Thesis Proposal:

Intelligent Tutoring Systems have Forgotten the Tutor:

Adding a Cognitive Model of Human Tutors

Neil T. Heffernan

Computer Science Department

Carnegie Mellon University

This thesis proposes to build a better intelligent tutoring system (ITS) for the domain of algebra symbolization by first studying which dialog techniques used by experienced tutors are effective, and then implementing an ITS capable of carrying out such dialog techniques. Traditional model-tracing ITSs have been effective for task domains that can be decomposed into constituent skills enabling each skill to be tutored separately. However, such a paradigm is problematic for tasks that cannot be decomposed into simpler parts. My completed work on student modeling shows that symbolization is such a task domain. Therefore, symbolization tutoring is likely to benefit from the employment of more human-like dialog techniques. The protocol analysis of experienced tutors will enable me to provide answers to important unanswered questions in the tutoring literature, such as: "What do tutors do in this domain?"; "Do tutors primarily give feedback on mistakes?"; "Do tutors show evidence of using their diagnosis of student misconceptions?"; "How much attention do tutors pay to motivational goals?"; and, finally, "How important is student initiative in a coached practice situation?" Answers to these questions will help to constrain the model of human tutoring that I develop. Based on preliminary protocol collection, I hypothesize that student learning will be correlated with the tutor's use of knowledge-construction activities, dynamic plan scaffolding, and reflection/generalization techniques. This proposed ITS will be fundamentally different from current model-tracing ITSs in that it will include a model of tutorial reasoning in addition to a cognitive model of student behavior. This separation of the tutorial model will enable the future testing of alternative theories of tutoring. Such testing is currently very difficulty due to the costs of training tutors. I propose that an ITS incorporating the effective techniques that an experienced tutor uses will outperform a traditional model-tracing ITS.

**Abstract: **

Introduction:

The Cognitive Model of Student Behavior:

The Domain to be Tutored: Symbolization

Difficulty Factors Assessment Methodology:

Generalization/Comprehension Hypothesis

Articulating Composition Hypothesis

How does the student model affect tutoring?

Identifying Missing Knowledge Components

Redesigning Current Scaffolding:

The Developmental Model

Questions & Hypotheses about Tutoring:

Relevant Prior Work and Questions about Human Tutoring

Hypothesis Derived from Protocol of Experienced Tutor

General Hypotheses about Tutoring

Specific Hypotheses about What Makes for Effective Tutorial Strategies

Proposed Research:

Protocol Collection of Experienced Tutors

Reflective Protocols:

The construction of the tutorial model

The Evaluation of the Effectiveness of Tutorial Strategies

The Proposed Intelligent Tutoring System:

Evaluations of the ITS.

Evaluation of Tutorial Strategies:

Contributions:

Timetable:

References:

Appendix A: A list of Observed Tutorial Operators

APPENDIX B: A Protocol of an Experienced Tutor

This thesis will result in the construction of an Intelligent Tutoring System(ITS) for the domain of algebra symbolization that will have a model of tutorial reasoning in addition to the standard technique of modeling student reasoning. This system will be able to carry on a dialog with students to help them solve problems. I have already built a model of student behavior and will discuss the completed research and the positive research results of that effort. The work I propose to do builds on, and an is enabled by, the model of student reasoning. In the second section, I discuss questions from the literature about the nature of tutoring, as well as hypotheses from preliminary protocol collection. In the third section I will discuss the proposed work that will be central to this thesis: the cognitive modeling of tutorial reasoning. The fourth section will discuss the evaluation of this research. I will conclude with a discussion of how a model of tutorial reasoning will enable hypothesis testing about what constitutes good tutoring.

We will first discuss the domain to be tutored, followed by the methodology we have used in analyzing this domain. We then conclude with the impact of this research.

The domain I am building an intelligent tutoring system (ITS) for is what I call *symbolization*. Symbolization is the skill to be able to take a problem situation, usually stated in words, and formulate a mathematical model, usually an equation. For instance, the problem *CS *shown in Table 1 is a traditional symbolization problem, and such problems are very difficult for students. Students in the second month of ninth grade algebra class have a success rate of about 30% on this problem.

International tests such as the Third International Mathematics and Science Study (Beaton *et. al.* 1996) have shown performance for United States eighth grade students on such items is under 50%. While students do poorly, it is becoming evident to educators that symbolization is a fundamental skill students need to be learning. In studying symbolization skills we have focused on algebra story problems but our results may also be relevant more generally to symbolization skills needed in using a calculator or programming a spreadsheet or other programming tasks. As these computational devices take over more of the symbol manipulation of algebra, symbolization deserves increasing instructional emphasis. We are trying to understand how students learn to symbolize.

I am using a difficulty factors assessment (Koedinger & MacLaren, 1997) as an efficient methodology for identifying the critical cognitive factors that distinguish competent from less competent symbolizers. I have studied a number of factors with two different assessments. The general methodology we have used is to identify factors we suspect of being critical factors that effect the difficulty of problems. Then we study with a factorial design the effects of each factor and any interactions that might exist. We have so far studied six different factors over the course of two experiments.

We have focused on the *translation* metaphor suggested by Paige & Simon (1979) for symbolization problems and have attempted to analyze whether students have more difficulty with the comprehension side or the production side. The comprehension side involves the process for reading the English text of the word problems and constructing the internal representations that would allow a student to solve the problem. The production side involves the processes of taking that internal representation and producing, in the language of algebra symbols, an expression that captures the meaning of the internal representation.

Much of the prior work (Aziz, Pain & Bna, 1995; Cummins *et. al.*, 1988; LeBlanc & Weber-Russell, 1996; Lewis & Mayer, 1987; Paige & Simon, 1979; Riley & Greeno, 1988) on word problem solving has focused on students' comprehension abilities. Cummins *et. al.* suggest "that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language." In addition, Nathan, Kintsch, & Young (1992) "claim that *symbolization* is a highly reading-oriented one in which poor comprehension and an inability to access relevant long term knowledge leads to serious errors." The general conclusion from the above research is that comprehension rules are key knowledge components students must acquire to become competent problem solvers.

Generalization/Comprehension Hypothesis

This brings us to the first naturally intuitive hypothesis about what makes algebra symbolization difficult. What we call the *generalization hypothesis* flows naturally from the previous literature that has focused on comprehension difficulties caused by abstract language. This hypothesis is that an important conceptual leap for students is to move from the concrete grounded world of arithmetic problems to the more abstract world of algebra problems with variables. According to this hypothesis, what is hard about algebra is the generalization ability needed to think about a problem with an abstract variable, as opposed to a concrete known.

But more recent research by Koedinger & Anderson (in press) found evidence that acquiring such comprehension skills is not sufficient for symbolization competence. They reported that in 36% [((result-unknown=55%) - (symbolize=35%))/55%] of problems that students comprehended well enough to find a numerical answer, students nevertheless failed to correctly symbolize them. We call the difference between the students’ performance on symbolization problems and on similar arithmetic problems the *symbolization* *effect*.

To determine if the symbolization effect is well-explained by the generalization hypothesis we (Heffernan & Koedinger, 1997) performed an assessment in which we compared students’ performance on symbolization problems (like problem *CS*) with their performance on *arithmetic symbolization* problems. We could turn problem *CA* into an arithmetic symbolization problem simply by changing the final question to "Write an single expression that shows all the computations you would need to do in order to find how much she can spend on each sister." The correct answer for this problem would be "(72-32)/4." Arithmetic symbolization problems are a special type of symbolization problem that have no variables. We found no statistically significant difference between students performance on these two types of problems. This suggests that the presence of a variable did not increase difficulty, thus calling into question the usefulness of the generalization hypothesis’s focus on the variable per se as the factor that causes the symbolization effect.

An alternative hypothesis was suggested by another factor we tested. We found what we called the *composition effect*. The composition effect is the difference in student performance on what we call *composed *problems, like *CS*, and their *decomposed* analogs (*DS1* and *DS2),* which are two separate problems from the student’s point of view, but for analysis purposes they are treated as a single problem and the problem is correct only if both parts are correct. The decomposed version of a composed problem is made by dividing the composed problem into two separate questions that each ask for one of the steps required to solve the composed version. Our results showed that there was a significant composition effect because many students could correctly answer both parts of a decomposed problem but fail to correctly answer the composed version. In other words, in this case, the whole is greater than the sum of its parts. The size of this composition effect overshadowed the small and not statistically significant difference, mentioned above, between arithmetic symbolization and algebraic symbolization.

Articulating Composition Hypothesis

We suspected that the cause of the symbolization effect was actually the composition effect. We will illustrate with an actual student who answers "72-m=n/4=" for *CS*. This student demonstrates that she knew the two steps she would need to perform if she had been given the arithmetic version *CA*. This student must learn how to correctly represent these two steps. She uses a placeholder variable, possibly because she does not know how to represent anything but a single operation at a time. For the expert it is obvious that "72-m" can be treated in the same way any number is treated. But the novice, who looks at "72-m" as a recipe and not as an object of reflection, does not know that she can operate on this quantity directly by adding parentheses and "/4". We call this missing skill *articulating composition* and we suspect that it is this difficulty that explains why symbolization problems are so much harder then arithmetic.

An important distinction between the *articulation composition hypothesis* and the *generalization hypothesis* is on where in the translation process a student has difficulty: the comprehension of the source language, or the production/articulation of the target language. The generalization hypothesis suggests that students are having difficulty on the comprehension side of the translation process. The *articulating composition hypothesis *suggests the difficulty is on the production side. To investigate if this "articulating composing hypothesis" is a good explanation, we performed a second difficulty factors assessment (Heffernan & Koedinger) in which we tested two factors: 1) arithmetic versus symbolization (*CA *versus *CS* and *DS* versus *DA*) crossed with 2) composed versus decomposed (*CS* versus *DS* and *CA *versus *DA*).

The results of this study showed that contrary to common belief and the emphasis of prior cognitive science research, students’ difficulty in algebra word problem solving appears less related to comprehension difficulties and more related to difficulties in producing symbolic expressions, particularly expressions that involve more than one operator. We presented a developmental model with three main transitions.. At the first non-trivial level, we saw about 10% of the ninth grade population, when sampled after one month of algebra class, were at the point that they could only do the easiest problem types which were decomposed arithmetic problems. We speculated that their poorer performance on composed problems might be due to working memory limitations or related to difficulties comprehending more complicated composed stories. The students at this level are evidence in favor of the *generalization* *hypothesis*.

About 1/3 of the students were at the next stage in the developmental model. These students were also able to solve composed arithmetic problems in addition to decomposed arithmetic problems, but were unable to solve any symbolization problems. The high number of such students point to the great difficulty of doing even simple decomposed symbolization problems. We considered four alternative hypothesis to explain this difficulty. We presented evidence that argued that the *generalization hypothesis* could only account for a modest fraction of this effect. The *articulation hypothesis* could do an approximately equal job in explaining this effect.

