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This study explored the use of writing to help students develop schemata knowledge for algebraic thinking. Eight seventh-grade pre-algebra students participated in a teaching experiment in which they solved algebraic problems related in mathematical structure. The problems involved contexts of growth and change and size and shape. Qualitative data analysis of students' writing indicated that students used all aspects of schemata knowledge-identification, planning, elaboration, and execution knowledge-to solve the algebraic problems. This article provides excerpts of students' writing and descriptions of ways they used these different aspects of schemata knowledge. The findings from this study support both the use of writing and the use of algebraic problems related in mathematical structure to help students develop tools they need to think algebraically. [PUBLICATION ABSTRACT]
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This study explored the use of writing to help students develop schemata knowledge for algebraic thinking. Eight seventh-grade pre-algebra students participated in a teaching experiment in which they solved algebraic problems related in mathematical structure. The problems involved contexts of growth and change and size and shape. Qualitative data analysis of students' writing indicated that students used all aspects of schemata knowledge-identification, planning, elaboration, and execution knowledge-to solve the algebraic problems. This article provides excerpts of students' writing and descriptions of ways they used these different aspects of schemata knowledge. The findings from this study support both the use of writing and the use of algebraic problems related in mathematical structure to help students develop tools they need to think algebraically.
Research in cognitive psychology provides a theoretical basis for students to develop schemata knowledge and the ability to think algebraically. What is algebraic thinking? Although it may include variables and expressions, algebraic thinking has a broader and different connotation than the term algebra. Kieran (1996) defined the term algebraic thinking as "the use of any of a variety of representations that handle quantitative situations in a relational way" (p. 4-5). Driscoll (1999) said that algebraic thinking could be considered to be the "capacity to represent quantitative situations so that relations among variables become apparent" (p. 1). Smith (2003) believed that for students to think algebraically they must be able to identify, extend, and generalize patterns in order to understand quantitative relationships. By combining these views of algebraic thinking, the definition used to direct this study was that algebraic thinking includes the ability to analyze and recognize patterns, to represent the quantitative relationships between the patterns, and to generalize these quantitative relationships.
Theoretical Framework
Developing Schemata Knowledge
Marshall (1995), a cognitive psychologist, stated that a schema is the means by which similar experiences are assimilated and consolidated in such a way as to be quickly and easily remembered. A schema allows an individual to organize similar experiences so that the individual can easily recognize additional similar experiences. Marshall emphasized that an individual does not memorize a schema; meaningful schema development means that an individual must understand concepts in order to recognize and construct patterns. Marshall suggested that there are four types of schemata knowledge: identification knowledge, elaboration knowledge, planning knowledge, and execution knowledge. Identification knowledge is pattern recognition. It is knowledge that contributes to the initial recognition of a situation, event, or experience and helps an individual solve a problem. Once individuals identify the basic situation, they then access the necessary elaboration knowledge for additional details.
Elaboration knowledge is knowledge that individuals possess that helps them to find in memory specific examples that can describe similar experiences (Marshall, 1995). Elaboration knowledge contains both verbal and visual information and enables an individual to create a mental model about how the current problem relates to a previous one. This knowledge helps when the individual needs to modify existing knowledge. Marshall stated that, coming together, identification and elaboration knowledge constitute a framework allowing the individual to form a tentative hypothesis about a situation and then to evaluate it-to see if the approach to the problem is sensible.
The next type of schemata knowledge Marshall described is planning knowledge. This knowledge helps the individual decide if a plan can be used that was tried on a similar problem-solving situation; however, this recognition of a similar situation does not necessarily mean that apian will come automatically. The individual must continue to use elaboration knowledge to lay out objectives and form a plan. After forming the plan, the individual then uses execution knowledge; the individual performs a skill, such as making a table, finding a smaller problem, or following an algorithm (Steele & Johanmng, 2004).
Using Writing in Mathematics to Access Students' Schemata Knowledge
When students use only procedures and algorithms to solve problems, it is difficult to access their schemata knowledge; therefore, if they are asked to write about how they thought about a problem or why they solved a problem in certain ways, teachers have a productive way to assess the depth of their knowledge, hi order to access and assess students' knowledge, however, it is important to ask them to answer particular questions or ask them to explain particular aspects of their thinking. Besides using writing as an assessment tool, asking students to give reasons for their actions in solving problems, to explain what they are doing and why, and to explain why their reasoning is correct are ways to help the problem solver adopt a more deliberative approach (Dominowski, 1998). This deliberative approach allows students to organize ideas, to develop new applications for knowledge, and to solve problems (Countryman, 1992).
In recent years the mathematics education community has increasingly recognized the importance of communication within the practices of doing, teaching, and learning mathematics (Boaler, 1997; Burton & Morgan, 2000). Burton and Morgan argued that an important part of learning to think mathematically is learning to take part in the discourses of mathematics; writing is one important form of communication or discourse. In addition, Morgan (1998) maintained that teachers should explicitly set about helping students to learn mathematical writing.
