The use of Langford's problem to promote advanced mathematical ...

The use of Langford’s problem to promote advanced mathematical thinking Thomas Lingefjärd Introduction College school students’ deficiencies in mathematical understanding and ways of thinking reach well back into the F-12 mathematics curriculum and “students’ mathematics education is in full swing by the time they enter college” (Steen, 1998). Unfortunately, many college courses also fail to address advanced mathematical thinking (Carlson, 1998). Consequently, it is important for college and university mathematics instructors to be aware of the nature of students’ deficiencies, the cognitive reasons for such deficiencies, and so be in a position to adapt introductory college mathematics instruction accordingly. The design of courses that facilitate transition to advanced mathematical thinking will require “a simultaneous focus on issues of pedagogy and learning alongside the challenging matters of content order and course organization” (Ferrini-Mundy, 1998). There is a visible trend in modern education that shows an increasing emphasis put on learning through problem-oriented or problem-based educational methods. The underlying idea is to improve the quality of students’ learning about complex problems or phenomena in the world through assignments that give rich opportunities for active investigation, analysis, and reflection. Such methods entail an increased use of a wide variety of different information sources. When studying mathematics at the tertiary level, many students use tools like graphing and symbol-manipulating calculators and a variety of sophisticated computer programs. Students also use assorted textbooks and other reference books, and many of them are likely to turn to their family members, friends, colleagues, or maybe neighbors as a reference group. One could argue that if students do seek information in a variety of ways, then the way these students study is close to the way people ordinarily work in many different professions. In many walks of life, people are valued for the everyday jobs or projects they do, their ability to work effectively with others, their responses to problem situations, and their capacity to find tools or information that will help them complete an assignment. In occupations as well as in modern studies, it is important and most likely beneficial for the individual to be open and flexible in approach. It is desirable and would be natural if examinations in mathematics could mirror that fact.

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Research question Is it possible to give students an assignment which would promote advanced mathematical thinking and at the same time gives rich opportunities for active investigation, analysis, and reflection? Is it possible to observe and validate such a transition?

Theoretical framework Mathematical understanding Since ancient times, people have been concerned about understanding (and lack of understanding) in connection with mathematics. Henri Poincaré (1970) underlined the ambiguity of the meaning of the verb: What is understanding? Has the word the same meaning for everybody? Does understanding the demonstration of a theorem consist in examining each of the syllogisms of which it is composed in succession, and being convinced that it is correct and conforms to the rules of the game? In the same way, does understanding a definition consist simply in recognizing that the meaning of all the terms employed are already known, and being convinced that it involves no contradiction? (Quoted in Sierpinska, 1994, p. 72) Sierpinska (1994) claims that researchers in mathematics education have different objectives when discussing the question of understanding mathematics. Some objectives are more pragmatic (to improve understanding), others are more diagnostic (to describe how students understand), and still others are more explicitly theoretical or methodological. What unites researchers is that they all have a theory of what understanding is, explicitly expressed or not. According to Sierpinska, there are at least four different theories or models of understanding in mathematics. To begin with, we have theories that are centered on hierarchies of levels of understanding. One such example is the van Hiele (1986) levels, but there are others. Second, we have models that describe understanding as a growing ”mental model,” ”conceptual model,” ”cognitive structure,” or something similar. The term cognitive structure comes from Piaget (see for example Piaget, 1978) and several authors refer to Piaget when constructing their model for the understanding of mathematics. Third, Sierpinska (1994) mentions models that look at the process of understanding as a dialectical game or interplay between two ways to apprehend

