Children’s perspectives on the engagement potential of mathematical problem tasks

English, Lyn D

This article examines elementary students’ perspectives on the engagement potential of particular mathematical problems and students’ views on their general classroom problem activities. Third-, fifth-, and seventh-grade children from different reform-oriented classrooms were individually interviewed about (a) how they would improve their classroom problem-solving activities and (b) the problems they find the most and least potentially engaging when presented with a range of routine and nonroutine problems. The children requested more relevant, meaningful, and interesting problem experiences in their classrooms, and the fifth and seventh graders requested more representational materials. The children’s criteria for determining potentially engaging and nonengaging problems primarily pertained to problem structure and perceived cognitive demands. The nonroutine examples that focused on important reasoning processes and did not involve computation had the greatest engagement potential, while the computational problems had the least appeal.

I like problem solving half and half. I don’t really like the questions where you have to write things down. I like it when you’ve got figures and things to move around. I like it when we have blocks, and you’ve got to solve these puzzles with the blocks and all that. (Nathan, Grade 7)

A good deal about the types of mathematical tasks students find engaging can be learned from the students themselves. Students’ opinions are important in reform efforts, particularly as educators try to design mathematical tasks that will engage students and maximize their learning (McLeod, 1994; National Council of Teachers of Mathematics [NCTM],1991; Smith, 1996). Worthwhile mathematical tasks are generally considered to be those engaging students’ intellect, capturing their interest and curiosity, developing their mathematical understanding and reasoning processes, and allowing for different solution strategies, solutions, and representational forms (English & Halford, 1995; NCTM, 1991; Stein, Grover, & Henningsen, 1996).

Meeting the challenge of designing and selecting such tasks requires teachers to have both content knowledge and student knowledge (Ball, 1993; NCTM, 1991; Simon,1995; Smith,1996). Content knowledge includes knowing what makes a worthwhile task, what it conveys about the processes involved in doing mathematics, and whether the task appropriately represents the intended concepts and procedures (NCTM, 1991). Content knowledge also encompasses a knowledge of the cognitive demands of a task, including the types of thinking processes required (Stein, et al.,1996) and the complexity of the mathematical relations that must be interpreted (English & Halford, 1995). The cognitive processes can range from the use of basic procedures and rules to the use of complex reasoning considered important in “doing” mathematics (e.g., framing and solving problems, looking for patterns, making and testing conjectures). The mathematical relations can be simple ones, such as those involved in basic addition and subtraction (i.e., relations between two parts and the whole amount) or more complex relations of the type found in rational number problems (for example, the distance from C to D is 24 kilometers. The distance from B to C is two thirds of the distance from C to D. The distance from A to B is three eighths of the distance from B to C. What is the distance from A to B?) In their study of reform-oriented classrooms, Stein et al. (1996) found that tasks which started out as cognitively demanding deteriorated during the course of implementation into tasks in which little or no mathematical cognition occurred. One of the major contributing factors was the inappropriateness of the task for a given group of students. In particular, students failed to engage in the task because they either lacked interest, were poorly motivated, or lacked prior knowledge. As Stein et al. noted, the prevalence of this factor highlights the importance of student knowledge; that is, knowing students sufficiently well to make informed decisions about the motivational appeal and difficulty level of classroom tasks.

