Language and Thought in Mathematics Staff Development: A Problem Probing Protocol

by Rita Kabasakalian - 2007

Background/Context:

The theoretical framework of the paper comes from research on problem solving, considered by many to be the essence of mathematics; research on the importance of oral language in learning mathematics; and on the importance of the teacher as the primary instrument of learning mathematics for most students. As a nation, we are not doing well enough in teaching our students mathematics, and we need to look to new perspectives about why this is happening.

Purpose:

This paper looks at a promising tool for enabling students who are not skilled readers to approach problem solving tasks. But in sharing its usefulness with teachers in professional development, the instrument uncovered specific and serious gaps in teacher content knowledge. Group discussion, using the protocol, helped to clarify some of those gaps.

Setting:

All three problem-probing sessions occurred during professional development activities for New York City Public School mathematics teacher leaders. The facilitator was Academic Director of a New Visions for Public Schools’ Middle School Mathematics Standards project, (MS)2, funded by NSF. The sessions were part of a variety of ongoing professional development activities, and occurred during the third year of a five year study. Therefore, the 12 to 20 participants of each session, though different from each other, all knew the facilitator.

Research Design:

The research design was qualitative. Teachers were presented with a written problem and directed not to solve it. The discourse that followed flowed of its own accord, with participants speaking as they thought and responded to the facilitator and/or the other participants. Records of speakers’ words were transcribed in one case by hand, and in two other cases by audio tapes. The discourse was then analyzed by the researcher from two perspectives: what mathematical misconceptions were revealed and how the talk helped to clarify them.

Conclusions/recommendations:

From preliminary explorations, the protocol seems an accessible and effective instrument for decoding and analyzing mathematics word problems by teachers who were not able to approach such tasks before. Also suggested was its potential value as an assessment instrument for teachers’ content knowledge. Further study is needed to determine what kinds of in-class supports teachers need to internalize the process and use this instrument on a regular basis with their students.

Mathematics consists of content and know-how. What is know-how in mathematics?

 The ability to solve problems (Polya, 1971, p. 574).

INTRODUCTION TO THE ISSUES

Success in middle school mathematics, notably in algebra, is considered to be a reliable predictor of later school success; yet many students in large urban school districts are not learning either mathematics content or know-how. This puts the future prospects of large numbers of urban children in jeopardy. Further, the relatively poor performance of American eighth graders in international mathematics assessments (TIMSS, 1995, as discussed in Kelly, 2002 ) predicts inadequate numbers of workers skilled in science and mathematics which could affect U.S. performance in global markets (Mathematics Equals Opportunity, White Paper, 1997). Part of the problem is that learning abstract mathematics concepts requires the ability to process language, an ability which large numbers of middle school students still need to learn.

To help middle school children learn these abstract ideas, teachers need to know elementary mathematics well enough to communicate in language that conveys concepts clearly. Further, the concepts must be conveyed so that students are enabled to form systems of mathematical thought.  If the teacher does not know the mathematics or the language for conveying the ideas clearly, the child is likely to be confused and may, with repeated frustration, turn away from mathematics altogether (Skemp, 1987; Davis & Hersh, 1981).

Skemp (1987, p. 21) asserts that  to know mathematics is one thing and to be able to teach itto communicate it to those at a lower conceptual level is quite another; and I believe that it is the latter which is most lacking at the moment. A vast amount of research attention has been directed both to the problem of an inadequate base of teacher knowledge in mathematics and to the more intricate understanding necessary for conveying its ideas to students. Ball, Lubienski and Mewborn list the following everyday activities in which teachers pedagogical content knowledge is required:

"

to puzzle about mathematics in an unanticipated idea or formulation proposed by a student;

"

to analyze a textbook presentation;

"

to decide to change the numerical parameters of a problem;

"

to make up homework exercises; to consider the relative value of two different representations in the face of a particular mathematical issue" (2001, p. 453).

All of these tasks are rooted in mathematics knowledge; they also occur in the actual experience of classroom practice. Much of this experiential knowledge is held by teachers in disparate notions, and often privately; it therefore does not become either systematic or part of the communal knowledge of teachers (Hiebert, Gallimore, & Stigler, 2002; Ball, Lubienski, & Mewborn, 2001). Teachers opportunities to learn are&a crucial feature of improving instructional capacity (Cohen & Ball, 1999, p. 15). These researchers argue that when teachers work in communal groups, sharing, discussing, and reflecting on their mathematics and pedagogical perspectives, both content and pedagogical content increase for all participants. Yet, as school districts attempt to deal with the constraints of time, money, and unrealistic demands for improving standardized test scores very quickly, providing time and place for teachers to work in study groups becomes a low priority.

PURPOSE OF THE PAPER

The purpose of this paper is to address the dynamic relationship between mathematics discourse and mathematics problem exploration as a tool for staff development both in and out of study groups. Specifically, the paper discusses a protocol developed in our Teacher Leader Seminars for talking deeply about the mathematics embedded in problems. Often, teachers feel constrained about revealing the gaps inor misconceptions abouttheir mathematics knowledge or analytic skills in a group setting. There is little in the research literature which addresses this combination of issues. The promising instrument which emerged from our effortsthe Problem Probing Protocoldoes just that. By design, it provides a safe venue for teachers to talk about the underlying mathematics in a problem, the language in which it can be illuminated, and the pedagogical strategies for conveying it effectively. Further, the protocol is elegantly simple, and can be used both by teachers themselves in study groups, and then directly with students in the classroom.

THE CONTEXT

To foster the practice of mathematics teachers working in groups, New Visions for Public Schools Middle Schools Mathematics Standards (MS)² project (funded by the National Science Foundation) helped to put in  place regularly occurring study groups in which teachers, with a teacher leader, would together improve their practice of mathematics. These study groups proved to be an effective vehicle for getting teachers to share and refine their teaching experiences; and in some instances, the level of mathematics discourse was dramatically raised in the schools. In addition, New Visions staff conducted regular seminars with regard to the goals of the project: improving content, pedagogy, and assessment, in the context of the study group model. Through these seminars, Teacher Leaders from several schools extended their effective learning communities beyond the boundaries of their schools, even generating and carrying out amongst themselves interschool staff development.

However, our observations of classrooms and study group meetings, as well as close interactions in seminars with the same teachers, pointed out some severe constraints of the study groups we helped to put in place.  Working in four of the most underserved districts in New York Citys public schools, we found that teachers did not necessarily know how to organize mathematics conversations within the group.  If that was the case, teachers content knowledge was unlikely to improve. Others (Supovitz,& Poglinco, 2001) have found that developing meaningful communities around instructional practice is not an easy task.  Groups may have a relatively easy time developing comfortable social interactions, but it is more difficult for them to develop sustained communities of practice around instruction (p. 7). The protocol described herein provides an accessible tool, a structure, for talking about mathematics ideas and instruction in those communities.

THE LITERATURE: THE VALUE OF LEARNING MATH THROUGH PROBLEM SOLVING

Solving mathematical problems has been a subject of investigation and speculation for many thinkers and philosophers from ancient times. It is no surprise, therefore, that middle school students find mathematics problems, particularly non-routine verbal problems, very challenging. Yet, good word problems embed within themselves a number of different math ideas. They therefore can serve as a powerful way to explore mathematics concepts and form connections among them.

