2000). Yet, if you give one moment of thought to this problem you will realize that a runner cannot keep up the same pace for 1000 meters as for just 100 meters. (halaman 185)

As another example, considers this problem: “there are 26 sheep and 10 goats on a ship. How old is the captain?” (as reported in Verschaffel et al., 2000). When this problem was given to first and second graders in France the vast majority gave an answer based on combining the numbers in the problem without any indication that the question was meaningless, and the result were replicated with a group of elementary school students in Switzerland (Reusser, 1988(. Students focused on “whether to add, subtract, multiply or divide rather than asking whether the problem made sense” (Vershaffel et al., 2000). Bransford and stein (1993) tried giving the sheep and goat problem to a fifth grader in the United States with the same result:

Much to our surprise and dismay, the answer given was 36. When we asked why, we were told: “Well, you need to add or subtract or multiply in problems like this, and this one seemed to work best if I add. (p. 196).

Based on findings such as these, schoenfeld (1991) concluded that students learn to engage in what he called “ suspension of sense-making” (p. 316) when they solve mathematics problems---that is, “suspending the requirement that the way in which problems are stated make sense” (p.316). furthermore, schoenfeld noted, “ there is reason to believe that such suspension of sense-making develops in school, as a result of school” (p. 316). Overall, there is convincing evidence that students need help in developing a productive disposition concerning mathematics.

IMPLICATIONS FOR INTRUCTION: TEACHING FOR PLANNING

When confronted with a new word problem, some students do not know what to do, even though they may know how to carry out the required arithmetic. Not knowing what to do reflects a lack of strategic knowledge, and in particular, a failure in planning. National assessment of mathematics achievement show that conventional instruction in word problem solving is not equipping students with the planning skills they need (Dossey, Mullis, Lindquist, & Chambers, 1988; Silver & Kenney, 2000).

How can we help students develop appropriate planning skills? A team of researchers called the Cognition and Technology Group at Vanderbilt developed a video-based program for helping students learn how to plan solutions to mathematics problems (Bransford et al., 1996; Cognition and technology Group at Vanderbilt, 1992; Van Hanegghan et al,. 1992). The materials consist of a series of video episodes of the Adventures of Jasper Woodbury, each lasting 15 to 20 minutes. In each episode, the character is faced with a challenge that requires mathematical problem solving, such as a trip, generating a business plan based on statistics, and using geometry meaningfully. Working in small groups, the students in the class solve the problem.

For example, in the episode “rescue at boone’s Meadow,” Jasper’s friend Larry teaches Emily hoe to fly an ultraligth airplane in scene that provides information about the palne’s payload, fuel capacity, fuel consumption, speed, and landing capabilities. Later, in a restaurant, Jasper tells Emily and Larry about his planned fishing trip to Boone’s Meadow,

Noting that a landing strip is located next to where he plants to park his car and that the hiking distance to Boone’s Meadow is 18 miles. On the way from the restaurant, Emily and Larry stop to weigh themselves. Jasper is next seen happily fishing at Boone’s Meadow when he discovers a wounded bald eagle and trough a radio is able to get this information to Emily. In the final scene, Emily is in a veterinarian’s office, where she learns about eagles and consult a map on the wall showing that no roads lead to Boone’s Meadow. The problem is how to save the eagle.

Working in groups to solve the problem, student must consider a range of possibilities and variables. For example, the ultralight aircraft needs to fly a greater distance than normal to get to the rescue site; to fly a greater distance, more fuel must be added, but this would increase the overall weight, which in turn would require a change in pilots. Although students can usually generate an answer within about 30 minutes, they are encouraged to work longer to develop a better solution. In the process of problem solving, they often review parts of the video to check or gather information. They also work on alternative versions of the problem. In all, they spend about a week on each episode.