Bruner (1986), a cognitive psychologist, maintained,"We teach a subj ect not to teach little living libraries on the subject, but rather to get a student to think mathematically for himself... to take part in the process of knowledge-getting. Knowledge is a process not a product" (p. 72). Many mathematics educators believe that students understand the process through writing. In fact, writing in mathematics was emphasized in The Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), which said, "Writing in mathematics can...help students consolidate their thinking because it require[s] them to reflect on their work and clarify their thought about ideas" (p. 61). Flores and Brittain (2003) found in their research that process writing helped students learn mathematics and establish a personal connection to new concepts.
Borasi and Rose (1989) suggested that writing plays a significant role in leamingmathematics because it helps students organize and clarify their ideas and learn to explain and give reasons for their ideas. The writing to learn movement in mathematics centers around the idea that students create mathematical knowledge by communicating their thoughts (Connolly & Vilardi, 1989). This idea is based upon a Vygotskian perspective (Zebroski, 1994) that writing creates a situation in which language and thought become dialectic. The act of putting thought into words causes the writer to make an effort to convey ideas and, thus, to create new knowledge. Writing is a way of verbalizing to oneself. This self-directed speech becomes inner speech (Vygotsky, 1962) and helps build understanding (Dominowski,1998).
Bell andBell(1985) stated that writing and problem solving use the same thought processes. In factPolya's (1981) four phases for solving a problem-understanding the problem (creating a mental picture of what is given, understanding the conditions, deciding what to look for); devising a plan (recognizing a similar problem that has been previously solved, thinking about how to use the data); carrying out a plan (computing, checking to see if steps are correct, proving correctness); and looking back (checking results, thinking about another way)are similar to the process of writing. Bell and Bell maintained thatthe writingprocess involves discovering a topic, deciding what to say about it, organizing how to say it, writing a draft, and finally revising and rewriting-all parts of Polya's four phases of problem solving.
Bell and Bell (1985) conducted a study of ninthgrade students to investigate whether writing in combination with problem solving had an effect on achievement. They found that writing made students more aware of their thinking-their metacognitionand increased their achievement when compared to students who didnot write to solve problems. Davidson and Sternberg (1998) found that writing built metacognition which, in turn, contributed to successful problem solving. Carr and Biddlecomb (1998) also found a link between writing, problem solving, and metacognition. In a recent study Pugalee (2001) investigated students' writing in mathematics. He analyzed 20 ninth-grade students' mathematical writing and found evidence of metacognition.
In regard to metacognition, however, Nisbett and Wilson (1977; cited in Marshall, 1995) found in their research that it was difficult to access metacognitive knowledge and to know when it is being activated because "[individuals may not be aware of their own problem solving and thus cannot describe accurately what they are doing or thinking" (p. 63). Because of this lack of awareness of one's own thinking and the difficulty in accessing another's metacognition, the focus of using writing in the present study was to access schemata knowledge, much in the way teachers assess students' conceptual and procedural knowledge. Indeed, Carr and Biddlecomb (1998) believed that writing helped individuals develop schémas for solving mathematics problems and that the development of schémas was key to understanding how metacognitive knowledge develops. The theoretical framework used in this study was Marshall's (1995) framework for developing schemata knowledgeidentification knowledge, elaboration knowledge, planning knowledge, and execution knowledge.
Solving Related Contextualized Problems Helps Students Build Schemata Knowledge
Silver (1981) found that good mathematical problem solvers use the mathematical structure of problems to generalize solution strategies. In his study, he found that by solving related problems, students transferred what they learned from one problem to another similarly structured problem. Davidson and Sternberg concurred (1998) with this idea and said that when students have experience solving particular types of related problems they are more likely to transfer solution plans between problems with the same solution process and can spend more time on higher-level planning.
Confrey (1997) suggested that one approach to helping students examine problems related in mathematical structure was a context-based approach. With this approach teachers help students see how algebraic ideas can emerge from real-world and concrete problem situations. By using the contextual approach teachers help students move from one problem to another similar problem and compare the structures of the problems. According to the Algebra Working Group (1997), students' understanding of algebraic concepts should be developed through activities embedded in specific contextual settings, such as growth and change, size and shape, and number patterns. These contextual settings also help students organize commonalties among situations, see relationships abouthowchange inonequantity is related to change in another, and make generalizations-all are important aspects of algebraic thinking.
