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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
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,
GUTSTEIN AND MACK442
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
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).
GUTSTEIN AND MACK444
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
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.
GUTSTEIN AND MACK446
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
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.
GUTSTEIN AND MACK448
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
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
GUTSTEIN AND MACK450
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.''
LEARNING ABOUT TEACHING FOR UNDERSTANDING 451
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
GUTSTEIN AND MACK452
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]
LEARNING ABOUT TEACHING FOR UNDERSTANDING 453
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.
GUTSTEIN AND MACK454
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
LEARNING ABOUT TEACHING FOR UNDERSTANDING 455
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.
GUTSTEIN AND MACK456
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.
LEARNING ABOUT TEACHING FOR UNDERSTANDING 457
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?
GUTSTEIN AND MACK458
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
LEARNING ABOUT TEACHING FOR UNDERSTANDING 459
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
GUTSTEIN AND MACK460
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).
LEARNING ABOUT TEACHING FOR UNDERSTANDING 461
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.
GUTSTEIN AND MACK462
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