Many thinkers have provided models for how to approach these types of problems. Among the most notable are those of Polya (1971) and Schoenfeld (1985). Two precepts underlie many of these models:  first, that heuristics, or protocols, are needed; and secondly, that solving problems in a social setting in which participants both buttress and challenge each others efforts is more effective than working alone. (Polya, 1971; Vygotsky, 1986; Schoenfeld, 1985; Resnick, 1994). Noddings (1989) addresses the relation of language to mathematics problem solving, and more particularly, the constraints imposed on mathematical probing by poor analytical reading skills.

The heuristics do indeed offer students ways to uncover text; however, these ways are embedded in the discourse of the researchers referred to abovediscourse that is generally unread by a classroom teacher. Rather, posted in many classrooms are very short lists of proceduresalmost sloganswhich are relatively inaccessible to people who are not experienced problem solvers or, more generally, skilled readers (Schwartz, & Riedesel, 1994, p. 75).

In a study on middle school childrens problem solving strategies in mathematics word problems, Stephen Pape (2004) breaks down the problem solving process into two separate parts: problem representation and problem solution.  The former depends on the students linguistic and semantic knowledge; the latter depends on mathematical knowledge. Work with middle school students and their teachers led him to believe that both of these groups were frequently unable to form coherent schemas that would give meaning to the words of a problem. They also did not have sufficient command of the mathematical knowledge. In addition to making sense of the text and then applying mathematics knowledge toward a solution, there is enormous anxiety, which imposes constraints both for the teachers and the students in approaching the task.

Given the widespread deficiencies in teachers content knowledge, and indeed confidence, it continues to be urgent to make problems accessible first to teachers with heuristics that take into account their excessive lack of content knowledge and inadequate skills in probing text. Pape (2004) suggests that teaching students to be strategic (in meaning-based approaches to problem solving) may require more explicit instruction than is often evident within socially constructive classroom environments (p. 213).

Experienced problem solvers have internalized ways of decoding, which come into play with more or less fluency, depending on experience, facility, and mathematical insight.  They may do the taking apart process very quickly in their heads, no longer conscious of the steps involved or, by contrast, methodically reread the problem, make lists, write equations, draw diagrams, trying one way and then another until a solution emerges. Inexperienced problem solvers and/or those who do not have the capacity to decode and process verbal language are likely to encounter several kinds of difficulties. They may not pay attention to the ideas in a sentence, but rather look for key words.  There have been those who directed students, even in routine problems, to look for words like and, difference, parts of, sum, times, as indicating that the way to solve the problem was to add, subtract, make fractions, or multiply. In effect, students were taught not to analyze the text as a whole when mathematics was involved. Often, the words that give clarity to a problem are the prepositions, the verb tenses, the comparative adjectives, the pronouns, or the number of separate events that must be accounted for. A possible trap is that poor readers are frequently inattentive or inaccurate in reading these seemingly unimportant small words. Inexperienced problem solvers may have difficulty sequencing the ideas given, thinking that they must all be understood at once or its impossible to understand anything. Experienced problem solvers recognize that a problem might take a long time, call up a number of false paths and corrections and, for some, cause a good deal of anxiety. It is the teacher who holds the key to making heuristics accessible to students (Skemp, 1987); yet middle school mathematics teachers are themselves both inexperienced and intimidated by mathematics problems (Ball, Lubienski, & Mewborn, 2001). That makes it more unlikely that they can enable students development in this complex cognitive activity.

THE RELATION OF LANGUAGE TO CHILDREN LEARNING MATHEMATICS AND THE ROLE OF THE TEACHER

Humans who are reared in nurturing circumstances are capable of learning a great deal about the world without formal instruction.  They learn through their senses and direct physical experiences.    Concepts are expressed in words.   Children, from their earliest attempts to name things, are constantly forming concrete concepts.   Such knowledge forms the basis of all school learning; it is profound and often unconscious.  The task of school learning is to engage this knowledge, make it conscious, and lead the child from the concrete to increasingly abstract thinking (Vygotsky, 1986; Skemp, 1987).

Abstract ideas, such as those which form the structure of mathematics, however, cannot be learned from the everyday environment.  These must be imparted from adults to children in words.  Students learn the formal logic of mathematics through teachers informed use of words (Vygotsky, 1986; Skemp, 1987). Required is the transfer of knowledge from someone who knows the mathematics to the child who comes with his/her own experiences and ideas of the world, and the language in which those experiences are framed.  That makes the learner largely dependent on the teacher; at worst, it exposes the student to the possibility of acquiring a lifelong fear and dislike of mathematics (Skemp, p. 18).

No one questions the idea that what a teacher knows is one of the most important influences on what is done in classrooms and, ultimately, what students learn (Fennema & Franke, 1992, p. 147). Not only must the teacher (of middle school students) thoroughly understand the basic concepts of elementary mathematics, but he/she must also be able to discern with accuracy which of these is not understood by the students; otherwise, a familiar picture emerges.  The student flounders in an unrelated set of meaningless (to him) statements; the initial anxiety and feeling that he himself is flawed leads to permanent math phobia. Skemp (1987) traces one root of mathematical illiteracy to the fact that as mathematics is frequently taught, students are not exposed to its elegance and power. Rather, emphases are placed on masses of details and procedures not connected to concepts or to any general structure.

Educational research in the past 15 years, according to Ball, Lubienski, and Mewborn, (2001) focused overwhelmingly on teachers knowledge and beliefs but only a small number of these studies explored how mathematics teachers knowledge affected their practice, and even fewer investigated how this knowledge or lack of it affected students learning.

The vehicle through which concepts are named, communicated, explored, and/or clarified, as related to other concepts, is language: that of the instructor and that of the learners. The need for meaningful classroom discourse is now universally accepted among educational researchers, and teachers are encouraged to use higher order questions. Redfield and Rousseau (1981) include the following as higher order: What do you mean?, How did you do that?  Why do you say that?  How does that fit with what was just said? Cazden (1988) modifies that argument by asserting that whether a given question engages the students thinking or not depends on the context.  What may seem to be probing questions may actually be closed, depending on, for example, whether the teacher has a definite answer in mind, whether the answer has just been given by the teacher, how the teacher reacts to the students response, and where the classroom conversation goes from there.

However, many teachers are not practiced in questioning techniques. Perhaps they do not know where a discussion could go in terms of furthering thinking; perhaps they are not astute in recognizing when students misconceptions can actually be used to advantage; perhaps they do not think clearly enough about the concepts being taught to clarify them for others.  Teachers who dont know how to make connections may be threatened by student questions, and may ignore and thus discourage an inquisitive student.  In any event, such sample discourse as presented in journal articles is of little practical value to an undeveloped teacher. This problem is adequately recognized, and scripted classroom protocols are sometimes advocated with the belief that the teachers assumed limitations can be virtually bypassed.

Neither strategy is adequate for enabling vital mathematics conversations in the classroom. Teachers need to be helped to think about their own thinking, to assess student thinking, and to engage in discourse that emerges from the work in the classroom. Good mathematics teaching involves a teacher who knows the mathematical concepts that students need, knows them in the context of a network of related concepts, knows what must be known before, and what will later be added to form a solid conceptual structure (Skemp, p. 18).