Purpose
The purpose of this study was to understand the ways that students used writing to develop schemata knowledge-identification, elaboration, planning, and execution knowledge-to solve algebraic problems related in mathematical structure. In order to achieve this purpose their writing was analyzed to understand these aspects of their schemata knowledge and to understand their algebraic thinking. Unlike direct instructional approaches, solutions to problems were not modeled beforehand. Even though students had learned different problem-solving strategies (i.e., making a table, working backwards, making diagrams) in their seventh-grade mathematics course, students decided which strategies they would use to solve each problem. Brenner et al. (1997) maintained that if students learn to incorporate different representations such as diagrams, concrete objects, tables, words, and equations, they develop rich approaches to solving algebraic problem situations. In this study the choice of problem contexts, size and shape and growth and change, had potential to lead to such multiple representations. Students could also search for number patterns that might be constructed from the two contexts. The question that guided the research was the following: In what ways do students write about and use schemata knowledge when solving algebraic problems related in mathematical structure? While investigating examples of ways students used writing to demonstrate schemata knowledge, the benefits of writing for developing algebraic thinking and accessing students' depth of knowledge was also explored.
Methodology
The Teaching Experiment
A teaching experiment research method was used for this investigation. In keeping with traditions of the teaching experiment, the investigation involved both interview phases and teaching phases (Cobb, 2000; Steffe & Thompson, 2000). In the interview phases, students were asked questions in which their use of schemata knowledge for solving related algebraic problems was explored. In the teaching phases, in order to help students construct schemata knowledge for solving similar problems and bring forth this knowledge in their solutions, particular problem assignments were manipulated based on students' solutions to earlier problems during the experiment. Polya's (1981) "solve a similar problem" strategy, in which students' repeated problem-solving experiences had identifiable features in common, was incorporated. The students solved progressively sequenced problems to encourage them to see relationships among algebraic situations. By using the teaching experiment methodology, instruction could be planned, adapted, and sequenced in such a way to help pre-algebra students construct effective problem-solving knowledge and develop their ability to think algebraically while the author also investigated the research question. Their teacher was involved in the planning and execution of this experiment. As Lesh and Kelley (2000) stated in their discussion of teaching experiments, this involvement by the teacher helps the researcher verify students' thinking. The teacher also acted as a witness to the teaching experiment, an idea that Steffe and Thompson (2000) proposed is important for validating students' thinking. It was important to understand students' knowledge for solving algebraic problems independent of the researcher ' s and teacher ' s mathematical understandings. The strength of a teaching experiment approach to research is that it helps uncover students' thinking because the design focus is conceptual analysis. Von Glasersfeld (1995) agreed that the process of conceptual analysis-continually looking behind what students say and do-makes the teaching experiment approach a powerful one for research.
Participants
The participants for this investigation consisted of 8 seventh-grade students enrolled in a pre-algebra mathematics class in a small Midwestern middle school. The curriculum up to this point had been the traditional pre-algebra curriculum from students' text. The text was a traditional one that had been used for several years and contained general mathematics, but also introduced the students to variables, symbolic expressions, and equations. The teacher in this class, however, had been using writing in mathematics all year. Students were accustomed to being asked to explain, both verbally and in writing, their strategies and mathematical procedures and explain why they chose their particular strategies and procedures. The teacher had encouraged them to make lists and tables to explain their thinking. She used a rubric to grade their work. This rubric included evaluation of their computations, strategies, and explanations. Students had not solved problems related to the ones used in this teaching experiment.
The students were above average in mathematical ability and, thus, in this school system they were placed into a pre-algebra course rather than a general seventhgrade mathematics course. The 8 students in this study were a subset of the 24 pre-algebra students in this class and chosen by the teacher to participate in this experiment because she believed they would provide a variety of approaches to solving problems. Five of the students were boys and 3 were girls. All students were Caucasian.
Procedures for Data Collection and Analysis
The students in this study solved eight generalizing problems-five linear and three quadratic problems. The teaching experiment lasted for 1 month; each problem instruction sequence lasted 2 or 3 days. Five of the problems are presented in this paper (see Steele, 2000, 2002, for the other problems). During the teaching phase of the experiment the author emphasized to students the importance of the process that led to their solutions. They were asked first to solve the problems individually. They were required to write about their thinking and to explain their approaches and generalizations. They could use manipulatives, graph paper, and any other materials in the classroom; no hints were given. It was important that students first reflect individually and write their reflections with no help from others, because this reflection helped the researcher understand their thinking and also helped the students fix firmly their own thinking about the problems.
Students' work was collected, read, and returned the next day. After receiving their papers they met in cooperative groups to discuss their problem-solving approaches. In these groups students read their descriptions, explained their thinking, explained their generalizations, and displayed their symbolic expressions. They listened to each others' strategies and solutions, compared their own strategies and solutions with those of others, and assessed whether they wanted to keep or change their approaches.
As students discussed their reasoning in small groups, the author and their teacher walked around, observed, listened, and asked questions. Some of the questions were as follows: How did you reason about the problems? How did your diagram help you? How did your table help you? Did you look for an easier way? Can you explain what the numbers represent in your expression? So you decided to try your expression on some larger numbers to see if it would work? After each session, field notes documenting observations and keeping track of new questions were added to a j ournal.