A large number of students were competent in composed arithmetic and decomposed symbolization, but not composed symbolization. In other words, a large number of students (40%) who could handle both difficulty factors(composition and symbolization) when presented separately, could not do any problems when both difficulty factors were combined. This was strong evidence that the *articulation* *hypothesis* was a good explanation for student performance, because the articulation hypothesis predicted exactly this sort of interaction between these two factors. To make the transition to composed symbolization problem students needed to learn how to correctly articulate how to combine the symbolizations of the two steps in a composed problem. This was particularly difficulty for students, even those who showed some proficiency in composed symbolization problems. We speculated that this was because the real difficulty of composed symbolization problems was that they required the students to know "how to treat an expression just like a number." A student might be able to write down the first step of "72-m", but unless he knows how to use that while writing down the second step, he will fail to correctly do the problem. If the student views and expression like "72-m" as a recipe then he will only be able to use the output. Such a student was the one shown above (who wrote "72-m=n/4=") who felt compelled to first assign the result of the first step to a variable before performing the second step. If a students can treat an expression as an object like any other number then they can indicate that you divide the result of "72-m" by 4 by writing the correct "(72-m)/4".

Our initial work (Heffernan & Koedinger, 1997) allowed us to identify suspected missing knowledge components that distinguish competent from less competent symbolizers. This lead us to speculate that if the hard part is composing steps together, then instruction focused on teaching students just the missing knowledge should improve their performance, even if divorced from practice on word problems. We do not have space for a full reporting, but in short, we trained 39 students on problems like "Let X= 72-m. Let B= X/4. Write a new expression for B that composes these two steps." We then looked for transfer from the learning of this skill to students’ improvement on algebra symbolizing. At the face of it, such problems seem totally unrelated to translating word problems to symbols. However, our cognitive analysis and difficulty factors assessments have identified substantial overlap in the skills required for these apparently unrelated tasks. In a single hour of training on this skill, we saw statistically significant increases in student performance on algebra symbolization as a result of training on symbolic substitution, even factoring out gains due to using parentheses correctly. This is possibly the strongest evidence we could present in favor of the *articulating composition hypothesis*. This result also supports the more general idea that there is greater difficulty on the production side rather then the comprehension side of translation, since these students improved without practice reading algebra word problems; the only practice they had was composing expressions together. While it may be that mathematically algebra symbolization is a *generalization* of arithmetic, cognitively it is more accurate to say algebra symbolization is the *articulation* of arithmetic.

Redesigning Current Scaffolding:

Another implication of this work is in the area of suggesting a redesign of current scaffolding techniques used in PAT (Koedinger, Anderson, Hadley, & Mark, 1995). PAT has a mechanism called the *Pattern Finder* shown below.

Ann is in a lake that is 2400 yards wide. She starts out 800 yards from the dock. She rows for "m" back towards the dock at a rate of 40 yards per minute. Write an expression for her distance from the dock.

**Tutor**: If the number of minutes was **2** minutes, how far would Ann be from the dock?

**Student**: 800-2*40

**Tutor**: If the number of minutes was **3** minutes, how far would Ann be from the dock?

**Student**: 800-3*40

**Tutor**: If the number of minutes was **4** minutes, how far would Ann be from the dock?

**Student**: 800-4*40

**Tutor**: If the number of minutes was **"m"** minutes, how far would Ann be from the dock?

**Student**: 800-m*40

Table 2. Pattern Finder dialog with idealized student.

The above dialog is based on the premise that students have an easier time doing arithmetic problems then algebra problems. Therefore ask student to do a few arithmetic problems and then let them generalize from the arithmetic problems. The implicit assumption is that the last step that requires the generalization from arithmetic to the language of algebra. But the implications of the work described above has shown that there is little difference between arithmetic symbolization and algebraic symbolization. This suggests that this scaffolding would not be particularly effective, since the task that it first asks students to do is about as hard as the target task of symbolization. Recently, Koedinger has confirmed this prediction by analyzing the performance of students on the individual steps of the pattern finder and found there is little difference between students’ ability to perform any of the last three steps shown in Table 2.

Currently I have a model written in the language of the Tutor Development Kit (Anderson & Pelletier, 1991) which was based on initially on the ACT-R (Anderson 1993) language. Since others (Bobrow, 1968; Rapp, 1986) have dealt with issues of parsing the actual problem statements, I am not focusing on that and instead have the model take in a high level representation of the problem statement. Such a representation is consistent with what we might imagine the output of a natural language processing system would output. When combined with the rules of our cognitive model, the representation is sufficient to solve the problems.

The actual representation includes two basic types of elements: quantities and relations. The quantity elements hold the English phrases for a quantity, the units, and the actual values and/or symbolic representation. The relations acts as connectors that link the quantities together in a network that is largely isomorphic to the parse tree for the final expression. The relations are directed, and typed (addition, subtraction, multiplication or division), and have multiple quantities as their inputs and a single quantity as there output.

Hall, Kibler, Wenger, & Truxaw, (1989) have argued that students reason using situational constraints from the problem statement (e.g. price is usually not a negative number, or the distance to the dock is decreasing over time). We found some support for students using such constraints in their symbolizing (Heffernan & Koedinger, in submission). This being the case, suggests that the model should be able reason to qualitatively and know certain "common sense" propositions about the domain such as the fact that the distance Ann is from the dock decreases as the number of minutes increase, or that most numbers are positive, or that one number is bigger than another. This will involve augmenting the model with some common sense knowledge and the rules to use that common sense knowledge.

The model also contains production rules that model both a student who works forwards from the givens to the goal as well as students who are guided by the goal to set appropriate subgoals. The rules are sufficient for solving any of the problems, as well as making many of the common errors students make. The current model is simple and is able to capture the major effects of arithmetic versus algebra symbolization. We anticipate being able to capture the effects we have observed in our difficulty factors assessments including the composition effect, and the effect of distractors in a plausible manner that will generate the typical errors students make.

We want this model to be able to be used for the purposes of being able to perform an error diagnosis on a student’s response as well as assisting in generating correct responses. We then want that error diagnosis to be passed to the tutor model which will be discussed below.

The first part of this section reviews questions from the literature about the nature of tutoring. The second section discuss preliminary hypotheses about these questions, illustrated with examples from an actual protocol of an experienced tutor.

We will now review some questions from the literature about what is the nature of tutoring. The first questions is quite general: "What is tutoring?" An answer to this question surely depends upon the domain of interest; this study will be focused purely on tutoring in the context of coached practice in which students learn a skill by practicing it and receiving assistance as needed from the tutor. Coached-practice tutoring probably covers a large amount of the type of tutoring that occurs in practice. For instance, the Learning Skills Center at the University of Pittsburgh employs about 60 undergraduates who work as tutors for other students dealing with the basic mathematics courses. This clearly indicates that it is a common belief that this sort of tutoring is beneficial to students.

"What happens in a tutoring session?" Merrill, Reiser, Merrill & Landes (1995) argue that the main thing human tutors do is to keep students on track and prevent them from following "garden paths" of reasoning that are unproductive and unlikely to lead to learning. Merrill *et. al.* pointed to the large number of tutors’ remarks that helped keep students on track while trying to learn Lisp programming. How a skill like symbolizing differs remains to be seen. Since these symbolization problems require answers that are only a few characters long, there might not be much danger of "garden paths" so the role of the tutor may be different.

So one division we can draw among perspectives is on the issue of how much tutors diagnosis. McArthur *et. al.* (1990) summarize the teaching literature on classroom instruction as emphasizing the amount of time teachers spend thinking about their students. But Putnam(1985, 1987) reports that tutors rarely determine the exact nature of misconceptions, or attempt to do so by asking diagnostic questions. McArthur *et. al.* argue that this approach is in direct contract to the diagnostic/remedial model implicit in much research on diagnosis (Brown & Burton, 1978) that is also prevalent in ITS systems. Many ITSs (Anderson, Boyler & Reiser, 1985; Anderson *et. al.*,1995) operate by using a cognitive model and pairing a particular incorrect action with a particular tutorial action. Therefore, "there is no adaptation to the current cognitive state of the learner other than the classification of his last step as an instance of a particular type of error" (Ohlsson, 1986, as cited in McArthur). McArthur et. al argues that such tutors are more like tacticians who respond to individual tutorial moves, and less like strategists who have overall plans for dealing with student’s misunderstanding. How much evidence of strategic planning versus tactical responses do we see in tutoring for this domain?

We will investigate 1) if this domain shows evidence of tutors diagnosing students errors and 2) whether they remediate misconceptions that might have lead to those errors. Some (Lepper, Aspinwall, Mumme & Chabay, 1990) have reported that human tutors do some diagnosis but do not report that diagnosis to the student; rather tutors use that diagnosis to influence what scaffolding strategies to use with the student.

How active are students in tutoring sessions? This is of particular interest because we desire to build a system capable of having a dialog with students; we must understand what types of student behavior are likely. If a primary characteristic of tutoring is tutors answering students’ questions then a different type of tutorial model will be appropriate. Lepper, Drake & O’Donnell-Johnson (1997) have reported that expert tutors make about 80-90% of their remarks in the form of questions. If this holds true for this domain we can feel more confident that we should be able to get substantial learning gains from a dialog that is primarily tutor -initiated. Even if our system is not totally capable of handling open-ended student initiative, by looking at protocols we will be able to classify the major types of student initiatives (asking for help, asking for a definition, etc.) and making the system capable of responding to those limited types.

Lepper *et. al.* also report that tutors spend time thinking about how to increase their students’ motivation. He reports that when instructional goals come in conflict with goals to increase student motivation, the goals related to motivation take precedence. Do we find evidence to support this claim?

Other relevant prior work is the work of VanMarcke & Vedelaar (1995) who have compiled a large catalog of domain general strategies. Clancey’s GUIDON (1983) was an early ITS that claimed to have tutoring rules. But those rules were quite primitive and could not adapt interactions to the needs of individual students.

McArthur *et. al.* (1990) states " the literature on effective tutoring indicates, at least, that tutoring is more than either global plans or a circumscribed response at a single moment in time. Neither notion alone helps us understand how a teaching goal (e.g., explain the distributive rule) becomes translated into a sequence of teaching techniques to satisfy that goal."

