25
Learning About Teaching for Understanding Through the Study of Tutoring Eric Gutstein DePaul University, 2320 N. Kenmore, Chicago, IL 60614, USA Nancy K. Mack National-Louis University, 200 S. Naperville Rd., Wheaton, IL 60187, USA In this study, we conducted a fine-grained analysis of an expert tutor’s (Nancy Mack) tutorial actions as she attempted, successfully, to help students learn fractions with understanding. Our analysis revealed that, as Mack tutored students in two different research studies, she took two types of tutorial actions previously unrecorded in the literature. By analyzing her actions using a methodology involving production rules, we suggest how her content knowledge, pedagogical content knowledge, and her knowledge of her students were interrelated and how they impacted on her instructional decisions and teaching actions. We also provide an example of how using production rules can be useful to discern some of the complexities involved in teaching and tutoring. Although teaching and tutoring are central to mathematics education, many of the intricacies involved in teaching and tutoring mathematics have eluded researchers over the years (Fennema & Franke, 1992; Koehler & Grouws, 1992). Consequently, mathe- matics educators have tended to describe teaching and tutoring generally as ‘‘complex processes’’ as it has not been easy to provide detailed descriptions focusing on critical factors. A number of researchers agree that teaching and tutoring are complex processes and suggest that understanding their complexities will provide insights into ways to improve and communicate about teaching (Davis, 1992; Koehler & Grouws, 1992). Ball (1993) further suggests that comprehending the complexities will provide insights into one of the most critical issues currently facing mathematics educators: How to teach mathe- matics for understanding. By understanding, we use the definition of Hiebert (1986): ‘‘The process of creating relationships between pieces of knowledge. Students understand something as they recognize how it relates to other things they already know’’ (p. 21). In recent years, researchers have approached these issues by closely examining teachers’ and tutors’ thought processes and knowledge (Clark & Peterson, 1986; Lampert, 1986, 1990; Schoenfeld et al., 1992; Ball, 1993). A growing number of studies 441 Direct all correspondence to: Eric Gutstein, DePaul University, 2320 N. Kenmore, Chicago, IL 60614, USA; E-Mail: [email protected] JOURNAL OF MATHEMATICAL BEHAVIOR, 17 (4), 441– 465 ISSN 0364-0213. Copyright C 1999 Elsevier Science Inc. All rights of reproduction in any form reserved. JMB

Learning about teaching for understanding through the study of tutoring

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Page 1: Learning about teaching for understanding through the study of tutoring

Learning About Teaching forUnderstanding Through the Study of

Tutoring

Eric Gutstein

DePaul University, 2320 N. Kenmore, Chicago, IL 60614, USA

Nancy K. Mack

National-Louis University, 200 S. Naperville Rd., Wheaton, IL 60187, USA

In this study, we conducted a fine-grained analysis of an expert tutor's (Nancy Mack) tutorial

actions as she attempted, successfully, to help students learn fractions with understanding. Our

analysis revealed that, as Mack tutored students in two different research studies, she took two

types of tutorial actions previously unrecorded in the literature. By analyzing her actions using

a methodology involving production rules, we suggest how her content knowledge, pedagogical

content knowledge, and her knowledge of her students were interrelated and how they

impacted on her instructional decisions and teaching actions. We also provide an example of

how using production rules can be useful to discern some of the complexities involved in

teaching and tutoring.

Although teaching and tutoring are central to mathematics education, many of the

intricacies involved in teaching and tutoring mathematics have eluded researchers over

the years (Fennema & Franke, 1992; Koehler & Grouws, 1992). Consequently, mathe-

matics educators have tended to describe teaching and tutoring generally as `̀ complex

processes'' as it has not been easy to provide detailed descriptions focusing on critical

factors. A number of researchers agree that teaching and tutoring are complex processes

and suggest that understanding their complexities will provide insights into ways to

improve and communicate about teaching (Davis, 1992; Koehler & Grouws, 1992). Ball

(1993) further suggests that comprehending the complexities will provide insights into one

of the most critical issues currently facing mathematics educators: How to teach mathe-

matics for understanding. By understanding, we use the definition of Hiebert (1986): `̀ The

process of creating relationships between pieces of knowledge. Students understand

something as they recognize how it relates to other things they already know'' (p. 21).

In recent years, researchers have approached these issues by closely examining

teachers' and tutors' thought processes and knowledge (Clark & Peterson, 1986;

Lampert, 1986, 1990; Schoenfeld et al., 1992; Ball, 1993). A growing number of studies

441

Direct all correspondence to: Eric Gutstein, DePaul University, 2320 N. Kenmore, Chicago, IL 60614, USA;

E-Mail: [email protected]

JOURNAL OF MATHEMATICAL BEHAVIOR, 17 (4), 441± 465 ISSN 0364-0213.Copyright C 1999 Elsevier Science Inc. All rights of reproduction in any form reserved.JMB

Page 2: Learning about teaching for understanding through the study of tutoring

suggest that rich descriptions of teaching and tutoring can emerge not only from

investigations that focus closely on teachers' and tutors' actions in general, but also by

focusing on them as they attempt to teach mathematics specifically for understanding

(Lampert, 1986, 1990; Schoenfeld et al., 1992; Ball, 1993; Fennema, Franke, Carpenter,

& Carey, 1993; Pirie & Kieren, 1994). In this study, we sought insights into teaching for

understanding by focusing on one tutor as she attempted to help students learn

mathematics with understanding in a complex content domain. Specifically, we focused

on Nancy Mack, one of the authors of this article, as she tutored third-, fourth-, and

sixth-grade students in two different studies focusing on students' learning of fractions

with understanding (Mack, 1990, 1995). We consider Mack to be an expert tutor in the

sense that she was successful in teaching for understanding Ð her studies provide

evidence that children were able to build on their informal mathematical knowledge and

construct meaning for symbols and procedures (Mack, 1990, 1995). [Informal knowl-

edge can be characterized generally as applied, real-life, circumstantial knowledge

constructed by the student that may be either correct or incorrect and can be drawn

upon by the student in response to problems posed in the context of real-world situations

familiar to him or her (Leinhardt, 1988).] While we appreciate that teaching and tutoring

are not the same, we believed that we could learn about teaching for understanding by

examining the complexities involved in Mack's tutoring.

Theories of teaching for understanding suggest that the development of mathematical

understanding is characterized by a dynamic, non-linear process of growth (Pirie &

Kieren, 1994), in which students come to understand ideas gradually over a period of time.

Researchers also suggest that the development of understanding is characterized by the

strength and number of connections an individual makes among ideas and between

different forms of representation (Hiebert & Carpenter, 1992). Thus, helping students

explicitly make connections is part of what teachers can do to help students develop

understanding (Hiebert et al., 1997). One way teachers do this is by carefully sequencing

sets of problems based on their content knowledge and knowledge of their students

(Hiebert & Carpenter, 1992; Hiebert et al., 1997). In this article, we examine Mack's

actions from these perspectives on teaching for understanding and look particularly closely

at how she sequenced problems.

The primary purpose of this article is to contribute to the developing theories of

teaching for understanding. We do this by describing in detail two aspects of what one

skilled tutor did in attempting to teach for understanding. Both of these aspects involve

Mack's selection of sequences of tasks. These particular actions have not been analyzed or

reported in the literature to this point. We also give a brief overview of her tutoring as a

whole in order to situate these specific aspects.