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understanding. The dialectical dualism may be illustrated by a concept considered as a tool in a problem-solving process and at the same time viewed as an object to study, analyze, and develop in a theoretical way. One well-known example is Skemp’s (1978) discrimination between instrumental and relational understanding. According to Skemp, instrumental understanding is what it takes to reach the right answer, while relational understanding means that you understand both what to do and why. Another way to describe this is as operational versus structural understanding (Sfard & Linchevski, 1994). The fourth type of understanding is a historical-empirical perspective in which the epistemological obstacles are united by today’s students (Sierpinska, 1994). Robert and Schwarzenberger (1991) claim that from a psychological perspective, it is meaningful to focus on tertiary students’ growing ability to reflect on their own learning of mathematics. They argue that advanced mathematical thinking includes the ability to separate knowledge of mathematics from meta-knowledge of mathematics, which includes, for instance, how correct, relevant, or elegant a solution is. They further advocate that students at this advanced level should have a great amount of mathematical knowledge, experience of mathematical strategies, and well-functioning methods together with aptitude for communicating those skills at least on a basic level. According to Robert and Schwarzenberger (1991), research shows that students vary greatly in this respect. Vinner (1997) extends the distinction between rote and meaningful learning with what he calls conceptual and pseudo-conceptual behavior. Conceptual behavior is characterized by the consideration of concepts, “as well as relations between concepts, ideas in which the concepts are involved, logical connections, and so on” (p. 100). In contrast, pseudo-conceptual behavior is based on rote learning, lack of reflection upon appropriateness of answers or reasons for errors, lack of underlying meanings assigned to symbols and words employed, use of superficial similarities, and “the belief that a certain act will lead to an answer that will be accepted” by an external authority (p. 115). Thompson and Sfard (1994) describe a similar distinction when they define “grasping the meaning” as “having the ability to think about the objects hiding behind the words” (p. 22). Learning through assignments Working on an assignment is an active learning process. Students are more likely to understand and retain material that they have used in an assignment to solve a problem, and it can serve very well as a bridge between concretization and abstraction. Mathematics is characterized by its abstract nature, and for the initiated, moving about within this abstraction is characterized by Devlin:

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When mathematicians define some abstract mathematical object or system as a “set of objects” satisfying certain properties, it usually doesn’t matter what the members of the set are; rather, what counts are the operations that can be performed on those members. In fact, even that is not quite right. The real interest is in the properties of those operations. (Devlin, 1997, p. 57) For the mathematics student or his or her instructor, whether in elementary school or college coursework, coming to grips with this abstraction is an educational challenge that continues to confront all those involved. But what happens to student’s learning when a group of students get the same assignment in mathematics? Is it self evident that the same learning process, the same evolution of their generalization and abstraction knowledge, the same active learning process undeniably occurs? Generalization and Abstraction Although related, the processes of generalization and abstraction have important distinctions. Both require an individual to look for commonalities, to isolate properties, and to stress certain features while ignoring others (Mason, 1996). Generalization involves an extension of an existing set of familiar objects or processes, while abstraction requires a shift of attention from the objects or processes themselves to the structure and/or relations among the entities. Focusing on the structure requires a mental re-construction to create a new object, itself subject to operations and having a set of properties (Dreyfus, 1991; Tall, 1991). Generalizing Generalizing is an integral part of classroom practice, where seeing the general in the particular lies behind most instructors’ examples and exercise sets. However, to be a successful instructional tool, both teacher and student must focus on the same aspects of the learning experience. Mason (1996) makes the distinction between “looking through” and “looking at” an object, such as “working through a sequence of exercises and working on these exercises as a whole” (p. 65). He cautions that while the instructor may see the general in a particular example, his or her students may only see the particular in what is offered as a general example. Mason illustrates how, for some, an example is an example of something, while, for the others, it is simply a totality in itself. The sum of the angles in a triangle is 180 degrees. What is the most important word, mathematically in that assertion? I suggest that it is … a. That tiny indefinite article signals the adjectival

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pronoun any, which in turn signals the adjective every, which refers to the scope of variability being countenanced. … Yet in many classrooms, it is the 180 that is stressed, presumably so that students will remember it. But failure to use this fact is rarely due to forgetting whether it is the 180 or some other number, but rather, due to lack of appreciation of the generality, the invariance, being expressed. (p. 67) Harel & Tall distinguish three different kinds of generalization which depend on the individual’s mental construction: 1. Expansive generalization occurs when the subject expands the applicability range of an existing schema without reconstructing it. 2. Reconstructive generalization occurs when the subject reconstructs an existing schema in order to widen its applicability range. 3. Disjunctive generalization occurs when, on moving from a familiar context to a new one, the subject constructs a new, disjoint, schema to deal with the new context and adds it to the array of schemas available. (Harel & Tall, 1989, p. 39) Abstraction An abstraction process occurs when students focus their attention on specific properties of a given object and then considers these properties in isolation from the original. This might be done, for example, to understand the essence of a certain phenomenon, perhaps later to be able to apply the same theory in other cases to which it applies. Such application of an abstract theory would be a case of reconstructive generalization – because the abstracted properties are reconstructions of the original properties, now applied to a broader domain. However, note that once the reconstructive generalization has occurred, it may then be possible to extend the range of examples to which the arguments apply through the simpler process of expansive generalization.

Methodology In a course on discrete mathematics, I used the following assignment as one part of the course exam for a group of prospective teachers of mathematics. The students were asked to present their solution in a written report, which clearly would be individual and not a copy of any classmates report. They were also permitted to use any kind of sources and help, as long as they referred to these sources in the report. The allotted time for the assignment was three weeks. All the 23 students produced written reports on the solution of the problem.