It has already been proposed that problem solving is an important vehicle for uncovering mathematics ideas and how they connect to each other. In general, there is widespread support of that idea. NCTM Standards of 1990 indicate that problem solving is one of the key goals of mathematics instruction; however, its implementation often falls short of its goals. Two examples of implementations which miss the mark are as follows: First, the allocation of problem solving to a 5 to 10 minute Do Now. In order to comply with that time constraint the teacher may do one of a number of things:

a.

ask if anyone has the answer, which is then shared with the class;

b.

work with students to get part of the problem solved and assign the rest for homework, in which case the answer and possibly the process will be shared with the class the next day;

c.

the teacher works out the process for the class.

If the problem is presented to students who are not experienced and/or confident problem solvers, they are unlikely to learn from any of the above procedures how to solve problems. They will not be able to generalize from superficial engagement with problems no matter how frequently such engagements occur. In working with mathematics teachers, it has been our experience that it takes an hour to an hour and a half to explore a problem in a way that makes a change in their thinking.

The second faulty implementation is one-shot professional development workshops. According to Ball, Lubienski, and Mewborn, (2001) [A] good deal of money is spent on staff development in the United States and although there is no shortage of in-service training for teachers, most training money is spent on sessions and workshops that are often intellectually superficial, disconnected from deep issues of curriculum and learning, fragmented and noncumulative (p. 437). In general, these do not make a deep enough impression to create change.  People have their accustomed ways of thinking. Adults have been thinking in particular ways for a long time; therefore, efforts to modify these idiosyncratic ways should be based on uncovering these ways of thinking, creating situations for doing so at frequent intervals in the context of that persons everyday experience.  In the case of teachers, interventions that are meant to change the ways they approach their practice must be repeated, over time, in various topics, and in the schools and classrooms. All of the above issues are reflected in the Problem Probing Protocol, a detailed discussion of which follows.

THE PROBLEM POSING PROTOCOL

HISTORY OF THE PROTOCOL:

As a teacher of middle school mathematics, I found that, in general, students were made very anxious by any kinds of mathematics problems in which words had to be interpreted. While some were poor readers, and some were second language learners, others did not have organizational skills that would enable them to impose a structure on the task.  I tried a variety of strategies from the research literature, but knowing what the problem was telling and/or asking required a degree of analytical probing that was more than my students could manage. If I worked with them on a problem, step by step, they were able to solve it; however, every problem seemed different to them and they were unable to internalize the frameworks and use them independently even after repeated efforts on my part.  Solving problems, perhaps the most important goal of mathematics education, remained inaccessible for most of my students whether they worked independently or in groups.

In the old tradition of If you cant beat em, join em, when a problem was posted for all to see, it occurred to me to say, This is a hard problem. What makes it hard?  In this acknowledgement of students experience, I seemed to have found an immediate antidote for their feelings of anxiety and ineptitude.  Naturally, they thought, if it was hard, how could they reasonably be expected to deal successfully with such a challenge. However, many were ready to suggest what was unreasonable about the problem: There are too many words, the words dont make sense, not enough information is given, those are hard numbers, I dont know how to work with fractions (or decimals, or division, or whatever). I dont see any question. All responses were acknowledged and duly written out. Of course, in that first step, they were beginning to take the problem apart.

OK, I said, Now find anything at all written in the problem. The anything was a literal anything, any detail, as long as it was written in the problem. Thus, even indifferent readers could find something to say.  And that was what happened. All responses were posted (thus honored), and gradually, a full picture of the problem was reconstituted.  But that was not necessarily enough.  The elements were listed as presented by the students but not necessarily in any order, and it was necessary to assess what was actually understood.  Thus, the next question: Tell the story of the problem in your own words.  This question was asked of a few students, who recounted what they understood. If details were left out or misconstrued, the next volunteer was asked whether he/she wanted to modify what was said. This, of course, didnt solve the problem, but it identified it and students could then consider strategies for the solution. Mayer (1992), in his analysis of mathematical problem solving and text processing theories, found two phases of this process: problem representation and problem solution. The protocol presented here addresses that first phase.

In working with middle school teachers as a professional developer, and indeed with school administrators, I found that their anxieties about non-routine, even routine word problems, mirrored the reactions of middle school children.  Understandably, it was worse for them because they, as teachers or leaders, were embarrassed, loathe to admit that the problems seemed beyond their capacity. I decided to use the same method as I had used with children, only I needed to provide a cover for the teachers embarrassment.  Thus, I asked not what they found difficult, but what their students would find difficult.  

THE PROTOCOL METHOD STEPS

The procedure is as follows: a mathematics problem is distributed to the group (of teachers, teacher leaders, coaches, administrators).  The problem can be exclusively verbal, graphic, numeric, and/or a combination of these. Instructions to participants are very simple:

1.

Do not solve the problem.

This directive is counter-intuitive. A more frequent experience for teachers (and administrators) is that they are given a problem to solve.  If on first glance the problem appears to have any off-putting aspects, the teacher or administrator (or a middle school student) puts the problem aside, and is sure that someone else will solve it. It is therefore not necessary to deal with the discomfort of feeling slower than ones neighbor, math-phobic, not good with words, or in any way deficient in mathematical understanding. The directive, Do not solve the problem, frees the teacher from that discomfort. Instead, the instruction is:

2.

Write what you think students would find difficult about this problem.

Since they are all teachers, this is familiar territory.  The task should be what teachers do with assurance.  They look at the problem with different eyes and begin to take it apart.  After a reasonable interval, the facilitator asks participants to share their findings and these are posted for all to see.  They are posted in the order given.  The list includes what the teachers may themselves find difficult (but not so identified) and what they assume students would find daunting.  This instruction also provides a model for teachers in working to help their students and sensitizes them in advance to students possible reactions.

3.

Now write anything thats in the problemany detail at all.

As before, after participants have had time to do that, the results are posted along with the original list.  By this time, each person has read the problem twice but the directive READ! was never given.  Is that important?  Yes, because many math teachers are not confident of their own skills at making meaning out of text. Further, it must be pointed out to teachers that they, when using this protocol with students, will also not use the word read because of its off-putting effects on middle school students who are not skilled readers.  Yet, by asking teachers to find whatever is in the text, we are teaching a method for unpacking text.

Each of the two first steps generates lively discussion and disagreement about the meaning of the text.  Participants keep being surprised that a seemingly simple problem turns out to be so complicated.  It is essential that the facilitator encourage and value all points of view so that a safe venue is created and preserved for participants. When the two lists are side by side, the concrete elements are plain to see, but not necessarily connected, particularly if there is verbal text.  In that case, one asks of a participant:

4.

Tell us the story of the problem.

The facilitator evaluates how well the story teller has synthesized the issues.  If the participant does not tell the story in a way that fits the problem, the facilitator may ask if anyone has another way of telling the story. If the problem is well understood, the teller often embellishes the verbal details, involving him/herself in the problem situation. If it is not understood, the hesitancy indicates where the obstacles lie and the facilitator enlists the other participants in addressing them.

5.

Work with a partner to solve the problem; write solutions or attempts on a problem sheets.