The first action, a digression, we define as a change from the mathematics topic that a

teacher and student(s) are discussing at a particular moment in time to a different topic,

with an eventual return to the initial topic. Digressions are sometimes referred to as

opportunistic teaching (or tutoring, e.g., Lewis, McArthur, Stasz, & Zmuidzinas, 1990),

but have not been discussed in the literature as a conscious action on the part of the

teacher to extend students' knowledge and conceptual understanding. For example, Lewis

et al. described two situations where digressions arose: (a) student errors or weaknesses

and (b) student suggestions or interests. However, they made no mention that tutors

digressed as an explicit means of strengthening students' understanding. The other action,

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exploring alternate-paths-not-taken, refers to situations in which Mack asked students

questions that required them to consider an alternate view that was in contradiction to

their answer, thinking, or reasoning. When Mack took these actions, it was always in a

situation after a student had answered a question correctly, and she wanted them to

consider a different view.

A secondary purpose of the article is to show how a particular methodology (one

using production rules, Newell & Simon, 1972) can be useful in analyzing and

understanding the complexities of teaching. We do this by showing how we used the

production rule formalism to gain insight into particular aspects of Mack's tutoring.

This present study was part of a larger project to develop a computer program to tutor

fractions (Gutstein, 1993). In order to develop the system, Gutstein undertook a

detailed study of Mack's tutoring using a methodology that involved creating

production rules. Gutstein was later joined in this effort by Mack herself in this

collaborative effort.

1. PRODUCTION SYSTEMS AND PRODUCTION RULES

A production system is a computer-based research tool used to investigate various

aspects of human and artificial intelligence. Researchers have used production systems to

model aspects of intelligence, for example, cognition (Anderson, 1983), problem solving

(Newell & Simon, 1972), and learning (Klahr, Langley, & Neches, 1987). These systems

have also been used to simulate the knowledge and decision making of experts in a

particular field (expert systems, Buchanan & Shortliffe, 1984). Production systems contain

production rules, which are written in the form of if±then statements.1 A production rule

will have one or more preconditions and an associated set of actions. If all the

preconditions of a rule are true at a given point in time, then the actions of the rule

may be taken. For example, the following rule is a possible way to represent how to add

equal-denominator fractions.

IF there is one arithmetic operation to perform,

AND the operation is ``+,''AND there are two numbers,

AND the first number is a fraction,

AND the second number is a fraction,

AND the denominators of the two fractions are equal.

THEN add the numerators of the two fractions,

AND create a new fraction as the answer,AND make the new numerator be the sum of the numerators,

AND make the new denominator the same as the first denominator.

The computer-based tutoring program Gutstein developed was a working production

system that improved its tutoring over time as it learned from the students who used the

system.2 As part of this research, Gutstein (1991) tried to formalize aspects of Mack's

tutoring knowledge using the artificial intelligence methodology of knowledge engineer-

LEARNING ABOUT TEACHING FOR UNDERSTANDING 443

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ing3 (Waterman, 1986) and developed production rules to approximate this knowledge.

This present research grows out of Gutstein's attempts to represent Mack's knowledge

using production rules. While having no illusions that one can capture the full complexity

of human teaching expertise in production rules (Lesh & Kelly, 1997), we show in this

article how production rules have the potential to effectively aid in providing detailed

descriptions of the complexities involved in the teaching/tutoring process as teachers and

tutors attempt to teach for understanding.

2. METHODOLOGY

2.1. Subject

Gutstein selected Mack as the subject of this study because she was an experienced

tutor whose research focused on her students' learning as she tutored them in a complex

mathematical content domain (fractions) (see Mack, 1990, 1995). Additionally, Mack

reported that as she tutored students, she attempted to help them learn fractions with

understanding (using Hiebert's definition) by building on their informal mathematical

knowledge. However, while Mack's research presented fine-grained analyses of students

learning fractions with understanding, it did not directly address the complexities involved

in her tutoring as she attempted to help students give meaning to fraction symbols and

procedures. Therefore, heeding the call of Romberg and Carpenter (1986) to integrate

research on student learning with research on teaching, we focused on analyzing Mack's

tutoring in two different research studies (Mack, 1990, 1995) in order to learn more about

teaching for understanding.

2.2. General Characteristics of Procedures Utilized in Mack's

Research Studies

Mack tutored eight sixth-grade students 11±13 times over a 6-week period in her first

study (1990). In her second study (1995), in which she used similar procedures, she

tutored four third-grade and three fourth-grade students twice a week for 3 weeks. She

combined clinical interviews with individual instruction in all sessions. Mack presented

most problems verbally and encouraged students to think aloud as they solved them. If

students failed to think aloud, she asked them to explain what they had been thinking as

they solved the problems. Mack provided concrete materials (fraction circles and strips)

for students to use and encouraged their use as long as students thought they were needed;

she also made paper and pencil available.

Mack focused instruction on critical mathematical ideas related to addition and

subtraction of fractions and on students' informal fraction knowledge. She drew on

students' informal knowledge and their responses to problems to design specific lessons.

Mack designed the lessons to be flexible with respect to the specific topics students

covered, the amount of time they spent on a topic, and the sequence in which they covered

topics. She also designed the lessons to allow for movement between problems she

presented verbally in the context of real-world situations and problems represented

symbolically (see Mack, 1990, 1995 for details).

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2.3. Data

Mack (1990, 1995) tutored 15 students in her studies. Each student had from six

to 13 sessions of 30 minutes each, over periods ranging from 3 to, 6 weeks, r for a total

of 131 sessions. Mack selected four students to represent variability with respect to their

ability to make connections between fraction symbols, procedures, and informal knowl-

edge. She selected two students (one from each study) who readily made connections

between fraction symbols, procedures, and informal knowledge and two students who

did not (also one from each study). She then provided Gutstein with their complete

transcripts, a total of 34 sessions.

Mack also provided a rational task analysis she constructed for addition and

subtraction of fractions that she used in her own research (see Appendix A). Mack

constructed this task analysis from her own understanding of fractions and her

research on children's learning of addition and subtraction of fractions. Her purpose

in constructing the task analysis was to identify critical mathematical ideas under-

lying addition and subtraction of fractions and possible connections between these

ideas. Additionally, Mack stated that she did not consider her task analysis to be

definitive, rather she considered it as a guide for possible sequencing of topics

during tutoring.

2.4. Data Analysis

Transforming Mack's transcripts into production rules and analyzing the rules for

emerging themes was an iterative and circular process. Gutstein (1991) initially selected a

random sample of the transcripts and created a production rule to represent each of

Mack's tutoring utterances or actions. Next, Gutstein showed the rules to Mack, and, with

her feedback, revised them. He continued to develop rules, show them to Mack, and alter

them based on their discussion. Over time, as the rules were developed and refined, we

looked for themes and patterns which we categorized into sets of tutoring actions (see

Table 1 below).

In creating the tutoring rules, Gutstein first designated the action part of the rule, i.e.,

he wrote down as specifically as possible what Mack appeared to be doing in a given

situation. He then focused on three questions to create the rule's preconditions: (a)

What was Mack doing? (b) When was Mack doing this? and (c) Why was Mack doing

this? Gutstein applied these three questions to each interaction and analyzed the

situations before and after Mack's action in terms of what appeared to be Mack's

assessment of the student's knowledge, the student's recent problem-solving history,

Mack's goals for the session (which she had written at the beginning of each

transcript), and her apparent immediate tutorial goal as it fit into her task analysis

(e.g., for a student to change an improper fraction to a mixed number). By attempting

to answer the first two of these questions Ð the what and when of each action Ð

Gutstein was able to specify the necessary preconditions to create a production rule for

each action Mack took. For the most part, the why of Mack's actions were high

inference questions, and we reserve that for Section 4 of this article. As Gutstein

examined instances in Mack's transcripts where she took similar actions, he altered the

preconditions of the newly created rules.

