The design of mathematics teaching

The design of mathematics teaching for upper level high school students

Nellie C. Verhoef

University of Twente, Faculty of Behavioural Sciences, Department of Teacher Training

Abstract

The goals of mathematics education and preferred teaching methods of upper level high school mathematics teachers were inventoried at the beginning of a period of instruction. 19 licensed Dutch mathematics teachers participated in the study. The data were gathered by semi-structured interviews. The data analysis was conducted independently by three mathematicians. Results of the study indicated no direct relation between the teachers’ goals of mathematics teaching and their preferred designs of instruction. Some teachers related their instructional designs to student characteristics. Half of the teachers reported choosing an abstract instructional design which did not match very well with the situated instructional approach reflected in their textbooks.

Keywords: Instructional design; Dutch mathematics teaching; Situated instruction; Abstract instruction

The aim of the present study is to investigate: (1) Which goals of math education teachers choose and based on these goals, (2) Which designs of instruction mathematics teachers consider for upper level high school students. Within educational curricula, mathematical knowledge is authoritative. It is a reliable predictor of access to higher education (CCSSO, 2007). Upper level high school students who wish to enrol for higher education have to choose electives. Either they can choose to enrol for arts and humanities, or for sciences. In any case these high school students need mathematics in their curricula for solving theoretical (mathematical) as well as practical problems.

Kilpatrick, Swafford and Findell (2001) emphasized that mathematics is a tool to understand and interpret the real world. They argued that those who can understand and apply mathematics have significantly enhanced opportunities to achieve success in continuing education and in daily life. The goals of mathematics education have typically been: (a) to understand mathematical concepts as well as mutual relationships and to think logically in order to construct mathematical proofs; and, (b) to apply procedures to solve mathematical problems as well as practical problems in different fields of application (Bransford, Brown & Cocking, 2000). The goal of mathematics education with a focus on understanding has been related to (a) mathematical concepts represented in units of instruction, (b) connections of these units in a structure, and (c) an emphasis on mathematical proof with logical arguments (Inglis, Mejia-Ramos & Simpson, 2007; Tall & Mejia-Ramos, 2007; Thurston, 1990). The goal of mathematics education with a focus on procedures to solve problems has been related to problem-solving skills and the application of mathematical techniques to achieve a practical purpose (Davis & Vinner, 1986; Jaffe & Quinn, 1993; Tall, 1990). How well does this twofold goal and conception of mathematics education work? A clear answer is not available. Studies with the focus on school mathematics instruction emphasized the importance of sense-making related to students’ learning activities (Ambrose, Clement, Philipp & Chauvot, 2003; Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti & Perlwitz, 1991; Fennema & Romberg, 1999; Hiebert & Wearne, 1993; Schoenfeld, 1988; Zohar & Dori, 2003). These studies demonstrated that memorization of facts or procedures without understanding often resulted in fragile learning. Conceptual understanding in relation with factual knowledge and procedural facility would make an alliance in powerful ways (Carpenter & Lehrer, 1999; Hiebert & Carpenter, 1992; Resnick & Ford, 1981; Schoenfeld, 1988).

These findings are in line with the assumption that high school teachers of mathematics will integrate their teaching methods with the goals of mathematics education. For this purpose the teaching methods were grouped into two categories: (a) starting with an abstract mathematical concept (in symbols) and solving abstract problems followed by practical worked examples and solving practical problems, or (b) starting with situated examples followed by solving practical problems in that situation and later generalizing to abstract mathematical concepts. Both approaches were theoretically founded and supported (Borovik & Gardiner, 2007; Bransford et al., 2000; Schoenfeld, 1992; 2006).