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Assignment Imagine the digits 1, 2, 3 in a sequence where each digit is used twice and arranged in the following way: 312132 Notice that there is one other digit between the ones (1), two other digits between the twos (2), and three other digits between the threes (3). a) Is it possible to arrange the digits 1, 2, 3 (each used twice) in other ways, so that these rules still hold? b) What will happen if you add two 4s to get a sequence of 8 digits? How many solutions will there be then? c) Investigate the problem for sequences of digits from 1 to 6, 1 to 7, 1 to 8, 1 to 9, and 1 to 10. Argue for your conjectures and conclusions!

The problem is known under the name Langford’s problem. Langford's problem is named after the Scottish mathematician C. Dudley Langford who once observed his son playing with colored blocks. Langford noticed that the child had arranged three pairs of colored blocks such that there was one block between the red pair, two between the blue pair, and three between the yellow pair, like so:

This solution is unique. Reversing the order is not significant, because all you have to do is walk around to the other side of a given arrangement and view it from that side. Langford added a green pair and came up with:

Generalizing from colors to numbers, the above became 312132 and 41312432. The problem has attracted mathematicians, computer scientists, and others, and is only solvable with the aid of computer programming for large n. The connection to Langford’s problem was not mentioned to the students in the study. Student responses The 23 students in this study were all in their second semester of mathematics in the teacher program for secondary mathematics at the University of Jonkoping. They had previous taken courses in algebra, geometry, probability and statistics in the program, and had also learnt how to use the computer software MS-Excel to solve various problems within discrete mathematics. The assignment was their third take-home exam in the course, and the previous two assignments had been modeled by the aid of MS-Excel. The students were informed about my purpose

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to extend their mathematical thinking from arithmetic thinking into advanced mathematical thinking and also about my intention to observe that transposition. A small group of student’s engaged MS-Excel also to solve Langford’s problem. Since Excel had proved to be helpful in solving other problems in the course, they tried hard to make the problem fit within the possibilities with Excel. The fact that the problem is not really suitable for Excel the way it was proposed was neglected by these students, since they selected a tool before analyzing the problem throughout. These students can be characterized as using expansive generalization. Student 1: I entered the figures in Excel and tried to make up a formula for the reorganization of numbers. When I didn’t find any formula that worked, I just put in the numbers for every possible combination in Excel. This worked fine for 4 digits, but was very tiresome and difficult for 5, 6, 7, and so on… digits. Student 2: I know it must be a combinatorial problem, but I can’t find the right formula in Excel. It drives me almost nuts, I have spent many hours on this problem by now, without getting anywhere. Another, much larger, group understood the connection to existing procedures and concepts within the course, such as calculations of sums of arithmetic series. This group of students managed to reach some valuable conclusions, but did not really solve the problem in full or manage to do enough generalization. All though there naturally were differences between the students, they all reconstructed ways of calculating arithmetic sums but only broadened and widened its applicability range to very small magnitude. These students can be characterized as using reconstructive generalization. Student 3: I have understood the problem like this. Number 1 is in position P(1) and P(1) + 2. Number 2 is in position P(2) and P(2) + 3. Number N is in position P(N) and P(N) + N+1. This yields an arithmetic sum and shows me that it is only possible to find solutions for 7 and 8, but not for any other numbers. Student 4: In order to get a complete number series, the formula 3N2 – N must be a multiple of 4. This happens when N = 4k and when N = 4k + 3, so for 7 and 8 digits there is at least one solution. Fortunately, 8 students actually managed to develop an impressing theoretical solution to Langford’s problem. Not only did they make the necessary connection to different concepts and procedures within the course, they also

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constructed a new general way to look at the problem, far beyond what was asked of them in the assignment. These students may very well be characterized as using disjunctive generalization. Student 5: I found that there is no solutions what so ever for 6, 9, and 10, but there are solutions for 7 and 8, and larger numbers. An interesting fact is that it works when the digit sum for the combination of digits is divisible with 4, and only then. Example: 2(1 + 2 + 3) = 12 and 2(1 + 2 + 3 + 4) = 20. Student 6: I have shown that there are at least two digits with the use of 7 and 8 digits which implies that there should be solutions also for 11 and 12 digits. When I continue, I find that there are solutions for 3, 4, 7, 8, 11, 12, 15, 16, 19, 20 digits, and etcetera. Finally, 4 students actually entered the digits 312132 into a search engine for Internet and got information about Langford’s problem that way. Nevertheless, the information they found turned out to be difficult to interpret and understand for someone who not had been working with it before, and these students had to ask for help in order to better understand what they had found on the web. Conclusions To me it was somewhat of a surprise that the students would handle the Langford’s problem so differently. It was also unexpected that the solutions would fit so nicely within the Harel & Tall’s generalization scheme. The problem obviously involves generalization (because it is not possible to solve otherwise), and gives the students possibilities to observe and identify one or more specific examples of behavior of Langford’s digits as typical of a wider range of examples embodying an abstract concept, something Harel & Tall label generic abstraction. There are three different principles connected to generic abstraction. The entification principle. This principle states that, for a student to be able to abstract a mathematical structure from a given model of that structure, the elements of that model must be conceptual entities in the student’s eyes; that is to say, the student has procedures that can take these objects as inputs. The necessity principle. This principle states that the subject matter has to be presented in a way to which learners can see its necessity. For if students do not see the rationale for an idea, the idea would seem to them as being evoked arbitrarily; it does not become a concept of the students.

