constructing pedagogical content knowledge from students’ writing in secondary mathematics

Mathematics Education Research Journal, Vol.3, No.],1991.

CONSTRUCTING PEDAGOGICAL CONTENT KNOWLEDGEFROM STUDENTS' WRITING IN SECONDARY

MATHEMATICS

L Diane Miller, Curtin University of Technology

One field of knowledge important to effective teaching ispedagogical content knowledge. With experience, teachersbecome aware of how students comprehend or typicallymisconstrue mathematical concepts, skills, and generalizations.They become aware of common misconceptions and "buggyalgorithms" constructed by students and of the stages ofunderstanding that students are likely to pass through in movingfrom a state of having little understanding of the topic to masteryof it. It is important for mathematics educators to find ways toexpedite the development of pedagogical content knowledge.The study reported in this paper suggests that in-class,impromptu writing prompts are one means by which teachers canget a clearer picture of students' understanding ornonunderstanding of mathematics in a relatively short period oftime.

For the past decade, mathematics educators and writing specialists havebeen combining their efforts and collaborating in research to examine thecognitive and affective benefits that students may directly derive as a result ofwriting in mathematics classes. Advocates have proposed various types ofwriting as being beneficial to students' development of higher order thinkingskills, improving attitudes, and reducing anxieties. Journal writing(Nahrgang & Peterson, 1986; Borasi & Rose, 1989) has been shown toaffectively benefit students by providing them an opportunity to express theiranxieties toward mathematics and the problems they encounter in the learningprocess. Waywood (1989).concluded that journal writing can also cognitivelybenefit students by leading them to summarise and reflect on the mathematicsthey are learning. Expository writing (Bell & Bell, 1985) has proven to be aneffective and practical tool for teaching problem solving. In general, Reuille-Irons and Irons (1989) suggest that writing activities "encourage children tobe creators of their mathematical knowledge. When they create their ownknowledge, they gradually build a picture of concepts and ideas that will beuseful in problem-solving situations" (p. 98). (For an overview of theprogress being made in a number of developmental projects on writing inmathematics classes, see Davison and Pearce, 1990.)

Constructing Pedagogical Content Knowledge 31

But what about benefits that the teacher may directly derive as a resultof reading students' writings? Data collected by Borasi and Rose (1989)suggest that teachers, too, may be equally affected in their teaching by readingtheir students' writings. There is a dearth of literature addressing the affectthe use of writing in mathematics classes has on teachers; specifically on whatteachers learn from reading their students' writings. The purpose of thispaper is to partially fill this void by examining potential teacher benefits ofusing writing in secondary mathematics classes.

Background to Current Research

Several sources in the literature suggest that the expertise involved inthe cognitive aspect of teaching can be leen as the merging of at least threefields of knowledge. One of these is called lesson structure knowledge(Leinhardt & Smith, 1985). Lesson structure knowledge includes the skilisneeded to plan and run a lesson smoothly, to create a smooth transition fromone lesson to the next, and to explain the material clearly. A second field iscalled subject matter knowledge (Leinhardt & Smith, 1985). No one candeny the need for a teacher to know the facts in a particular domain or thealgorithms, heuristics, and principles which operate on those facts.

A third field of knowledge important to effective teaching is calledpedagogical content knowledge. According to Shulman (1986), teachersshould possess "the understanding of how particular topics, principles,strategies, and the like in specific subject areas are comprehended or typicallymisconstrued, are learned and are likely to be forgotten" (p. 26). Inmathematics, like any other subject area, effective teachers should be aware ofthe conceptual and procedural knowledge that students bring to the leaming ofa topic. They should be aware of common misconceptions constructed bystudents and of the stages of understanding that students are likely to passthrough in moving from a state of having little understanding of the topic tomastery of it.

Experienced teachers know that pedagogical content knowledge is, inpart, constructed over time through teaching experiences. For example, afairly common misconception developed early in the study of algebra is that7x means 7 plus x rather than 7 times x. Experienced teachers know this andsometimes review the counterexample in their instructional comments toemphasize the correct meaning of 7x. The importante of pedagogical contentknowledge to effective teaching is just beginning to arouse the interest ofmathematics education researchers. The findings of Peterson, Fennema,Carpenter, and Loef (1989) suggest that teachers' pedagogical contentknowledge may be importantly linked to teachers' classroom actions and,ultimately, to students' classroom leaming in mathematics.