Does participating in the jasper series affect students’ planning skills? To examine this question, 10 classrooms of fifth and sixth graders (Jasper-trained group) received instruction in three or four Jasper adventures for 3 to 4 weeks, whereas 10 matched classrooms (control group) received their regular instruction, which focused on word problems. To measure their planning skills, all students took a planning test before and after instruction, consisting of problems such as shown in figure 5-14. As you see, the test included questions about how to plan a solution to a word problem (question 1) and how to break a problem into parts (question 2). Although both the Jasper-trained group and the control group scored about the same on the pretest (i.e., achieving scores of approximately 20 % corret), the Jasper-trained group on the posttest (i.e., achieving approximately 40% corret compared to 25% for the control group). These results show that training with the jasper adventure resulted in a large improvement in students’ planning performance, whereas conventional training in word problems did not .

Why does the jasper series improve students’ planning skills? The Adventure of jasper Woodbury is based on three principles that distinguish it from conventional mathematics programs:

Generative learning: students learn better when they actively construct their own knowledge rather than when they passively receive information from the teacher.

Anchored instruction: students learn better when material is presented within an interesting situation rather than as an isolated problem.

Cooperative learning: students learn better when they communicate about problem solving in groups rather than when they communicate about problem solving in groups rather than when they work individually.

Although it is not possible to isolate the features of the jasper series that are responsible for its effectiveness, the program is based on a combination of generative, anchored, and cooperative methods of teaching and learning. In summary, as one reviewer noted, the jasper series “uses videodisc computer technology as a vehicle for changing the fabric of instruction, away from transmission and toward active problem solving in realistic contexts” (Lehrer, 1992, p. 287). Additional research is needed, but this promising project encourages the development of other programs based on the same instructional principles. (halaman 187)

This section examined both heuristics that help students devise solution plans and obstacles that may impede successful planning. Heuristics for planning include using a related problem, restating the problem, and breaking the problem into subgoals. Obstacles to planning include difficulties in finding a related problem, reliance on meaningless solution procedures, and a failure to persevere. How can you provide strategy training? It is important for students to recognize that there may be more than one right way to solve a problem and that finding a solution method can be a creative activity. Students need to be able to describe their solution methods and to compare their methods with those used by other students. Some researches have been successful in explicitly teaching strategies for problem solving, such as asking students to write a list of operations (or a number sentence) necessary for solving a problem, to list the subgoals needed in a multistep problem, or to draw a conclusion based on the partial completion of a solution plan. Sample multiple-choice items for some of these tasks are given in the third section of figure 5-1. In summary, devising and monitoring a solution plan are crucial components in mathematical problem solving. Students and teachers should pay as much attention to process (i.e., their solution strategy) as to product (i.e.,the final numerical answer). Research is needed to verify the preceding suggestions for improving students’ strategic planning skills mathematics. (halaman 188)

As shown in table 5-1, beliefs are another important kind of knowledge underlying the cognitive processes of planning and monitoring. For example, try the items in table 5-7. These the items are taken from a questionsnaire intended to measure students’ beliefs about mathematics learnings ( i.e., 17 items like the first 3 in table 5-7) and students’ beliefs about themselves as mathematics learners (i.e., 11 items like the last 2 in table 5-7). When Mason and Scrivani (2004) gave this questionnaire to fifth graders attending elementary school in Italy, they found that students did not have strong positive beliefs about mathematics learning or being a mathematics learner. To improve the students’ beliefs, Mason and Scrivani developed 12 1.5-hour mathematics lessons (treatment group) intended to help students see themselves as competent mathematics learners who possessed useful problem-solving strategies, who appreciated that there are many ways to solve a problem, and who were able to spend the time needed to solve nonrountine problems. The treatment group showed a large pretest-to-posttest gain on the questionnaire, where as the control group did not. Importantly, the treatment group also showed a large pretest-to-posttest gain in solving math problems, whereas the control group didi not. Overall this study encourages the idea that is possible to help students develop productive beliefs about mathematics. In a recent review of research, muis (2004) concluded that students’ beliefs about mathematics are related to their mathematical problem solving performance and importantly there is encouraging preliminary evidence that such beliefs can be changed through instructional interventions.

