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).

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?

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).

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).

References

Ball, D. L. (1993). Halves, pieces, and twoths: Constructing representational contexts in teaching fractions. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research (pp. 157-196). Hillsdale, NJ: Lawrence Erlbaum.

Baroody, A. J. (1993). Problem solving, reasoning, and communicating (K-8): Helping children think mathematically. New York: Macmillan.

Carpenter, T., Fennema, E., & Romberg, T. A. (1993). (Eds.), Rational numbers: An integration of research (pp. 13-47). Hillsdale, NJ: Lawrence Erlbaum Associates.

Charles, R. I., Mason, R. P., & Martin, L. (1985). Problem-solving experiences in mathematics: Grade 4. Menlo Park, CA: Addison-Wesley.

Dossey, J. A., Mullis, I. V. S., Lindquist, M. M., & Chambers, D. L. (1988). The mathematics report card: Are we measuring up? Trends and achievement based on the 1986 National Assessment (NAEP Report No. 17-M-O1). Princeton, NJ: Educational Testing Service.

English, L. D. (1996). Children’s problem posing and problem-solving preferences. In J. Mulligan & M. Mitchelmore (Eds.), Research in early number learning (pp. 227-242). Australian Association of Mathematics Teachers.

English, L. D. (1997). The development of fifthgrade children’s problem-posing abilities. Educational Studies in Mathematics, 34, 183-217.

English, L. D. (in press). Children’s reasoning in solving relational problems of deduction. Thinking and Reasoning.

English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, NJ: Lawrence Erlbaum.

Franke, M. (1988). Problem solving and mathematical beliefs. Arithmetic Teacher, 35, 32-34.

Franke, M., & Carey, D. A. (1997). Young children’s perceptions of mathematics in problemsolving environments. Journal for Research in Mathematics Education, 28(1), 8-25.

Garner, R., & Alexander, P. A. (1989). Metacognition: Answered and unanswered questions. Educational Psychologist, 24(2), 143-158.

Gholson, B., Smither, D., Buhrman, A., & Duncan, M. K. (1997). Children’s development of analogical problem-solving skill. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 149-189). Hillsdale, NJ: Lawrence Erlbaum.

Greer, G., & Wantuck, L. (1996). Menu of problems. Mathematics Teaching in the Middle School, 1(8), 637-640.

Kroll, D., & Miller, T. (1993). Insights from research on mathematical problem solving in the middle grades. In D. T. Owens (Eds.), Research ideas for the classroom: Middle grade mathematics (pp. 58-77). New York: Macmillan.

Lester, F. (1994). Musings about mathematical problem-solving research: 1970-1994. Journal for Research in Mathematics Education, 25(6), 660-675.

Marshall, S. P. (1989). Affect in schema knowledge: Source and impact. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving: A new perspective. NY: Springer-Verlag.

McLeod, D. B. (1989). Beliefs, attitudes, and emotions: New views of affect in mathematics education. In D. B. McLeod & V. M. Adams (Eds.), Affect and mathematical problem solving: A new perspective (pp. 245-258). NY: Springer-Verlag.

McLeod, D. B. (1994). Research on affect and mathematics learning in JRME: 1970 to present. Journal for Research in Mathematics Education, 25(6), 637-647.

National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, VA: Author.

Schoenfeld, A. ( 1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Eds.), Handbook of research on mathematics teaching and learning (pp. 334-370). NY: Macmillan.

Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journalfor Research in Mathematics Education, 2 7(5), 521-539.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145.

Smith III, J. P. (1996). Efficacy and teaching mathematics by telling: A challenge for reform. Journal for Research in Mathematics Education, 27(4), 387-402.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Wheatley, G. (1997). Reasoning with images in mathematical activity. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 281-296). Hillsdale, NJ: Lawrence Erlbaum.

Wood, T., & Sellers, P. (1997). Deepening the analysis: Longitudinal assessment of a problemcentered mathematics program. Journal for Research in Mathematics Education, 28(2), 163-186.

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

Copyright School Science and Mathematics Association, Incorporated Feb 1998

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