The primary focus of pre-service teacher education programs is onsubject matter knowledge and lesson structure knowledge. The beginningteacher's first concerns in the classroom are generally behavioural

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management and lesson preparation. Only with time does the construction ofpedagogical content knowledge begin. If we accept the premise thatpedagogical content knowledge is an important ingredient to effectiveteaching, then practicing teachers, mathematics educators, and researchersmust ascertain how the construction of pedagogical content knowledge can bebeneficially expedited. The results of a study conducted in the United Statesand reported in this paper, suggest that the use of in-class, impromptu writingprompts can promote the construction of pedagogical content knowledge forboth experienced and inexperienced teachers.

The Study

In-class, impromptu writing prompts - the type of writing activityutilised in the study - are simply-worded statements or questions directingstudents' thoughts to the explanation of a single concept, skill orgeneralization. For example, a teacher may give the following prompt to aclass of year nine students. "Why do we say that division by zero isundefined?" Prompts can also encourage students to express their attitudesand/or anxieties about mathematics or problems they encounter while learningmathematics. For example, "Telt me (in writing) one thing you like aboutlearning mathematics and one thing you do not like about learningmathematics."

Three secondary mathematics teachers from a large metropolitan highschool in the United States agreed to participate in a study to examine thepotential teacher benefits of using impromptu writing prompts in mathematicsclasses. This paper will focus on two of these teachers. One teacher wasrelatively inexperienced as a second year teacher; the other had more thanfifteen years experience teaching secondary mathematics. They both taughtfirst year algebra. Successful completion of first year algebra is a graduationrequirement of the school district participating in this study. Generally, year 9students (fifteen years old) enrol in first year algebra. One class of students(25-30 students per class) for each teacher participated in the study.

Students were generally asked to write at the beginning of a classunless the prompt asked about something that occurred in class that day.They were given five minutes to read the prompt, formulate their responseand write it on paper provided by the teacher. Initially students were asked towrite during three out of every four instructional periods. As the studyprogressed, the number of writing tanks were reduced to about three perweek. Students were not rewarded for writing nor penalized for not writing.Thus, every student did not write every day.

Approximately fifty-five prompts were used in the study. Twenty weredirected toward the affective domain; for example, attitudes toward andanxieties about mathematics, and feelings about class and topics beingstudied. Ten queried students' attitudes toward their responsibility inlearning; that is, what determined whether or not they did their homework andhow they prepared for an assessment. Twenty asked students to explain a


mathematical concept, skili or generalization. Occasionally students wereasked to write on a "fun topic" or to write anything they wanted to write on atopic of their choice. One prompt in this category asked students to identifytheir favourite single digit number and to explain why they liked it. Anotherprompt said "Write about any topic you want to today. Whatever you want tosay, write it."

Teachers generally read each set of writings the day they werecollected. Once per week the teachers were asked to reflect upon what wasbeing leamed from reading the students' writings and to write about theirpredominant impressions. As the study progressed, the teachers' reflectionsbecame more evident in oral discussions than in their writing.

The students' and teachers' writings were collected fortnightly for onesemester along with notes taken during the collaborative team meetings andduring discussions with individual teachers. Since the focus of this discussionis on how writing prompts can expedite the development of teachers'pedagogical content knowledge, the remeinder of this paper will concentrateon selected prompts which asked students to write about their understandingof mathematical content.

An interpretive research methodology (Erickson, 1986) wasimplemented in this study. Documentary materials consisted of students'responses to timed, in-class impromptu writing prompts, teachers' writingsabout what they were learning from the students' writings, and field notesderived from discussions with individual teachers and during meetings of theresearch team. As suggested by Glaser and Strauss (1967), a thoroughanalysis of the students' writings was completed. These writings weresearched for recurring pattems that contributed to the identification andexplanation of ways in which impromptu writing prompts enhanced theteachers' ability to assess how well students were comprehending themathematics they were studying. In many respects, this was an exploratorystudy since very little research has been undertaken to examine benefits forteachers in using writing in mathematics classes. Thus, it was deemedinappropriate to search for conclusive empirical evidence which would allowthe researchers to predict teacher benefits from using impromptu writingprompts in a mathematics class. More relevant to the research team was aninvestigation of what teachers can leam from reading their students responsesto impromptu writing prompts.