LEARNING ABOUT TEACHING FOR UNDERSTANDING 445

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After he initially created the tutoring rules, Gutstein gave them to Mack, met with

her five times (2 hours each time) over several months, and had multiple one-hour

phone conversations with her to discuss and refine the rules. We separately and

together examined the transcripts and therules. Mack explained her actions by

referring to her instructional goals, rational task analysis, knowledge about how

students learn fractions with understanding, and beliefs about learning and teaching.

Using stimulated recall, we had Mack discuss the situations preceding her actions,

and revised the preconditions accordingly. The process of having Mack explain her

tutoring in great detail so that we could approximate her tutoring knowledge using a

production rule representation led us to more deeply understand her actions and their

interconnections.

All discussions with Mack were audiotaped and documented by field notes. To validate

the rules, Gutstein interviewed individually three other mathematics educators (including

Mack's thesis advisor) to whom he showed transcript portions with their associated rules.

These discussions were audiotaped and transcribed verbatim, and Gutstein used these to

corroborate and modify the rules when necessary. Additionally, Mack applied the tutoring

rules to sample transcripts she had not yet analyzed to determine if the rules accurately

described interactions between herself and the students. In all cases, Mack found that the

tutoring rules provided valid descriptions.

As a simplified example, after one of Mack's students (Ted) had solved a problem in a

real-world context involving one minus four-fifths, Mack wrote 1 ÿ 4/5 on his paper and

asked him to solve the problem. Ted replied, `̀ It's three-fifths.'' Mack then turned over his

paper and asked him to mentally solve a problem she posed verbally involving eating four-

fifths of one whole pizza. In terms of what Mack was doing, she was removing the

symbolic representation and asking Ted a corresponding problem in a real-world context.

The when in this situation occurred after he was able to correctly solve the problem in a

real-world context, but unable to when it was represented symbolically. Mack's retro-

spective answer as to why she took this action came during the analysis phase of this work;

TABLE 1. Specific Elements of Mack's Tutoring (See Gutstein, 1993 for Details)

1. Introduced and emphasized conceptual knowledge before procedural knowledge;

2. Introduced and emphasized concrete representations before symbolic ones;

3. Helped students connect ideas across different representational systems;

4. Helped students connect ideas within the same representational system;

5. Promoted cognitive dissonance for discrepant answers in different representational systems;

6. Helped students to induce general concepts and principles;

7. Had students talk as least as much as she did;

8. Used every question for the dual purposes of assessment and instructional guidance;

9. Turned questions back to students for them to answer/solve for themselves;

10. Initiated discussions that departed from correct answers and not just incorrect ones;

11. Used analogy/comparisons to previous problems;

12. Infrequently modeled problem solving, gave explicit instructions, explained ideas, or corrected students'

answers;

13. Explored alternate-paths-not-taken;

14. Leaped forward to more advanced problems, then fell back to intermediary positions;

15. Digressed to related topics;

16. Decomposed problems into smaller and simpler ones, then reassembled them.

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she reported that she believed he could solve the problem if presented verbally in a real-

world context, and she wanted him to solve it represented symbolically in an appropriate

and meaningful way. (We define meaningful ways here as students' own ways that make

sense to them, through visualization, mental mathematics and number sense, using

manipulatives, and/or by using paper and pencil and inventing alternative representations

and methods, rather than by previously learned rote, symbolic procedures.) She also said

that she thought Ted was thinking of the `̀ 1'' as one-fifth rather than one whole (and was

subtracting one-fifth from four-fifths). Thus, one can represent this interaction by the

simplified tutorial rule `̀ If a problem is given in symbolic form and the student's response

is incorrect after having solved it in a real-world context, then re-give the problem as a

word problem within a real-world context.'' Later, we present much more detailed

production rule analyses of tutoring situations.

3. FINDINGS

3.1. An Overview of Mack's Tutoring

In this article, our focus is on two aspects of Mack's tutoring, digressions and exploring

alternate-paths-not-taken. In order to contextualize these facets within her tutoring

approach as a whole, we first summarize and present, without evidence, a brief overview

of her tutoring (see Gutstein, 1993 for details). Many of the specific aspects of Mack's

tutoring have been discussed elsewhere, both in her own reports and in those of other

researchers. For example, Mack (1990) describes how she helped children connect their

informal mathematical knowledge with their knowledge of symbolic procedures by

shifting representational systems as she gave them problems to solve. Brown (1993)

discusses how Mack related the influence of rote procedures students had previously

learned to their capability to build on their informal mathematical knowledge, and Hiebert

(1993) discusses how Mack decided what type of problem to give students based on their

responses. These reports provide valuable insight towards a comprehensive theory of

teaching for understanding, but they do not discuss the full range of tutoring actions that

Mack took in teaching for understanding. We attempt to briefly provide such an overview

in this section.

In our discussions (and in her published reports, 1990, 1995), Mack stated that she had

two goals for each tutoring session: (a) to comprehend how students learned to add and

subtract fractions with conceptual understanding, and (b) to ensure that students learned as

much as possible about addition and subtraction of fractions and the concepts underlying

these operations. She also related that she viewed teaching as a problem-solving process

that involved students and teachers as cooperative problem solvers drawing on both

students' prior conceptual knowledge and on teachers' knowledge of the content and of

their students' knowledge.

In her tutoring, Mack appeared to base her instructional decisions primarily on two

bodies of knowledge: her content knowledge (in the sense of Shulman, 1986, including

pedagogical content knowledge) and her knowledge of her students. She drew

constantly and interdependently on her instructional goals, multiple aspects of her

knowledge, and students' responses, thoughts, utterances, and beliefs to determine

tutorial actions and appropriate directions for instruction. Her tutorial strategies were

LEARNING ABOUT TEACHING FOR UNDERSTANDING 447

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informed by her task analysis, but each interaction with a student was determined

ultimately by her understanding of what the student did and did not know at the

particular moment. This demanded great flexibility on her part and was a knowledge

intensive task Ð it was `̀ teaching as problem solving'' (Carpenter, 1988) in a

complex domain.

Mack focused instruction on the development of students' conceptual understanding of

fraction symbols and procedures, used discourse to assess their knowledge and help them

learn, and guided them through sequences of topics towards increasingly more difficult

material. Each question and problem had the dual purpose of assessing what students

knew and helping them learn. Her continual assessment of her students' knowledge Ð its

interconnections in particular Ð allowed her to build on what they knew and assisted

them in developing conceptual understanding of fraction symbols and procedures. She

helped them create relationships between their knowledge components and build

connections into new, related realms.

Mack led students to extend concepts and ideas within and across representational

systems; when students had inconsistent answers in the different systems, she pushed

them to resolve the conflicts for themselves. She introduced concrete contexts before

symbols and helped students induce general principles as ways to help students

develop meaning for symbols and to develop and understand their own procedures.

When students made errors or had questions, rather than correct or answer them

explicitly or model solution strategies, Mack used analogy and comparisons to

students' previous work, decomposed and simplified problems, returned the questions

to the student with some hint, or asked another question or problem. Her conversations

with her students were about their answers, thoughts, beliefs, and explanations, and she

often used students' correct answers as departure points for extended discussions,

including exploring alternate paths students had not taken. Lastly, while she changed

topics when she was convinced a student knew an area well enough, she altered the

topic sequence by digressing to related topics and also skipped topics and then fell

back to intermediary ones as needed. All these allowed students opportunities to

discuss and reflect on their thinking and work. These tutorial actions, taken as a whole,

helped students learn with understanding (under Mack's operational definition) and

support existing theories of teaching for understanding (Hiebert & Carpenter, 1992).