During the interview phase of the teaching experiment each student was formally interviewed four different times using the same problems. The same types of questions as those during group work were asked. To determine if students saw similarities in the problems, questions were asked such as, "Do you see arelationship to any problems you have already solved?" A journal of field notes documenting observations from each interview was also kept. Using the field notes, observations from cooperative groups, interviews, and students' work, instruction was adapted when choosing the next day ' s problem. The next problem situation was slightly different but closely related in mathematical structure to the previous one. This cycle continued until the end of the 1-month teaching experiment. All cooperative group sessions and interviews were audiotaped and transcribed. All students' written work that contained their diagrams, tables, and writing was collected and kept. Data were systematically and rigorously organized, analyzed, classified, and consolidated using the Developmental Research Cycle (Spradley, 1980) to determine patterns and themes. Using this method of data analysis to "determine its parts, the relationship amongparts, and their relationship to the whole" (Spradley, 1980, p. 85) a portrait of ways students developed and used schemata knowledge emerged.
Findings and Discussion
Excerpts from students' writing will be used to discuss the findings. These excerpts will show classifications of schemata knowledge to demonstrate the ways students used the four types of schemata knowledge-identification, elaboration, planning, and execution knowledge (Marshall, 1995). Examples from all 8 students' writing will be provided to demonstrate the four types of knowledge. These demonstrations of schemata knowledge will be embedded within the context of the problems they solved. The work of these students was chosen because it was representative of how students approached solving the problems.
The first problem in the teaching experiment students solved was the shaded squares problem (see Figure 1 ). Because this was the first problem in the teaching experiment, none of the students mentioned any relationship to a previous problem they had solved. Tommy, one of the seventh-grade students, rather than discussing any identification to a previous problem began by elaborating on-demonstrating elaboration knowledge-how he solved the problem. As Marshall stated, this knowledge helps the individual find in memory specific examples of actions to use to solve certain types of problems. It contains both verbal and visual information and enables an individual to create a mental model about how to approach the current problem.
Tommy elaborated on the patterns as he used his execution knowledge-knowledge for which an individual performs a skill, such as making a table, finding a smaller problem, or following an algorithm (Marshall, 1995). He solved the problem by drawing diagrams of smaller problems (squares with different numbers of shaded squares) and made a table. He used his table to elaborate-to identify patterns (numeric relationships) between the number of squares, the number of shaded squares, and the number of unshaded squares (see Table 1).
First, I found out that every time you go up by 1, like from 2x2 to 3x3, you add 4 more shaded squares. I found out that there is always 2x2 less or 3x3 less or 4x4 [the square numbers in the third column] less on the inside than there is in the whole shape.... The formula is N - (# of squares on a side - 2) x (# of squares on a side - 2) = # of unshaded squares. Then you just take (# of squares on a side) χ (# of squares on a side) - unshaded squares = shaded squares.
Tommy used his execution knowledge to calculate the numbers of shaded and unshaded squares. As he elaborated, he articulated well and used identification knowledge to recognize some of the key numeric patterns in the problem and establish the relationship between the number of shaded and unshaded squares. Elaborating on the problem was key to identifying the patterns (using identification knowledge). As Marshall stated, this knowledge includes such ideas as pattern recognition and contributes to the initial recognition of a situation, event, or experience. This identification of the patterns helped Tommy make the generalization needed to think about the problem algebraically. Although he drew diagrams to solve the problem, he mainly used his table of values (generated by drawing diagrams and filling in corresponding data values) to establish numeric patterns and then to extend the patterns to formulate a generalization. After experiencing that his table helped him fmdpatterns for his generalization, Tommy continued to form plans of mainly focusing on the numeric patterns in his tables to solve the rest of the problems in the teaching experiment.
Another student, Cathy also began by elaborating on how she solve the problem: "In a 25x25 grid, 96 squares would be shaded. I got this because there are 25 squares to a side. But 4 squares, the corner ones, are shared so you don't count them twice." She then demonstrated her planning knowledge-the type of schemata knowledge that helps the individual recognize an action that may be used in a particular problemsolving situation. Marshall explained that this recognition does not necessarily mean the solution comes automatically because an individual must continue to elaborate in order to form a plan. Because the students in this class were accustomed to drawing diagrams when they solved problems, it was not surprising when Cathy made a plan that involved using diagrams. She wrote, "I will use a smaller example to tell what I mean." She then demonstrated her execution knowledge (see Figure 2 and Table 2).
Cathy continued to elaborate about why she executed particular actions to solve the problem. She constructed the idea of shared corners and subtracting out the shared or common components.
If you count the corners twice, you get 12. If you don't, you get 8. The corners are shared and so you only count them once. So knowing this information, the formula is (N * 4) - 4 = # of squares. You times n by 4 because a square has 4 sides that are all equal. You subtract 4 because the corners are shared. Without subtracting 4,1 would have counted the corners twice and my answer would be 4 too high.
Through elaborating on and identifying the patterns Cathy tried to make sense of the relationship between the number of shaded squares and the number of squares on the side of the figure. Making sense of the quantitative relationship between the two variables in the problem was essential to making both a verbal and symbolic generalization. In addition, elaborating on the problem helped Cathy evaluate and explain why her solution was correct.