I have performed about 20 preliminary student tutoring sessions that have enabled me to try out different techniques and begin to come up with a categorization of tutorial utterances. These sessions have all been of the coached-practice variety. These sessions, combined with my background as an secondary school mathematics teacher, have provided some intuition about components of good tutoring. But my intuition is just that: unsupported intuition. Therefore, I will now discuss some hypotheses to the questions raised in the previous section with the help of an actual protocol I collected from an experienced tutor ( a female middle school teacher tutoring a male seventh grade student). This particular session was about one hour long and is reproduced in its entirety in Appendix B. I have attempted to come up with a categorization for tutorial utterances as shown in Appendix A. I refer to these categories as the tutorial "operators" as they represent the major choices that the tutor has to choose between when deciding how to assist a student. In Appendix A I give each operator a name of the form Op*x* , a brief description, and point out some real examples and cite them by line number as they appear in Appendix B. In Appendix A, for illustrative purposes, I offer an invented canonical example all for the same sample problem. These categories for tutorial utterances are likely to change as more protocols from experienced tutors are collected and analyzed. In the next section I will discuss some tentative hypotheses derived and illustrated from this single hour of tutoring. In the last section of this proposal I discuss how I will gather evidence to investigate these hypotheses.

General Hypotheses about Tutoring

First we consider the general question of "what does a tutoring session look like?" Tutoring seems to be a highly interactive process offering plenty of confirmation, as reported by Merrill et. al (1995). Sentences tend to be short and in a 1-hour session lead to about 400 back-and-forth utterances. The tutor spends very little time on the problems the student gets correct (8 out of 17), but the tutor spends longer on problems the student has difficulty. But Merrill’s claim that tutors want to get students quickly on track is not supported; for example the fifth problem that begins at line 17 goes on until line 146 which is about a span of ten minutes. The tutor did not simply tell the student the answer. The tutor made the student work and tried to show the student that his answer would not work (Op 17, line 62).

Consistent with McArthur *et. al.*’s results, the tutor seems to engage in more strategic then tactical interactions. There are a variety of multi-step plans the tutor engages in. A hypothetical example to illustrate the point is shown below.

Tutor: Try the following problem "Sue made 72 dollars by washing cars to buy holiday presents. She decided to spend "m" dollars on a present for her mom and then use the remainder to buy presents for each of her 4 sisters. She will spend the same amount on each sister. How much can she spend on **each** sister?"

1 Student: I don’t know

Tutor: OK, let me ask you an easier question. Suppose the amount she gave to her mother was "8" instead of "m", what would the amount each sister would receive?

2 Student: 16 dollars

Tutor: Correct. Can you show the math for how you got that?

3 Student: 64/4

Tutor: That is correct. Can you now show how you got the 64?

4 Student 72-8

Tutor: Good. Now write how you got the 16 again, but this time use 72-8 in the place of 64.

5 Student: 72-8/4

Tutor: Great, but you forgot one thing. See if you can figure it out.

6 Student: (72-8)/4

Tutor: Good. Now, let’s go back to the original problem and ask if instead of 8 dollars it was "m" dollars that she gave to her mother.

7 Student: (72-m)/4

Tutor: Correct! Next problem..

Table 3: Proposed new instance based strategy

Note that in the above example the fact that a human tutor is not impeded from carrying out an overall plan of scaffolding by analogizing off the student’s calculation of an instance based on m=8. The student fifth response is an error, and the tutor remediates that error and then gets back to the overall plan. If the student’s second response had been 72-8/4 the tutor could pick right up with that and carry on as in the above dialog. The tutor transcripts shows that she uses several different types of plans that sometimes involve a number of steps including operators number 1, 7, 11, 12, 16, 17, 18, 19 and 20 (all shown in Appendix A) The reader is encourage to take moment to look over these examples and how they are evidenced in the actual protocol as cited with each example.

In general, tutors try to see what the student did wrong and then break the problem down to expose the bit of reasoning the student misses. But often the tutor responds in ways that most ITSs do not. One instance of this is when the tutor makes comments that reflect (op 4 &5) on earlier problem-solving, sometimes even after a correct answer. Most ITSs do not provide feedback at any times other then when the error actually occurs. But I hypothesize that tutors often reflect at moments when the student is not taxed with the problem of keeping all the information associated with the problem in working memory. At these points the student is more likely to be able to successfully generalize (op3). At other times the tutor provides "feedforward" (as opposed to feedback) before the student actually errs(op6). This may be due to the tutor anticipating an error that would interfere with current issues.

Other aspects of the tutor’s dialog reveal that the tutor is sensitive to more than just the bugs the student makes. The tutor refers to previous explanations and examples (op2). The tutor also will remind the student what they have accomplished (op13). The tutor also comments on repairs (op23) the students attempt to try to fix their wrong answers, and the tutor does not treat these new answers in the same way she would treat a first attempt. For example, in line 253-268 (note the line numbers refer directly to Appendix B), the tutor first gives negative feedback, and on the second attempt the student gets the correct answer, but, possibly due to the fact that the tutor may suspect the student guessed correctly, challenges the student to defend the answer (op4). All of these examples bring up context to which a human tutor is sensitive but most current ITSs are not. Some ITSs have been redesigned (Moore, 1996) to make the dialog generated to be sensitive to what has been said already. These concerns are sometimes simply linguistic concerns done to make the system sound more human-like and not necessarily for pedagogical purposes per se.

Another important consideration when looking at tutorial utterances is possibly whether the utterance is directed at getting the student to choose the correct goal (line 37-41) versus assuming they have the correct goal and tutoring them on the procedure to accomplish that goal. Catrombone (1996) has pointed out the utility of worked examples that highlight the subgoals the students need to set, and an analogy might be found in good tutoring. The tutor seems to show evidence of trying to get the student to generalize (op12 and op9) by stating a general instance of what the student just performed (see line 246).

The literature has also debated if tutors do diagnosis, as well as if they remediate misconceptions? At this early point, the evidence agrees with Lepper in that there are many operators (Operators numbered 2, 5, 6, 8, 10, 11, 14, 15, and 17-25) that the tutor uses that appear to be selected for particular errors, but few of these actually report a diagnosis (like operator 15). The few instances of direct feedback of a diagnosis seem to be in response to slips (operator number 21). This is possibly due to the fact that the tutor does not want to spend time remediating the error, possibly because the tutor might think the student already knows the correct procedure but made a careless mistake.

Next we consider if Lepper *et. al.* (1997) claim that tutors pay attention to motivational goals is supported by the evidence. An inspection of the transcript reveals that very few comments seem to relate to motivation. The few comments seem to be at the beginning or end of a problem when the tutor says on line 11 that she is looking for a harder problem since the student got the first three problems correct. Other comments related to the student obviously indicating frustration and the tutor notes this at line 403. But, overall, few remarks seem to be attempts to prop up the student’s motivation. Nevertheless, the tutor indicated in a post session interview that motivational concerns were important, so possibly her implicit theory is that the student’s awareness of his own learning is what will lead to motivational gains.

The next question is, "How important is student initiative?" This dialog suggests that student initiative might not be particularly important. There are few instances of the student asking a question or changing the topic of conversation. The instances seem to be simple questions for clarification (line 18). This initial result is encouraging for the hope of building an ITS capable of assisting student that does not need sophisticated natural language comprehension abilities to deal with open-ended student initiatives. I believe that we can build a model of tutorial dialog that has a simple comprehension mechanism. This simplicity will be enabled primarily by the ITS only asking questions that it can understand the answers to. Most of these questions will be asking for mathematical expressions. Some questions might include asking the student to respond with a noun phrase such as "What does the ‘40*m’ represent?"

Specific Hypotheses about What Makes for Effective Tutorial Strategies

I hypothesize three types of tutorial strategies will be particularly effective and correlated with learning. These three hypotheses all fall under the single hypothesis that good tutors "ask more and tell less" (the phase "ask more and tell less" is from the Center for the Study of Tutoring’s grant proposal.) Specifically I think certain types of asking are more effective then telling; knowledge-construction activities, dynamic plan scaffolding, and reflection/generalization techniques.

The first characteristic I discuss is knowledge-construction activities, which I define as attempts by the tutor to get the student to construct a piece of knowledge from parts that the student already knows. One example of knowledge construction is a case-based explanation like the following hypothetical dialog inspired by observed instances (like in line 32, 104 and 258).

**Student**: I don’t know

**Tutor**: OK. Let’s suppose we knew that d=9 dollars. Can you calculate the cost of the shirt?

**Student**: 6 dollars

**Tutor**: How did you get that?

**Student**: I got that by subtracting 3 from 9.

**Tutor**: Good. Now how did you get the 3?

**Student**: I took one-third of 9.

**Tutor**: Good. How do you write in mathematical notation one-third of 9.

**Student:** 1/3*9

**Tutor**: Good. Now can you put this altogether and state the expression for the cost of the shirt if d=9?

**Student**: 9-1/3*9

**Tutor**: Great. Now can you answer the original question of how much the shirt cost if it’s D and not 9?

**Student:** d-1/3*d

**Tutor**: Great.

This dialog illustrates an example of teaching the student to be able to learn how to write algebraic symbolizations with variables by scaffolding off of the students ability to do arithmetic problems. In this way the students is more likely to avoid shallow learning of rules because his knowledge of symbolization is tied to his existing knowledge of arithmetic. I hypothesize that this sort of knowledge-construction is probably particularly important on skills that are not easily decomposable. A skill might not be easily decomposable if it represents a single production rule in a cognitive model. A second reason a skill might not be easily decomposable is if when you decompose it the parts are easier then the whole, indicating some difficulty is lost when the problem is decomposed. This happens to be the case with symbolization as discussed in the first section of this proposal. This suggests that symbolization is prime candidate for knowledge-construction activities that are not simply decomposing the problem into its parts.

Another knowledge-construction activity is to try to get the student to see that his answer violates implicit constraints of the problem (e.g. line 375 and 252 of Appendix B). For instance, on line 374 when the tutor asks the student "Which is longer?" the tutor is not simply issuing a buggy diagnosis of the student’s answer. The tutor goes on to connect the student’s answer to the semantics of the problem causing the student to realize his error.

Another example of a knowledge-construction activity is shown at line 359. Here the student has committed an error that the tutor realizes she can get him to see by pointing out that if the student looks at the units of the numbers involved, then the student will realize that the expression couldn’t give the desired answer. This is a knowledge-construction activity as it build off of the student’s knowledge that if you subtract inches from inches you get inches.

Yet another knowledge-construction activity is when the tutor does not simply tell the student how to calculate elapsed time but gets him to construct that knowledge for himself. In lines 240-248 the student starts out by incorrectly symbolizing how to compute elapsed time. The tutor ask him to calculate elapsed time for some made up numbers, which the student is able to do. Then the tutor asks the student to generalize from that answer the general principle (line 246-247). Then at the end of the problem on line 248 the tutor encourages the student to reflect on such problems to not make that same error again. Below we will discuss other example of this last step of reflective behavior that can occur after a student has already given a correct answer. I hypothesize that this whole micro-plan is more effective then simply telling the student how to calculate elapsed time.

A final knowledge-construction activity is illustrated by Operator 18, when the tutor tries to get the student to use his knowledge about evaluating expressions as well as his knowledge of order of operations to realize he has made a mistake of not including parentheses. The protocol shows on line 161 such a knowledge-construction question. Interestingly on the next problem the student uses parentheses even when they are unnecessary and the tutor brings this up in a reflective moment after the student has already answered correctly. Again, this aspect of reflection will be explored below.