In starting with abstraction, it was supposed that the origin of mathematics is a structure. Gardiner (1996) argued that students would be familiar with that structure. Mathematics instruction should teach students to remember (learn by rote), to reckon (to calculate accurately) and to reason (to think mathematically). In this conception of mathematics education, effective mathematics teaching should reflect mathematics as inescapably abstract from the very beginning of the instruction. Instruction should concentrate on identifying symbolic techniques, such as algebraic procedures. These should easily and most often routinely be used in solving problems. This type of instruction is related ‘to knowledge that is rich in relationships’ and ‘to rules or procedures to solve problems’ (Hiebert & Lefevre, 1986, p.3; p.7). In fact, there is no fixed order in the acquisition of mathematical skills versus concepts (Rittle-Johnson & Siegler, 1998). The mathematical knowledge should be understood and the procedures for solving problems will follow automatically. Consequently, problems in different domains - that are instances or contain instances of a mathematical concept - can be solved. Entities and their relationships can be perceived (recognized) as an instance of a mathematical concept. The procedure to solve the problem as a sequence of mathematical operations can be carried out. In order to solve a problem in a context, the procedure can be decomposed and de-contextualized.

The advantage of abstract instruction to learn mathematical concepts are the acquisition of fundamental concepts and the possibility to learn to argue strictly logically, based on axioms in the formal domain of mathematics. Abstract instruction is primarily symbolic. Different mathematical concepts could be clarified in such a way that the students can develop the concepts. On the one hand abstract problems can be varied and different problem solving methods can be used. On the other hand comparable problems can be solved by one recognizable standard method. Abstract instruction is adaptable to new tasks (Skemp, 1976). A disadvantage of abstract instruction is that less-talented students, who may lack the fundamental model of mathematics as a symbolic enterprise, may not understand the concepts from symbolic problems that lack a direct and easily visualized relationship to familiar experience (Huntley, Marcus, Kahan & Miller, 2007). That means that the students do not see any relationship with a job situation or with problems they have to solve in their everyday life may struggle with abstract mathematics. Unfortunately, they may be unable to apply what they have learned and not appreciate the everyday value of mathematics.

Starting with examples provides a context in which mathematical knowledge and skills may more easily develop. Schoenfeld (1992) mentioned: "Mathematics instruction should provide students' understanding of important concepts in the appropriate core content. Instruction should be aimed at conceptual understanding rather than at mere mechanical skills, and at developing in students the ability to apply the subject matter they have studied with flexibility and resourcefulness" (p.32). This vision of mathematics instruction focuses on meaningful situations and problems. For all students, and especially for weaker students, the benefits of mathematics are likely to become clear. Mathematics is applicable in concrete as well as in abstract situations. The use of a mathematical concept or a problem-solving procedure in different contextual examples is powerful (Vinner, 2007). Making the practical context explicit may strengthen the development of the relevant mathematical concept or procedure. The problem-solving procedure may be applicable, at least in part, to other comparable situations, and such near-transfer tasks provide a rich source of assessment problems.

Skemp (1972) argued that this type of situated instruction proceeded from and resulted in a process of mathematical generalization as a sophisticated and powerful activity. On the one hand, mathematical concepts developed in a process linked to a large number of diverse situations (Verhoef & Broekman, 2004; 2005). On the other hand, situated instruction indicated that in order to correctly formulate generalizations, learners should need to abstract from a specific content (abstract or concrete) and single out similarities, the structures and the relationships (Kruteski, 1976). The theory was strengthened by the movement that instruction should be situated and that apprenticeship training was most effective (Brown, Collins & Duguid, 1989). The underlying assumption was that mental representations were incomplete and that thinking exploited the features of the world in which one was embedded, rather than operating on abstractions of it.

The advantages of situated instruction for learning and understanding mathematical concepts by using contexts (both concrete and abstract) as central thinking model, were supposed to be the intrinsic motivation and the interest of all students. The context challenged the students — for the less-talented students as well — to explore and to discover (Silver, Ghousseini, Gosen, Charalambous & Font Strawhun, 2005). A disadvantage of this design of instruction was formulated by Anderson, Reder and Simon (2000). They mentioned that knowledge, which is too tightly bound to specific, narrow contexts would hinder transfer of that knowledge. In addition, the author supposed that not all contexts are equally motivating for students.