This part of the process is still emerging, and we continue to experiment with it. The dialogue cited later in this paper emerged from one group of 12 participants in which the facilitator remained part of the process throughout.  In our analysis of these data, it seemed clear that group members were taking a larger role with each other as the process continued, suggesting that the facilitators role had become less necessary in the solution part of the process.  Therefore, in later staff development sessions, we assigned participants to work in groups of four or five, with each group presenting a charted solution to the plenary group.  In that situation, we noted that not all participants were actively involved.  Thus, most recently, we directed people to work in groups of two.  Sometimes, in protest, some of the four or five people at a table asserted that they all agreed on the solution and that one solution sheet was adequate.    However, our insistence on the pair arrangement has resulted in widely divergent approaches to the problem even from two pairs sitting at the same table.

6.

Present your small groups strategies and solution to the large group.

The participants genuine engagement with mathematical ideas emerges most strikingly in the presentation phase.

 

PROBLEMS INVESTIGATED USING THE PROTOCOL

An important concern is the choice of a problem.  For working with teachers, the problem chosen should be one which addresses basic mathematical issues that relate to the middle school curriculum, but the problems should present challenges which can be uncovered in conversations. Three separate problems are considered in the text in the following pages. The challenge in the first problem lies in its grammatical construction.  It will be addressed briefly below.

Problem I: Matt averages 80% on a test, but on his most recent test he achieved only ¾ of his average. In that test, Matt got 9 answers correct.  How many problems were there on the last test?

Mathematics in the Problem

In response to the question, What would students find difficult? participants, coaches of K8 mathematics teachers, pointed out that students might not be clear about what average means, that there are different forms of numbers, percents, fractions, and whole numbers, and that it would be difficult to interpret the meaning of such varied numbers in the problem. However, the issue that generated the most discussion among the coaches was interpreting the language: averaged 80% on a test. The participants argued about whether the 80% average referred to one test or more than one. One participant offered the interpretation, Matt generally gets around 80% on a test.  Another gave the example that she averaged $5.00 for a cab ride to work; others clarified further that an average results from adding several test grades and dividing by the total number of tests. However, members of another pair persisted in their argument that the 80% referred to one test, ignored the word average altogether, and vigorously defended a completely different and inappropriate solution to the problem.

These kinds of arguments do not typically emerge in staff development sessions in which teachers (coaches) are told what to do and how to do it, with the result that their own views of  the mathematics and their understandings and misconceptions are left intact.

Problem II:  On the number line shown, locate one billion.

0___________________________________________________one trillion

Mathematics in the Problem

The challenge in this problem, in which there are very few words, lies in abstracting some basic premises of our number system: knowledge of place value, the relationship of the place value table to the number line, patterns, and in this case, patterns of powers of 10.  It also requires recognition of what can be perceived visually and what must be logically deduced because it cannot normally be perceived.

Knowledge of place value seems at first to be so commonplace as to be trivial.  But place value is associated with a table devised for convenience, which actually impedes understanding.  The columns are represented as equal intervals; however, when based on the values of the numbers they stand for, each column to the left should occupy a space ten times as wide as the column on its immediate right.  Of course it would be impossible to illustrate these relationships accurately on a sheet of paper, so practical considerations demand the display of columns of equal width. Many people hold on to that frequently viewed image, never having clarified it. Understanding the difference between the typical visual representation and the idea of powers of 10 is an abstraction. It requires a logical leap, which we assume happens.  This assumption proves, in many cases, to be false.

The number line is a graph of the x-axis, which is used for many comparative purposes. Its rules are simple: equal spaces should represent equal intervals unless otherwise indicated, and items to the right are larger than items to their left. Depending on the values represented by endpoints, the line is divided into appropriate equal intervals. This is one of the skills required in graphing and one to which teachers must pay attention. The concrete expression of a number line is a ruler with units of 1 to 12, 1 to 36, even 1 to 100 as on a meter stick. Some more refined concrete movement is made when students can count 10 millimeters in a centimeter, but that concept should be specifically taught with the physical material in hand.  However, the fact that a given length of line can represent an interval of 0 to 10, 0 to 1000, or 0 to 1 trillion is a large abstract leap for some teachers.

THE DIALOGUE

The extended dialogue which follows took place on the first day of a three-day (MS)² Summer Institute.  There were 12 teacher participants: one primary facilitator, who is referred to in the dialogue as Facilitator Rosa, and other project staff members who contributed to the discussion from time to time.  Those contributors are designated Facilitator Patrick and Facilitator Jeannie. Teachers have been given arbitrary names so that the dialogue is more readable.

Answers to question, What might students find difficult about this problem?

Francine: Students see large numbers as interchangeable; they all mean the same.

Benito: Its hard to determine intervals for dividing the line.

Josef: Its hard to know how many zeros are in one trillion.

Nero: Its difficult to visualize large numbers.

Carmen: The relative comparison between zero and 1 billion.

Winnie: Students are used to seeing single integer number lines so they dont understand that small spaces can represent large intervals.

Answers to question, What information is in this problem?

Nero: The starting point is zero.

Winnie: The end point is 1 trillion on the given line segment.

Carmen: At the end of the problem the number line should have 0, 1 billion, 1 trillion.

Facilitator Rosa: How could you talk about this problem?  How would students get this information?

Josef: Reading it, looking at information.

Facilitator Rosa: Did I ever say, Read the problem?

Facilitator Rosa, while presenting the problem, includes comments on pedagogy and empathetic ways to talk to students.  While this problem does not have many words, the probing method teaches the decoding of text without saying so:

Facilitator Rosa: I didnt say read it (the problem) because kids respond negatively. By asking whats hard, kids have to read the problem at least twice, but I never said read.

Teresa: (Weve been taught) the 5-steps of problem solving. Kids were taught this:

1) read 2) Re-read 3)Circle words. Now youre telling us another way.

Facilitator Rosa: I think its better not to say read because adolescent kids who have problems reading are likely to become anxious at that instruction.

Carmen: It doesnt work.

Facilitator Rosa: As they continue to work as a group, kids help each other construct the problem.  By the time every student adds his/her ideas a picture emerges.  You must respect every idea contributed and put it on the board.

Now, how will we solve it? Volunteers? Be brave!

I hope someone comes up with an incorrect placement because its the most constructive starting point.

Several comments have been made by Facilitator Rosa to create and to model for teachers how to create a safe environment for sharing:

Latisha:  (draws number line and locates 1 billion far to the left on the number line)

0_____._________________________________________1 trillion

1 billion

Facilitator Rosa :How did you do it?

Latisha:  I thought how many numbers are between 0 and 1 billion and how many numbers are between 1 billion and 1 trillion. So I placed it closer to 0.

Facilitator Patrick: Why did you put it at that spot?

Latisha:  From 0 to a million equals from a million to a billion.

Latisha chose an answer that seemed at first to be correct, but when prodded, revealed a misconception about the magnitude of a million as compared with a billion, which is 1000 millions:

Francine: I put mine farther to the right. (draws a number line and locates 1 billion somewhat to the right of the center of the line).

0___________________________________._________________1 trillion

Billion

Francine: I used place value to decide.

Latisha: Oh, I caught my error.

Latisha, whose original choice was more nearly correct, was quick to agree with Francine, revealing further that she had a partial understanding of the relationships of the numbers, but did not hold it securely and was therefore willing to change without understanding the concept any better.