A second characteristic of tutorial dialogue is dynamic plan scaffolding. I hypothesize that students have two separate problems when learning a new domain. They need to learn the valid operators they can use as well as the planning rules to guide the selection of the operators. It seems that experts have a few general rules, while novices start by learning small and overly specific rules. Subgoal reification is a topic that Corbett & Anderson (1995) has shown to be helpful and is inspired by the idea that in order to encourage students to learn the rules for planning a solution, you should ask them to make their subgoals explicit. This domain also has similar possibilities for reification, like the final problem in the transcript when the tutor and tutee work first on finding the "rate of burning" before trying to symbolize other aspects. The tutor demonstrates a form of subgoal identification on lines 379 and 381 when she asks the student what he has computed and if that subgoal will help him achieve the goal of finding the rate (which is itself a subgoal). The work by Corbett & Anderson as well as Catrombone (1996) suggest that focusing attention on subgoals can lead to learning. Current ITSs usually fail to provide assistance in the planning of appropriate subgoals. But Corbett & Anderson’s computer-programming ITS has taken the approach of forcing the student to reify his subgoals. But it is questionable whether students treat the subgoal questions like separate parts of a problem and therefore don’t achieve the full potential benefit. I hypothesize that it is useful to have a dynamic and flexible subgoal reification scheme in natural language. It might be better to allow the student to attempt the problem first, and only if they fail to ask them to reify their subgoals. So the proposed ITS might intelligently use subgoal reification only when necessary.

A third characteristic of human dialogue that I hypothesize leads to effective learning is encouraging the student to generalize and reflect. Line 246 show an example when the tutor restates the student correct answer but does so in general terms. We have already seen an example above where the tutor got the student to learn the general procedure for calculating elapsed time, by coaching the student to generalize from a single calculation of a particular elapsed time. Another example of generalizations is on line 401 when the tutor confirms an answers and states it in general terms. Another example of post-problem reflection occurs on line 229 when the student gives a correct answer but the tutor encourages him to reflect on it by challenging his answer. A very similar situation occurs on lines 225-268 when the tutor gets the student to write the correct answer but spends a long time afterwards making sure the student understands why. In summary I hypothesize specific types of tutorial strategies or operators will correlate with student learning. I anticipate it is these strategies that will be the ones I implement in the tutorial model described below, but, will look for unexpected strategies as well.

I will now review the proposed research starting with the collection of protocols of experienced tutors, followed by the construction of a tutorial model, and planned evaluations.

I plan to collect protocols of 50 minute tutoring sessions of experienced tutors while they tutor students. I plan to use experienced mathematics teachers who teach basic algebra and thus have spent considerable time instructing symbolizing. Unfortunately, there are not predetermined ratings for tutors as there are on chess experts. Therefore, I use the term "experienced" as I do not plan to verify their "expertise." I will use pretests and posttests to allow me to gauge the amount of learning that takes place. I have already tested these assessments and found that I, myself, am capable of getting substantial learning gains from students in 50 minutes time. The experienced tutors I plan to use will most likely come from the pool of teachers who are associated with the current PAT project. The time of these tutors is valuable, and will be the determining factor in the logistics of designing the experiment. An anticipated arrangement is to collect protocols of the tutors, while they tutor one of their own ninth grade students after school in their own classroom, while I record the session on videotape. I anticipate collecting approximately 15 hours of tutoring protocol from 3-4 experienced tutors. I have chosen to have more than one experienced tutor to get a broader coverage of actions that tutors perform.

The nature of the material has also been tested. I have compiled a list of problems and broken them down into sections. The tutors will be free to tutor as they like, with the constraint that they must use the problems given to them and also make sure that a student is able to do a problem in a section on his own, before moving on to harder problems. This mastery criterion will be imposed on the tutors so that we can better compare the performance of these tutors with a traditional CAI approach as described and motivated in the section below on the planned evaluation. I feel that some of the superior performance of human tutors is their ability to do good problem selection. This factor is not a factor I wish to study. Therefore, I will control for this by requiring both the humans and later the experimental computer programs to all use the same curriculum sequencing algorithm.

I will collect some reflective protocols from each tutor. I will have the experienced tutor give a think-aloud protocol as he views the tape of the tutoring session.

Once the protocols are collected I will be analyzing them to get answers to the questions I presented above. In particular, interest will be paid to categorizing the different sorts of things our tutors say, similar to what I present in Appendix A. I anticipate it will be relatively easy to categorize the types of things tutors say, but what will be harder, is identifying the reasoning behind such utterances. For instance, some utterances that are part of a larger plan. More generally, the problem is that we only have direct evidence of what the tutor said, not what he thought that lea him to decide to say that particular utterance. I therefore plan to use the reflective protocols to help distinguish between possible explanations for the tutor’s behavior.

I plan to implement the tutorial model in some production rule language such as ACT-R (Anderson, 1993) or possible a reactive planning system. Having a concrete tutorial model is desirable as it provides a very explicit formulation of a theory of tutoring. The model will possibly choose tutorial actions based on contextual variables, some of which are in the student model. For instance, if the student makes a mistake on a rule the first time he is asked to try that rule the tutor will probably respond differently, then if the student has had an opportunity to practice this rule in the past. Other contextual variables the student model is expected to provide to the tutorial model include the most likely diagnosis of the student’s answer.

The Evaluation of the Effectiveness of Tutorial Strategies

Through the use of pretest and posttest I will be able to determine what specific skills students learned during their tutoring session. I will then be able to analyze what strategies the tutor used. I will look for correlations between different tutorial strategies and learning. I will seek to find evidence to confirm my initial hypothesis outlined above with regard to knowledge-construction activities, dynamic plan scaffolding, and reflection/generalization techniques. This initial hypothesis might prove wrong, and I might find new tutorial strategies I did not expect and find they are well correlated with learning. This motivates collecting protocols from a number of different tutors.

Once the tutorial model is complete, then the two major components of the ITS are finished. Final components include giving the ITS a curriculum that includes problems to choose from broken up into different sections. The ITS will use the same mastery learning algorithm that the human tutors were required to use (do the problems in each section in order until the student can get a problem correct on his first attempt.). Therefore, an additional component will actually carry out this problem selection.

Evaluations of the ITS.

I propose to have 4 control conditions as a means of better understanding the reasons for any possible improvement. The first control condition of the expert tutors has already been described.

I will compare the performance of our system to a second control condition of a traditional CAI approach. Such an approach is what is typically used in schools today and is simple to author and contains no more intelligence than a slightly automated slide show. The behavior of the CAI control will be to use the same mastery learning problem selection criterion as the human tutors used. But rather then try to dialog with the student when the student has trouble it will simply tell the students whether he is correct or not, as well as a "canned" explanation of what the correct procedure is for solving the problem. We will have one of our experienced tutors write these canned explanations.

The strongest control condition will be the existing PAT ITS modified slightly so that it only teaches symbolization (currently it also teaches graphing functions and symbol manipulation as well).

In summary, to the degree that this system does better than the CAI control, it justifies the effort of building ITSs. The degree to which the program surpasses the performance of PAT will be a measure of the improvement gained by incorporating effective tutorial dialog strategies.

Having a system with an encapsulated model of tutorial reasoning will allow future exploration of hypotheses about what sorts of tutorial reasoning and interactions best lead to learning. Currently, experimentally testing normative theories of tutoring behavior is very difficult for it requires training humans to act in specific ways in accordance with a given theory. This thesis will solve this problem by allowing the testing of competing theories of tutoring to be tested by building models of the competing theories and measuring the relative effectiveness of the system. At the same time, the existence of this system will force theories of tutoring to be specific enough to guide tutorial interaction.

It would be easy to go one step beyond simply testing specific hypotheses about what makes for good tutoring and build a system that measures the effectiveness of the different tutorial operators to see which are mostly likely to cause the student to learn the desired skill.

I conclude with a discussion of the anticipated contributions of this thesis.

1) This work will have provided an analysis and cognitive model of symbolization, a fundamentally important skill.

This analysis revealed that contrary to current common belief and prior cognitive science research, student’s difficulty with algebra word problems appears less to do with comprehension difficulties and more related to difficulties with producing symbolic expressions. This work led to the identification of a skill of composing expressions that we showed to be a missing knowledge component for many students, and we found that if instructed in it, they would transfer this knowledge to symbolization problems.

2) This protocol collection will provide answers to questions that have been hotly debated in the literature as to what tutors do and identify some of those things they do that are correlated with learning.

One question that will surely be answered is how well our tutorial sessions fit into a schema of diagnosis and remediation as so often used in intelligent tutoring systems, or whether a more sophisticated theory of tutorial interaction is required. I will also be able to report which tutorial strategies are well correlated with learning.

3) This work will have built the first model of tutorial action based on protocol analysis.

Expertise in tutoring provides an interesting new domain to try to cognitively model because it involves thinking about thinking. In fact, we are modeling the thinking of someone who to some degree has to attempt to model what the student is thinking. Not only is tutoring interesting as an object of study for cognitive science, it is of great practical interest. Having a system with an easily modifiable theory of tutoring will allow for experimentation that is ordinarily impractical. This thesis will demonstrate this method with at least one study of the utility of different tutorial strategies.

4) This thesis will seek evidence to confirm the hypothesis that modeling effective tutorial strategies can lead to better tutoring over model-tracing ITSs.

With the appropriate experimental controls I will be able to compare the new ITS’s potential gain against a tight control condition that few ITS experiments have been so rigorously compared against. Often ITS are compared against weaker controls of classroom instruction. This work will compare the ITS performance against a strong CAI control condition offering immediate feedback and individual mastery learning. I will also be able to provide a tight analysis of the actual benefit actualized as compared to the potential benefit as measured by comparing experienced human tutors with the CAI condition.

2-4 Month to collect tutoring protocols

3-6 Months to perform protocol analysis and develop a model of tutoring

3-5 Months to implement the theory of tutoring in a cognitive model

2-4 Months to beta test the ITS and run the experimental condition and computer control condition.

2-4 Months for data analysis and writing document

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Appendix A: A list of Observed Tutorial Operators

What follows is a list of tutorial operators that might be incorporated into the model of tutorial reasoning I propose to build. I have attempted to define types of tutorial interaction. I note any use of these operator in Appendix B and cite them by line number. The categories are not mutually exclusive. To make this list easier to browse I have provided an idealized example in the context of the following problem:

Ann is in a lake that is 2400 yards wide. She starts out 800 yards from the dock. She rows for "m" back towards the dock at a rate of 40 yards per minute. Write an expression for her distance from the dock.