Knowing what students consider to be engaging and meaningful mathematical tasks is particularly important as teachers attempt to implement problem activities that are more cognitively complex than the traditional, low-level computational tasks (Stein et al., 1996). Such knowledge can be particularly empowering for teachers in their reform efforts (Smith, 1996), yet studies exploring students’ perspectives on different mathematical tasks have declined since the last decade (Lester,1994). According to earlier research, students traditionally have viewed mathematical tasks as those in which they apply learned rules or procedures, with little requirement for in-depth thinking (Dossey, Mullis, Linquist, & Chambers,1988; Franke,1988). Problems considered to take longer than a few minutes to solve usually have not been well received by students, reflecting a belief that mathematical problems should not be time consuming (McLeod, 1989; Schoenfeld, 1992). Now, with the current implementation of reformoriented activities, students are showing broadened perspectives on mathematics and on mathematical problem tasks (Franke & Carey, 1997; Wood & Sellars, 1997). These recent studies, however, have concentrated on activities dealing with numbers and elementary computation. Little attention has been given to how students perceive a range of problem tasks, including the more cognitively complex examples that call upon various reasoning processes and general problem-solving strategies. Knowing the types of problems that students consider potentially engaging, as opposed to what teachers and researchers consider engaging, supports construction and implementation of more appropriate and meaningful problem experiences (cf. arguments of Stein et al., 1996). This issue was addressed in the present investigation, together with students’ perspectives on their classroom problem solving. More specifically, the following questions were explored: 1. How would third-, fifth-, and seventh-grade students from reform-oriented classrooms improve their classroom problem activities?

2. Given a selection of routine and nonroutine problem tasks, which do the children consider to be (a) the most potentially engaging and (b) the least engaging? What criteria do they use in making their decisions? Method Elementary school children from Grades 3 (N = 32), 5 (N = 32), and 7 (N = 29) participated in the investigation. Their mean ages were 8.1 years, 10.2 years, and 12.0 years, respectively. The children were selected from three state and three nonstate schools, located in diverse neighborhoods of a major city in Queensland, Australia; a range of ethnic backgrounds was represented. At each grade level, the children were drawn from three classes across the schools. The learning environments of the classes reflected the current spirit of curriculum reform, comparable to that of the United States. For example, all classes made appropriate use of concrete materials and other representations in solving computational problems, with students encouraged to explore different approaches to solutions and to share their ideas and opinions. There were, however, some differences between the grade levels in the nature and extent of the problem-solving experiences implemented. The thirdgrade teachers generally placed greater emphasis on computational problem solving than on nonroutine problem solving, while the fifth- and seventh-grade teachers included more nonroutine problem solving in their curriculum. There were also some within-grade differences, with respect to the extent of nonroutine and collaborative problem solving undertaken. For example, one seventh grade teacher placed a strong emphasis on exploring a range of nonroutine problems but did not facilitate a lot of collaborative work. In another classroom, the two seventh-grade teachers involved the children in considerable collaborative work but less exploration of nonroutine problems. The two teachers in the third seventh-grade classroom gave the children many opportunities for collaborative problem solving and also presented them with a range of mathematical problem situations, including openended environmental investigations. The participants from each grade level were chosen on the basis of their responses to two problemsolving assessment instruments administered on a whole-class basis. One assessment instrument focused on routine computational problems, while the other comprised nonroutine problems. The selected children displayed one of three forms of problem-solving competence: competence in both routine and nonroutine problem solving, competence in either routine or nonroutine problem solving, or competence in neither. The children were selected in approximately equal numbers from each of the classrooms and were interviewed individually by a research assistant with graduate qualifications in mathematics education. The interviews addressing the present issues proceeded along the following lines: Today we are going to have a talk about problem solving. I’ve got lots of different problems for us to talk and think about. You won’t have to actually solve them, though. You will just be telling me some of your ideas about problems and problem solving.

First of all, let’s think about the different math problem activities you do in class. What do you think your teacher could do to make these problem activities more appealing (enjoyable) and more worthwhile for you, so you would really want to do the activities?