Cathy ' s plan for working out this problem was one of drawing diagrams of smaller-sized squares, analyzing the diagrams for general patterns, organizing this data in a table, identifying the patterns, and writing a generalization in word and symbols. Tommy'splan was similar; however, he analyzed the numeric patterns in the table when finding the generalization, rather than analyzing the diagrams. Tommy was not able to create a symbolic generalization as Cathy did. They both elaborated on the problem, made a plan, and executed the plan, then identified the patterns-all parts of schemata knowledge that Marshall found in her research.
The sequence of instruction continued with a similar problem. It was hoped that students would elaborate on the new situation and identify a problem that was similar in mathematical structure to the shaded squares problem as they solved the triangle dot problem (see Figure 3). Cathy accessed in her memory the elaborationknowledge she needed to recognize the similarity of this problem to the previous one: "I looked at this problem and realized it was exactly like the problem I did last time only with a triangle instead of a square." Cathy then identified sharing and subtracting as the aspect of the problem that she thought related it to the previous problem. By identifying the sharing attribute in this problem she was able to draw from her schemata knowledge to subtract out the shared components. She then executed calculations and created a symbolic generalization.
The reason there are only 12 dots in the triangle...and 5 dots to a side is because you don't count the corners twice. You multiply the number of dots on a side times 3 because there are 3 sides in a triangle, and you subtract 3 because there are 3 corners in a triangle and they are shared. So for a 13-dot triangle there will be 36 dots. (13 * 3) - 3 = 36. So the formula is (N * 3) - 3 = # of dots.
In her written explanation of how she solved the problem Cathy clearly explained how she used identification knowledge from the shaded squares problem to solve the triangle dot problem. By elaborating on the number of dots along one side of the triangle and how these dots could be used to predict the total number of dots, she identified the pattern that each corner dot was shared between two sides. She modified her previous generalization to write an expression for the new problem. As Cathy described how the two problems were related she drew on both verbal and visual interpretations (elaboration knowledge) of the shaded squares problem and modified them for the new problem. She solved for specific cases (using execution knowledge) to explain why her generalization was correct. This evaluation provided more evidence of her elaboration knowledge.
For later problems, Cathy continued to manipulate different diagrams for each problem to find shared components to subtract them out and to generalize the patterns she identified into verbal statements and symbolic expressions. Her utilization ofher schemata knowledge gave her the ability to generalize across contexts. In fact, for all of the problems in the teaching experiment, Cathy found shared components to subtract when drawing diagrams.
Jane also identified the similarity in mathematics structure between the triangle dot problem and the shaded squares. She demonstrated her elaboration and identification knowledge: "When I read this problem I remembered a formula that I saw used on a problem I did just recently." Then she used her planning knowledge and executed her plan: "I decided to try this formula on the 5-dot triangle." She calculated (using execution knowledge) and wrote, "(5 * 3) -3 = 12. That formula worked so I decided to try it on another triangle. (6 * 3) - 3 = 15. It worked for this triangle too. So the formula is (n * 3) - 3 = # of dots." Jane then elaborated on how she thought about the problem and why the two problems were similar. To elaborate she used more execution knowledge (she drew the diagram in Figure 4). She wrote, "Because of the overlap. I timesed « by 3 because n is the number of dots on one side and there are 3 dots that don't overlap. I subtracted 3 because the three corners overlap." Instead of the idea of sharing that Cathy used, Jane called it "overlap." However, she also subtracted the overlapping components. Also, by checking specific cases to explain why her answer was correct she continued to elaborate by evaluating her generalization.
Steve saw a connection between the two problems, as well. With his narrative he demonstrated clearly how identification and elaboration knowledge are intertwined. Once Steve identified the basic pattern situation, he then accessed the necessary elaboration knowledge for additional details. When he reasoned about the triangle dot problem, he explained,
When I was reading and writing the problem I realized that it was almost similar to the shaded square problem. The first thing I saw was that each side went down by 1. So for the 5 dot, one side had 5, one had 4 and the other had 3. So the first side would be n, then n - 1 and n - 2 because of the over laping [sic] corners.
Marshall ( 1995) stated that, together, identification and elaboration knowledge constitute a framework that allows the individual to form a tentative hypothesis about a situation and then to test it (using planning knowledge). This framework was also evident in these students' writing about their solutions. Elaborating on the physical structure of the problem helped Steve formulate and carry out a plan (even though he did not actually state his plan). Steve discussed how he used his execution knowledge to solve the problem and make a generalization: "Iusedthismethodforthe 13 dottriangle and got 36 dots. Now here is the general expression for any number, n + (n - 1) + (n - 2) = total."