That Mack was successful in having her students learn with understanding is

documented in her own research (1990, 1995). Table 1 contains 16 specific tutoring

actions that encapsulate Mack's tutoring and that represents one person's actions in

attempting to teach for understanding.

While the overview of Mack's tutoring may be useful in cataloging one tutor's

teaching for understanding in one `̀ package,'' it does not provide the type of analysis

needed to more fully understand what the actions are, and why and when a tutor might

choose to use them. Nor does it make explicit either the relation-ship of teachers'

decision making to their knowledge, nor the relationship of their content knowledge to

their pedagogical content knowledge and knowledge of students. For that, we closely

examine two of these actions. Our goal is to provide a detailed look at aspects of one

person's teaching for understanding, as well as to show how our use of a production rule

formalism to analyze Mack's tutoring provided us with insights about some of the

underlying issues involved in teaching for understanding.

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3.2. Digressing to Related Topics to Help Students Extend Knowledge

In Mack's two studies (1990, 1995), all the students came to instruction with informal

knowledge related to adding and subtracting like fractions without regrouping. For

example, they were able to solve problems like, `̀ If you have three-eighths of a pizza

and I give you two-eighths more of a pizza, how much pizza do you have?'' One of her

instructional goals was to help students extend their knowledge to increasingly complex

problem situations so they could solve in meaningful ways problems represented

symbolically involving unlike denominators and regrouping (e.g., 5 1/3 ÿ 1 1/2). To

achieve this, Mack used her rational task analysis to structure fraction topics around the

following sequence of increasingly complex addition and subtraction problems: (a) add

and subtract like (denominator) fractions without regrouping, (b) one minus a fraction less

than one, (c) one plus a fraction less than one, (d) a whole number greater than one minus a

fraction less than one, (e) subtraction of like mixed numerals, (f) addition of like mixed

numerals, (g) addition and subtraction of unlike fractions without regrouping, and (h)

addition and subtraction of unlike mixed numerals with regrouping.

Mack attempted to guide students in extending their knowledge by presenting them

with problems appropriate to what she believed they understood. As we illustrate below,

Mack drew not only on her knowledge of the content and the sequence of addition and

subtraction problems resulting from her task analysis, she also drew on students' responses

and her knowledge of students' thinking. Thus, while Mack led all students through

roughly the same sequence of addition and subtraction topics, the specific sequence of

fraction topics was non-linear and varied substantially by individual. This varied, non-

linear sequence of topics resulted from three related actions Mack frequently employed as

she attempted to help students extend their informal knowledge of fractions: digressing to

related topics; decomposing problems students found too challenging into one or more

simpler problems; and leaping ahead to more advanced topics and then falling back to

more intermediate ones to help students when needed. We focus on digressions in this

section. For us, a digression is a situation that can be characterized by the following:

1. The teacher (or tutor) has a specific instructional goal in mind for a student (e.g.,

learning a particular idea, concept, fact, procedure, or relationship) and her actions are

focused on achieving the goal.

2. At a particular point in time, the teacher shifts her instructional goal and her actions

reflect the shift in goal. That marks the beginning of the digression.

3. After a period of time that may last from a few seconds to several minutes, the teacher

resumes the pursuit of the original goal, often picking up where teacher and student(s)

left off. That marks the end of the digression.

We make no stipulation on the content of digressions in general, as they look different and

have various functions depending on the teacher (Lewis et al., 1990).

Mack led students to related topics via digressions to extend students' understanding of

various concepts as they encountered them in addition and subtraction problems. More

specifically, Mack sequenced topics based on the types of addition and subtraction

problems we describe above; however, when problems involved concepts such as

equivalent fractions, mixed numerals, and improper fractions, Mack often digressed to

LEARNING ABOUT TEACHING FOR UNDERSTANDING 449

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these topics. (We note that when Mack did digress to equivalent fractions, she did not

constrain students to just simplify fractions as one could interpret from the transcript

portion below.)

The following protocol from Todd's (a fourth-grade student) second session illustrates

how Mack used digressions. The text in parentheses is part of Mack's original transcript;

text in brackets is our commentary. During this session, Todd had struggled through

solving a real-world problem involving adding two-fourths and one-fourth before Mack

continued with the following problem.

Mack: You have seven-tenths of a sausage and pepperoni pizza and you eat two-tenths of

that sausage and pepperoni pizza for lunch, how much do you have left?

Todd: Okay, seven, five pieces left.

Mack: Okay, five pieces left, what's the fraction name for that?

Todd: Umm, ten, so, yeah, ten-fifths left of a pizza.

Mack: Almost.

Todd: One-fifth left.

Mack: (Covered paper and repeated problem.)

Todd: Five, five-tenths, because I ate five of the ten.

(Mack asked Todd to write the problem; he wrote 7/10 ÿ 2/5 vertically.)

Mack: Did you eat two-fifths of the pizza?

Todd: Two pieces. (Mack had Todd work with fraction strips. He put out seven of the

tenths pieces and put out two-tenths, thus, he realized that he needed 2/10.

Referring to the five, he continued): That's how much I had left. If I had seven

and I ate two, I had five left (wrote 5/10).

(Mack asked Todd if he could write the problem like he had written 2/4 + 1/4 = 3/4, which

he had written vertically on his paper, or write it horizontally like 2 + 1 = 3.)

Todd: (Wrote 10 ÿ 7 ÿ 2 = 5.) If I had ten pieces minus seven then I ate two, two, I

would have five.

[At this point, Mack initiates a digression to equivalent fractions.]

Mack: If you have five-tenths, what's another name for how much you have? Can you

think of another fraction name for how much you have?

Todd: Two-halves? Because, umm, ten is two fives, I have two-halves.

Mack: Okay, so if you had five of the ten, would you have two-halves?

Todd: No, I would have one-half.

[At this point, Mack ends the digression and goes back to addition of like-denominator

fractions less than one.]

Mack: You have three-eighths of a sausage and pepperoni pizza and I give you four-

eighths more of a sausage and pepperoni pizza . . .Todd: (Interrupting Mack) Seven-eighths of a pizza, 'cause four plus three is seven, and

I have eight pieces and all I have left is one more piece left.

As we tried to understand Mack's actions and their source, we posed the three questions

that guided much of our analysis: What was she doing (when she took digressions)?,

When did she take them?, and Why did she take them? (which we leave for Section 4).

The first question essentially asks for a definition of a digression. For us, that is

relatively straightforward. As we state above, a digression in a tutoring situation is a

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switch for a period of time from the current problem type to another topic that is related in

some way, with a return to the original problem type. However, it was when we examined

the when question, that our use of the production rule formalism gave us insight into the

fine-grained details. To design production rules, one tries to determine the precise

conditions under which an action can occur, since every precondition must be satisfied

for the rule's actions to occur. Together, the five preconditions below answer when Mack

initiated digressions.