This is the first evidence of confusion between the number line and the place value table.  The place value table is represented as a mirror image of the standard table and identical to the number line.  This misconception persisted throughout the session.  When it was not clarified by the discussion process, the facilitator raised the issue near the end of the session. Facilitator Rosa did not attempt to control the discussion, continuing to foster free talk.  Neros comment demonstrates.

Nero: Im thinking of decimals and place value so I divide the line into 3 (equal) parts.  

________________.___________________.__________________________________

O                         M                                      B                                 T

1 million = 10⁶

1 billion = 10¹²

1 trillion = 10¹⁸

Neros diagram introduced two new misconceptions, one in the context of the mirror image place value table, and the other that of decimals.  Since 0, million, billion, trillion are equally spaced on the number line, they are similarly represented on his list as equally spaced powers of 10.  The issue of decimals was not followed up. However, Neros evaluations of the powers of 10 reflect training in another culture.  In some European and African schools, our billion is called a milliard (10⁹) and 10¹⁸ is called a trillion. This point was unknown to the writer at the time of the discussion. Nero has not resolved for himself the difference in terminology, but one can see how such misnomers can mislead students. Since this was not clarified, the discussion continued with Winnies understanding of Neros language; her comment repositioned the dialogue:

Winnie: I have a question. Are there 18 zeros in a trillion?

Carmen: No.

Teresa: It has 12 zeros.

Winnie: Shouldnt it be 1 billion is 10⁹?   1 trillion is 10¹². I put a billion close to a trillion because its a big number, but realized that a trillion is 1000 times more than 1 billion.  There are a thousand billions in one trillion.

Facilitator Rosa recognized that Winnie had a clear idea of the problem but did not want either to cut off discussion for the others or to minimize the importance of what Winnie said.  She therefore used a strategy which acknowledges a student somewhat ahead of the group.

Facilitator Rosa: Could someone restate what Winnie just said?

No one restated the problem but attempts were made by Facilitator Rosa to examine a simpler problem.  She suggested a number line from zero to a million, locating 1000. This proved to be no help.

Latisha: Mathematically, I can see how to go from one to the other, but kids would be confused, so before we can place it on the number line, the confusion is between million, billion, and trillion.  I dont see it automatically.  I have to do the multiplication.  Seeing it on the number line would be an abstraction because I cant visualize it.

Facilitator Rosa: Instead of a trillion, what number can we start with for kids?

Winnie: Where would 1 be on a line of 0 to 1000?

Facilitator Patrick: Where would 100 be?

Facilitator Rosa: Which of those questions, Winnies or Patricks, is easier?  

Carmen: Divide the number line into 10 parts to locate 100.

Facilitator Rosa: Where would you put 1?

Carmen: Very close to 0.

Facilitator Rosa: Could I change this to the original problem?

The facilitator tried to determine whether the participants could generalize from the simpler problem, but found that more intervening steps were necessary.  Winnie, ahead in the grasp of the mathematics, was instrumental in helping her peers move with her:

Winnie: If we look at the number line as in the original problem, the 1 (on the line from 0 to 1000) is 1 billion.

Facilitator Rosa: If each interval mark was 100 before (on line from 0 to 1000) lets go to a new value for that interval. Let each interval mark equal 1000, then what would the endpoint be?

Francine: 10,000.

Facilitator Rosa: Does that do it?  (solve original problem)

Cora: No, Rosa. So what do we do?

Benito: Up it.

Facilitator Rosa: Lets make it 100,000 (for each interval on the line).

Francine: That (the end point) would be 1 million.

Facilitator Rosa: Does that get us where we want to go?

Cora: No. Still not there yet.  

Carmen: Make each line (interval mark) 1 million&The endpoint is 10 million.

Latisha: I understood the number concept, but how to break it up on a number line rather than a number itself....Its not just solved by adding a few more zeros. Its hard for us as adults, so (we can) understand why kids find it hard.

Winnie: Its hard to apply the same idea on a small scale to larger scale numbers.  I used some comparisons to go through the process.  Its troubling to see that its hard to apply with larger numbers.  It feels like a whole new problem, but it isnt.

Facilitator Rosa: What power of 10 is 1 trillion?

Benito: 10¹² equals 1 trillion.

Facilitator Rosa: What (power of 10) is 1 billion?

Cora: 10⁹. So if each line is a million, were not there yet.

Carmen: Change the end point of the number line to figure out each interval, then keep upping the numbers.

Carmen: Lets make each (interval) line a billion, 10⁹.

Winnie: If each line is a billion, whats the end point of the number line?

Latisha: 10 billion.

Teresa: When we went from 100 to 1000, we increased by 10; when we went from 1000 to 1 million we increased by 1000. I see my errors if I use this approach.

Cora:  Once we break it down to a simpler problem, 0 to 1000&if we stick with 1000 its easier. We dont get confused.

Facilitator Patrick: Maybe we can see a pattern of powers of 10 that will tell us weve reached another number 1000 times greater.  A thousand is 10³. Is there such a pattern?

Facilitator Rosa:&(writing). So, continuing with Facilitator Patricks point:

1 thousand = 10³

1 million =   10⁶

1 billion = 10⁹

1 trillion = 10¹²

Teresa: Kids get confused. They need to see the in between steps. The trick on the number line is to show increasing by 10.

Facilitator Patrick: If they know benchmarks, kids will understand.

Teresa: I dont agree.

Facilitator, following Teresas suggestion, listed the values of each power of 10. Then:

Facilitator Rosa: So where do we start on the line to get to 1 trillion?

Cora: 100 billion, 10¹¹.

Facilitator Rosa: Wheres one billion?

Cora:  Very close to zero. (ANSWER)

A particularly interesting phenomenon is the conversation which continued by its own momentum after the answer was arrived at publicly. The teachers continued to be engaged in the mathematics.   Striking, also, is the complexity of these comments as compared with those at the start of the process:

Teresa: OK, so we broke it up into smaller parts and found the answer. What if&we didnt start off with intervals of 100 billion. If we didnt use intervals, would the answer be the same if we used individual numbers? Would the graph look the same?

Francine: We found that 1 billion is close to zero if going from 0 to 1 trillion but isnt you space different if the number line goes from 0 to 100 billion? If I have to convert the graph from 0 to a trillion, would it still look the same if we dont use intervals of 100 billion?

Nero: What if your graph looked like this?

_________________________________________________________

10⁰     10¹     10²     10³&&&&&&&&&&&&&&&       10¹²

Winnie: Are you saying that the spaces are the same?

Nero: It should be on a scale 100 times the space.

Winnie: Because kids dont understand that it increases 100 times.

Carmen: Were changing the intervals on the number line. Can the line mean anything? Thats confusing for kids. Its (such) an abstraction.

Facilitator Rosa: What is the relationship between the number line and the place value table?

Teresa: The place value table is a way of representing the number line.  You can draw the place value table to represent the group of the numbers youre talking about-a shorter representation of the number line.

Latisha: The number line represents numbers; place value represents relationshipthe number of zeros in rounding.

Benito: The place value table shows the size of a number, but the number line shows relationship of numbers.

Facilitator Rosa:  Lets illustrate this and then well understand it better. Take the number 58.