As Marshall (1989) suggests, by removing the requirement to formally solve the problems, it was hoped that a more relaxed atmosphere would be created, thereby reducing any negative emotional reactions that might inhibit the children’s responses. Each child was then presented a set of problems comprising a mix of routine and nonroutine examples, as shown in the appendix. The problems were representative of effective problems, displaying those features described in the introduction. Several of the problems were chosen from well-known teaching resources (e.g., Baroody, 1993; Charles, Mason, & Martin, 1985; Greer & Wantuck, 1996), while some had been used extensively in other problem-solving studies conducted by the author (e.g., English, in press). In an effort to present problems suitable for the grade levels, problems A2, J, and K were used only with the seventh graders, while problems G, H, and I were presented only to the third and fifth graders. Although openended investigations and student-generated problems are important in the curriculum (Silver & Cai, 1996), the present investigations focused only on “preformulated” problems. The problems were presented on individual cards and displayed randomly before the child. Where necessary, the problems were read for the child. The following instruction was then given: “Have a look at each of these problems. If you had to solve these problems, which problem or problems would you most want to solve?” Then, “Why did you choose that problem?” After responding, the child was next asked which of the problems he or she would least want to solve and why. The interviewer then asked what changes the child would make so that he or she would want to solve that problem. Results Research Question 1 – Improving Classroom Problem Activities. There were some interesting trends across the grades in the children’s responses to this question; however within-grade differences were not clearly discernible. The grade level trends are presented in the data of Table 1.

Although the children’s classrooms were reform oriented, the third and seventh graders, nevertheless, felt that problem solving could be made more interesting and more relevant for them. Included in the children’s responses were requests for more interesting or exciting words or contexts, the inclusion of children’s names in the problems, and “problems on the things we like.” For example, Gareth (Grade 7) explained, “Do problems on the things we like and get away from all the things like fruit and the usual stuff.” Surprisingly, Gareth came from the classroom environment considered to exemplify curriculum reform (e.g., the children had experienced a range of problem situations including out-of-class investigations, were encouraged to explore different approaches to solution, and were presented activities that challenged and broadened their thinking). During a related class discussion in the same classroom, Hayley commented, Well, sometimes I think that problem solving is pretty boring, but one way you could make it better is like, you could add some humor to it and make it fun. Like, make it exciting so we would look forward to it more. Instead of everyone sitting down and doing an activity sheet, you know, picking up our pencils and working a problem… that’s pretty boring…change the process, like the problems where we go outside to do things or where we sit in groups on the floor. That’s much better. Both the third- and fifth-grade children frequently requested a reduction in difficulty level, with their suggestions including, “Give us easier problems,” “Do part of the problem for us,” and “Give us more information.” Not surprisingly, the third-grade children also asked for different problems to solve, that is, different from their usual computational examples. Several children requested particular problems that they liked, such as puzzles.

Also of interest in the children’s responses were the requests from each of the classes in fifth and seventh grade to increase the amount of representational material in their problem activities. The children suggested the inclusion of more diagrams, illustrations, and hands-on materials as a means of enhancing their problem experiences. This finding was mirrored in the children’s selections of potentially engaging problems, as discussed next.

Research Question 2 – Selecting Most and Least Engaging Tasks

The problems children chose as potentially most engaging and least engaging are shown in Table 2. From the data, distinct trends in the children’s choices were displayed across the grades. An unexpected finding was the uniformity in the children’s selections, irrespective of their particular classroom experiences.

The problems involving deductive reasoning (see appendix, Al and A2) were selected by the fifth- and seventh-grade children as the most potentially engaging examples. Surprisingly, A2 had more appeal than Al for the seventh graders, even though it involves more complex reasoning processes than Al. The most common reason for selecting these problems pertained to the structure of the problem (50% and 59% of fifthand seventh-grade responses, respectively). Their reasons here included the following: “I like the way there are clues; they give you some help.” “The way the problem goes, it lets you use elimination procedures.” “The problem allows you to use a diagram.” The children also saw the problem as “just like a puzzle” and “like a mystery you have to solve.” Several children commented on the cognitive demands of the problem; that is, they considered it to be a challenge for them (“It really gets your mind working, and I like that”). The third-grade children who chose this problem type also considered it to be challenging and “fun,” with a few simply saying, “I like that sort of problem.” These deductive problems were also chosen as most engaging by other fifth-grade students in an earlier study (English, 1997).