What Three Prompts Yielded

The Commutativity PromptStudents sometimes view mathematics as isolated parcels of

infonnation which must be memorised for "the test" and then forgotten so thatnew parcels can be memorised and the cycle continue. Mathematics is a studyof relationships, but is often presented as independent packets of knowledgewhich are not related to other mathematical concepts or "real world"

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applications. Sometimes teachers fail to provide experiences which helpstudents construct mathematical relationships, other times their attempts arenot successful. Evidence of a failed attempt surfaced when the first yearalgebra students in this study were asked to write about their understanding ofcommutativity. When implementing an interpretive research methodology inan educational setting, the contextual background is crucially significant to thesense-making process of the data being examined and conclusions drawn.The following paragraph will provide the reader with the background for thisparticular prompt.

Three months into the school year, an assessment ásked students todemonstrate an understanding of the Commutative Property for Addition andthe Commutative Property for Multiplication by stating the properties inabstract form and labeling statements that represented each property; forexample, students were asked to identify the property represented by 5 + 7 =7 + 5. A majority of the students correctly quoted the rules in their abstractform (for example, for real numbers a and b, a + b = b + a and a x b = b x a)and labeled statements as the Commutative Property for Addition and theCommutative Property for Multiplication. During a team meeting, someoneasked if the students would be able to apply what they knew aboutcommutativity to operations other than addition and multiplication. The twofirst year algebra teachers indicated that they had briefly commented ondivision and subtraction in class and that exercises in the students' textbookhad allo provided counterexamples using division and subtraction. The teamdecided to ask the students to respond to the following prompt.

You have studied the Commutative Property for Addition andMultiplication for real numbers. Not all operations are commutative.Explain why each of the following operations are not commutative:(1) Division (2) Subtraction (3) Raising a number to a power

The teachers were confident that not all students would produce an acceptableresponse. However, they were equally confident that the majority of theirstudents would respond acceptably. Fifty papers were collected from twoclasses. The students' responses are listed in Table 1.

Four of the fifty students (8%) produced acceptable responses to allthree operations. (Marked acceptable by the teachers, two research assistants,and a mathematics education researcher.) Eleven (22%) acceptably respondedto division and nine (18%) demonstrated an understanding of why subtractionis not commutative. The teachers were most disappointed in the students'responses to division and subtraction. While the students had worked withnumbers raised to a power and had adequately demonstrated they knew whatax meant, the teachers had not discussed the operation in terras of beingnoncommutative. The query raised in the prompt asked students to relateprior knowledge; that is, their understanding of addition and multiplicationbeing commutative, to an unfamiliar context, raising a number to a power.


Table 1Responses From Students

Responses deemed acceptable by the research team

Division -(a) division is not cummutative because 3 i5 is not equal to 5y3(b) because you can't switch them around because you will come out with a

wrong answer(c) because they don't have an order and you can't change them around(d) Because you can't change them around — —(e) you cant change the numbers around 3)450 # 450)3

see in addition you can change the order 3 + 5 = 5 + 3 multi 3 x 5 = 5 x 3(t) It just wont work out 2

3)6 but 6)3.0(g) Because in division you come up with a different answer(h) you can't reverse the numbers and come out with the same answer(i) isn't cummutative because you can't switch numbers around to make it easier

844 # 8

(j) Because if you have 10+5 =2 you cant say 5 y 10 because it zQnt wQrk out(k) if you change the numbers around the answer will change 8)10 10)8

2. Subtraction -(a) subtraction is not commutative because 3 - 2 is not equal to 2 - 3(b) same for division if you swich them around and try to subtract your going to

come out with a wrong answer(c) has to be in certain order(d) They have to be in a certain order(e) It doesn't work out either 6 - 3 = 3 but 3 - 6 = -3(f) the number can only be done one way to come out with that answer(g) when you subtract if swich the numbers you get a different number 8 - 5 = 3

5-8=-3(h) Because if you turn a number around like 3 - 2= 5 it won't work out as 2 - 3(i) If you change the numbers the answer will change 5 10

-10 --

3. Raising a number to a power -(a) raising a number to a power is not cummutative because 3 5 is not equal to 5 3(b) cant example 32 = 9 23 = 8(c) Because if you have a number like 2 3 = 8 it will be different if you had 3 2 = 9(d) If you change the number the answer will change 53 35

Responses deemed unacceptable by the research team:

1. Division -(a) a=b-c=a a=b=c justchangethevariablearound(b) - A=B=B=A(c) 1

7)770

(d) commutative is not a division because the rule don't say it(e) 1

6)660

(f) because you divide(g) it doesn't work 2 _,^

4)8 4)2.0

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Table 1 continued

(h) I don't understand why the following operations are commutative 48 - 672+83+5=5+3 53•5.53 102.100= 100.102

(i) you can't divide 4)2 and get a whole numberG) 2

4)8(k) GQd made juhat way(1) 4)8 = 4)8 Because commutative property has nothing to do with division(m) ygdivid^(n) 5)30 6)30 _(o) 10)100 100)10(p) For division if you divide the larger number into the smaller number the

answer will not be even (ex. 9010)900

2000

but if it is done the other way, you have to add dismials (ex. .0111900) 10.0000

1000EQ

1000-900

1

2. Subtraction -(a) a - b - c = a + b - c just change the variable around but subtract(b) A-B=B-A(c) 3-4=1(d) commutative is not a subtraction because the Tule don't say it(e) 2-1=1(f) Because you subtract(g) 5-3=2doesn'tdoiteithermaybe 5-3=-2(h) ex. 12 - 6 = 6 - 12 = 6 It wouldn't work out like that because you can't

subtract 12 from 6(i) 4-3=3-4(i) 4

-22

(k) God made it that way(1) 6- 12 = 12 - 6 because you can't subtract 12 from 6(m) Because you subtract differently than adding(n) you subtract(o) 7-2=2-7(p) 7-2=5 5=2-7

3. Raising a number to a power -(a) r•r•r•r•r•r•r = r7(b) the same as above (see division d. & subtraction d.)(c) 77 a•a•a•a•a•a•a(d) Because you don't(e) doesn't work. I don't know how if it works(1) + 43(g) It just doesn't work. I don't know why. It just wont.(h) 23(i) God made it that way(j) you use power(k) 52.2 = 2.52


The students' writings to this prompt contributed to the development ofthese teachers' pedagogical content knowledge. While the majority of thesestudents had successfully quoted rules in abstract and had recognized their usein specific examples, they did not really understand the concept ofcommutativity. The experienced teacher expressed her disappointment butwas not shocked by the results. The second year teacher was bothdisappointed and surprised by the results. In one team meeting, the secondyear teacher kept reiterating that she had discussed this in class and theirtextbook had given theet counterexamples using division and subtraction.Generally, a teacher cannot assume that because students can state rules in theabstract that they really understand the principle or can apply it to anunfamiliar context. This may seem like a trivial, universally known fact inteaching, but how often do teachers proceed with the curriculum assumingstudents know more than what they really do. Seventy-eight percent of thefifty students returned unacceptable responses, blank papers, or papers with"Don't know" written beside each query. Clearly a majority of these studentscould not relate what they knew about commutativity of addition andmultiplication to division, subtraction, and raising a number to a power.

Other than being reminded that students can quote rules withoutcomprehending their content, the teachers' pedagogical content knowledgealso benefited in other ways. For example, look at the students who madefalse statements about basic facts in subtraction: 5-3 = -2; 12-6 = 6-12 = 6;4-3 = 3-4; 7-2 = 2-7. Once again, the inexperienced teacher was surprised bythese errors. Her comments indicated that she believed her students knew that5-3 was not -2; that 6-12 was not 6; that 4-3 was not equal to 3-4; and that 7-2was not equal to 2-7. But, she had no explanation why these statementsappeared in their writing. A comment made to the mathematics educationresearcher was "It's like they're not thinking about what they're writing.They're just putting down things that don't make sense." Perhaps this is anexample of students manipulating symbols and not thinking about what thesesymbols mean in the process of trying to make sense out of something new tothem. Very little is known about the processes of assimilation andaccommodation as students are constructing new knowledge in mathematics.Many of the students' statements were nonsense; but was that because theirthoughts were in disarray as they were trying to relate prior knowledge to anew context and make it meaningful? In the short term, these teachersdecided to return to commutativity, emphasizing its meaning and reillustratingwhy it was applicable to some operations and not to others. In the long term,they each commented that future lessons on this topic would be revised toemphasize the meaning of commutativity rather than the memorization of anabstract rule.