Solution Execution

WHAT IS SOLUTION EXECUTION?

Once you understand the title problem presented at the beginning of this chapter and devise a plan for solving it, the next major component is to carry out your plan. For a problem like the title problem, you need to be able to carry out arithmetic operations such as 7.2 X 5.4 = or 0.72 X 432 = . as you can see, problem execution requires procedural knowledge (i.e., knowledge about how to carry out procedures such as addition, subtraction, division, or multiplication).

( chapter 5 learning mathematics 189)

The acquisition of computational procedures involves a progression from naÃ¯ve procedures to more sophisticated procedures, and from tedious application of procedures to automatic application. In summary, as children gain experience, their procedures become more sophisticated and automatic. With wxperience students develop a collection of procedures that can be selected for various computational problems.

RESEARCH ON SOLUTION EXECUTION

Development of Expertise for simple Addition as an example of the development of expertise in computation, let’s consider a child’s procedures for solving single-column addition problems of the form,

m + n = . ,

where m and n are single-digit positive integers whose sum is less than 10.

Fuson (1982, 1992) has identified four major stages in the development of computational expertise : counting-all, counting-on, derived facts, and known facts. The caounting-all procedure involves setting a counter to 0, incrementing it m times, and then incrementing it n times. For the problem 2 ¬+ 4 = , the child might put out one finger and say “1”, put out another finger and say “2”, pause, put out a third finger and say “3”, put out a fourth finger and say “4”, put out a fifth finger and say “5”, and put out a sixth finger and say “6”.

The counting-on procedure involves setting a counter to m and incrementing it n times. For the problem 2 + 4 = , the child might put out two fingers and then say “3, 4,5,6,” as each of four additional fingers was put out. One version of this approach is what Groen and Parkman (1972) call the “min model”, which involves setting a counter to the larger of m or n and then incrementing the counter by the smaller number. For the problem 2 + 4 = , the child might put out fingers and then say “5,6,” as each of two additional fingers is put out.

The derived facts procedure (also called decomposition) involves using one’s knowledge of number facts to figure out answer for related problems. For example, the first number facts that a child learns are usually the doubles, such as 1 + 1= 2, 2 +2=4, 3+3=6, and so on. For the problem 2+4= , a student might say

Their fingers. Another method is to measure the time it takes to solve addition problems. Figure 5-15 summarizes the counting-all and counting-on (min version) procedures, with boxes representing actions and diamonds representing decisions. For example, we could make the following predictions concerning response times for each procedures. For the counting-all procedure, response time should be a function of the sum of m + n. for the

FIGURE 5-15

Counting all and counting on (min) procedures for simple addition

Source: from thinking, problem solving, cognition (2nd ed.) by Mayer © 1992 by W.H.freeman and Company. Used with permission

( chapter 5 learning mathematics 191)

Problem 2 + 4 = or 4 + 2 = , the child must increment a counter six times. For the min version of the counting-on procedure, response time should be a function of the smaller number (m or n ). For the problem 2 + 4 = or 4 + 2 = , the child must increment two times. For the derived facts procedure, response time should be the same for all problems since the child is simply “looking up” the answer in memory.

To determine which procedures children use as they begin formal instruction in computation, Groen and Parkman (1972) asked first graders to answer a series of single column addition problems. Their response time performance could best be described by the min model of the counting-on procedure. Figure 5-16 shows the response time for problems that min models says require 0 increments (such as 1 + 0 = or 5 + 0 = ), 1 increment (such as 5 + 1 = or 6 + 1 = ), 2 increments (such as 5 + 2 = or 6 + 2 = ), 3 increments (such as 5 + 3 = or 6 + 3 = ), and 4 increment (such as 5 + 4 = or 4 + 5 = ). As shown, response time generallyincreases by about one third second for each additional increment in the value of the smaller number. Thus most problems seem to be solved by setting a counter to the larger number and incrementing it by the smaller number. However, you might note that there is some evidence that doubles (0 + 0 = , 1 + 1 = , 2 + 2 = , and so on) were answered rapidly regardless of the number of increments; this suggests that doubles might already be well-memorized number facts (requiring a known facts procedure), whereaqs other problems require a counting procedure. ( section 1 learning 192).