The problems requiring spatial reasoning (C1/C2) also appealed to the children, especially the third graders. Their reasons for choosing such a problem were rather limited, however, with 50% of them responding to the effect, “I really like this sort of problem,” and 20% referring to the perceived simplicity of the problem. Two of these third graders stated that the problem presented a challenge for them, while two considered they were “good at those problems,” and one child referred to the appeal of the visual representation. The fifth graders’ responses to this problem were comparable to the third graders’. In contrast, 45 % of the seventh graders referred to the challenge the spatial problem presented, with the remaining seventh graders referring to the appeal of the visual representation and to the opportunity for experimentation (“I like experimenting with things to try and find the right answer”). The scheduling problem (K) also emerged as a potentially engaging problem for the seventh graders. Problems which involve executing a sequence of moves to get from the given state to the goal state have been a popular task in psychological studies of problem solving (Gholson, Smither, Buhrman, & Duncan, 1997). The children who chose this problem mostly liked the challenge it presented (“It really makes you use your brain”) and the opportunity it provided for different approaches to solution (“There are lots of different ways you can figure it out.” “You can use trial-anderror.”). The problem was also perceived as an interesting one to tackle, the absence of numbers appealing to a couple of students.

The problems children considered to be the least engaging generally reflected the perceived mathematical complexity of the problem. The combinatorial reasoning problem (B) held limited appeal for the third and fifth graders, in particular, in spite of the problem’s seemingly real-world context. The problem was considered “too hard,” with “too many things to do” or “too many things to think about,” and too time consuming. A couple of children also claimed, “I get mixed up with these problems, and I sometimes miss things out.” Similar responses to these combinatorial examples were found in other studies conducted by the author (English, 1996).

The computational word problems (D and El/E2) were also rarely chosen as potentially engaging examples. Mainly, the children considered the problems “too easy” and “too boring” (“All you have to do is subtract.”). A couple of seventh-grade children commented that they did not like doing the required computational work. The computational reasoning problems (F-J) also did not appear engaging, especially problems F and J. Overwhelmingly, the children saw problem F as “confusing,” “too complicated,” having “too many words,” and comprising “too much to work out” and “too much to think about.” Only one child, a seventh-grader, explained that she did not understand the problem and did not know how to solve it. In suggesting what they would do to make this problem an engaging one, the children mainly referred to ways of making it less demanding. The third- and fifth-grade children asked for hands-on materials to be supplied to facilitate solutions and also suggested reducing “the amount of working out you have to do.” In addition to these recommendations, the seventh-grade children suggested changing the structure of the problem (“Make it like those problems where you have the clues” [Problems Al/A2]). A few children suggested changing the context of the problem (“Make the problem more interesting than rows; use something else”). Despite its visual representation, Problem I lacked appeal for the third and fifth graders, mainly because some children considered it “too complicated” and “too confusing,” with “too much to work out” and “too much to think about.” Others stated that they had difficulty in understanding the problem or did not know how to solve it. To enhance the problem’s appeal, the children suggested supplying cut-outs of the dominoes, rearranging some of the dominoes (i.e., providing part of the solution with the problem), and adding more information to clarify the problem.

Given the substantial research on children’s difficulties with fractional concepts (e.g., Carpenter, Fennema, & Romberg, 1993), it is not surprising that Problem J lacked appeal for the seventh graders. The children perceived the problem as difficult and confusing, with several of the children commenting that they did not like the fractional computation entailed and that they were “not good at fractions.” Their suggestions for making the problem appealing included changing the fractional amounts to whole numbers (“like say that the distance is 5 times”), providing a better diagram, making the problem “easier to understand,” and reducing the amount of “working out” to be done. Conclusions Although the present findings are limited to a particular sample of elementary students, they do raise a number of issues worth considering in designing classroom problem experiences. First, it is important to take into consideration how our students perceive the mathematical problem activities we present them in class. This consideration includes knowing what students find interesting, relevant, and meaningful, as well as how they perceive the cognitive demands of a given problem task. The suggestions for classroom improvement offered by the present group of students indicate the need to Design problem activities around our students’ interests and experiences and encourage their input in doing so.