The first precondition has two components: (a) that the answer the student gave for the

current problem `̀ lend itself'' to taking a digression, and (b) that such a digression be

`̀ reasonable'' for the particular student at the specific time, given the answer. In the above

problem, Todd answered five-tenths, which satisfied the first constraint that the answer be

a candidate for digressing to equivalent fractions. Whether or not a given response lends

itself to digressing to a related concept is purely a question of the mathematics itself and is

manifested in the tutor's content knowledge (i.e., her task analysis). However, whether the

digression is reasonable at the moment depends on the tutor's view of what the student

knows and understands at the particular time. Consider a hypothetical situation in which

Todd had solved an isomorphic problem with the answer of seven-tenths. Seven-tenths

does lend itself to digressing to equivalent fractions (= 14/20 = 21/30 = . . . ). However,

it probably would not have been reasonable for Todd at that point in his development,

given his apparent state of knowledge as exemplified by his difficulties in the protocol

and his earlier difficulty with the problem 2/4 + 1/4. Thus, in this precondition, one can

see Mack using both her content knowledge and her knowledge of her students in an

interrelated way.

The second precondition has to do with the choice of the particular digression topic

itself. Equivalent fractions were not the only possibility. One could conceivably ask for the

decimal equivalent of five-tenths, or for that matter, how many fractions equivalent to five-

tenths are there? Both are mathematically appropriate questions, as are many others in

certain circumstances, and encompass concepts related to the original one in an enlarged

task analysis. What constrained Mack's choice to equivalent fractions? The issue here is

whether the possible digression topics are `̀ close enough'' within the tutor's task analysis

to the original topic, again, given the particular student and the specific situation. In Todd's

case, equivalent fractions was reasonable, the other two topics we list were not. This

precondition too illustrates the interrelationship of content knowledge and knowledge

of students.

While it is necessary that potential digression topics be close enough to the original topic,

another constraint exists as well. The third condition is that the digression topic must not

already be so `̀ well known'' by the student that there would be no value in the digression.

For example, if a student had just solved a problem like 4 3/6 ÿ 1 3/4 and answered 2 3/4,

easily using an invented solution, finding equivalent fractions, and regrouping, there might

be little point in asking for another fraction name for the three-fourths part of the answer.

One would probably infer (if the student had not already demonstrated it on this or other

problems) that the student could produce answers to this question. From the tutor's point of

view, the digression might have little value. As a further example, for Todd, the question

`̀ what is ten minus five'' is also close enough within the task analysis to be a potential

digression question (meeting the second precondition), but one could probably assume that

for him, a fourth grader, that topic would be too `̀ well known.''

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The fourth precondition also focuses closely on Mack's knowledge of her students.

While tutoring, Mack would occasionally ask a student `̀ what's another name for . . . ,''

she and the student would digress, and at the return to the original topic, the student would

be lost. Taking a digression entails making a reasoned decision that the student will have

an overall gain in her knowledge and will not be set back by the side trip. This is based on

the assessment that the student's knowledge of the original topic is `̀ sufficiently well

known'' so that she can withstand some side explorations and come back without having

to regain too much lost ground. In Todd's case, that seemed to be valid, since he solved the

next problem with little difficulty.

The final precondition is that the student's answer to the problem must be right.

Mack only initiated digressions when students answered problems correctly. We note

that this is clearly different from the discussion of Lewis et al. (1990) of opportunistic

tutoring which described tutor-initiated digressions for the purpose of addressing student

errors or weaknesses.

These five conditions: (a) that the student's response `̀ lend itself'' to taking a

digression, for the particular student; (b) that possible digression topics be `̀ close enough''

in the task analysis to the original topic (again, for the specific student); (c) that the

possible digression topics not be so `̀ well known'' by the student that taking a digression

would have little value; (d) that the original topic be `̀ sufficiently well known'' by the

student to allow for the digression to take place without causing more pain than gain; and

(e) that the student's answer to the current problem must be correct Ð taken together

constitute the if part of the production rule we developed to represent Mack's decision

making when initiating a digression. All conditions must have occurred at the same time

for Mack to have initiated a digression in this or a similar situation. The then part of the

rule is for Mack to choose an appropriate digression topic, generate a specific problem,

and initiate the digression.

Our use of production rules to analyze the when of Mack's digressions helped clarify

some of the details of how her content knowledge and her knowledge of her students were

related. All but the last precondition of the digression production rule rely on Mack's

knowledge of her students while the first two depend on her content knowledge as well.

Thus, one can see how both types of knowledge are interrelated and necessary to help

explain her actions, and neither is sufficient by itself.

Mack initiated digressions to related topics with all the students in her two studies in a

manner similar to the way she interacted with Todd; she provided students with problems

appropriate to what they understood and attempted to extend their understanding. More

specifically, Mack presented students with fraction addition and subtraction problems, and

when new or related concepts were embedded in the problems, Mack led students to

explore these concepts and then return to the problem from which they digressed. These

actions usually proved to be effective in helping students extend their knowledge to give

meaning to more complex problems and problems represented symbolically. On the

average, Mack initiated one to two digressions a session with each of her students.

There were times when her digressions were apparently inappropriate because

one of the five preconditions was violated. (It was from these type of inappropriate

tutorial actions, among others, that the self-improving tutoring system of Gutstein,

1993, learned.) For example, in the following protocol, the fourth precondition was not

met, that the student's knowledge of the original topic was sufficiently well known to

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withstand the digression. Mack thought that the student's knowledge of the original

topic was stronger than it actually was. Thus, when they returned to the original topic,

he was lost.

[This is from Tony's (a sixth grader) fourth tutoring session (Mack, 1987). Tony had

just spent a considerable amount of time and had finally solved 4/5 + 3/10, as a word

problem, with a large amount of difficulty. Mack had him write down that he had

exchanged 4/5 for 8/10. Tony then wrote 8/10 + 3/10 = 11/10.]

Mack: What's another name for 11/10? [initiates the digression to change an improper

fraction to a mixed number]

Tony: Umm (pause) 3/4.

Mack: No, let's think about it. Is 11/10 more than one or less than one?

Tony: Less, more than one, well because 10/10 is one, and we added one more, so that's

11/10.

(Mack had Tony put out one unit strip and then get out the tenths strips and place as many

tenths strips next to the unit as needed to see that ten-tenths was the same as one whole.

She then had him take out as many more as he needed to make eleven-tenths and then

asked him what he had with the unit strip and the extra tenth. He told her `̀ 1 1/10,'' but

then continued.)

Tony: But how did we get 4 and 3, 4/5 with the 3/10 to be 11/10? [clearly still confused]

3.3. Exploring Alternate-paths-not-taken to Extend Students'

Knowledge

The other aspect of Mack's tutoring that we discuss is what we call exploring alternate-

paths-not-taken. This refers to situations in which Mack asked students questions that

required them to consider an alternate view that contradicted their answer, thinking, or

reasoning. The following transcript portion shows several examples of Mack exploring

these alternatives with a student. As before, our comments are in brackets, and text in

parentheses is part of Mack's original transcript. This transcript portion is from Tony's

sixth session.

[Start of the lesson.]

(Mack put out unit strips and sixths for Tony to work with. Tony showed six-sixths. Mack

then put out one-sixth more at the end of six-sixths.)

Tony: It's six-sevenths.

Mack: Six-sevenths.

Tony: Wait, no, seven-sixths.

Mack: Why seven-sixths?

Tony: 'Cause there's seven of sixths.

(Mack related the problem situation to the first session when Tony said that four-fourths

and one-fourth more was five-fifths.)

Tony: I said five-fourths didn't I?

Mack: No, you didn't, you said five-fifths. Now how come today this is five-fourths and

this is seven-sixths instead of being five-fifths and seven-sevenths? [exploring

alternate-paths-not-taken]

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Tony: Because I used to not know what that was (wholes).

Mack: What what was?