The Ratio PromptA second prompt that was beneficial to the construction of the teachers'

pedagogical content knowledge was the following:

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In working the following problem, write down not only everything youdo, but everything you think about in coming up with your answer.

A painter uses four cans of paint to paint six rooms. How many roomscan be painted with six cans of paint?

Once again, the contextual background for this prompt is helpful ininterpreting what the teachers learned from reading the students' writings.The first year algebra classes covered ratio and proportion early in theacademie year (September). Several problems had been worked wherestudents had solved for an unknown term in the proportion. A few wordproblems had been solved in class and assigned for homework, but thegeneral assessment had provided students with a structured proportion with amissing term and the students were asked to solve for the unknown; for

example, 4

x The preceding prompt was given in late October. The12 =— 'teachers were wanting to ascertain how many students would use a proportionto solve the problem.

Forty-eight papers were collected. Ten (21 %) were returned eitherblank or with "Don't know" written beneath the prompt. Twenty students(42%) submitted unacceptable responses. Fifteen of these twenty said thecorrect answer was eight. Six students submitted eight as an answer butprovided no explanation for their solution. Nine of these fifteen studentsprovided a rationale similar to the following response:

Because 6 is two more than 4, I will add 2 to 6 and get 8. 6 cans of paintwill do 8 rooms.

Eighteen students (37%) submitted an acceptable response. Six ofthese 18 said the answer was 9 but provided no explanation for their solution.One student wrote the following:

4 cans for 6 rooms6 cans for 9 rooms I looked for something half way between.8 cans for 12 rooms

Seven students answered 9 as the number of rooms and provided anexplanation similar to the following:

Half of 4 is 2. 4 cans can paint 6 rooms. 2 cans can paint 3 rooms.If you have 6 cans, you are only adding 2 more to 4so 3 + 6 = 9

Four students answered 9 with the following work submitted:


1-1124)6

12x6 = 2x6 =9

No one used a proportion to solve this problem. Rather than using a solutionprocess which was relatively new to the students, they had chosen to rely onother means of solution which were more meaningful to them.

The teachers' pedagogical content knowledge benefited in two waysfrom the students' writings to this prompt. First, they realized that studentsare reluctant to adopt new methods of solution when their prior knowledgeand experiences work for them; and second, the writings support theexistence of a misconception or buggy algorithm which several studentsapplied (for details on buggy' algorithms see Davis, 1982; Davis, Jockusch,& McKnight, 1978). That is, when one number is increased by two, theother number in the problem is increased by two. Once again, theexperienced teacher was not surprised by this error while the inexperiencedteacher was. Both teachers indicated that future lessons on solving problemsof this type would include comments about the buggy algorithm.

The Variable PromptThe third prompt presented in this paper provides additional examples

of students' misconceptions in first year algebra. The teachers had introducedthe concept of variable early in the year and had emphasized that any letter canbe used to represent an unknown quantity in a problem. They were satisfiedwith students' ability to translate simple statements into mathematicalexpressions; for example: "Write a mathematical sentence for the following:Four tienes a number equals twenty-eight." The majority of the students weresuccessfully writing 4x = 28 or 4n = 28, etc. The following prompt wasgiven in mid-November, three months into the school year:

Pretend you are trying to explain why "x" in the first and "y" in thesecond equation will be the same number. Pretend you are trying toexplain this to a ten year old. Write how you would go about it.

4x = 284y=28

Twenty-five papers were collected from the second year teacher's class. Fourblank papers were retumed. Four students submitted an acceptable response.The following response reflects the general statements made by these fourstudents. "If you multiply 4 by x, you get 28. If you multiply 4 by y, youget the the same answer. Since you are multiplying by the same four, x and yhave to be the same number because you get the same answer." Four otherstudents attempted an answer, but submitted unacceptable responses. For

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example, one student wrote `Because you can use any variable in an equationas long as you keep everything else straight along in the equation." Twostudents submitted responses which were classified as tautologies by theresearchers. One student wrote "x & y could stand for anything but theystand for the same thing here." Four students wrote what the researchersclassified as nonsense; one example being, "I tell hem or her to read it."

Seven students submitted responses that proved enlightening to theinexperienced teacher but did not surprise the experienced teacher. During ateam meeting, the experienced teacher indicated that she knew fromcumulative years of experience that students made these types of errors.These seven responses, as written by the students, are listed below. Adiscussion of their significance follows.