Implement abroad range of problem experiences. Include more representational materials, such as diagrams, illustrations, and hands-on materials. Determine how students perceive the difficulty level of problem tasks and address this accordingly. These recommendations are, of course, not new to mathematics educators. What is important, though, is that they have come from the students themselves, operating in several different reform-oriented classrooms. It is particularly interesting to note the fifth and seventh graders’ request for more representational materials. This result serves to remind educators that the use of hands-on and other representational materials should not decline with increasing grade level. The children’s responses to the second question provide us with valuable insights into the types of problems that appear to have engagement potential. The children’s criteria for selecting potentially engaging and nonengaging problems primarily pertained to problem structure and to perceived cognitive demands, with mathematical complexity being a key deterrent. The context in which a problem was set was rarely mentioned as a criterion for an engaging problem. Of the set of problems presented to the children, the nonroutine examples that involved reasoning processes but not computation had the greatest appeal. The deductive reasoning problems were seen as especially engaging, with the presence of clues giving them an air of mystery, as well as allowing students to apply specific strategies (e.g., elimination) or to construct tables or diagrams (e.g., a matrix). It is pleasing to observe the appeal of these problems, given the importance of deductive reasoning in students’ mathematical development (English & Halford, 1995; Kroll & Miller, 1993).

Likewise, the children’s positive responses to the spatial reasoning problems and to the scheduling problem reflect the important role these examples play in their mathematical development (Gholson et al., 1997; Wheatley, 1997). The seventh graders liked the cognitive challenge of these problems, as well as the opportunity for experimentation, while the third graders were attracted to the visual structure of the spatial problem. Although the seventh graders saw the scheduling problem as challenging, they were keen to tackle it largely because it enabled multiple approaches to solution. The children’s responses to the computational problems clearly indicate lack of engagement potential, at least for the present sample of children. The computational word problems were not sufficiently challenging to attract the children, with the perceived “boring” structure of the problems also being a deterrent. The computational reasoning problems held the least appeal, mainly because they were seen as having complex mathematical structures and making excessive cognitive demands. Although these computational examples represent just one of several approaches to developing and extending students’ mathematical understandings, they do have an important place in the curriculum and warrant attention (Baroody, 1993; NCTM, 1991). Their lack of appeal highlights the importance of encouraging students to share their perceptions, understandings, and opinions as they tackle these different problem situations. While the present findings add to our knowledge of children’s perspectives on various mathematical problem tasks, there are some limitations to be acknowledged. These limitations need to be addressed in a more controlled research study. The first limitation is that the investigation relied on the children’s oral responses and explanations, frequently to questions not previously considered. This was especially so for the third graders, who frequently found it difficult to express their ideas as clearly as the older children; this could have masked their actual perceptions and sentiments (as suggested in Garner & Alexander, 1989). Second, although all of the classrooms were reform oriented, the teachers’ approaches to problem solving were identifiably different. While this added an interesting and significant dimension to the present findings, the use of more comparable classrooms might yield further insights into children’s perspectives on mathematical problem tasks. A third limitation pertains to the selection of the present problems. Although there was a broad range of examples, the inclusion of other important problem types, such as open-ended investigations, hands-on problem explorations, and peer-generated problems, might have made an interesting difference to the findings. Finally, different insights might have been obtained had the children been asked to actually solve the problems, although the removal of this requirement was part of the investigation design. A follow-up study needs to explore whether children’s chosen problems actually do engage them mathematically when implemented.

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Lyn D. English

Queensland University of Technology

Author Note: Correspondence concerning this article should be addressed to Lyn D. English, Centre for Mathematics and Science Education, Queensland University of Technology, Victoria Park Road, Kelvin Grove, Brisbane, Australia, 4059. Electronic mail may be sent via Internet to L.English@qut.edu.au

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