Another problem that students solved was the staircase problem (see Figure 5). This problem was different in mathematical structure from the previous problems; it was the first problem that dealt with a quadratic function. Previous problems had all been linear functions. Would students continue to use the schemata knowledge they had developed for solving problems? Would they continue to draw diagrams? Find answers to smaller problems? Use tables? Or would their plans and executions of their plans become more trial and error? They would not be expected to identify this first quadratic problem with the linear functions, and in fact, none of the students identified this type of problem with any they had previously solved. However, they did all begin by drawing diagrams of smaller problems.
In the staircase problem Zeke executed some diagrams of different heights of staircases and counted the number of squares (see Figure 6). In his elaborations on his diagrams he demonstrated that finding a pattern in the diagrams had become a part of his plan.
I started by trying to find a pattern by making a staircase that [was] 5 steps high. Then a staircase with 6 steps high, than 3 steps high. I couldn't find a pattern so I started to make [an] 18-step high staircase. I did it different that time. I started with 18 and worked backwards [see Figure 7]. When I did that I noticed a pattern that worked. It was N + (N - 1) + (N - 2) + (N - 3) + (N - 4)... + (N - N). So that number of blocks in a [sic] 18 step staircase is 171 blocks.
Figure 5. The staircase problem.Figure 6. Zeke 's diagrams for the staircase problem.Figure 7. Zeke's diagram for the 18-step high staircase.
During his elaboration he was able to identify this descending relationship in the staircase to identify and execute a generalization. To solve the problem Zeke was able to evaluate his plan of drawing and elaborating on smaller diagrams and change directions when his plan was not a productive one for him.
Josh, another student, executed a diagram (see Figure 8) similar to Zeke ' s, but he continued down from 18 steps to 1. Before he counted or sought to identify the patterns in the diagram, his plan was to use a formula he knew. He observed an overall triangular shape in his diagram and recognized that the number of squares would help him find the area of the triangle:
First, I tried to use the formula of finding the area of a triangle [b x h divided by 2]. When I used that formula for the staircase that's 18 blocks high I noticed that the formula was of [sic] by 9 blocks. [He then used the same diagram to elaborate on smaller step heights]. Then, I tried the same formula for the staircase that is 4 blocks high+was off by 2 blocks. Then, I studied the formula after adding the extra blocks on by changing the formula to (18 x 18) ÷ 2 + 9=171 blocks.I did the same to the 10 block staircase, by changing the formula to (4 x 4) ÷ 2 = 10 blocks. Then, instantly I noticed that the amount that I added on to the formula was 1/2 of the # of the highest stack of blocks. For example, 1 added 9 onto the 18 block high staircase + added 2 onto the 4 block high staircase. So I changed my formula to(18xl8)-2 + (18÷2) = 171[and] (4x4)÷2+(4÷2) = 10 blocks. So my expression for a staircase N blocks high is (N x N) ÷ 2 + (N ÷ 2).
Josh solved the problem by elaborating on how the number of steps in each height of staircase could give the correct numerical answer when he solved the formula for the area of a triangle with different sized staircases. In his explanation he even used such elaborating-type words as studied and noticed. When he completed his computations he identified a pattern and created his symbolic generalization.
Rebecca solved the problem by using the given diagram ( see Figure 5). She elaborated and explained how she executed her solution:
I noticed as I drew the diagram above that there are 4 along the longest bottom row, and 4 in the highest column, but I must remember not to [count] the shared block twice [the corner block], because there is [sic] only 7 blocks. Because it is set up this way, Fm lead [sic] to believe in an 18 step figure the blocks would go 18 along the bottom+ 18 up, but the total would be 35. But I know that isn't the total because it is j ust the bottom+top most blocks. Next I know the row in the 18 figure will progress as 18 blocks in a row, 17 in the next, then 16,15,14... 1, because the small figure goes 4, 3, 2, 1 in blocks.
Rebecca identified the corner block as shared and reminded herself of its importance. She then stated her plan: "Now, to figure out the rest of the # of blocks in the figure I will add 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, + 1 to 35." She continued to elaborate and stated how she identified the pattern,
I start with sixteen because I already added in 1 block from every row. I added and found that there were 171 blocks in the 18 step figure. To find the total for any figure you need to take N, the # of blocks in the bottom most row and add N - 1 to it, then N - 2 and keep going up until you add just 1. Then you have the total # of blocks.
Her generalization was the same as Zeke's but she had identified the patterns in a different way. Rather than separately considering each column of steps, she examined the perimeter blocks on the height and base of the staircase. Zeke, Josh, andRebecca all solved the problem by adding up the numbers of blocks in each column of their staircase. In other words, they built up the staircase. This "building up" plan became characteristic of the plans of several students when they solved the quadratic problems.
When students solved a similar problem in mathematical structurethe diagonals of a polygon problem-three of them (Mike, Jane, and Josh) evaluated the patterns and identified it with the staircase problem.