Parkman and Groen (1971) also found that a min model best fit the performance of adults. However, the time needed for an adult to make an increment was one-fiftieth second, compared to one third second for first graders. Because it is unlikely that a person can count silently at a rate of 50 increments per second, Parkman and Goren (1971) offered an alternative explanation: for almost all problems, adults have direct access to the answer in their memories (i. e., on most problems, adults use a known facts approach), but on a few problems they fall back to accounting procedure. Ashcraft and Stazyk (1981) accounted for the performance of adults by assuming that they must “look up” answer in a complicated network. Thus, according to this view, adults apparently use some version of a known facts approach (also called retrieval), whereas first graders seem to be using some version of a counting approach. More recently, LeFevre, Smith-Chant, Hiscock, Daley, and Morris (2003) provided evidence that some adults report using procedures other than retrieval on some simple arithmetic problems, particularly involving multiplication. For example, on trials when they said they used counting (e.g., for 3 X 5 = , they said “5, 10, 15”) response time depended on the number of counts, but on trials when they said used retrieval (e.g., for 3 X 5 = , the answer just came to them) response time did not depend on the number of counts. It appears that people have a variety of strategies available to them, although most adults use retrieval most of the time for simple arithmetic problems.

Selection of Addition Procedures Students seem to progress through a series of mathematical discoveries, inventing progressively more efficient procedures for solving simple arithmetic problems. In a research review, Hopkins and Lawson (2002) distinguished between two views of how this cognitive development happens. According to a traditional view, the procedures develop in a step-by-step linear progression, with counting-on replacing counting-all, and then decomposition replacing counting-all, and then retrival replacing decomposition. According to an overlapping waves view, students progressively add more sophisticated procedures to their repertoire and increasingly use them. In short, we can ask: does a child’s procedural knowledge consist mainly of the more efficient methods that have replaced the earlier ones or of an ever-increasing collection of procedures ranging from the least to the most mature? To answer this question, Siegler (1987) asked kindergartners, first graders, and second graders to solve 45 addition problems, such as “ if you had 8 oranges and I gave you 7 more, how many would you have? “and” what is 8 plus 7?” after giving each answer, the students were asked to describe verbally how they solved the problem.

Children reported using five kinds of procedures: guessing (or not responding), counting-all, counting-on (min version), decomposition (also called derived facts), and retrieval (also called known facts). Table 5-8 shows the percentage of time students at each grade level used each of the procedures. Most children reported using at least three procedures, and at no age was one procedures? Siegler (1987) found that on problems for which children reported using a counting-on procedure, the min model was a good predictor of their solution times, but on problems for which they reported using other strategies, the min model was not a good predictor. In reviewing these result, Siegler and Jenkins (1989) concluded that “these and a variety of other data converged on indicating that children used the strategies they reportednusing and that they employed them on those trials where they said they had” (p. 25). (chapter 5 learning mathematics 193)

Overall, these result suggest that students build an arsenal of addition procedures and choose procedure independently for different addition problems. Interestingly, children use a retrieval approach (which can also be called known facts) for easy problems, but for hard problems they rely on what Siegler and Jenkins (1989) call” back-up strategies” (such as decomposition or counting). In deciding whether to use counting-all or counting-on (min version), students are more likely to use counting-on (min version) when one of the addends is smaller, such as 9 + 2 = , than when the two addends are close in value, such as 5 + 6 = . as you can see, this choice makes sense because it is easier to use a counting-on (min version) when one addend is smaller. Siegler and Jenkins (1989) argue that rather than using a single procedure for all addition problems, children “behave adaptively …in choosing among alternative…strategies” (p. 29).