As well as in other countries, Dutch textbook writers were mainly influenced by the situated instructional approach (Greeno, Collins & Resnick 1996; Sierpinska & Lerman, 1996). The Dutch approach was originated from Freudenthal’s (1973; 1983) instructional-design theory and rules of Real Mathematics Education (RME). Freudenthal emphasized mathematical manipulations as human activities. He was convinced that education had to be narrowly connected with student’s real life experiences. Freudenthal (1991) propagated the principle of students’ re-invention of mathematics as mathematicians did before. The RME approach substantially influenced the instructional designs in all Dutch primary and secondary school textbooks.

As a consequence, textbooks for upper level high school mathematics showed a multitude of situated problems to illustrate and support the conceptual development of mathematics. Textbooks such as Numbers and Space (Reichard et al., 2003) and Modern Mathematics (Boer et al., 2004) have an 87 percent market share. These textbooks support the two different Dutch mathematical upper level high school programs: (a) a program with a focus on arts and humanities, and (b) a program with a focus on the sciences. Arts and humanities students’ textbooks contain well illustrated real life assignments. Science students’ textbooks also have exercises with symbols in a formal mathematical language. They have to prepare themselves for studying mathematics or a science major at a higher academic level.

The textbooks for both categories of high school students showed the same situated teaching methods for the acquisition of meaningful and transferable knowledge. The use of the textbook supposed that the students are encouraged to discover strategies to solve theoretical (mathematical) as well as practical problems.

At the Dutch teacher training institute, the design of situated instruction became a point of departure and was widely practiced. High school teachers soon adopted the approach. At the moment, however, both enthusiasm and complaints about the situated instruction approach are made public. The time consuming use of situated problems may result in a lack of time for practicing functional, operational mathematical skills. This might be a problem for high school students, who want to major in one of the sciences or applied sciences. Their knowledge is possibly too strongly connected with contexts and specific situations. Too little practice of general procedures may result in a lack of automated skills. Moreover, it might be difficult to show the development of the abstract mathematical concepts from situated examples (Van de Craats, 2007). As a consequence students’ knowledge base may have a lack of useful mathematical concepts and their mutual relationships.

Although the textbooks use a preferred teaching method, it is possible to model context-abstraction as a dichotomy. The question is what mathematics teachers will do in practice? They are ultimately responsible for the instructions of the subject matter for their students. They play a crucial role in shaping the students’ learning processes. It is supposed that mathematics teachers develop their own teaching method, which is a result of their teaching experience, conviction, identity and commitment over the years (Escudero & Sanchez, 2007; Fullan, 1998; Holmes, 1998; Korthagen & Vasalis, 2007; Palmer, 1998; Ryan & Deci, 2002; Verhoef & Terlouw, 2007). Another possibility is the development of mathematics education in collaboration with classroom teachers, educational researchers, and instructional designers who share the common goals of understanding and improving the teaching and learning of mathematics (Magidson, 2005). In this study it is supposed that math teachers will gradually develop knowledge in favour of a situated instructional approach for realizing the goals as a result of the use of the textbooks.

19 upper level high school teachers participated; six of them were women and 13 were men. The participants were selected pseudo-randomly from a list of the Association of Dutch Math Teachers (e.g., numbers 1, 51, 101, and so on). This Association is the only one for upper level high school teachers in the Netherlands. Members of this Association are typified as being involved and active. The teachers received a phone call describing the study and asking whether they were willing to be interviewed. If they were unable to participate, numbers 2, 52, and so on, received a phone call. Of those who participated, the ages varied between 28 and 61 with an average age of 52. 16 teachers had acquired a master’s and three a bachelor’s certificate. All had acquired a certificate for teaching mathematics for upper level high school students. The average number of years of practice was 21.2. 89 percent used textbooks based on a situated instructional approach. The teachers are tagged by capital letters (A until S).

Semi-structured interview. The interview protocol contained two main questions: (1)Teachers’ goals of mathematics education split up into understanding math concepts using visual, graphical or numerical representations; or procedures to solve problems using algorithms and calculations, and (2) the decision of the teaching method at the start of the instruction sub divided into abstract concepts and procedures to solve abstract problems with recognition, calculations exactly and argumentations; or situated worked examples with meanings and contexts.