Tony: This and this was (pieces and unit). Now I learned what they are, now what the

real names for them are . . . Now I know equivalent fractions, I know how to get

the denominator the same, and I know how to add fractions, and I know other

names for the equivalent fractions, and I know the names of these (the pieces).

(Mack then gave the problem: `̀ If you have a board that's three feet long and you cut off

two-fifths of a foot, how much do you have left?'')

Tony: What's two-fifths? I've never had that fraction before.

Mack: Well, what do you think two-fifths would look like? If we're talking about fifths?

Tony: It's five of it here, five of it there (indicating fraction strips, Mack answered

`̀ yes''), oh umm, (pause) two (pause) two and three-fifths.

Mack: Very good, how'd you figure that out?

Tony: Well, okay, we want five and we want, and we know, okay, take those two away

(points out two whole strips) and we still have two left, have two feet, now I've

gotta figure out the fraction part of it. So if I take two-fifths away from five-fifths,

I'll just have two take away, I mean five, five take away two, and then I came out

with three-fifths.

(Mack suggested Tony work problem on paper; Tony wrote 3ÿ2/5. Mack asked Tony

how he could show his work on paper. Tony said he'd bring his manipulatives into class if

his teacher would let him. Mack had Tony rework through the problem with

manipulatives, recording in symbols what he did at each step, and asked him if he

could make one whole with five-fifths. Tony found fifths (first tried eighths) and traded

one in for five-fifths, and had two units and the five-fifths. Mack then asked him `̀ What's

another name for three?'' and Tony responded `̀ Two and five-fifths.'' He then rewrote the

problem as 2 5/5 ÿ 2/5 = 2 3/5.)

Mack: How come you didn't subtract five minus five? The first day you told me you

subtract those denominators. [exploring alternate-paths-not-taken]

Tony: I don't know. I just sort of looked at it from here. I never figured this out in

my head.

(Mack suggested he figure it out. Tony thought the denominator was the top number;

Mack explained what a mixed numeral was.)

Mack: Now why didn't we take the five minus five? [continuing to explore alter-

nate-paths-not-taken]

Tony: 'Cause you can't, it's, you'd be left with zero. You can't have a denominator

of zero.

Mack: Could I add five plus five and get ten? [i.e., can you add denominators? This is a

further continuation of the exploration.]

Tony: Yeah, I suppose you could because you could reduce it, or umm, estimate it.

Mack: Well suppose I have five-fifths, and I was adding two-fifths to it, what would

I have?

Tony: Seven-fifths.

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Mack: How come I don't have seven-tenths? [further exploration . . .]Tony: 'Cause you don't add the two denominators.

Mack: Why?

Tony: 'Cause it's a rule . . . it's sort of like adding ones and tens, you can't do that

(referring to place value notation).

(Mack suggested looking at the pieces to figure it out Ð in the board problem, he had put

five-fifths together with two-fifths. Mack had Tony write the problem. Mack said Tony

had previously said that he could get seven-tenths [i.e., he had earlier thought one added

both numerator and denominator when adding fractions, thus, this is a further exploration],

but she didn't see seven-tenths.)

Tony: You can't get seven-tenths. (pause) Isn't it sort of like you said, you can't add tens

and ones?

(Mack told Tony to look at pieces and the size of the pieces.)

Tony: They're fifths, that means umm, umm, you'd have to cut these in half

each time.

Mack: Then would it be seven-tenths?

Tony: Yeah.

Mack: It would be seven-tenths if I cut each one in half?

Tony: Uh huh.

(Mack suggested he use tenths to see.) [He changed 2/5 and 5/5 to tenths.]

Mack: Okay, now do I have seven-tenths?

Tony: In this? No you can't. There's not seven pieces, there's ten-tenths, that's one.

Mack: Are you sure there's ten-tenths?

Tony: (Counted to eight, then to six) One . . . eight, one . . . six. There's fourteen.

(Mack stressed he was not changing the size of the piece, if he did, he would be finding

equivalent fractions.)

Mack: Two-tenths is the same as one-fifth.

Tony: And seven-fifths is the same as fourteen-tenths (pause), oh yeah! cause seven plus

seven is fourteen and five plus five is ten.

Mack: Okay.

Tony: Now I know that I know this stuff.

[Mack then gave him a subtraction problem and the session continued.]

In this protocol, Mack first explored an alternate-path-not-taken when Tony correctly

said `̀ seven-sixths'' after Mack placed that amount in fraction strips on the table. Mack

asked him, `̀ Now how come today this is five-fourths and this is seven-sixths instead of

being five-fifths and seven-sevenths?'' The second instance is the extended conversation

that started with Tony correctly solving the (word) problem, 3 ÿ 2/5. Mack asked him,

`̀ How come you didn't subtract five minus five? The first day you told me you subtract

those denominators?'' After a brief discussion, she returned with the question, `̀ Now why

didn't we take the five minus five?,'' to which Tony responded, `̀ You can't have a

denominator of zero.'' Tony's response left open whether he believed that one could

subtract denominators if the result were non-zero, or how one treated the denominators

when adding fractions. Mack investigated that by asking, `̀ Could I add five plus five and

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get ten?,'' in reference to adding denominators. His response appeared unclear, and she

asked directly, `̀ Well suppose I have five-fifths, and I was adding two-fifths to it, what

would I have?'' When Tony answered correctly, she asked, `̀ How come I don't have

seven-tenths.'' He answered that you couldn't add denominators, she asked why, and he

told her it was a rule. From that point, Mack continued the exploration, asking him to use

manipulatives, counterposing his answer in symbols against his answer with fraction

strips, and discussing notions of equivalence.

As with digressions above, we first examine the what of these explorations. These are

tutor-initiated questions that cause students to consider answers that contradict their

current ones. These alternative answers were sometimes students' previous, incorrect

answers (as in the first example above) or sometimes answers reflective of possible

misconceptions that the particular student might have (e.g., adding denominators when

adding fractions). Sometimes they related to an incorrect conception of, or reasoning

about, a topic different from the one with which the student was grappling, but that was

related to the current topic. Rather than have students explain their correct answers Ð

which Mack almost always did Ð these explorations challenged students to argue

why an alternative path was incorrect. In all cases, the explorations were in the form of

`̀ How come it's not . . . ,'' if not in those exact words, and they always represented

incorrect reasoning, explanations, or answers. Some of the explorations were short

interchanges, and some were extended; this varied depending on the student's response

to the questions.

In considering the when of these explorations, we found that four preconditions

comprise the if portion of a production rule that approximates Mack's actions. The first

precondition is that the answer the student gives to the current problem must be correct.

Mack only initiated explorations and counterposed incorrect or inappropriate reasoning,

explanations, or questions to her students' answers when they were right.

The second precondition is that the particular concept that the tutor and student are

discussing must be a significant mathematical idea. The most common topic of Mack's

explorations concerned ideas of what one does with denominators (and why) when adding

and subtracting fractions. She never initiated exploring an alternate-path-not-taken when

the topic was one that was peripheral to the main content areas of her task analysis. For

example, Mack did not initiated explorations when students correctly solved a problem but

had made earlier computation or careless errors on the same problem type. Thus, implicit

in Mack's choice of situations in which to initiate explorations was a level of importance

she attached to the various conceptual units involved in learning to add and subtract

fractions; that is, her content knowledge played an important role in her decisions.