Op1: Concrete Instantiation Analogy

The first step is an instantiation request and is found in lines 32, 104, and 258. The second and third steps are both requests for the student to articulate how they got an answer. An example where that was tough for the student is at lines 45, 51, 56, and 65. The last step is a request to show the two evaluation steps in a single expression and is found in lines 70 and 151.

Tutor: That is not right. Let me ask you an easier question. Suppose the number of minutes she had been rowing was "3" instead of "m". What would the distance to the dock be?

Student: 680

Tutor: Correct. Can you show the math for how you got that?

Student: 800-120

Tutor: That is partially correct. How did you get the 120?

Student 40*3

Tutor: Good. Now write how you got the 680 again, but this time use 40*3 in place of 120.

Op2: Tutor makes reference to dialogue history

On line 282 the tutor asks the student to recall a fact that they used several problems previously on line 216. Line 306 shows the tutor using "Once again, what is..." (Moore (1996) addresses issues related to generating natural language with the natural human quality of being sensitive to what has been previously said in the dialog.) On line 312 the tutor points to a previous problem to encourage the student to use that as an example.

Tutor: You have made the mistake of calculating speed using multiplication several times today; try to remember how you calculated speed on that problem about the Indy 500.

Op3: Encourage the student to generalize

On line 183 the tutor challenges the correct but unnecessary use of parentheses. Line 401 shows the tutor confirm a correct response and at the same time trying to introduce vocabulary. Line 246 is an example where, after the student states how to compute an instance, the tutor states the procedure in general terms. Line 170 is an example where the tutor suggests that a generalization is possible but doesn’t ask the student to do the generalization.

Tutor: Good. Now what if the speed wasn’t 40 but was "s." What’s the answer now?

Op4: Reflection after the correct answer

On line 229 the student is challenged on how he used one of the numbers given in the problem, which he had mentioned he did not know how to use (earlier on line 201) so the tutor came back to reinforce. On line 169-170 the tutor adds additional comments after the student has already arrived at the correct answer.

Tutor: Good. You didn’t make the mistake of using division for the fractional relation as you have a few times today. Try to remember that on future problems.

Op5: Reflection on, or Challenge of, a correct answer if the tutor suspects guessing

On line 253 and 255 the student quickly corrects his initial wrong answer with the other easy to guess alternative, so the tutor spends considerable time (lines 255-268) challenging his understanding of that answer, presumably due to a suspicion that he doesn’t understand why his answer is correct.

Tutor: How do you know the answer is 800-40*m and not 800+40*m?

Op6: Feedforward (given before the student makes an anticipated mistake)

On line 227 the tutor anticipates a missing parentheses mistake and possibly chooses to help the student avoid that problem because of the rather lengthy dialog the tutor and tutee are engaged in.

Tutor: Now, during the last few problems you have made several mistakes on problems that ask you to figure out the amount of a fractional discount. This upcoming problem is another of that type. Try to remember what operator you needed on problems like this (like the problem about selling T-shirts).

Op7: Requesting that the student recall information by either questioning and or hinting

Line 306-310 shows the tutor asking the student to recall a definition, and then when the student fails, switching to hinting by providing part of the definition "The word total has to be there?" Line 384 & 282 are recall requests both followed by line 388 and 284 which are hints towards that information. Line 192 is a similar request for recall.

Tutor: How do you calculate distance traveled when given a speed and time?(question form)

Tutor: How do you calculate distance. Remember d=r*t. (hint form)

Op8: Challenge the student’s answer

Line 219, 221, 223 and 225 challenge parts of the students answer, rather than say it is wrong. Line 221 even challenges part that is correct. Line 75 and 106 are other examples of this very general style of delivering what is usually negative feedback. Line 242 begins with a rhetorical question about the procedure a student uses, that is presumably asked to focus the student’s attention on the part of the answer that is wrong, and not to actually elicit a response as shown by the fact the tutor immediately asks a follow up question.

Tutor: Are you sure about that plus?

Op9: Ask the student to identify the name of a quantity represented by a symbol (or expression). Include follow up clarification if the student is not specific enough.

Line 210 and 212 are examples. This does not just apply to numbers given in the problem. On line 24-27 the student has apparently used "2/3" to stand for "two-thirds of the cost of a shirt" and the tutor asks the student to identify what the "2/3" represents?

Tutor: Your answer of 2400+40*m is not correct. But the 40*m is part of the answer. Before we go on, can you please tell me in words what the 40*m represents?

Student: The start distance

Tutor: From where?

Student: the dock

Tutor: Good.

Op10: Ask a student to identify the symbol(or expression) that represents a quantity (vice versa of above)

On line 82 the tutor asks the student to recall from the problem statement the symbol used to represent a quantity.

Tutor: What is the speed of the rowing?

or

Tutor: In your expression "800+40*m" what part represents the distance rowed after "m" minutes?

Op11: Correct a bug by referring to the implicit semantics about the relative size of numbers

Line 374 and 252 both show this technique of correcting a bug for the wrong ordering of subtraction arguments by asking "Which is larger?"

Tutor: Your answer of 40m-800 is wrong. Which one is supposed to be larger?

Op12: Stating the general quantitative relationships in words. This is like generalization, but can occur anywhere in the problem

Line 216-218 and 246 are examples and line 282 is a request for such a statement.

Tutor: So the distance she has left to row is equal to the distance she starts from the dock minus the distance she has rowed back towards the dock. Now that we have that figured out, Let’s go back to the original question of how far she has left to row

Op13: If the student might have forgotten what work he has already accomplished then remind the student what steps they have already completed

Line 407 is an example of the student having completed a subgoal of finding the slope and then being reminded to think about what the original question asked for.

Tutor: Good. You now have told me that the amount she made at bagging groceries was 5*h and the amount she made delivering newspapers was (30-h)*7. What is the total amount she earned at both jobs?

Op14: Positive Feedback on parts that are correct

Lines 315, 334 and 326 are just a few of the many examples explicit positive feedback in the form of "OK". Lepper et. al. (1997) argued that a primary characteristic of good tutors was their indirect style including negative feedback and positive feedback. The implicit positive feedback on line 183 is an example where the positive feedback for the answer was simply in the form of indicating the student should move on to the next problem. Line 362 seems to show an extreme example where the tutor gave positive feedback for just a small kernel of a correct response.

Tutor: Good. You now have told me that the amount she made at bagging groceries was 5*h and the amount she made delivering newspapers was (30-h)*7. What is the total amount she earned at both jobs?

Op15: Simple Feedback on an identifiable bug category

Tutor: Your answer is missing parenthesis

or

Tutor: You reversed the order of the subtraction sign.

or

tutor: You calculated the amount she rowed, not the distance she has left to the dock

Op16: Ask the student to figure out what sub-goal to set

An example where the tutor asks the student what information is needed for a sub-goal is at line 286 See Catrombone, (1996) for an argument that helping the students see the sub-goal structure is of crucial importance. Line 37 contains an instance of the student having set too large a goal (trying to find the cost of all 4 jackets), and the tutor points this out by asking the student what the goal for the problem is (cost of a single jacket).

Tutor: That is not right. Let me get you to simplify the problem for yourself. In order to figure out the distance she has left to the dock, what quantity do you have to first figure out? (Answer: "the distance Ann has rowed after "m" minutes.")

Op17: Socratic Technique showing a contradiction from a student’s error

Line 32 is an example showing a successful use of a Socratic technique that makes the student aware of an error in his answer by asking him to evaluate an expression. Line 260 is another example of the tutor showing the student that his symbolization does not agree with his arithmetic computation. Line 359 is an example where the tutor points out a conflict to the student in that their expression does not give an answer in the units the student was expecting.

Tutor: Let’s take a look at your answer of 800-m. What does your answer predict the distance Ann is from the dock will be after 3 minutes?

Student: 800-3=797.

Tutor: OK good. I agree that that is what your answer predicts. But let me ask you to now consider how far Ann rows in 3 minutes when she is going 40 yards per minute?

Student: 120 yards

Tutor: So what does that predict should be Ann’s distance from the dock after 3 minutes?

Student: 680.

Tutor: Good. That is correct. You now see that your answer of "800-m" could not be correct, since it predicts she will be 797 yards form the dock after 3 minutes, which is not correct. Please take another guess as to what you think her distance from the dock is after "m" minutes.

Op18: Order of Operations Sub-Dialogue

Line 161 is an example of the tutor engaging the missing parenthesis errors possibly on problems that require parenthesis.

Tutor: Your answer of "70-m/4 "is wrong because it is missing parenthesis. Try to fix it.*[correct answer is (70-m)/4]*

Student: 70-(m/4)

Tutor: Suppose you knew what value "m" had. What is the first math operation you would perform in your head? (subtract or divide)

Student: subtract

Tutor: Correct. But the answer you wrote above "70-(m/4)" would have which math operation done first?

Student: subtract

Tutor: No. Remember that what is inside parenthesis is done first. Try again. What math operation would be done first in your answer 70-(m/4)?

Student: divide

Tutor: Correct. Now see if you can put parenthesis around 70-m/4 to make the subtraction happen first.

Student: (70-m)/4

Op19: Teach students how to do unit analysis

On line 359-361 the tutor attempts to use unit analysis to convince the student that his answer must not be correct.

Tutor: Your answer of 800-40/m has a problem. Let’s look at the 40/m that you probably mean to be the distance she has rowed so far. What are the units on the 40?

Student: yards per minute

Tutor: Good. What is the unit on the "m"?

Student: minutes

Tutor: If you divide yards per minute by minute what do you get?

Student: yards per minute squared

Tutor: Correct. So you see your answer of "40/m" couldn’t possible be correct because it gives a number in the wrong units. You want an answer in yards. Try again at what you think is the correct expression for Ann’s distance form the dock.

Op20: Coach the student to realize distractor numbers are not needed

The student only made one error involving a distractor number, and the tutor early on coached the student to realize this error in lines 37-41.

Tutor: In your answer 2400-40*m you used a number that is not needed. Which number do you think is not needed?

Student: 2400

Tutor: Good- Now that you have identified that 2400 is not needed, take another guess at the distance she is from the dock.

Op21: Slips and other mistakes that tutors don’t dwell on

Line 278 is an example where the student wrote the expression with the units in the expression and the tutor did not dwell on the error and simply told the student to leave off the units inside expressions. Possibly, some errors would be explored with tutoring if a novice made them but those same errors might be interpreted as slips if a more advanced student made them.

Tutor: Your answer of 80-40*m looks like it might be right but you used 80 instead of 800. Please fix your answer.

Op22: Tutor focuses attention on a previous answer that was more correct

This appears to happen on the many different answers the student gave between 17 and 145, but is hard to identify.

Tutor: Your answer of 800+40m is close. Try again.

Student: 2400+40*m.

Tutor: No- you are getting colder. Let us go back to your first answer of "800+40*m" and work from there. Now what is...