19 of 22 teachers, who received the phone call, responded affirmatively. During that call the appointment for an interview was made. All interviews were held in 2008. Each interview took about 45 minutes. The interviews were recorded. A secretary worked out the recorded 19 interviews in a typewritten text (an example is given in Table 1).

Table 1

Typewritten text (teacher K)

Note. The italic written sentences’ parts are used as to label teacher’s answers at the choice of the goal of math education. The underlined written sentences’ parts are used to label teacher’s answers at the choice of the start of instruction to attain these goals.

Table 1 indicates that teacher K emphasizes understanding math concepts as a goal of math education and an instructional design that starts with abstractions. Teacher K’s answers at the interview questions were written reproduced and sent back for approval (Table 2).

Table 2

Example of a letter to a teacher (teacher K). The content of the letter comprised (a) the school data (name, address and type of high school, indicated by the Dutch standard abbreviations), (b) the teacher’s personal particulars (name, age, degrees, teacher licenses and job history), and (c) the reproduced answers to the interview questions. In this example only the reproduced answers to the interview questions are shown.

Note. The italic written sentences’ parts are used as to label teacher’s answers at the choice of the goal of math education. The underlined written sentences’ parts are used to label teacher’s answers at the choice of the start of instruction to attain these goals. The bold sentences’ parts are used to gather unlabeled remarks.

Table 2 repeats the indications of Table 1. The reproduced answers were sent back for approval to the nineteen teachers each. They react if necessary (Table 3).

Table 3

The written approval of a teacher (teacher K)

Table 3 shows that teacher K agrees with the written report.

Reliability. Three raters, all mathematicians, marked and labeled characteristic expressions in each teacher’s interview transcript separately. They shortened and summarized notable parts of sentences. After the raters had separately marked and labeled the characteristic expressions, the Fleiss' (1971) kappa coefficient indicated over 70 percent agreement between the raters. This is a substantial level of agreement (Landis & Koch, 1977). In case of disagreement, the raters’ pre-determinations were gathered and discussed. The raters decided to try to get more teacher information by sending an extra personal written question by email to those about whom there was disagreement among the raters. The raters asked to clearly choose for a goal of math education by understanding mathematical concepts or by learning procedures to solve problems, in relation with a choice of teaching method at the start by abstraction or by situated worked examples. Two out of 19 teachers gave a final decision after the reproduced question by email. Three out of 19 could not make a final choice. The teachers motivated their choices, dependent on a student population. The remainder was discussed intensively again by (1) consulting the taped interviews, and (2) checking teachers’ first considerations. Finally, this resulted in agreement between the raters.

Semi-structured interviews. In total 58 goal statements were gathered. One teacher (C), who makes four goal statements, could not make a final choice. Therefore the remained 54 goal statements were categorized: 16 goal statements about only understanding mathematical concepts, and 12 goal statements about only procedures to solve problems; 12 goal statements about understanding mathematical concepts followed by procedures to solve problems, and 14 goal statements about procedures to solve problems followed by understanding mathematical concepts. In Table 4 characteristic goal statements of mathematics education related to their decisions of the teaching method at the start of the instruction are shown. Quite often, the teachers included different sub goals in their complete goal statements. All goal statements, including those for sub goals, were categorized into:

The rows indicate the different goals of math education: (i) understanding; or (ii) procedures to solve problems. In the columns the choices of teaching method at the start of instruction are shown: (i) abstractions; or (ii) situated worked examples. The teachers who make a characteristic goal statement are underlined marked by capitals behind the brackets. The other capitals behind the brackets indicate equivalent goal statements.

Table 4

Teacher’s characteristic goal statements related to the chosen teaching method in different categories (after Thurston, 1990; Jaffe and Quinn, 1993)

Note. Only those teachers (marked by capitals), who make a statement are included. Some teachers indicate their statements into different groups like arts and humanities students versus science students. These statements are marked by special group indications into brackets behind the statements.