The diagonals of a polygon problem asked, "How many diagonals are in an 8-sided polygon? Write an expression to represent how many diagonals are in an N-sided polygon." (NCTM, 1997)
An eighth student's (Mike) work in the teaching experiment demonstrates this identification to the staircase problem and shows another example of "the building up" plan. To solve this problem Mike made diagrams of smaller problems. He did not actually state this plan; drawing figures of smaller problems seemed to be an implicit plan. He drew several polygons (see Figure 9), then made a table to help him look for patterns between the number of diagonals and the number of sides on a polygon (see Table 3). This execution knowledge also helped him elaborate on the data; he listed his data and also included columns that explained how the used he data to obtain the number of diagonals of a polygon. He found the pattern using identification knowledge and evaluated his generalization by solving for specific cases.
Figure 9. Mike's diagrams for the diagonals of the polygon problem (three-sided to seven-sided figures).Table 3Mike's Table for the Diagonals of the Polygon Problem
After executing a solution for polygons from a triangle to a hexagon, Mike then elaborated on how he had executed a plan for finding a generalized symbolic expression and clarified how his work made sense:
First I tried to find a pattern in the sides, diagonals, and diagonals per vertex that I didn't need to carry out to the target # [8 in this case]. The formula that I came up with was
(sides - 3) * (sides ÷ 2) [see column 3].
The way I found this was the (sides - 3) equals lines [diagonals] per vertex. The (sides ÷ 2) is the number of pairs that equal to the maximum number of lines [diagonals] you can draw from each vertex. In a [sic] octagon the sides minus three is five. Then you multiply it by 4 because there are 4 groups of 5 and you get twenty. That formula works for all polygons.
Mike drew diagonals from each vertex and identified the pattern of increasing numbers of diagonals coming from each vertex as the number of vertices increased. He then developed the idea of pairs of diagonals as he counted. In the table he added and divided to obtain the numbers in the diagonals column as a way to explain the process of drawing the diagonals and keeping count as he went along. He built up the diagonals as he went from each set of vertices and then used the table to help him complete the symbolic expression. Mike again used a building-up plan to find his generalization for another problem-the roads problem:
In the Asphalt Kingdom each city is connected to the other cities with a road of smooth blacktop. In order to make it simple for people to get around the Asphalt Kingdom there is a road connecting each city with all of the others. This happens for every city in the Asphalt Kingdom. For example, 10 years ago the Asphalt Kingdom only had 4 cities with 6 roads to connect them so there was a direct route from any city to any other city. Today the Asphalt Kingdom has grown to have 14 cities. How many roads does it have now? Write a general expression using N that the king, King Steamroller, can use to determine how many roads he will need to have for any given number of N cities. (D. Johanning, personal communication, May 9, 1999)
Mike evaluated the patterns and identified the relationship between the diagonals of the polygon problem and this new problem. Once again he elaborated on the problem by executing a diagram to build up the number of roads in this new scenario to seek to identify a pattern. When Mike solved the roads problem he remembered how he had solved the diagonals of a polygon problem and wrote, "First I remembered how to get a formula on the number of diagonals of any shape [the previous (s - 3) * (s ÷ 2)]." He then elaborated on the relationship: "Since the roads were connecting all of the cities the formula (N ÷ 2) x (N - 3) + N gives you the total number of roads."
He demonstrated execution knowledge by performing a calculation,
(N ÷ 2) x (N - 3) + N
7 x 11 + 14 = 91
He elaborated on how he developed this symbolic expression and seemed to make a case again for his actions:
I got this formula by taking the number of pairs that equal the number of sides (cities) and multiplying them by the number of diagonals take away the 3 from each vertex (n - 3) then you add the number of sides onto that because you already had 14 roads and you get the number of lines (roads). There are 91 roads in the asphalt kingdom.
He then explained how he had drawn figures for each number of cities until he got to the six-city diagram and then remembered this was how he had solved the diagonals of a polygon problem. He pointed to the six-city diagram that he had drawn (see Figure 10) and said,
I divided it in half because there's three pairs. Then you multiply it by N minus three, which is the number of diagonals that come out of each side. It's the number of diagonals out of each vertex. There's three coming out of here and there's six points. And you have to add the perimeter.
With this last excerpt he demonstrated that he identified a relationship between the diagonals of a polygon problem and the roads problem, but also saw how it was different because of the perimeter roads. Using elaboration knowledge Mike modified and adapted the symbolic expression he had created from the diagonals of a polygon problem, (n - 3) * (n ÷ 2), to meet the real life situation of already having one road to each city on the perimeter of the cities (six roads in this particular case). He elaborated on how the relationship of these diagrams compared to the diagrams he had drawn for the diagonals of the polygon problem and used the same building-up plan that had allowed him to generate his old equation by adding n for the perimeter roads.
Conclusions
This article has provided a discussion of ways in which students used writing to describe how and why they solved problems. Through this experience students learned to communicate their schemata knowledge by explaining and justifying their solutions. Explaining their approaches in writing helped students perceive mathematics as a meaningful sense-making activity. Their writing samples demonstrated that students were able to provide clear reasons for their choices of actions to find solutions. This opportunity to explain their thinking in writing helped them develop conceptual knowledge, knowledge that according to Hiebert and Lefevre (1986) is rich in relationships and connections linking relationships. This conceptual knowledge and sense making cannot be generated by procedures learned by rote.