As another example of how students select addition procedures, consider the simple addition problem, 8 + 3 = . if you are like most adults, you simply retrieved the answer (i.e., 11) from you long-term memory. However, elementary school children use a variety of procedures to solve simple addition problems including decomposition to 10 (i.e., 8 + 3 = 10 + 1 = 11) or counting-on (for 8 + 3, start with 8 and increment it three times, as 8,9,10,11). Torbeyns, Verschaffel, and Ghesqueire (2004) asked high achieving second graders, low achieving second graders, and third graders in Belgium to solve simple addition problems and indicate the methods they used. Table 5-9 shows that high-achieving second graders were more likely than low-achieving second graders to use retrieval as their solution procedure, whereas low-achieving students were more likely to use decomposition to 10 or counting on as their solution procedure. Third graders performed just like the high-achieving second graders. These result suggest that students tend to shift from more attention demanding procedures (such as decomposition and counting-on) to less attention demanding procedures, studens have more working memory capacity for other aspects of mathematical problem solving. However, consistent with the overlapping waves view, all three strategies were avaible for all learners.

As you can see from Siegler and Jenkins’s (1989) study as well as more recent work (shrager & Siegler, 1998; torbeyns et al., 2004), there is convincing evidence for the overlapping waves view (Hopkins & Lawson, 2002). The overlapping waves view is based on the idea that children often have multiple procedures available, can invent new ones,

( section 1 learning 194)

experience competition between inefficient procedures (such as counting procedures) and efficient ones (such as retrieval), and gradually develop retrieval as their main procedure for simple addition. Complex Computational procedures once a child has achieved some level automaticity in carrying out simple procedures (e.g., single-column addition or subtraction), these procedures can become components in more complex computational procedures. For example, solving a three-column subtraction problem such as 456-321 =

Requires the ability to solve single-digit subtraction problems such as 6-1 = , 5-2 = , and 4-3 = . the procedure for three-column subtraction is summarized in figure 5-17, where the boxes represent processes, the diamonds represent decisions, and the arrows show where to go next. As shown, one of the skills required to use this procedure is the ability to carry out single column subtraction (e.g., see step 2c).

Figure 5-17 diagrams the procedure that children are supposed to acquire; however, some students acquire a flawed version. For example, a student may have a procedure for three-column subtraction that contains one small “bug” (i.e., a procedure with one or more bugs in it) may be able to answer some problems correctly but not others.

As you can see, the student who solved these problems obtained correct answers for two out af the five. A more precise way of characterizing the student’s performance is to say that the student is using a procedure that has a “bug” in it : at steps 2a, 2b, and 2c, the student subtracts the smaller number from the larger number, regardless of which one is on top in the problem statement. Brown and Burton (1978) have argued that a student’s knowledge of subtraction procedures can be described by listing which bugs (if any) are found in the student’s processes. This example involves a very common bug, which brown and burton call” subtract smaller from larger.”

According to Brown and Burton (1978), errors in subtraction may occur because a student consistently uses a flawed procedure, not because a student cannot apply a procedure. To test the idea, Brown and Burton gave a set of 15 subtraction problems to 1,325 primary school children, and developed a computer program called “BUGGY” to analizye each student’s subtraction procedure. If all the student’s answers were correct, BUGGY would conclude that the student was using the correct procedure (shown in figure 5-17).

If the student made errors, BUGGY would attempt to find one bug that chouldaccount for them. If no single bug could be identified, BUGGY would evaluate all possible combinations of bugs that could account for the errors. Table 5-10 shows some of the most common bugs; for example, 54 of the 1,325 students behaved as if they had the “smaller from-larger” bug. Although the BUGGY program searched for hundreds of possible bugs and bug combinations, it was able to find the subtraction procedure (including bugs) for only about half of the students. The other students seemed to be making random errors, were inconsistent in their use of bugs, or may have been learning as they took the test. Thus Brown and Burton’s (1978) work allows for a precise description of a student’s procedural knowledge-even when that knowledge is flawed.