On place value table it looks like this:

  

Tens	Ones
5	8

Lets locate 58 on the number line using Benitos definition. (Draws number line from 0 to 60 using intervals of 10.)

Teresa: Get to 50 and count over 8.

Facilitator Rosa: The place value column tells us how to read a (particular) number.

Winnie: Imagine if there were no place value table.  There are still numbers.  How would we indicate a spot a location on the number line?

Latisha: How could we identify a number if it doesnt have a place value?

Cora: How did they do what they did without zero?

Problem III: Find  [    ]  if  5/9  of   [    ]  = 1/8

MATHEMATICS IN THE PROBLEM

A number of aspects of this problem could pose difficulties for middle school students, as listed below.  A more basic difficulty emerged much later, in the analysis of the data. Exploration of this problem, therefore, will have two phases:  the first, what we looked for in the presentation as teacher-participants experienced it; the secondunforeseen complexities of the problem itselfwill be taken up in the discussion of the findings, later in the text.

Phase A: Potential difficulties for students

use of the prepositions if and of;

a seemingly odd pair of given fractions,

a variable in two places on the same side of the equal sign and no variable to identify the fraction on the other side,

inherent difficulties with division of fractions,

the potential for an algebraic solution, and

equivalence of fractions.

The teachers dialogue as it emerged in the protocol confirmed that these were important issues.

THE DIALOGUE

The following dialogue took place in a training session for staff developers of middle school mathematics teachers.  There were 14 staff developer participants present and one primary facilitator.  For clarity, the participants will be referred to as teachers or participants so as not to confuse the terms staff-developer and facilitator.  As cited elsewhere, in this problem participants were assigned to work in pairs during steps 5 and 6 of the protocol; that is, during the solution and presentation phases.

Answers to question, What would students find difficult about the problem?

What goes into the boxes?

What to do first?

Do you need to fill in one or two boxes?

What is the meaning of value?

Will they understand divided by?

Is x a variable or a multiplication sign?

Are we looking for the same number or different numbers?

Of and if are confusing.

Answers to the question, What information is in the problem?

5/9 of x

Two missing values or unknowns

Is 1/8^th the answer?

Equals 1/8 means what is on the left is equal to 1/8.

It seemed from these lists that some participants did not understand the problem clearly. The lack of clarity emerged further from the directive,

Answers to the question, Tell the story of the problem.

How much more do I need to give to the 5/9 side so that it is equal to the side with 1/8?

5/9 times what number will give me 1/8?

5 divided by 9 times what will give you 1 divided by 8?

How many 5/9 makes up 1/8?

1 divided by 8, the quotient would be 5 divided by 9.

There was still confusion expressed by some participants in explaining how the numbers relate to each other. Further confusion emerged when some teachers attempted to describe the problem in real-world terms.

Mary brought a cake and Marys mom gave her some amount which will give you 1/8 of a cake but she is not to eat it all.  She can only eat 5/9 of that some amount.

If I had 1/8 of a cake, and you divided among 5/9 of my friends, how much would each get?

ANALYSIS OF TEACHERS WORK

It is, in general, a challenge to express division of fractions problems accurately in story form. Since, after discussion, none of the algebraic stories was acceptable to all, the issue of finding a real-world story was set aside temporarily. Participants were directed to work in pairs to solve the problem and present their solutions on chart paper.

In the solutions, all used some form of division of fractions algorithm, and all knew mathematics well enough to get the correct answer to the equation. However, the degree of clarity in the use of mathematics procedures varied.  In some cases, the deviations were minor, and the step-by-step process still revealed the mathematics ideas.  Some of the solutions might leave a learner with questions about why are we doing this.

Teachers responses shown in figures A G represent examples of their knowing the mathematics and inventing shortcuts based on that knowledge. Under each solution are listed the elements which would make the mathematics clearer for students. These are explained on the rubric page that follows the solution.

Solutions  A G posed by teachers:

Figure A: missing rubric items 4 and 5

Figure B: missing rubric items 1, 2, 3a, b, c, d (variable is ignored)

Figure C: missing rubric 5

Figure D: missing rubrics 3c and 3e

click to enlarge

Figure E: missing rubrics 1, 3b, 3c, 3d

Figure F: missing rubrics 1, 3b, 3c, 3d (variable is ignored)

Figure G: Problem recast and logical procedure is followed until&(see explanation, next page)

RUBRIC FOR COMMENTING ON TEACHERS SOLUTIONS AS THEY RELATE TO PEDAGOGICAL STRATEGIES

1. The problem should first be written as presented.

2. The variable must be identified.

3. Each change should reflect a mathematical justification for making the change.

a. To solve for an unknown quantity, the unknown quantity must be by itself on one side of the   equation.

b. To isolate the unknown quantity, use those operations which will relocate the other numbers on the same side of the equation.

c. To relocate the other numbers, see what operation connects the variable to those other numbers and do the opposite operation.

d. Since an equation means that the numbers on each side of the equal sign must have the same value, if you perform an operation on one side you must perform the identical operation on the other side.  There, do what you did on one side to the other side.

e. In general, it is clearest for the variable to remain on the same side of the equal sign through all the steps of the solution process.

4. Operation signs of some kind must be used with numbers.

5. To lessen confusion, if you use a letter to represent an unknown quantity, x or y, for example, you should not use the same notation as an operation sign.

Figure Number	Rubric Items absent from the Solution
A	4, 5
B	1, 2, 3a, b, c, d (variable is ignored)
C	5
D	3c and 3e
E	1, 3b, 3c, 3d (variable is ignored)
F	1, 3b, 3c, 3d (variable is ignored)
G	Problem recast and logical procedure is followed until the solution step, when unexplained rule is applied.   This would be acceptable for elementary school students, but since the problem was presented algebraically, an algebraic process is appropriate for solution in middle school.

Shortcuts are effective cognitive strategies when invented by users who grasp the mathematical logic. However, when presented to students for whom the logical structure has not yet been formed, the shortcuts are likely to be confusing. Just as teachers, in studying student work, must ascertain what next steps will help their students to be better mathematicians, so the facilitator of this protocol must design next steps for teachers to be more effective communicators. Two issues remained unresolved at the end of the session. The first was unclear representation of the algebra by some of the participants. The second was the confused real world meaning revealed through attempts to tell the story. It was decided that focusing on the algebra was the easier task, and therefore, it was done in the following session.   Exploring strategies for unpacking the equivalence relationships so that a story could be clearly constructed is woven into later sessions and not reported here.

At the follow-up session, one week later, participants were asked to ascertain the mathematics ideas that underlay solution of the problem.  They were puzzled by the question.  The teachers didnt know what was meant by a math idea.  After some examples by the facilitator, they came up with the following (in the order given and, as before, in participants own words):

One side of an equation has the same value as the other side.

The inverse of multiplication is division.

Multiply both sides of an equation by the same number leaves the equation the same.

A number multiplied by its reciprocal is always one, except for zero.

A piece of a piece is a smaller piece.

The sum of the parts is equal to the whole.

Get the variable alone on one side of the equation.

You get rid of the number with the variable by using the inverse operation.