Students also built understanding of procedural knowledge-knowledge that Hiebert and Lefevre stated is composed of the formal language or symbolic representation of mathematics. Most importantly, students developed connections between their conceptual and procedural knowledge by analyzing and recognizing patterns, representing and generalizing quantitative relationships in patterns, and generalizing quantitative relationships toward a formal symbol system.
By requiring students to write as much of their thinking as possible as part of the problem-solving process, they were helped to consolidate their schemata knowledge and give convincing reasons for their solution strategies. They learned to include the important steps, so that there were no gaps in reasoning. Driscoll (1999) stated that when students memorize in order to work out arithmetic and algebraic algorithms they are not likely to make connections between their computations and algebraic symbols. In contrast, in this study students created their own symbolic expressions that were connected to contexts of the diagrams they drew. By asking students to write about their thinking and to state their generalizations in their natural language, most were able to then write their generalizations symbolically. Through explaining in writing the generalizable patterns in relationships between the quantities in the problems, they made their algebraic thinking explicit. This explicitness helped them develop the schemata knowledge needed for solving similar algebraic problems.
Piaget (1952) believed that for schemata knowledge to develop students need repetition, recognition, and generalization. Instruction in this teaching experiment was adapted and sequenced in order to get at these three components. The repetition of related problems helped students recognize patterns (use identification knowledge) and compare types of problems with previous actions to solve them (use elaboration knowledge). The recognition of patterns helped them form plans (use planning knowledge) to execute solutions (use execution knowledge). Students in this study used all aspects of Marshall's (1995) model of schemata knowledge. Students did not necessarily use the different types of knowledge sequentially as Marshall found in her work with young children who were learning to solve arithmetic story problems. Students in the present study usually applied execution knowledge and then used identification knowledge to observe the patterns. Elaboration knowledge, however, appeared all along the sequence.
As in Marshall's (1995) work, visual representations became powerful aspects of elaboration knowledge and were intrinsically intertwined with identification knowledge. If students had previously solved related problems, they activated their identification knowledge first-it preceded other types of knowledge. Elaboration knowledge was most often used by students and seemed to be the most powerful in helping them solve problems and create their verbal and symbolic generalizations. Elaboration knowledge helped them modify their schemas to new related problems situations. By elaborating on their execution knowledge (e.g., diagrams, tables, solutions for specific cases) and identification knowledge (e.g., pattern finding) students evaluated their generalizations and created strong problem-solving strategies.
Marshall (1995) found in her research that planning knowledge was the most difficult to assess. She believed that planning knowledge was often difficult to separate from the other types of knowledge. The same results were found in this study. The students often did not explicitly state their plans (even though some did); often assumptions had to be made about their planning knowledge because of their particular actions. Because creating a plan to solve a problem is the most important part of the solution process, perhaps when writing their explanations the plan is so implicit that students do not actually identify that they have a plan, even though they provide evidence in their actions that they have a plan in mind.
Students' conceptual understanding was a result, not only of their writing, but also of the concrete algebraic problems related in mathematical structure. This design of instruction engaged students in ways that were different from thinking used in computing and solving equations. Within this instructional design, students created their own representations and invented algebraic notation from the patterns in variables that they had analyzed and created in the problems. This experience provided an opportunity for students to have ownership of difficult concepts. In this case the concepts were algebraic, but helping students invent their own notation can apply to other mathematics concepts, as well. Notation is a tool; tools need to be understood in the contexts in which they are used. Writing about related problems provided that context for these students. Because these students understood the contexts from which their notation came, they related their notation (tool) to how it was used.
Implications
This study supports both the use of writing and the use of structurally related concrete algebraic problems, especially problems that encourage the use of diagrams to help students develop the tools they need to think algebraically. Students' writing provided a powerful source to access their schemata knowledge. In addition, by using schema theory from cognitive psychology as a way of designing a sequence of instructional episodes, students become active learners. Students created their own detailed problem-solving schemata knowledge by constructing it for themselves, not by having the teacher first model the solutions. Questions for future research to extend this study might include the following: How might teachers help students develop schemata knowledge in other aspects of mathematics or other aspects of algebra? How might teachers use writing to access students' knowledge of other mathematics concepts? Carr and Biddlecomb (1998) believed that a primary problem in mathematics is students' failure to plan and evaluate. This teaching experiment was one successful approach that helped students develop the ability to perform both of these important aspects of solving problems. It is suggested that teachers consider using schema theory and writing in their instruction.
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Diana Steele
Northern Illinois University
Editors' Note: Correspondence concerning this article should be addressed to Diana Steele, Dept of Mathematical Science, Watson Hall 357, Northern Illinois University, DeKalb, IL 60115.
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Copyright School Science and Mathematics Association, Incorporated Mar 2005