Role of Automaticity in Procedure Exection suppose I showed you a number on a computer screen, such as 159, and asked you to say the number (i.e., “one hundred fifty nine”). If you are like the children in a study by skwarchuk and Anglin (2002), your answer took less than 2 seconds. Automaticity refers to being able to complete a cognitive task without devoting conscious attention, and is a key element in the development of reading skill (as described in Chapter 2). Similarly, in early mathematics learning students develop automaticity in number naming skill. To examine this issue, Skwarchuk and Anglin asked students in grades 1,3,5, and 7 to read numbers presented on a computer screen. Figure 5-18 shows that the time to name a number decreases from grade 1 to grade 7; for example, hundreds fall from 5.3 seconds in grade 1 to 1,8 seconds in grade 7, and tens fall from 3.1 seconds in grade 1 to 1.0 seconds in grade 7. The significance of this research is that as students become more automatic in their number naming skill, they can free up cognitive capacity to develop more sophisticated cognitive processing. For example, when you can count effortlessly, you may have the cognitive capacity available to discover the benefits of shifting from counting-all to counting-on. When you can retrieve basic number facts without expending much mental effort, you can devote your attention to monitoring how well you are carrying out a more complex procedure such as three column subtraction. In short, as lower procedures become automated, they enable students to use them as components in more complex procedures.

IMPLICATIONS FOR INSTRUCTION: TEACHING FOR EXECUTING

How can we help students build a useful base of procedural knowledge? For nearly 100 years, drill and practice has been the dominant instructional method for teaching arithmetic procedures. In drill and practice, a student is given a simple problem and asked

To give a response, such as “what is 2 plus 4?” if the student gives the correct response, the student receives a reward, such as the teacher saying, “right!” if the student gives the wrong response, the student receives a punishment, such as the teacher saying, “wrong!” when you use flash cards, with the question on one side and the answer on the other, you are learning by drill and practice. When you sit in front of a computer screen thet presents problems and gives feedback, you are learning by drill and practice. When you answer a series of exercise problems in a textbook and then check your answers, you are learning by drill and practice.

Although drill and practice can be an effective method for teaching procedural knowledge, it may not be the only worthwhile method. A major problem is that learning procedural knowledge (such as how to add and subtract) can become isolated from conceptual knowledge (such as what a number is) so that mathematics becomes a set of meaningless procedures for students.

Case and colleagues (Case & Okamoto, 1996; Griffin, Case, & Capodilupo, 1995; Griffin, Case, & Siegler, 1994) argued that the learning of basic arithmetic procedures must be tied to the development of central conceptual structures in the child. According to this view, the most important conceptual structure for learning arithmetic procedures is a mental number line.case et al. developed a test of student’s knowledge of a mental number line that included their ability to compare two numbers, to visualize the number line, to count, and to determine the magnitude specified by number word. When they gave the test to 6-year-olds of low socioeconomic status (SES). Nly 32 % demonstrated an acceptable knowledge of the number line; however, 67% of high-SES 6-years olds demonstrated such knowledge. More importantly, 25% of the low-SES children and 71% of the high-SES children could solve simple addition problems, such as 2 + 4 = .

Why do some students have difficulty with simple addition? According to Case and colleagues, the source of the difficulty is that students lack a representation of a mental number line. If this premise is true, the instructional implication is clear: Teach students to construct and use mental number lines as a prerequisite for learning arithmetic procedures. This is the approach taken in a math readiness program called “Rightstart” (Griffin & Case, 1996; Griffin et al., 1994, 1995). The program consist of 40 half-hour session in which students learn to use a number line by playing a series of number games. For example, in one game two students each roll a die and must determine who rolled the higher number. The student who rolled the higher number then moves his or her token along a number line path on a playing board. The first student to reach the end of the path wins. These games promote skills such as comparing the magnitude of two numbers, counting forward and backward along a number line, and making one to one mapping of numbers onto objects when counting.