The third precondition is directly connected to Mack's assessment of her students'

knowledge. Every time Mack initiated an exploration, it was not only when the student

had solved a problem correctly, but it was also when she assessed that the student's grasp

of the particular concept was relatively weak. In Tony's case above, both of her

explorations departed from points where he solved the problems correctly, but where

Mack assessed that his knowledge was somewhat fragile. She knew this because he

had previously solved both problems incorrectly and also from working closely with

him. The transcript provides evidence that his knowledge of why one does not add (or

subtract) denominators when adding (or subtracting) fractions was weak and rule-based

at best.

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Precondition four is somewhat different in that there are two possibilities for the

precondition. (In an actual production system, these two possibilities would be

represented in two different rules.) The first case is when Mack initiated the exploration

based on a student's previous, incorrect answer. The precondition is that a student's

previous, incorrect answer must represent a misconception contradictory to the current

one. For example, when Tony correctly stated that there were `̀ seven-sixths'' (of

fraction strips) showing, Mack counterposed that against his earlier, erroneous answer

of `̀ five-fifths'' when there were five-fourths showing. Here, the significant mathema-

tical idea is that fractions represent a quantity with respect to a specific unit; Tony's

earlier answer of `̀ five-fifths'' was evidence that he did not initially understand that the

unit was represented by only the four-fourths. Rather, he appeared to think of the unit as

however many pieces were on the table. That misconception was contradictory to the

correct understanding, and Mack took the opportunity to place the two conceptions in

opposition to each other. Her focus on relatively recent, previous misconceptions

suggests further her concern with, and attention to, the fragility and robustness of

students' knowledge.

The second case for this precondition occurred when Mack initiated the exploration

even though the student had not previously answered the same problem (or problem

type) incorrectly. In these situations, Mack explored alternate paths touching on

significant mathematical ideas that were related to or part of the current concept. In

Todd's second session, Mack gave him the (word) problem of 2/4 + 1/4, which he

eventually solved correctly. Mack asked him to write it, which he said he did not

know how to do. Mack then gave him a word problem involving 2 + 1, which he

wrote vertically as a sum, then she asked him again to write 2/4 + 1/4. He finally

wrote it correctly and vertically. Mack asked him, referring to the numerator in his

answer of 3/4, `̀ Now how did you get the 3?'' He answered, `̀ Because 2 + 1 is 3.''

Mack asked him, `̀ Now how come you have a 4 down here (referring to the

denominator in his answer)? How come since you went 2 + 1 is 3, you didn't go

4 + 4?'' Todd had not previously added denominators when adding fractions, nor

indicated he might believe that, yet Mack still initiated this exploration. Thus, the

precondition may be stated that the student's answer must lend itself to an exploration

of an important mathematical idea that is connected to the current concept. The

connection of the alternate path concept to the current concept is a relationship within

Mack's task analysis, and the decision to explore an alternate path in this situation is

an indication of her sense of the student's knowledge.

These four conditions: (a) that the current answer be correct, (b) that the current

concept be a significant mathematical idea, (c) that the student's grasp of the current topic

be relatively weak, and (d) that either a previous, incorrect answer represent a

misconception contradictory to the current one or the current answer lend itself to an

exploration of an important idea connected to the current concept Ð make up the

preconditions for the production rule we developed to represent Mack's actions when

exploring alternate-paths-not-taken. All conditions must have occurred at the same time

for Mack to have initiated an exploration in this or a similar situation. The then part of the

rule is for Mack to choose an appropriate topic to explore, generate a specific question,

and initiate the exploration. The preconditions here also demonstrate the connections and

interrelations of content knowledge and knowledge of students' conceptions.

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Mack explored alternate-paths-not-taken with all the students in her two studies in a

manner similar to the way she interacted with Tony. She used their correct answers as

departure points for explorations that challenged students to justify why their answers

were correct and to explain why the ideas Mack posed were not. These actions usually

proved to be effective in helping students effectively communicate their rationale and

broaden their knowledge.

4. DISCUSSION

In this study, we present a fine-grained analysis of two specific aspects of an expert tutor's

actions as she taught fractions for understanding. Our focus has been on clarifying the

what and when of specific actions involved in such teaching. In that sense, this article

contributes to the developing theories of teaching for understanding (Hiebert & Carpenter,

1992; Hiebert et al., 1997).

By developing and analyzing the production rules to represent Mack's actions, we

gained some understanding of how her content knowledge, pedagogical content knowl-

edge, and knowledge of students were interrelated. Shulman (1986) defines pedagogical

content knowledge as `̀ . . . the understanding of how particular topics, principles,

strategies, and the like in specific subject areas are comprehended or typically miscon-

strued, are learned and likely to be forgotten'' (p. 26). Mack's pedagogical content

knowledge is indicated, among other ways, by her specific tutoring actions. For example,

that she builds on students' informal mathematical knowledge reflects her knowledge of

how children develop meaning for symbolic procedures Ð an aspect of pedagogical

content knowledge. Each production rule's preconditions also embody Mack's knowledge.

As we point out in our findings, her content knowledge and knowledge of her students are

involved in each precondition, sometimes singly and other times in concert. Taken as a

whole, each rule embodies all three types of knowledge in an interdependent way that

suggests how these interrelationships played a critical role in guiding her instructional

decisions and tutorial actions (Brown, 1993). These findings support the theory of tutoring

of Schoenfeld et al. (1992), which suggests that tutorial actions depend on a complex set of

interrelationships between the tutor's instructional goals, content knowledge, pedagogical

content knowledge, and knowledge of students' thinking with respect to the content

domain. They also support the contention of Shulman (1986) that teachers' content

knowledge, pedagogical content knowledge, and knowledge of students' thinking all play

a critical role in teaching for understanding.

These findings have implications for in- and pre-service teacher education. A

substantial body of literature suggests that many teachers' knowledge is weak in at least

one of the areas Shulman (1986) discusses (Fennema & Franke, 1992). Therefore, it is not

clear if or how teachers whose knowledge is weak in any one area may consistently

generate appropriate questions to promote the development of students' understanding. If

teachers (in- and pre-service) cannot generate appropriate questions on their own, might

they be helped by engaging in experiences that guide them in examining sequences of

problems, identifying how one problem varies from another, and identifying when and

what types of digressions could be taken to related concepts? Might helping them analyze

the what, when, and why of specific teaching actions, including their own, assist them?

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Could these types of experiences not only help teachers determine appropriate questions in

complex content domains, but also simultaneously help them strengthen their content

knowledge, pedagogical content knowledge, and knowledge of students' thinking?

Investigations into these issues are needed to gain insights into ways to help teachers in

their efforts to teach for understanding (Borko et al., 1992).

The analytic framework we used in this study suggests that production rules can be

an effective way to gain insights into the tutoring process and gives an example to the

mathematics education community of how this methodology may be useful in under-

standing some of the intricacies of teaching. The process of designing production rules

provides a way to analyze in depth the specific conditions under which particular

actions can occur. Furthermore, specifying preconditions helps make `̀ the tacit

explicit'' and may yield a picture of the interconnections between a tutor's actions,

content knowledge, pedagogical content knowledge, knowledge of students' thinking,

and ultimately, epistemology. This is the case even if one is not concerned with the

formal creation of production rules, nor interested in computer simulations. In fact, as

an unexpected outcome of this study, Mack used the production rule framework to

examine and better understand her own tutoring. This suggests that this analytic

framework may be accessible to others to help them deepen their own understanding

of the teaching/tutoring process. Accessibility of analytic frameworks such as this are

particularly important at this time. A number of researchers (Ball, 1993; Fennema et

al., 1993; Lampert, 1990) suggest that gaining insights into one's own teaching plays

a critical role in struggling with issues related to teaching for understanding. Thus,

production rule analyses may prove valuable to teachers, tutors, and mathematics

educators as they attempt to find effective ways to understand and actualize teaching

mathematics for understanding.