Table 4 indicates that some teachers relate their preferred choice of a teaching method to a student population, arts and humanities students or science students. Three teachers emphasize understanding as essential for arts and humanities students. They mention practical problems and their familiarity with applications in contexts to stimulate imaginableness. Two other teachers accentuate problem solving skills for science students with respect to scientific studies. They mention calculations, proves, and problem solving actions by valid arguments as essential.

In Table 5 the teachers’ goal statements of mathematics education related to their decisions of the teaching method at the start of the instruction are summarized. The rows indicate the different goals of math education: (i) understanding; or (ii) procedures to solve problems. In the columns the choices of teaching method at the start of instruction are shown: (i) abstractions; or (ii) situated worked examples.

Table 5

Categorization of teachers, based on teachers’ goals and their instructional statements

In Table 6 sixteen goal statements of those teachers who only mentioned understanding and twelve who only mentioned the use of procedures to solve problems are related to the decision of the teaching method at the start of instruction.

Table 6

Number of occurrence of teacher’s goal statements related to the chosen teaching method in different categories (after Thurston, 1990; Jaffe and Quinn, 1993)

Note. Situation in the title is an abbreviation of situated worked examples

For twelve statements of teachers who first mentioned understanding followed by a goal statement about the use of procedures to solve problems, the relationship with the teaching method statements are shown in Table 7. In the same table the relationship for the remaining fourteen statements of those who first mentioned procedures followed by understanding is shown as well.

Table 7

Number of occurrence of teacher’s goal statements with a follow up related to the chosen teaching method in different categories (after Thurston, 1990; Jaffe and Quinn, 1993)

Note. Situation in the title is an abbreviation of situated worked examples.

4. Discussion

The results of this study show that teachers have different goal priorities. Eight teachers emphasize the understanding of mathematical concepts as their goal priority. An analysis of their goal statements reveals that they accentuate the understanding of concepts and their mutual relationships and of logical thinking in order to prove abstract mathematical suppositions. Ten teachers mention the use of procedures to solve mathematical problems as their first priority of mathematics education. The analysis of their statements shows that they emphasize the training of problem-solving skills and of special mathematical techniques. The results make clear that based on the number of participants, neither goal is given priority by any significant majority of mathematics teachers. Much the same is true for the goal statements that the teachers make: a total of 28 statements show understanding of mathematical concepts. A total of 26 statements include the training of skills and techniques. Since understanding of the content of a curriculum is generally seen as a condition for successful education, it is striking that so many statements are devoted to the training of mathematical skills and techniques. From the responses, it becomes clear that many of those who emphasize the understanding of mathematical concepts also include the use and practice of procedures in their teaching. These teachers accentuate to attain the goals of understanding mathematical concepts and their mutual relationships, to think logically in order to prove abstract mathematical suppositions, by using mathematical technical skills as procedures to solve problems. And those who emphasize the use of mathematics and practicing procedures ask attention for logical thinking, use of arguments, strategies for problem solving, puzzling, and discovery for the construction of mathematical knowledge. These teachers stress to attain the goals of applying procedures to solve mathematical problems as well as practical problems in different fields of application, by using mathematical concepts. From their emphasis on procedures and thinking the author supposes that this reflects their view on the development of mathematical generalization by sophisticated and powerful activities. Their goal is that the students should try to grasp the algorithms in such a way that they can transfer them to solve a whole category of problems. They highlight meaningful instructional designs of math education. But this is only possible if the mathematical concepts that define the entities (mathematical objects) and their relationships are constructed during their problem solving activities and become integrated in the student’s cognitive structure. The development of algorithms as a trick should be avoided as one of the teachers so clearly reports. Of course the well-defined concepts can and should be explained and illustrated in a well-designed instruction in advance, but most probably this will only partially help the students to understand the use of these concepts and their integration into cognitive structures. The meaning and the integration of this meaning into the student’s cognitive structure becomes clear by their use and reflection on activities.