Op23: Tutor comments on the repair the student attempted

*The lines 174-182 show a student who was able to get the two separate parts, and then got tutored simply on the repair to combine the two expressions together. In general, this requires additions to student model to model the process of creating new answers by modification of existing answers*.

** **Student:** **-800-40*m

** **Tutor: No, that is not correct. See if you can see for yourself what is wrong and try again

** **Student: -800+40*m

** **Tutor: No. You changed the expression to be an expression that gets larger as "m" gets larger, but in fact the distance to the dock should decrease as the number of minutes she rows increases.

Op24: Levels of Specificity

This operator is orthogonal to the above operators, since all the tutors remarks can be more or less specific.

Tutor: 1) Your answer has 2 errors

2) You are missing parentheses

3) You need parentheses around the 30-b.

4) Your answer should be 5*b + 7*(30-b)

Op25: Be able to differentiate a close answer from a very wrong answer, from an unintelligible one

This operator is also orthogonal to the above operators. Presumably, most of the above operators can be made more or less helpful in the degree that they get more specific in hinting as to what the student should do.

Tutor: No, but that is close. I can see you only made one mistake.

versus Tutor: No that is not correct. Let me help you...

versus Tutor: I am sorry, but I can’t understand what you wrote. ...

Op26: Engage the student to try to diagnosis what the student was thinking

Line 20 and 24 are both examples where the tutor asks the student first for simple clarification, and then for an explanation as to how the student arrived at a part of the answer.

Tutor: Why do you say 40/m?

APPENDIX B: A Protocol of an Experienced Tutor

In the following transcription the student’s written remarks are indicted in brackets. The tutor is an experienced current middle school math teacher. The student is a seventh grade male student who is a student in the tutor’s classroom. The student had a list of problems in front of him and each problem’s text is reprinted (__underlined__) when the student reads the problem. The student had a blank sheet of paper on which he wrote his answers. Generally, his paper included only attempts at symbolization with a few accompanying words possibly indicating the units. The session lasted approximately one hour. The session consisted of 17 problems, 8 of which the student answered correctly on the first try. This transcription was made from a video tape of the session. Pauses are indicated with colons and one colon indicates about one half of a second pause.

2. TUTOR [Opening remarks and asks student to read out aloud and begin]

3. STUD [Reads problem] __Mike starts a job at McDonald's that will pay him 5 dollars an hour. Mike gets dropped off by his parents at the start of his shift. Mike works a "h" hour shift. Write an expression for how much he makes in one night?__

4. [writes "h*5=how much he makes"]

5. TUTOR That’s' right number. #2

6. STUD [reads problem]* *__Mary opened a new music store. She got CDs delivered on her first day. She got 5 truck loads of CDs delivered. Each truck that arrived dropped off 12 boxes. Each box she received had "c" CDs. She sold CDs for 11 dollars each. How many CDs were delivered that first day?__

7. [writes "5*12*c=# of CD’s"]

8. TUTOR OK

9. STUD [reads problem] __Ann is in a rowboat in a lake that is 2400 yards wide. She is 800 yards from the dock. She then rows for "m" minutes back towards the dock. Ann rows at a speed of 40 yards per minute. Write an expression for Ann's distance from the dock.__

10. [writes "800-11*40"]

11. TUTOR OK I know there are some harder ones in here. :::::

12. TUTOR Read out aloud so we know what's wrong.

14. STUD [reads problem] __Ms. Lindquist is a math teacher. Ms. Lindquist teaches 62 girls. Ms. Lindquist teaches "f" fewer boys than girls. Write an expression for how many students Ms. Lindquist teaches.__

15. [writes "62+(62-f)= # of students"]

16. TUTOR Good. All right. Number of students. Cruising!

17. STUD [reads problem] __Mark went to the store to buy 4 jackets that cost "d" dollars each. When he got there the store was having a sale of 1/3 off the usual prices. How much did each jacket cost him?__::::::

18. STUD Are the jackets all the same price?

19. TUTOR Yup. Each.:::::

[writes "(d-2/3)*4="]::::::::::::

20. TUTOR What's that?[points at the subtraction sign] Is that a subtraction?

21. STUD That's two thirds

22. TUTOR You mean that subtraction?

23. STUD Yeah, that’s a subtraction of two thirds.

24. TUTOR OK. Can you.. Is two thirds.. What does two-thirds represent?

25. STUD 33 percent

26. TUTOR Right.. Or.. I mean. Not, Not, what is it in percent but what does it represent?

27. STUD 2/3 of the whole price.

28. TUTOR Is that what you are subtracting?

29. STUD Yeah

30. TUTOR Are you subtracting two thirds of the whole price?

31. STUD Yes

32. TUTOR What if.. What if the jacket costs 9 dollars, How much would you be subtracting?

33. STUD Wait hold on ::: Oh that would be wrong.

34. [crosses out previous line and writes "(d-33%)*4 = money spent"]:::::::::::::::::::::::::::::::

35. TUTOR Um. So if instead of "d" dollars it cost ten dollars, what would the answer be? So if it cost 10 dollars what would the answer be? Not ten dollars. Ten dollars is bad; what if it cost 9 dollars? What would the answer be.

36. STUD Three. No- it would be six dollars.

37. TUTOR It would cost six dollars. Is your expression gonna get that? Really, what are they actually asking for?

38. STUD Um.. Home much did each jacket cost?

39. TUTOR Right.

40. STUD Then this would be...

41. TUTOR That's not necessary [ Tutor points to the "4"]. So would your expression work?

42. STUD [crosses out the incorrect "4" from his answer of "(d-33%)*4" ] Yeah, I think so

43. TUTOR Try it. How would you get 6 using your expression. Because your expression should give me that answer that you know it’s right.

44. STUD It would be nine minus 33% equals ::::[writes "9-33%=6"]:::: 33% is three, yeah, so the number would be 6.

45. TUTOR So how did you get that three?

46. STUD Well one third of nine is three.

47. TUTOR Right. You have to tell your reader that. That's what your expression is all about. It’s telling the person to do that. You haven't told the person to do that. That you.. You know instinctively to do it but you haven't told the person to do that

48. STUD So I should right 33% percent as one third? [writes "33%=1/3"]

49. TUTOR Well.. But are you just gonna subtract one third or are you gonna subtract.33333 Is that what you are subtracting?

50. STUD Well I am only subtracting like :::

51. TUTOR Yeah! No! - so what are you really subtracting. How did you calculate that?

52. STUD Well. :::::::::::::::::::: Well if the whole thing is nine.

53. TUTOR Ah uhm

54. STUD And making 33% plus 33% ,33%, equal 100%

55. TUTOR Equals one hundred PERCENT

56. STUD Uhm. Yeah, 33% could equal three

57. TUTOR How did you calculate that? ::: Just doing that little piece, what would you do to get that three. What did you do in your head?

58. STUD I transfer 33 to 1/3 and

59. TUTOR [interrupts ] Right- oh- you could have. And you never really needed the 33, you could have just used the third.

60. STUD [continuing] and divided nine by three which equals three,

61. TUTOR Which equals three.

62. STUD But I am not using one I am using two

63. TUTOR Right

64. STUD equals 6 [wrote 9/3=3*2=6]

65. TUTOR So, So you could.. So you did. Is that how you got your six? Oh did you get your six this[points to the expression "9-33%"] way. This is different. Here you are using subtraction and here you multiplied by two. So you need to pick one way or the other and write a complete expression

66. STUD OK.

67. TUTOR Because either way is fine

68. TUTOR We could even right both. Let’s write both. Let’s write an expression this way [points at "9/3=3*2=6"] and write an expression this way[points at "9-33%"].

69. STUD OK.

70. TUTOR But you have got to tell your reader to do this thing [points at "9/3=3*2=6"], because that's the thing you forgot to tell your reader.

71. STUD OK. ::::

72. TUTOR The person who is gonna use your formula, because you are making a formula.

73. TUTOR Because if somebody else came along, and used this they would go nine- minus 33%, or nine minus point 333. What is nine minus 1/3?

74. STUD Nine minus one third is 2/3

75. TUTOR No. Nine minus one third is eight and two thirds. Right?

76. STUD Well, one third.... So I have to mention that minus, Um, one third, of whole number

77. TUTOR Of what whole number

79. STUD Of the whole

80. TUTOR Right which is?

81. STUD [wrote "9-1/3 of whole=2/3"]

82. TUTOR Yeah, exactly, which is nine. Which actually is what in the original problem?

83. STUD Um, "D"

84. TUTOR "D", So how would we write the expression this way?

85. STUD This way?

86. TUTOR Yup, so skip a line...with using d. now write the expression

87. STUD Minus ::::::::[wrote "d-1/3=2/3="]:: equal :::2/3 :::which equals :::I don't know

88. TUTOR Nine minus one third. Does that give you the answer

89. STUD well no

90. TUTOR No, because that gives you eight and two thirds

91. STUD Yeah

92. TUTOR What do you have to do. Because if you have to do it you have to right it down for your expression?

93. STUD Say, it’s of the whole number

94. TUTOR of the whole number. You need to write that. Let’s get rid of that by rewriting d minus :::::: and of ::: and of means what

95. [writes "d-1/3 of whole number"]

96. STUD Times

97. TUTOR Times! So you can use one of your minus, plus [operators I guess]:::: So now let’s rewrite this one. This one looks great, but let’s rewrite it using just letters and times symbols instead of the words. "D minus"

98. STUD I think it.. Would it be that way?

99. [writes "d-1/3"]

100. TUTOR Ahhum, But don't forget the, That's only this part. That's one third of the whole number. And then you got to go. Don't forget this part [points at the "d-" I assume] "d" minus

101. STUD Where does that go in? [that I presume refers to the "d-"]

102. TUTOR Where do you think it goes in?

103. STUD I would just disregard that part

104. TUTOR Well what is this gonna give you[points at d*1/3 I think] Let’s say d is nine, what's this gonna give you

105. STUD Um. 6

106. TUTOR One third of nine?

107. STUD No, it’s gonna give me three

108. TUTOR It’s gonna give you three, Which you know is not the right answer. Right? What do you want to do with that three?

109. STUD Times the other two

110. TUTOR Oh- OK

111. STUD I means d times 2 equals 6 [writes =d*2=6]

112. TUTOR Well what's d? D is nine.

113. STUD Oh yeah

114. TUTOR You don't want to multiple d times two, what do you want to multiple by times two

115. STUD Nine...if you are using...

116. [drops pencil in exasperation and holds head]

117. TUTOR Well let’s go back to the third. This third, When you right 33% that's just the percent, that's not the quantity. You want to subtract the quantity, not the percent, or the fraction. You don't want to subtract a third you want to subtract a third of the whole number, or a third times the whole number, SO instead of writing just 33% or just one third, you want to right THAT[ points at 1/3d], because that's the quantity not just the fraction, because the fraction just tells you what part, it doesn't tell you exactly how much. A third could be a third of two thousand, it could be a third of seven. So let try. So you want to take your number and subtract 33% but not 33%, you want to subtract 33% OF[emphasized] the whole number. So how would you write that.

