Background Information and Conceptual Framework

Central to recent mathematics education reform efforts is increased attention to mathematical reasoning and sense making (NCTM, 2000). Reform documents describe school algebra as (1) representing and reasoning about problem situations, (2) transforming and operating on algebraic symbols, and (3) generalizing and justifying. The curriculum reforms set the stage for adjusting the content and changing the focus of traditional algebra courses and emphasize making algebra concepts accessible to elementary and middle school students. One of the big ideas in early algebra is inductive reasoning (Becker and Rivera, 2006). Christu and Papageorgiu (2007) characterized inductive reasoning as making generalizations (i.e. deriving new knowledge) from the finite incomplete number of initial cases. Haverty, Koedinger, Klahr, & Alibali, (2000) identified that inductive reasoning involves (1) data gathering, (2) pattern finding, and (3) hypothesis generation. Cañadas and Castro (2007) further delineated inductive reasoning processes, explaining that engaging in justifications is essential to inductive thinking and closely relates to hypothesis generation. Operational definition of inductive reasoning used in this study (Table 1) draws on Haverty's et al. and Cañadas and Castro's descriptions.

  <>Table 1. <>Processes characterizing inductive reasoning¹

Inductive Reasoning Process	Description	Operational Description
1.Data Gathering	Activities indicative of collecting, organizing and representing data.	Recognizable by the use of different strategies to collect or organize cases (e.g., tables, diagrams, lists)
2. Pattern Finding	Activities indicative of conducting investigations and analysis of collected data	Recognizable by one's ability to observe particular cases (organized or not) and to identify the next, unknown case
3.Conjecturing	Activities indicative of constructing, proposing and  validating hypothesis	Recognizable by hypothetical statements about the pattern based on one's analysis of regularities in discrete number of initial cases. Recognizable by arguments used to validate the truth of the conjecture

¹ based on Haverty et al., (2000) and Cañadas and Castro (2007)

Inductive reasoning plays significant role in the K-8 mathematics; facilitates problem solving, learning and the development of expertise (Haverty et al., 2000).Several studies (e.g., Rivera & Becker, 2007; Richardson, Berenson, & Staley, 2009) examined pre-service teachers' inductive reasoning in the context of analyzing figural or numerical patterns. Little attention, however, has been paid to pre-service teachers' inductive reasoning in problem-solving contexts. The goal of this research was to characterize pre-service teachers' inductive reasoning while solving contextually-based tasks examining:

(1)   Strategies pre-service teachers use to collect, organize and represent data

(2)   Pattern finding activities identified in pre-service teachers' problem solutions

(3)   Conjectures pre-service teachers formulate and justifications they provide

(4)   Inductive reasoning pathways that characterize successful and unsuccessful generalizations

Method

Participants were 21 pre-service teachers enrolled in the Problem Solving and Reasoning course; first in a 3-course mathematics sequence for K-8 teachers. The objective of the course was to deepen K-8 pre-service teachers' problem-solving ability, capacity for logical thought, appreciation and understanding of mathematics.

The data consisted of 105 solutions to five investigative tasks. The pre-service teachers had two weeks to complete each task and were explicitly asked to explain and justify mathematical claims they made. They were asked to carefully record their thought processes, questions they asked themselves, conjectures they made and the results of testing them. The goal for the pre-service teachers was to document progress of their thinking rather than to provide a final, clean, solution to each task. Each task facilitated inductive reasoning, fostered construction of new mathematical ideas, required thinking about possible algebraic formulas, and did not involve any rote manipulations. Figure 1includes an example of one investigation.

 

Investigation 1: 1. It's a hot summer day, and Eric the Sheep is at the end of a line of sheep waiting to be shorn. There are 50 sheep in front of him. Being an impatient sort of sheep, though, every time the shearer takes a sheep from the front of the line to be shorn, Eric sneaks up two places in line. How many sheep will be shorn before Eric? Find some way of predicting how many sheep will be shorn before Eric if there are 50 sheep in front of him. 2. Eric gets more and more impatient. Explore how many sheep will be shorn before Eric if Eric sneaks past 3 sheep at a time. How about 4 sheep at a time? 10 sheep at a time? 3.When someone tells you how many sheep there are in front of Eric and how many sheep at a time he can sneak past, describe how you could predict the answer. 4.What if Eric sneaks past 2 sheep first, and then the shearer takes a sheep from the front of the line? Does this change your rule? If so, how? Why? 5.The farmer hires another sheep shearer. There is still one line, but the 1st and 2nd sheep in line get shorn at the same time, then Eric sneaks ahead. Explore what this does to your rule. Explain your answer.