As to the teaching method the results of the study show that the teachers either prefer abstraction or situated instruction. The number of teachers that choose for either method is equal. Moreover the results do not show a significant relationship of teaching method with either of the two goal categories. A close analysis of the statements however makes clear that in some cases a trend for a preferred teaching method may be supposed. These trends are only found in the statements of those teachers who only mention either understanding of concepts or the training of mathematical procedures as their single goal or sub goal. The results of the study indicate that to increase understanding of the use of logical or valid arguments for a mathematical proof, the instruction should be abstract. The same trend is shown in case of training mathematical techniques. For illustrating and training mathematical techniques the teaching method should be abstract. If a more complex goal of mathematics education is mentioned, the choice of teaching method does not show any possible trend in the choice for one of the two methods. It is supposed that if the teachers are forced to make a choice for either an abstract or a situated approach, any prototypical example of an instruction may come across. Based on that example the choice is made. The results of this study indicate that the choice of an appropriate teaching method will probably be made at the level of a sub goal.

Although the interview question about the teachers’ goals of mathematics teaching was completely open and general, the teachers’ answers show that they sometimes describe their goals for the population for which they are teaching, either arts and humanities students or science students or both. The results of the interviews show that five teachers differentiate the goals to the population involved. From these results a clear difference in goals cannot be made. For science students the teachers indicate the importance of manipulation of symbols as a basis for thinking, the use of mathematical concepts, calculations, puzzling and discovery, the use of the structure of mathematics itself, the use of valid arguments, and the combination of proves and calculations. For arts and humanities students the teachers also emphasize understanding, use of arguments and practicing procedures. But instead of only working within the domain of mathematics, for the arts and humanities students they strongly emphasize the usefulness of mathematics: procedures and strategies should make sense.

The results of this study further show that the teachers use both designs of instruction to the same extent despite of the fact that a substantial majority of the teachers use ‘situated’ textbooks. They either prefer abstract or situated instruction. The teachers provide some indications how generalization and abstraction may be realized. Statements such as: “generalize step-by-step to formulas, examples should be followed with trial and error actions with math concepts, concrete situated worked examples should be followed by abstraction and demonstrate worked examples with numbers and figures” may give some direction. Many teachers who prefer the use of sufficient situated worked examples probably assume that this will lead to generalization and abstraction, although detailed instructions that will foster the needed cognitive process are not clearly described. More research is needed in order to be able to design a more detailed instruction.

Given that Dutch textbooks are mainly based on a situated instructional design, this study clearly demonstrates that math teachers have developed their own preference for a teaching method. It is supposed that the method used is related to a sub goal of mathematics education. The author therefore rejects the use of one single teaching method.

The results of this inventory do suggest that teacher training programs should include practicing both instructional design approaches. If the goal of mathematics education is to understand its concepts, the only use of situated worked examples will not be sufficient to reach this goal, because the salient and emotional properties of the context may hinder the understanding of the abstract concepts. The understanding of abstract mathematical concepts however does not indicate a successful situated use in different fields of application. The teachers’ help is necessary to clarify math concepts’ properties and relationships in different contexts. Then the probability that the students will be able to recognize mathematical concepts and its relationships will increase. The only use of abstractions does not guarantee successful use of mathematics in different fields of application, because the symbols may be meaningless for students. The teachers’ support emphasizes possible representations of the symbols as well as formal, logical expressions with symbols. The students will try to imagine possible objects that are represented by the symbols. It is unclear how situated instruction promotes the development of abstract mathematical concepts and how the students can use abstract mathematical knowledge to solve practical problems. Mosvold (2008) compared math teachers’ use of textbooks based on situated instructional approaches in Norway, Denmark and Japan. His research findings show that Dutch math teachers use their textbooks in a traditional way, without a relationship to situated instruction. The students solve problems like textbooks’ tasks. They fill in sub parts of real-life connected problems. Realistic Mathematics Education was not visible in the classroom practices. Therefore, it is recommended to direct further research activities on these two issues.

Acknowledgements

The author would like to thank prof. dr. S. Dijkstra, University of Twente, Faculty of Behaviour Sciences, Department of Instructional Technology for his support and guidance and prof. dr. J. Michael Spector, University of Georgia, Department of Learning and Performance Support Laboratory for his constructive comments and for linguistic editing.

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