118. TUTOR That's the

119. STUD Yeah

120. TUTOR You did this

121. TUTOR That's 33% percent. This is a good part of your expression. So how would we write the whole expression? there are two different ways. And you have played around with both of them. Let’s stick with this one for now. This one you take d, your whole amount, and what do you want to subtract. What exactly do you want to subtract?

122. STUD I want to subtract, Actually I want to add one third to it

123. TUTOR You want to add one third? Oh, You want to take a third and then another third. and add the two together, or multiple by two

124. STUD Add one third of the whole number

125. TUTOR OK- So write that down. So you want to take

126. STUD I want to take the 1/3 of d.

127. [writes 1*3d +1/3d "]

128. TUTOR Ahhum. Good. And you got the one third of d not just one

129. STUD Times, I mean plus, another third of d

130. TUTOR Good, and that's one way to right the expression

131. STUD which equals two-thirds

132. [ adds "2/3d=?" to the line with 1/3d+1/3d ]

133. TUTOR Right

134. STUD times d

135. TUTOR Right

136. STUD equals

137. TUTOR Good, and that is one way to write the expression and that's a perfectly good way. If you know a third is coming off, you know two thirds is left.

138. STUD Yeah

139. TUTOR You could have also used this way, which you originally started. You could have done "D" minus a third of d, because that also gives you two-thirds of d. But you can't do, You would have to say "d minus 1/3 of 3, you can't just say d minus a third because you can't just subtract a third. A third of what? So you have to have the "of what" part. Either of those expressions are great! [tutor writes "d-1/3d"]

140. TUTOR See it?

141. STUD Yeah

142. TUTOR This[points at d-1/3 presumably] is where you fell apart, because you wanted to subtract a third, but you just can't subtract a third but you can't just subtract a third of something, you have to subtract a third OF something, and that "of" triggers multiplication. See it

143. STUD Yeah, that's how I would have done it in mind, because I understand what I mean

144. TUTOR Right, Exactly, but when you are writing an expression you are writing it for any old person who comes and uses that formula. Right? Or , more specifically, but in your case, you will probably be writing this for a computer. If you are writing this for a computer your computer has to.. you have to be VERY specific for your computer, or it’s not gonna do what you want it to do. Right?

145. STUD Yeah

146. TUTOR OK, let’s try number 6

147. STUD [reads problem] __Sue made 72 dollars by washing 6 cars to buy holiday presents. She decided to spend 32 dollars on a present for her Mom and then use the remainder to buy presents for each of her s sisters. She will spend the same amount on each sister. How much can she spend on each sister?__

148. STUD Now uhm ::::::: times thirty two equals :::::: 40 writes "72-32=40 "]

149. TUTOR Remember that we are trying to write an expression, we are not trying to do any work.

150. STUD Yeah, divided by "s" equals money spent on my sisters [continues on writing a single line by adding the "/s" onto the line above to get "72-32=40/s=money spent on sisters"]

151. TUTOR Now write one big expression that shows everything, without having , without having done any work

152. STUD Can I just use like numbers or letters,

153. TUTOR Yeah, WELL use the numbers, but here. You actually.. Remember how I talk about all the time, this[?] is not really equal to this. Are these two things really equal? [points to the "72-32" and the "40/s"]

154. STUD Ah NO

155. TUTOR [repeats his no] Kind of yucky, so let’s down here right one expression, without doing any math. Pretend you forgot how to subtract. Can you right an expression with doing any of the subtraction, division, or multiplying?

156. STUD Yeah, I think so

157. TUTOR That shows the whole thing?

158. STUD :::::::[writes 72-32=a/Stud] ::::::

159. TUTOR Right, instead of a, let’s just use this. Pretend

160. STUD Oh I get it :::: it will be 72 minus 32 equals something minus thirty two divided by s.

[writes 72-32=72-32/s = money spent"]

161. TUTOR so really you don't want to put the equal sign, you just want to write that Now!, what is being divided by s? because order of operation says you divide before you subtract. So what is really being divided by s?

162. STUD Ah [writes parenthesis]

163. TUTOR right, now you wanted to write this equals, because you want to go one step at a time. Bad habit, because then you get equal signs between things that aren't equal. So here

164. STUD So here I really don't need this [??]

165. TUTOR Right. You really want this [??], and then we do...then we work down, to show that everything above it is equal to [inaudible]

166. STUD It’s just because I can't write on this [student crosses out "72-32=" that was part of the "72-32=72-32/4"

167. TUTOR So you see what I mean?

168. STUD Yeah

169. TUTOR [talks over the student] Yeah, that's like forgetting to capitalize at the beginning of the sentence. It’s just yucky. OK Laughs

170. TUTOR What if someone said it wasn't thirty-two dollars, this doesn't give me the right answer. I want to spend 30 dollars on my mother. Now you can change it easily. Here, it not as easy to change, because you don't know where the 40 came from. OK, number 7what does number 7 says

171. STUD [reads problem] __John and his wife Beth have been saving to give the 5 children presents for the holidays. John has saved 972 dollars for presents and Beth has saved "b" dollars. They give each child the same amount. Write an expression for how much each child gets.__

172. STUD The same amount between them, or the each amount between just one person.

173. TUTOR It doesn’t matter which way.. If each person gives them the same amount, there gonna get the same amount at the end, right?

174. STUD Ah, yes. So it would be 972 divided by 5 equals = a [wrote "972/5"]

175. TUTOR Just leave, because well make one big expression.:::::

176. STUD "b divided by 5" wrote onto 972/5 = b/5"

177. TUTOR So how much does each kid get?

178. STUD Ah. :::::::They get :::::::They get h+c? I don't understand this one part

179. TUTOR Hoe much are they gonna take in?

180. STUD They are gonna take in this divided by 5 and this divided by 5

181. TUTOR AND

182. STUD Ah Plus, So it would be :::::::::writes "(972/5)+(b/5):::

183. TUTOR Next one. Do you need those parenthesis there?

184. STUD Not really

185. TUTOR Why not?

186. STUD Ah, because division get done first left to right

187. TUTOR Right so those aren't necessary [(b/5)+(972/5)]this one here, those were. You need to tell your person you got

188. STUD Because you have to minus the two numbers first

189. TUTOR right, you got to do the subtraction first

190. STUD [reads problem]__Bob left at 3 P.M. and drove 550 miles from Boston to Pittsburgh to visit his grandmother. Normally this trip takes him "h" hours, but on Tuesday there was little traffic and he saved 2 hours. What was his average driving speed?__

191. STUD ::::Well ::: Ah :::: so he save two hours :::ahum:::::::::

192. TUTOR Do you know how to calculate average driving speed?

193. STUD I think, but I forget

194. TUTOR Well average speed, as your mom drove here did she drive the same speed the whole time.

195. STUD No

196. TUTOR But she did have an average speed. How do you think you calculate he average speed?

197. STUD It would be h hours divided by 550 miles an hour.

198. TUTOR So which way is it? It’s miles PER hour. So which way do you divide?

199. STUD It would be 550 divided by h

200. [write 550/h=mph"] OK so now, that's how you calculate miles per hour. So now how about for this problem? Read the problem again. Because you got the right idea. You know how to calculate average speed. But what exactly do you have to do for this trip

201. STUD Um. Well he save two hours, but I don't know how that is important

202. TUTOR Well how do you calculate... Not for Bob but for your mom, how did you calculate what her average speed was driving to CMU this morning?

203. STUD Ahm, I guess you would I would have done it 550 divided by h

204. TUTOR yeah [even though the 550 is not for his mom?] That's how you calculate average speed but what exactly is it? 550 represent what?

205. STUD Miles per hour

206. TUTOR No.

207. STUD Oh 550 miles

208. TUTOR Right

209. STUD Divided by h

210. TUTOR Which represents?

211. STUD Miles per hour

212. TUTOR No what does h represent?

213. STUD Hours

214. TUTOR Hours! So what are you getting? What are you dividing by what?

215. STUD Oh miles divided by hours.

216. TUTOR Right TOTAL miles divided by

217. STUD Total hours

218. TUTOR So let’s calculate it for this guy, That's exactly the concept, TOTAL miles divided by TOTAL hours [writes "550/h"]

219. TUTOR Is that what it is?

220. STUD Yeah

221. TUTOR Is 550 the total miles? [neat!]

222. STUD Yes

223. TUTOR Is h his total hours?

224. STUD Yes

225. TUTOR Is it??

226. STUD Oh no h-2

227. TUTOR OK- right this again and write it correctly so that order of operations and stuff works

228. STUD [Writes "550/(h-2)"]

229. TUTOR Exactly, so where did the 2 go in?

230. STUD The two hours he saved on traffic

231. TUTOR To calculate the total hours, so good.

232. TUTOR How we doing, we got lots of time. All right thinking harder. These are pretty good. Let’s try number nine. Okay

233. STUD Okay

234. TUTOR [Laughs and mentions hard work]

235. STUD [read problem]__Julie was trying to raise money to help fight Cancer. She got 7students to each donate "s" dollars and "t" teachers to each donate 10dollars. Write an expression for how much she collected?__

236. TUTOR Number 10, or no number 9

237. STUD [writes 7*s +t*10=money to fight cancer"]

239. TUTOR Good. Next problem

240. STUD [reads problem] __Cathy took a "m" mile bike ride. She rode at a speed of "s" miles per hour. She stopped for a "b" hour break. Write an expression for how long the trip took?__

241. STUD uhm :::::::::::::::::::::: writes "s/m+b"::::::::::::::::::::::::::::::::::

242. TUTOR How do you calculate the amount of time it takes you? If your, if your, if your riding at, let’s make it simple. If you are riding at 20 miles per hour, OK and you go 100 miles, how many hours did that take you?

243. STUD Um 5

244. TUTOR 5 and how did you get that 5? How did you use the numbers 100 and

245. STUD 100 miles divided by miles per hour

246. TUTOR So you took the miles and divided it by the [garbeled, probably "speed"]

247. STUD Miles divided by s plus b equals time [writes m/b+t ]

248. TUTOR Right, OK, whenever I get these.. did you see how I had to stop and think? I have stop and think for these to? so I always remember to stop and think, which way do I have to divide, because I know I have to divide, which way? OK? So you have to figure out which that is? OK number 11

249. STUD [reads problem] __Debbie has two jobs over the summer. At one job she bags groceries at Giant Eagle and gets paid 5 dollars an hour. At the other job she delivers newspapers and gets paid 7 dollars an hour. She works a total of 30 hours a week. She works "b" hours bagging groceries. Write an expression for the total amount she earns a week.__

250. STUD ::::::::::::::::::::::::::::::::::writes b*5+(30-b)*7::::::::::::::::::::::::::::::::::::::

251. TUTOR Beautiful, excellent, good work That one was tough

252. STUD [reads problem]__Michael starts a business selling lemonade. He buys 35 dollars worth of supplies including lemons, pitchers, cups and advertising. He sells a 16 ounce glass of lemonade for 2 dollars. If he sells "g" glass of lemonade, how much profit will he end up making__

253. STUD ::::: 2 dollars for each cup:::35::: minus 2 * g [writes "35-2*g"]

254. TUTOR Which number do you want to be bigger? Which number is gonna be bigger? 35 or 2*g?

255. STUD Ah.. ::: well let’s see what it’s gonna be Else he loses money and [writes "2g-35"]

256. TUTOR So this would calculate how much money he lost? Now if you do this, will this calculate how much money he lost?