A packet of photographs of the previous sessions solutions was distributed and participants were asked to work in groups to reexamine them and choose the one that most clearly revealed the mathematics ideas. Each group then rewrote and modified the solution selected. Two of the modified solutions are below (insert figures H and I) and are typical for all groups except one. They were significantly clearer than original solutions and only rubric item 5 was absent. Members of one group defended their original solution, Figure E.

Figure H

click to enlarge

Figure I

click to enlarge

Phase B. Findings in Problem III.

Teachers solving their problems in small groups presented a broader array of strategies and beliefs about mathematics, as compared with the solution in problem I.  For example, some believed that finding a common denominator was an advantageous first step in division of fractions problems. Others argued that the use of common denominators was an incorrect procedure. Another participant argued in favor of the just-invert-and-multiply strategy.   Absent from these arguments was the expressed need to articulate, for students, the axioms and properties that justify the procedures.

In the follow-up session we found that the teachers, if probed, knew how to articulate the underlying mathematics ideas.  They seemed, in general, unable to realize how important it was to do so in teaching. This problem has widespread implications for how we move children into algebraic thinking.  It is generally considered good mathematics pedagogy to allow students to solve problems in many different ways, encouraging their invention of idiosyncratic solutions.  Good mathematicians do that, and the middle school teachers weve been talking about produced shortcuts, and even inventive solutions to the problem under discussion.  However, algebra is a highly structured discipline.   Some students can figure out its properties from constructive methods, but most students need to be taught the structure of the beautiful, orderly world of elementary algebra.

In examining the data afterwards, it became clear that we had set aside what was a major difficulty of the problem. The third step of the protocol, tell the story of the problem, had served to pinpoint its mathematical complexity.  The participants, experienced teachers of middle school mathematics, were unable to describe a real-world situation that fit the numbers in the problem.  On the other hand, by whatever method, they were easily able to find a correct numerical answer. Yet, at the next session, we focused on procedural knowledge rather than on the conceptual complexities of the problema decision that in retrospect seems questionable.  It does, however, enable us to see the value of the third step of the protocol.  It did not catch, in the first two steps, the true complexity of the problem.  But, in the analysis of responses to the third step, the final sense-making step, an alarm went off. How many problems do we present to children in elementary and middle school that pose such conceptual, but not procedural difficulties? In selecting the problem from a collection of problems for middle school children, the researcher did not recognize that this problem was of this variety. Further, the third step of the protocol was considered an addendum, a synthesis of what would have been essentially uncovered in the first two steps.

Some of the stories presented by participants were:

Mary brought a cake and Marys mom gave her some amount which will give you 1/8 of a cake, but she is not to eat it all.  She can only eat 5/9 of that same amount.  

Whats wrong with the above story? In the case described, Marys mom gave her 1/8^th of the cake.  In the problem as given, an unspecified part of the cake was given to Mary. She ate 5/9 of that unspecified amount, which turned out to be 1/8^th of the cake. In this case, the unspecified amount (9/40) does not figure in the explanation.

Another story is offered by a different participant:

If I had 1/8 of a cake, and you divided it among 5/9 of my friends, how much would each get ?

Lets see what that could mean.  It asks a different question.  He starts with 1/8 of a cake and divides it among five of his nine friends.  The nine in this case is irrelevant because it refers to the number of friends he has rather than the cake.   Also, 1/8 in this case as well, is less than the amount of the cake started with, though it is the amount eaten by the five friends.

After struggling with these responses and the non-story responses that were offered, the researcher afterwards constructed a story with the elements given by the above teacher-participants:

At a party our table was given a small part of a cake.  We divided the piece we were given into nine parts.  However, only five of those nine parts were eaten.  Those eaten actually represented 1/8 of the entire cake.

How much of the original cake was given to our table?

What makes that story so elusive? Superficially, if 5/9 of some number is 1/8 (of the whole), then that some number, by the division of fractions algorithm, will be 9/40, or the entire piece from which 5/9 is taken. But that 9/40 is itself part of the whole.  Further, none of the fractions is an obvious factor of the others, nor does there seem to be a relevant common multiple. The following geometric representation (constructible only after the numerical answer is found) clarifies further. There are 40 cells which make up the whole. Nine of the cells are marked with x. Five of those nine are marked with xx. Those five represent 1/8 of the total, or the part of the cake eaten.

xx	x
xx	x
xx	x
xx	x
xx

How important is it that children be able to construct a story that enables a problem in mathematics to make sense? That is the very essence of the current arguments around traditional as compared with constructivist methods, an essence that unexpectedly hit us in the face in this problem.  Does that mean that such problems cannot be included in the mathematics curriculum for adolescents or their teachers?  Surely, the answer is no.  In the history of the development of mathematics, words rather than symbols expressed the thinking of the great minds.  Gradually, those words were replaced by symbols and the use of symbols enormously enlarged mathematical possibilities.  In teaching adults to work effectively with children and adolescents, we need to go backwards and recreate the development of mathematical ideas by finding words that uncover the symbolic processes.  This is not a trivial task, but is instead a necessary bridge toward, ultimately, enabling students to work effectively with symbolic language.

The original question, What would students find difficult about this problem? did not uncover the true difficulties. However, the third step of the protocol, tell the story of the problem, or synthesizing and making sense of all its elements, made clear that the problem was, in general, not understood by the participants.  The time constraints of our session led us to bypass that issue and have the teacher-participants solve it with a numerical answer.  When we reviewed the procedural answers submitted, it was clear that the procedures part would not be clear to their students, so participants were encouraged to re-present their solutions in clear algebraic form.  Only later analysis of this data called attention to the fact that no participants could put the problem correctly into real world terms, or, alternatively, make sense of the problem.

DISCUSSION

The Problem Probing Protocol is not a method for solving problems, or a challenge to those many excellent ones that exist.  Rather, its value lies in its providing a structured way for teachers, through talking together in groups, to uncover their own mathematical thinking, and thus, to make mathematics ideas clearer for students.  The conversations recorded here point out issues that remain confusing to teachers who have been practicing for a number of years. In many instances, these professionals would not acknowledge that their content knowledge, or knowledge of how to communicate mathematics, is inadequate.  The Problem Probing Protocol discussed in these pages can contribute significantly towards improving that condition. The following exploratory strategies were employed:

"

What would students find difficult about this problem;

"

What information is in the problem;

"

Tell the story of the problem;

"

Work with a partner in solving and recording the solution;

"

Present and defend the solution to the larger group.

These methods enabled the teachers to reflect on the mathematics as their students might, one step removed from their own sensibilities. That distance made it possible for them to feel safe in volunteering what they knew and believed.  It also caused the participants to focus on what students might think.  In the process, other teachers listened to them, commented on ideas presented, and gradually modified positions and understandings.  And some left with unanswered questions that previously they would not have thought to ask. In some ways, the last point is the most significant.

The role of the facilitator is an important one, and should be clarified here. In the early phases of professional development with a group of teachers, the facilitator models the protocol.  As teachers internalize the procedure, the facilitators role diminishes, and, it is hoped, that it ultimately disappears.   The time frame in which this happens varies depending on the collective content and pedagogical knowledge in a given group.