Figure 1. An example of investigative task used in the study (adapted from Driscoll, 1999)

The data were analyzed qualitatively using a priori and open coding. Guided by operational definition of inductive reasoning (Table 1) pre-service teachers' solutions were first analyzed to identify and code the three inductive reasoning processes. Open coding (Miles & Huberman, 1994) was used to provide further characterization of each process. Finally, patterns were sought in the identified characteristics of inductive processes across all problems and participants.

Results and Discussion

Given the limited space of this proposal only selected results are presented.

Multiple dimensions of inductive reasoning activities were identified.  Data gathering activities were characterized with respect to (a) initial experiences, (b) tools, and (c) organization (Figure 2).

 

Initial Experiences	Developing sense of problem situation	Reenacting (focus on end result)
		Sense seeking (focus on pattern finding)
Tools	Representing information	Physical representations Diagrams Tables Graphs
Organization	Organizing information	Tools used: Single or multiple Organization: Systematic or not

Figure 2. Data Gathering Activities

For example, initial data gathering experiences were characterized as reenacting or sense seeking. While sense seeking supported uncovering underlying patterns, reenacting focused on end-results, as revealed in pre-service teachers' reflections:

I started by using blocks to represent the sheep. I tried with 25 … but I … got confused and just ended up picking out 50 blocks and counting out the whole set, 2 at a time, in order to do the problem. I acted out and saw that Eric jumped 17 sheep before he was at the front of the line. Solving the problem with cubes I was able to see right away that 17 sheep will be shorn before Eric. However, only … [later on] I started looking for a pattern. It was definitely easier after I started at 1and went up than first starting with 50 and counting down. I actually had to go to only about 10 sheep to see my patterns. (PST 3)

Pattern finding activities were characterized as (a) connecting, and (b) extending (Figure 3). Both were associated with seeking an understanding of the relationship between cases within a given problem situation or the relationship between classes of problem situations.

 

Connecting	Focusing on connections between individual cases within a single problem situation	(a)     Seeking changing attributes among cases within the same problem (b)     Seeking invariant attributes among cases within the same problem situation
	Focusing on connections between classes of problem situations	(a)     (a) Seeking changing attributes between classes of problem situations (b) Seeking invariant attributes between classes of problem situations
Extending	Extending cases within a given problem situation	Thinking about cases within more general structure  than originally presented
	Extending classes of problem situations	Thinking about problem situation within more general structure than originally presented

Figure 3. Pattern Finding Activities

For example, connecting was identifiable through actions in which pre-service teachers explicitly pointed at similarity between individual cases or classes of problems. Connecting cases within the same problem situation is illustrated in PST 19's solution (Figure 4) and the accompanying explanation “for every 3 sheep in front of Eric, the number of sheep shorn [before Eric] increased by 1.”

Figure 4.  Connecting cases within the same problem situation (PST 19)

Local and global hypotheses (Figure 5) were identified. Local hypotheses were characterized as statements about cases within specific problem situation. Global hypotheses were characterized as statements about classes of problem situations. Both were associated with justifications categorized as (a) example-based, (b) relationship-based, (c) partial, or (d) procedural.

 

Local hypothesis	Hypothesizing about cases within single problem situation
	Validating hypotheses about cases	(a) example-based justification; testing the hypothesis with specific examples, establishing validity by trial and error   (b) relationship-based justification; focusing on the relationships between cases  (c)Partial justification; validating only selected elements of the hypothesis (e.g., partial validation of a rule)   (d) Procedural-justification; explaining the meaning of symbols used in description of general rule or restating the rule in words
Global hypothesis	Hypothesizing about classes of problems
	Validating hypotheses about classes of problem situations	(a) example-based justification; testing the hypothesis with specific examples, establishing validity by trial and error  (b) relationship-based justification; focusing on the relationships between cases  (c)Partial justification; validating only selected elements of the hypothesis (e.g., partial validation of a rule)  (d) Procedural-justification; explaining the meaning of symbols used in description of general rule or restating the rule in words

Figure 5. Conjecturing activity evident in analyzed solutions

For example, global hypothesis expressed an observed pattern in general terms as illustrated in PST 7's solution in reference to the rule  in “skip 2” class of problems: “For any number of sheep in front of Eric you always have to round up.”

The results characterize multiple dimensions of pre-service teachers' inductive-reasoning in problem-solving contexts. The data revealed pre-service teachers' overall limited ability (and perhaps inclination) to justify. At the same time the evidence suggests that justifications provide support for constructing more successful generalizations. To prepare pre-service teachers for challenges of early algebra instruction and provide them with tools for helping K-8 students reason about mathematics that arises from various problem situations teacher preparation programs need to focus on helping pre-service teachers develop ability to construct valid arguments and increasing their awareness of the role justification might play in constructing successful generalizations.   

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