This brings us to the issues of Mack's instructional goals and pedagogical philosophy,

although the discussion here is preliminary. Throughout our work, we were interested in

why Mack took the actions that she did. On the surface, her digressions interrupted the

sequence of fraction topics in her task analysis, seemed disjointed and out of place,

occasionally confused students, were highly unusual tutoring techniques, and had little

apparent purpose. Why did she use them and why so consistently? From inspecting the

topic before and after her digressions (which were always either identical or almost so),

and from examining the preconditions for digressions, we believe that the why was that

Mack was explicitly trying to help students make connections between different mathe-

matical concepts. We have identified four, non-mutually exclusive reasons for why Mack

wanted students to make these connections: (a) to help students strengthen a related piece

of knowledge, (b) to provide a conceptual basis for a future topic that they would

encounter, (c) to help students make explicit some knowledge they had used in solving

problems, and (d) to help students solve the particular problem on which they were

working. Of these, the last was the least frequent, and according to Mack, the least

important. In the protocol in the digression section of the article, the main purpose of the

digression was to help Todd strengthen a related piece of knowledge by helping him make

a connection between his original problem (subtracting like-denominator fractions where

both were less than one) and finding equivalent fractions. Thus, we can generalize about

Mack's digressions to say that their primary purpose was to help students extend and

broaden their understanding. This broadening seemed to help lay the groundwork for

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students to subsequently deepen their understanding of the digressions topics as well; later

transcripts provide evidence of Todd's ability to use equivalent fractions in real-world

contexts, to change improper fractions to mixed numbers, and to solve subtraction word

problems using fractions.

We also considered the why of Mack's explorations of alternate-paths-not-taken. We

have identified four reasons why Mack took these actions: (a) to cause students to reflect

on the discrepancies between their current answer and one they had previously given so

that they might themselves reconcile the differences;4 (b) to help lay the conceptual

groundwork for a future, related conversation, (c) to help students make explicit some

knowledge by themselves explaining why a conception or reasoning alternate to their own

was inappropriate or incorrect, and (d) to strengthen students' confidence and ability to

think critically by challenging them with the negation of the ideas that they had expressed.

All four reasons are in evidence in the transcript in the exploring alternate-paths-not-taken

section. Thus, similar to her reasons for taking digressions, her primary purpose in taking

explorations was to help students extend or broaden their understanding.

The frequency of digressions and exploring alternate-paths-not-taken suggests that

Mack thought it important to students' learning that they understood the interrelation-

ships between different fraction topics. Since the transcripts show that Mack explicitly

and regularly helped students make connections, we infer that this was a central aspect

of her perspective on how students construct knowledge. [We believe that teaching and

tutoring actions are guided by underlying belief systems and philosophies of knowl-

edge (Davis, 1992).] This inference was substantiated by Gutstein's interviews of

Mack, as well as by her writings in which she explains her view of understanding by

referring to Hiebert and Carpenter's (1992) characterization of understanding as

involving the process of creating relationships between pieces of knowledgeÐrelating

new knowledge to things already known.

Although Mack's actions were effective with students in individualized instructional

settings, it may be possible, and feasible, in a whole class or group setting to utilize

instructional actions such as digressing to related concepts and exploring alternate-

paths-not-taken. Investigations are needed to see how and if the specific aspects of

teaching for understanding in individualized settings that we discuss in this article can

aid the development of students' understanding in whole class and group settings.

As a final note, we wish to broaden the particular perspective we present here to say

that we believe teaching for understanding, as well as learning with understanding,

involves more than just cognition. Teaching is a sociocultural process that not only

entails cognition, but also complex beliefs about knowledge construction, pedagogy,

assessment, and content, as well as about culture.5 Thus, a fuller theoretical under-

standing of mathematics teaching needs to combine cognitive perspectives with

investigations of sociocultural issues (Cobb, 1994; Gutstein, Lipman, Hernandez, &

de los Reyes, 1997). Although this study does not explicitly address these issues, we

hope that findings such as ours can contribute to theories of teaching for understanding,

with the caveat that a cognitive perspective can only provide part of the answers.

Furthermore, the research focus needs to be broadened for equity concerns as well, to

help ensure that all children learn mathematics (Ginsburg, 1988; Secada, 1991). If we

seriously accept the notion that the strength of our understanding is characterized by our

grasp of multiple representations and viewpoints, and of the interconnectedness of our

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knowledge (Hiebert & Carpenter, 1992), then it seems essential that mathematics

educators view studies like the present one in conjunction with research into the

sociocultural context of learning as a way to achieve a more general and deeper

understanding of successful teaching for all students.

The significance of this study is that it provides a detailed window into some

aspects of the complex nature of tutoring and may help us learn more about teaching

(and tutoring) for understanding. As mathematics educators consider investigations such

as this alongside of and complementary to studies that span the areas of culture and

cognition (e.g., Lave, 1988; Saxe, 1991), or that specifically deal with the social context

of learning from an equity perspective (e.g., Ladson-Billings, 1994), what we learn about

teaching for understanding may contribute to our knowledge of successful teaching

in general.

Acknowledgment: An earlier version of this paper was presented at the Annual

Meeting of the American Educational Research Association, San Francisco, CA, in

April 1995. The order of authorship is alphabetical; the authors view this paper as a

collaborative effort to which they both contributed in different ways. The authors would

like to express their thanks to Thomas Carpenter and Ellen Ansell for comments on

earlier versions of this article. This article is based on portions of Gutstein's doctoral

dissertation completed at the University of Wisconsin-Madison under the direction of

Jude W. Shavlik.

NOTES

1. Production rules represent most of the knowledge embodied in a production system.

Particularly for this paper, we distinguish between production rules, which, at a minimum, can

approximate specific condition±action pairs, and a production system, which represents a more

comprehensive entity including the production rules, memory issues (search, index, retrieval), and

control mechanisms to choose the appropriate rule(s) to use in a given situation. In this paper, we

focus on production rules.

2. The `̀ students'' who used the system were actually computer simulations of human students.

The system learned using machine learning techniques (Michalski, Carbonell, & Mitchell, 1983), a

branch of artificial intelligence research. The purpose of the research was to see if a computer

tutoring program could learn in ways that a human tutor might through his or her tutoring

interactions with students. See Gutstein (1993) for details.

3. A knowledge engineer attempts to extract knowledge from a domain expert. This is usually a

long-term, iterative, and collaborative process characterized by interviews and conversations, think-

aloud protocols, stimulated recall, analyses of written work and solved problems, and feedback from

the expert as the knowledge engineer develops the knowledge base.

4. We note that Mack did this often, in other contexts as well. See the table of her tutoring

actions, #5: Promoted cognitive dissonance for discrepant answers in different representational

systems, and #11: Used analogy/comparisons to previous problems.

5. We define culture as the ways in which a group of people make meaning of their

experiences through language, beliefs, social practices, and the use and creation of material objects.

Culture is continually being socially constructed; the culture of specific racial, ethnic, or gender

groups, for example, cannot be reduced to static characteristics or essences, and individual identities

are constructed through the intersection of racial, ethnic, class, gender, and other experiences

(McCarthy, 1995).

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APPENDIX A. MACK'S RATIONAL TASK ANALYSIS

FIGURE 1. Nodes represent meaningfully known knowledge chunks (i.e., with conceptual understanding).

Arcs represent different ways in which children acquire the chunks.

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