Does number line training help students learn arithmetic procedures? To answer this question, researchers (Griffin & Case, 1996; Griffin et al., 1994, 1995) gave one group of low SES first graders the Rightstart training (treatment group), whereas another group of similar children received their regular mathematics instruction (control group). First, there was overwhelming evidence that number-line training helped students build conceptual knowledge of number lines. On a posttest of number-line knowledge, 87% of the treatment group and 25% of the control group demonstrated skill on number line tasks such as determining which of two numbers was smaller. Second, there was evidence that number line training helped students learn arithmetic procedures. On a posttest with simple addition, 82% of the treatment group and 33% of the control group gave correct answers.

(chapter 5 learning mathematic 199)

Third, treatment students were more successful than control students in learning mathematics in school: 80% of the treatment group and 41% of the control group mastered first-grade mathematics units on simple addition and subtraction. Griffin and Case (1996) noted that” a surprising proportion of children from low income North American families—at least 50% in our samples—do not arrive in school with the central cognitive structure in place that is necessary for success in first grade mathematics, “so”their first learning of addition and subtraction may be a meaningless experience” (p. 102). However, they argued that this problem can be overcome with a relatively modest instructional program aimed at promoting the conceptual knowledge that underpins arithmetic procedures.

In reviewing the Rightstrat program, Buer (1993) argued for the importance of connecting procedural and conceptual knowledge: Without this understanding [of the mental number line] [students’] basic number skills remain recipes, rather than rules for reasoning. If they don’t understand how number concepts and structures justify and support these skills, their only alternative is to try to understand school math as a set of arbitrary procedures. Why arithmetic works is a mystery to them… for mathematics to be interrelated in instruction. (p. 90)

Number line training is an important demonstration of the value of helping students make connections between arithmetic procedures and number concepts.

As you can see, some children enter elementary school with number sense, as indicated by their understanding of how counting works. For example, suppose I put five dolls on a table and ask you to count them. Gelman and colleagues (Gelman & Gallistel, 1986; Gelman & Meck, 1983) specified five principles of counting:

Some students enter elementary school with strong counting skills such as these, whereas others do not. Are these differences related to early mathematics learning in elementary school? To answer this question Aunola et al. (2004) tested finnish children on counting skills in preschool and mathematics achievement at six times during elementary school through Grade 2. Table 5-11 shows the correlation between counting skills in preschool and subsequent mathematics achievement in elementary school was high for each of the six times students were tested. In addition, students who scored high in counting skill in preschool showed a slower rate of math learning in elementary school. The result suggest that the development of counting skill-a key component in number sense-should be a major goal af early mathematics learning, although experimental studies are needed to test this idea.

Two important aspects of numbers sense are counting and number identification, as exemplified in table 5-12. If we teach students about these basic components of number sense, can we help them improve on them, and will they also amprove on prearit metic tasks such as the addition-subtraction tasks shown in the bottom of table 5-12? To address this question, Malofeeva, Day, Saco, Young, and Ciancio (2004) provided some preschoolers in a head Start program six 20-to-25 minute sessions in which they learned counting and number identification skills (treatment group). For example, they practiced counting forward and backward, numeral recognition and naming, answering question about the order of numbers (e.g., what comes after 4?), and identifying errors in counting. Other preschoolers in the same school received an equivalent amount of training in a nonmathematical topic (i.e., insects). The top two portions of figure 5-19 show that the modest training in counting and number identification skills—totaling less

Than 3 hours—resulted in substantial improvements, whereas the control group did not show a substantial improvement. Importantly, children who received training in the basic skills of counting and number identification also showed substantial improvements in solving addition and subtraction word problems, whereas the control group did not. This study shows that it is possible to teach the basic skills of number sense, and that learning them transfers to tasks requiring arithmetic thinking.