This protocol is a learning experience for facilitators as well as participants and should lead to their increased ability to reflect, understand, and change. For example, Teacher Winnies critical part in helping others to understand the relationship among powers of 10 in problem II was taken as a signal that the facilitators future role should be curtailed in the solution part of the process.  Thus, in later sessions, participants worked together in pairs without the facilitators intervention, as in problem III.   The facilitator might not have been pleased that some participants were unwilling to alter their views even in the face of contradictions; but teachers also felt displeased when faced with questioning of concepts and ideas that they considered known or routine. A facilitator must not lose sight of the fact that real change happens in the context of trusting relationships built over time among people who dont always agree.

The coming together of practitioners for analyses of their work is an established practice in all professions.  Only lately has its importance for teachers been recognized.  The timeworn image of a teacher, isolated behind classroom doors, to sink or swim with little outside support, is neither necessary nor beneficial. Yet, the opportunity for study groups to meet in schools, under the direction of teachers, has not taken hold widely enough.  The Problem Probing Protocol offers teachers a useful tool for structuring and strengthening their coming together for mutual development in mathematics in two ways: it offers staff developers and then, by example, teachers, a simple but formal procedure for entering a discourse on any math problem; and secondly, the discourse enlarges the topic under discussion, enabling participants to make more connections and to become more aware of their own mental processes.

There is no one cure for such a widespread problem as our failure to teach mathematics to so many students who both need to and are able to learn it.  Much more must be done to uncover teachers thinking and address their confusions.   Modeling the protocol in the teaching of teachers and articulating the steps involved, in the context of non-judgmental listening, is a promising vehicle. The process needs to be regularly repeated over relatively brief intervals of time so that it is internalized and analyzed by the participants.  As it becomes theirs to ponder, they might choose to use it for encouraging student talk in their own classrooms.

Protocols provide formal structures for the ways in which people in various situations talk to and behave with each other. Why do we need these in education?  Isnt it enough that teachers talk to each other about their work as they normally do?  No! asserts McDonald (2003). The kind of talking needed to educate ourselves cannot rise spontaneously and unaided from just talking.  It needs to be carefully planned and scaffolded (p. 4).

McDonald (2003) writes further that protocols may encourage an environment for learning (by educators and their students) based on the theory that knowledge is socially constructed.  That is, encounters with other peoples understanding enables [sic] learners to gain and deepen their own understanding.  Moreover, along with John Dewey, we believe such learning environments foster democracy as well as cognition.  They encourage learnerswhether first graders, graduate students, or colleagues in professional educationto appreciate the value of diverse ideas and deliberative communities (p. 7). The Problem Probing Protocol fosters democracy in other ways.  It provides an approach to mathematics problem solving to teachers of students with undeveloped reading and analytical abilities. Teachers who transfer this protocol for their students use will enable those students in developing important thinking, reading, analytical and mathematics skillsskills that will greatly enhance their life prospects.

SUMMARY AND CONCLUSIONS

In preliminary explorations, we examined the effectiveness of a three-part protocol which provides a simple, formal tool for enabling analysis of mathematics word problems.  The text reports in detail the use of the problem probing protocol with two groups of middle school teachers.  It is particularly valuable for use with unskilled readers and their mathematics teachers, enabling them to form, in socially constructed learning situations, schemas for understanding and approaching solutions of the problems.  The protocol also suggests potential value as an assessment instrument for teacher content knowledge.  In our sessions, teachers deep misunderstandings of basic mathematics concepts were revealed, and then clarified by the comments and questions of other participants.

The protocol is especially effective in addressing two aspects of the problem solving process: making meaning out of a problems words, and reducing the extreme anxiety which some teachers and many students experience when confronted with a mathematics problem.

In a focus group held for participants of our study, teachers who had been with the project for more than two years reported that the protocol was part of their regular teaching practices, and that they engaged in much more problem solving with their students.

Further study is needed to determine to what extent teachers exposed to this approach ultimately use it with their students, and then to what extent it changes student performance.  How much exposure and what kinds of in-class supports do teachers need in order to internalize the process and substitute it for their established ways of thinking about problems?

References

Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers mathematical knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (4th ed.). New York: Macmillan.

Cazden, C. B. (1988). Classroom discourse: The language of teaching and learning. Portsmouth, NH:  Heinemann.

Cohen, D. K. & Ball, D. L. (1999). Instruction, capacity, and improvement. Consortium for policy research in education research report series, RR-43. University of Pennsylvania, Graduate School of Education.

Davis, P. & Hersh, R. (1981). The mathematical experience. Boston: Houghton Mifflin.

Fennema, E., & Franke, M.L. (1992). Teachers knowledge and its impact. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp. 147-164). New York: Macmillan

Hiebert, J., Gallimore, R., & Stigler, J. W. (2002). A knowledge base for the teaching profession: What would it look like and how can we get one. Educational Researcher, (31)5, pp. 3-15.

Kelly, D. L. (2002). The TIMSS 1995 International Benchmarks of Mathematics and Science Achievement: Profiles of world class performance at fourth and eighth grades. In Educational Research and Evaluation, Vol. 8, No. 1, pp. 41-54.

Mathematics Equals Opportunity. (1997, October). Middle school: Getting on the road to challenging mathematics and science courses. (White Paper, Oct. 20, 1997) www.ed.gov/publ/math/part 4.html.

Mayer, R. E. (1992). Mathematical problem solving: Thinking as based on domain-specific knowledge. In R. E. Mayer (Ed.), Thinking, problem solving, and Cognition, (pp. 455-489). New York: Freeman.

McDonald, J.P., Mohr, N., Dichter, A., & McDonald, E.C. (2003). The power of protocols. New York: Teachers College Press.

National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, VA.

Noddings, N. (1989). Preparing teachers to teach mathematical problem solving. In R.I. Charles & E. A. Selver (Eds.), The teaching and assessing of mathematical problem solving. Reston, VA: Lawrence Erlbaum for NCTM.

Pape, S. (2004). Middle school childrens problem solving behavior: A cognitive analysis from a reading comprehension perspective. Journal for Research in Mathematics Education, (35)3, pp. 187-219.

Polya. (1971). How to solve it, a new aspect of mathematics method. 2nd edition. Princeton, NJ: Princeton University Press.

Redfield, D. L. & Rousseau, E. W. (1981). Review of Education Research, v 51, n2, pp. 237-45.

Resnick, L. (1989). Treating mathematics as an ill-structured discipline. In R. I. Charles & E. A. Selver, (Eds.), The teaching and assessing of mathematical problem solving. Reston, VA: Lawrence Erlbaum for NCTM.

Schoenfeld, A. H. (1985). Mathematical problem solving. London: Academic Press.

Schwartz, J. E. & Riedesel, C. A. (1994). Essentials of classroom teaching: Elementary mathematics. Needham, MA: Allyn & Bacon.

Skemp, R. R. (1987). The psychology of learning mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.

Supovitz, J. A. & Poglinco, S. M. (2001). Instructional leadership in a standards-based reform. Report by Consortium for Policy Research in Education, Philadelphia, PA.

Vygotsky, L. S. (1986). Thought and language. Cambridge, MA: MIT Press.

Cite This Article as: Teachers College Record Volume 109 Number 4, 2007, p. 837-876 https://www.tcrecord.org ID Number: 12867, Date Accessed: 11/27/2021 6:40:49 PM Purchase Reprint Rights for this article or review