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The parallel principle. When instruction is concerned with a “concrete” model, that is, a model which satisfies the entification principle, the instructional activities within this model should be designed to parallel the processes that will later apply within the abstract structure. This will mean that the instruction potentially involves only an expansive generalization, in which the concrete model is manipulated in a generic way. But it is designed to lay the seeds for a much easier reconstructive generalization at a later stage when the abstraction of the formal concept occurs in a corresponding abstract manner. I argue that Langford’s problem is such a problem that the parallel principle applies. Obviously all the students were able to use their own procedures to take the objects of the problem as inputs. Since the students are all in the mathematics teacher program, the necessity principle holds. I found that the problem provided the students with excellent possibilities to expand their generalization in a totally unrestrictive way. Since the parallel principle encourages a generalization of the procedure to be passed from the examples to the abstract concept by a process more parallel to an expansive generalization, the properties must be reconstructed in the abstract context. Consequently the passage from generic abstraction to formal abstraction remains one requiring reconstruction, but a reconstruction with potentially less cognitive strain. (Harel & Tall, p. 41). I still do not know if I should be happy that 8 out 35 students managed to reach that abstract level, or sad that not all of the students got there. The fact that the vast majority of the students preferred to put a lot of effort into finding specific solutions on their own is nevertheless very encouraging. It proves that one very well can use alternatives to more traditional ways of examining mathematical performance and understanding, and thereby promoting active investigation, analysis, and reflection. References Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In A. H. Schoenfeld, J. Kaput, & E. Dubinski (Eds.), Research in Collegiate Mathematics. III (pp. 114-162). Providence, RI: American Mathematical Society. Devlin, K. (1997). Mathematics: The Science of Patterns. New York: Scientific American Library. Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 25-41). Dordrecht: Kluwer Academic Publishers. Ferrini-Mundy, J. (1998). Response to core curriculum in context: History, goals, models, challenges. In J. A. Dossey (Ed.), Confronting the Core Curriculum: Considering Change in the Undergraduate Mathematics Major

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(pp. 15-16). Washington DC: Mathematical Association of America. (MAA Notes No. 45). Harel, G. & Tall, D. (1989). The General, the Abstract, and the Generic in Advanced Mathematics. For the Learning of Mathematics, 11 (1), (pp.38–42). Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra: Perspectives for Research and Teaching (pp. 65-86). Dordrecht: Kluwer Academic Publishers. Piaget, J. (1978). Success and understanding. London: Routledge and Kegan Robert, A., & Schwarzenberger, R. (1991). Research in teaching and learning mathematics at an advanced level. In D. Tall (Ed.), Advanced mathematical thinking. (pp. 127-139). Dordrecht: Kluwer. Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification – the case of algebra. Educational Studies in Mathematics, 26. 191-228. Sierpinska. A. (1994). Understanding in mathematics. London: Falmer Press. Skemp, R. (1978). The Psychology of Learning Mathematics (expanded American ed). Hillsdale, NJ: Erlbaum. Steen, L. A. (1998). Core curriculum in context: history, goals, models, challenges. In J. A. Dossey (Ed.), Confronting the Core Curriculum: Considering Change in the Undergraduate Mathematics Major (pp. 3-14). Washington DC: Mathematical Association of America. (MAA Notes No. 45). Tall, D. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 3-24). Dordrecht: Kluwer Academic Publishers. Thompson, P. W. & Sfard, A. (1994). Problems of reification: Representations and mathematical objects. In D. Kirshner (Ed.), Proceedings of the Sixteenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp. 3-34). Baton Rouge, LA: Louisiana State University. Vinner, S. (1996). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics, 34. 97-129. Van Hiele, P. M. (1986). Structure and insight. Orlando: Academic Press.

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