Although some researchers have been successful in teaching number sense using a scripted set of lessons delivered by researchers, you might wonder what happens when regular classroom teachers try to infuse teaching of number sense into their regular classroom routine? To address this question, Arnold, Fisher, Doctoroff, and Dobbs (2002) provided training to preschool teachers were given an activity book with 85 number activities they could choose from, and they received a 2-hour training on how to foster number sense using the activities. For example, one of the counting activities involved building caterpillars using felt circles:

Make a lot of felt circles that can be used on a flannel board. Two of the circles should have faces on them, as the circles will be used to build caterpillars. Children are divided into two teams and take turns rolling a die. The number on the die is the number of circles the child adds to their caterpillar (each caterpillar starts with just the face circle placed on the flannel board). After every child has had a turn, children count the number of circles making up each caterpillar to determine which one is longer. (Arnold et al., 2002, p. 770)

The activities were designed to each recognizing written numbers, one-to-one correspondence, comparison of two numbers to determine which is larger or smaller, and relating counting to quatity. Students were tested on early mathematics skills (such as counting 10 blocks, displaying 4 fingers, and telling what number comes after “23, 24”) before and after a 6-week period in which teachers infused the number activities into the regular program (treatment group) or used the regular program (control group).

Figure 5-20 shows that the treatment group produced a 47% increase, whereas the control group produced a 10% increase. The effect size attributable to the training was d= 0.44. which is a medium effect. Overall, there is encouraging evidence that a modest infusion of number sense activities delivered by regular preschool teachers can have a significant impact on mathematics learning.

What can you do to improve training in computational procedures? This question was addressed early in the history of educational psychology by Thorndike (1922), who argued for the importance of practice with feedback. Thus, to acquire skills in solving computation problems students need practice in solving computation problems as exemplified in the last portion of figure 5-1. In addition, students need feedback on whether their answers are correct. This advice has become well accepted in educational psychology and is amply supported by research, such as in Chapter 7. However, more recent research shows that students tend to develop new arithmetic procedures by using their previously learned procedures and conceptual knowledge should be tied to a learner’s conceptual knowledge by making computation more concrete.

CHAPTER SUMMARY

Let’s return one final time to the tile problem described at the opening to this chapter. To solve that problem, a person needs several kinds of knowledge: linguistic and factual knowledge for problem translation, schematic knowledge for problem integration, strategic knowledge and beliefs for solution planning and monitoring, and procedural knowledge for solution execution.

A review of mathematics textbooks and achievement tests reveals that procedural knowledge is heavily emphasized in school curricula (Mayer et al., 1995). For example, students are given drill and practice in carriying out computational procedures. In this chapter, we refer to this type of instruction as solution execution. However, systematic instruction in how to translate problems, how to make meaningful representations of problems, and how to devise solution plans is not always given. Problem translation involves converting each stantement into an internal representation, such as a paraphrase or diagram. Students appear to have difficulty in comprehending simple sentences, especially when a relationship between variables is involved, and students often lack specific knowledge that is assumed in the problem (e. g., the knowledge that a square has four equalsides). Training in how to represent each sentence in a problem is an important and often neglected component of mathematics instruction.

Problem integration involves putting the pieces of information from the problem together into a coherent representation. Students appear to have trouble with unfamiliar problems for which they lack an appropriate schema. Training for schematic knowledge involves helping students recognize differences among problem types. Solution planning and monitoring involve devising and assessing a strategy for how to solve the problem. Students appear to have trouble describing the solution procedure they are using, such as apelling out the subgoals in a multistep problem. Strategy training is needed to help students focus on the process of problem solving in addition to the product of problem solving. In addition, students often harbor unproductive beliefs, such as the idea that a problem has only one corret solution procedure or that problems do not make sense. Students need interventions that help them form more productive beliefs about mathematics learning.

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