257. STUD No

258. TUTOR What if he sold 5 glasses what's the answer gonna be.

259. STUD Ten

260. TUTOR That would be ten. What's ten minus 35?

261. STUD Oh I got it., money made, then 35

262. TUTOR No let look at this I think you can. This should work for both, because what's the answer? If he sold 5 glass what would the answer be?

263. STUD Um - minus

264. TUTOR What does that represent?

265. STUD Well you could just take off the minus and it would be how much money he lost

266. TUTOR Exactly. So that minus just represents lost. So this works for both of them

267. [writes 10-35= -25"]

268. TUTOR So the math even at negative number, those negative numbers allow it to tell us all the answer, even with one expression, that's why those negative numbers are so cool.

269. STUD [reads next problem] __A jacket that normally cost d dollar goes on sale for 2/3 of its original price. How much does the jacket cost on sale?__

270. STUD We already did this, but :: two thirds time d

271. [writes 2/3*d]

272. TUTOR And the only thing I am gonna tell you is that when you write it like that you have to put parenthesis, because otherwise it looks like 3 times d or you could write it like this

273. STUD Actually I think...

274. TUTOR which is the same thing

275. TUTOR point 66666 which is 66%, but this way is always better.

276. STUD [reads problem] __Rebecca makes a "h" hour car trip. For 3 of those hours it was raining and Rebecca drives at 40 miles per hour. The rest of the time it was sunny and she drove 55 miles per hour. Write an expression for the total distance Rebecca drives.__

277. STUD h-3 times 55 :::: would be ::::: plus ::::::::: 3 times 40 [writes "(h-3)*55mph + 3*40mph"]

278. TUTOR Right and usually with the expression we don't put the units in

279. STUD [reads problem] __John drove 300 miles to grandmother's at 30 miles per hour. He drove back at 20 miles per hour. He drove a total of 600 miles. What was his average speed?__

280. STUD 600-30+20 divided by :::::::::::::: two :::::::: no this parts wrong ::: writes 600-[(30+20)/2] and then scratches out the 600-"

281. TUTOR Right

282. TUTOR That [points at (30+20)/2"] looks great but it doesn't work. OK You would think it would, you are just averaging, but it doesn't work. What did we define average speed as earlier?

283. STUD Um

284. TUTOR Had the words total in it. Had to do with totals.

285. STUD Ah total : Ah total miles plus 50, plus total miles per hour

286. TUTOR No, no. To calculate miles per hour what do we need?

287. STUD Miles and hours

288. TUTOR Miles and hours. We need TOTAL miles and what else?

289. STUD Total hours

290. TUTOR Exactly

291. STUD So:::

292. TUTOR So for his complete trip

293. STUD So seeing it’s 600, it has to be half, 300 miles could be the first half

294. [writes "300"]

295. TUTOR You have to deal with this idea of total

296. STUD I am gonna figure out the hours for each half of the trip and then add them together.

297. TUTOR Exactly

298. STUD Uhm ::::::::::::::::::::

299. TUTOR You can, pick an easy problem and figure it out

300. STUD :::::::::"writes divided by 30=10 hours 300/20=A"::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

301. TUTOR Remember that you don't need to do the calculations because I am gonna ask you at the very end to actually just write the expression without doing the math. OK.

302. STUD So it would be 10 plus A? mumbles

303. [writes "10+A" and then adds in front of it "600*"]C; is that how you calculate miles per hour

304. STUD Ah no

305. STUD Mumbles [writes "(10+A) * 600"]

306. TUTOR That just the exact same thing you had before, expect you are that you are just multiplying "A" times 600 but I suspect you want to do the whole thing? How did we calculate. Once again what is the definition of average speed? Average miles per hour?

307. STUD Um

308. TUTOR Tell me out loud. Definition of average speed

309. STUD :::

310. TUTOR It has to do with total. The word total has to be there.

311. STUD :::

312. TUTOR How did we do it over here?[points to previous problem where speed was calculated]

313. STUD 550 total miles per hour total?

314. STUD Ah, total miles

315. TUTOR OK

316. STUD Then h-2 total hours

317. TUTOR Right, so we took, total miles and did what?

318. STUD Divided Ah

319. TUTOR Divided it by total hours writes ["600/(10+A)"]OK beautiful. You weren’t given ten and given "a". What were you given? \So the person that is reading this doesn’t have any arithmetic already done.

320. STUD 600 divided by 30, no it’s 300 divided by 30..write 300/30

321. TUTOR Good that's the tens; plus three hundred divided by 20 adds "+300/20"

322. TUTOR OK

323. STUD ::::600 divided by that then divides by putting the 600/ in front of it. for a final expression of "600/(300/30+300/20)"

324. TUTOR Yup, and after next year you won't right it with that division

326. TUTOR OK the next one is a good one.

327. STUD [reads problem] __A car salesperson is paid a base salary of $200 per month plus the additional amount of money in commissions for each car she sells. She sold four cars last month and received "x" dollars that month. If she sells "h" cars this month, write an expression showing how much she earns this month.
What's her commission?__

328. TUTOR Good question.

329. STUD Oh I have to find this out don't I

330. TUTOR I think so

331. STUD Um x [minus] that

332. STUD It doesn't say how many car. Ah 4 cars last month so it would be x-200 divided by 4

333. [he puts in the parenthesis only after writing "x-200/4"]

334. TUTOR OK

335. [he has written (x-200)/4 = commissions" and then adds a "c" before the commission apparently to indicate that commission’s will have the variable "c" stand for it]

336. STUD Equals commissions or "c". So this month would be 200+c*4 = salary

337. [writes 200+c*4=salary]

338. TUTOR This month?

339. STUD Yeah

340. TUTOR Read it again.

341. STUD This month, or she sold h cars

342. [scratches out the 4 and replace it by "h"]

343. TUTOR You want to skip a line so you aren’t messy

344. [starts writing "200+c*h" and tutor interrupts after the "c" to say

345. TUTOR OK now is C something that was given to you?

346. STUD No

347. TUTOR Where did it come from? What numbers

348. STUD So have to do this [scratches out the "200+c*h" he just wrote]

349. TUTOR You have to rewrite the WHOLE expression.

350. [writes 200+(x-200)/4*h=salary]

351. TUTOR Good

352. TUTOR Let’s see if that last problem is any harder

353. STUD [read problem] __A candle that has been burning for 2 minutes is 8 inches long. Three minutes later the candle is "x" inches long. Assume that the candle will burn at this same rate. Write an expression for the height of the candle after the candle has been burning for "m" minutes.__

354. STUD So assuming that for each minute it’s x inches long it’s a little tricky

355. TUTOR Ahum

356. STUD :::::::: x inches divided by 8inches equals

357. [writes "xinches/8 inches=rate of burning " {does this quantity name the rate or the amount burnt?}]

358. STUD rate no it would be minus, [changes the minus 8 to divided by 8"] inches is rate of burning

359. TUTOR Inches minus inches is gonna give you answer in what units?

360. STUD Inches

361. TUTOR Inches, is that a rate unit?

362. STUD It would be one inch per minute

363. TUTOR Yes, that’s what you want. so the per stands for what operation?

364. STUD divided by. so I was right!

365. [starts to change the minus back to a dividing sign]

366. TUTOR No, no, no, but wait a second, you want inches divided by inches?

367. STUD No

368. TUTOR That's gonna give you

369. STUD

370. TUTOR You are on the right track because you want inches per [[left a sentence completion]

371. STUD Yes, I was think If I divided by minus x inches by minus 8 inches it would be the inches per minute

372. TUTOR No what does that give you?

373. STUD That would be x-8

374. TUTOR Which is longer, x or 8?

375. STUD x

376. TUTOR [Read the problem again]

377. STUD "assuming the candle has been burning for 2 minutes is eight inches long. Three minutes later the candle is x inches long."

378. STUD Oh three minutes later. so it would be five minutes "assume that the candle will burn at the same rate. Write an expression for the height of the candle after the candle has been burning for "m" minutes. :::::

379. TUTOR Now you got a good start. I like what you did but what does it give you?

380. STUD It give me uh um the number of inches the candle has burned in the time of three minutes

381. TUTOR Right! So will that help you find a rate?

382. [To the line that now reads "8 inches-x inches=rate of burning" he adds "in 3 minutes" which is improper]

383. STUD Yeah

384. TUTOR Yeah because rate is what? How is rate defined?

385. STUD Its’ ::: I don't know

386. TUTOR You just told me

387. STUD It’s the time or something like that

388. TUTOR It’s got a "per" in it

389. STUD Yeah

390. TUTOR So what is it for this situation?

391. STUD The amount of candle that has burned in so many minutes

392. TUTOR Right. So and candle burned is measured in what in this problem

393. STUD Inches

394. TUTOR Inches. it could be measured in grams of wax or something. But in this problem it is measured in inches. So we want our rate to be what? Inches [left time for completion]

395. STUD um. wait, if it’s going less it will go x

397. TUTOR Yeah-- that you can go ahead and fix, so that's gonna give you inches. How are we gonna have to fix that to give us the rate?

398. STUD And we can divide eight minus 8 by three to give us one minute

399. TUTOR That’s gonna give us the rate. Exactly!

400. STUD Eight

401. TUTOR The unit rate!

402. STUD Eight divided by oh eight. [on a new line he writes 8inches/" and stops ] I mean to say x times 8.[ changes the line to say "x*8inches/"] Oh that's wrong.

403. TUTOR I can see you are getting tired, this is our last problem.

404. STUD Eight minus x inches :: divided by [writes (8-x inches)/3=rate in 1 minute of burning"]

405. TUTOR Right

406. STUD Three equals rate for one minute mumbles

407. TUTOR OK so what's the actual question?

408. STUD m times (8-x) inches divided by three [writes "m*(8-x)/3"]

409. [Note: this is the amount that has burned not the end height of the candle but the tutor accepted the student’s answer.]

410. TUTOR Because rate times time gives you inches,

411. TUTOR Beautiful! Those are hard. Those are good ones.

End of transcript