(a) All letters of the alfebeat are = to 1 that is why they are the samenumber.

(b) Any letter is 1. S is 1, A is 1, B is one, C is 1. So x and y willbe one too. Any letter will be one.

(c) The x and y are the same because they are alpabets and allalpabets equals the same.

(d) x = 7 y = 24 Since the 4 and 28 are the same you divide tofind x and subtract to find y because x comes before y.

(e) 4x = 28 x is the ungiven number so you subtract 28-4x = 24

(f) If 4x = 28 x = 24 and 4y = 28 y = 24 Both of theproblem is the same just y and x will = the same

(g) I would say that you have to find out what x and y is.

The first two students clearly have the misconception that any letter of thealphabet is equal to one. These students did not sit near one another so theteacher concluded that they arrived at this misunderstanding independent ofeach other; that is, they did not copy prompt responses in class. The nextexample (c) is interesting because it alone suggested that the student believedall letters in the alphabet were equal or represented the same number. Oneinterpretation of this response is that x and y are the same in these twoequations: 5x = 30 and 3y = 9. This turned out to be an incorrectinterpretation. The teacher talked to this student during seatwork one day andlearned that the student was trying to explain that when everything else is thesame, then any letter in the alphabet can be used to represent the unknownnumber. At this stage, the student was much better able to communicatemathematical knowledge verbally than in writing. However the teacher'sdiscussion with the student was initiated by the student's writing, thus givingcredit for the growth in pedagogical content knowledge to the writingexperience.

The responses of the next three students suggest that they think therelationship between a coefficient and the variable is additive/subtractive.Example (d) reflects an attempt at justifying the interpretation, although not


enough is provided to attempt an analysis of the student's thoughts. Thestudent submitting (f) is consistent with the misconception. The last examplesuggests that the student is not ready to accept that x and y are the same. Theexplanation provided is to find out what they are, but no attempt is made tosolve the equations.

The inexperienced teacher found these misconceptions surprising. Onher traditional types of assessments when students were asked to solve for xor y in an expression like 4x = 28, she had assumed an answer of 24 had beena "careless error." Her own writings summarized her feelings: "The writingsgave me an opportunity to look at each students' work individually. I leamedthat the words "yes" and "I understand" are not precise answers. Studentssometimes thought they understood but their writing indicated that they didnot understand."

The experienced teacher was not surprised at the misconceptions whichsurfaced in the students' writings. The writings were beneficial to theexperienced teacher because they helped her to assess which individualstudents held what misconceptions more quickly than using traditional wholeclass discussions. An excerpt from the experienced teacher's writings amplysupports this assertion: "For me as an experienced teacher, I found that I gota clearer picture much sooner of how the students were thinkingmathematically. I was able to locate specific problem areas and makenecessary corrections that I may not have been aware of had we just solvedproblems." Fieldnotes from several team meetings indicated that theexperienced teacher was having some of her pedagogical content knowledgereconfirmed by the students' writings while the inexperienced teacher wasusing students' writings to help construct her pedagogical content knowledge.

Discussion

Of the twenty prompts designed to assess individual students'understanding of mathematical concepts, only one proved nonbeneficial to thedevelopment of the teachers' pedagogical content knowledge. It askedstudents to copy a problem worked in class during seatwork or at home andexplain "step by step in your own words" what was done to arrive at a finalanswer. Inevitably, students copied problems "step by step" using onlysymbols with no explanation in words about the process they used or thethoughts they had while working the problem. The inexperienced teacherused this prompt six times until one student wrote: "To be fully/totallyabsolutely honest w/you. Im sick of this [prompt]." The teacher stoppedusing the prompt.

Space does not permit a detailed analysis of all the prompts used in thestudy. The results of other prompts are detailled in Miller (1990), Miller (inpress) and Miller and England (1939).

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Much of what was learned by the teachers from reading the students'writings served to reconfirm pedagogical content knowledge previouslyconstructed by the experienced teacher while supporting the development ofnew pedagogical content knowledge for the inexperienced teacher. For botteteachers, the writings helped to identify individual students' misconceptionsand nonunderstanding much more quickly than through the traditional wholeclass discussion process.

Questions raised by critics of the study have been "What's so uniqueabout impromptu writing prompts? Can't teachers leam the same things byasking students questions in class and by putting prompts of this type onassessments?" While teachers can ask open-ended, thought provokingquestions in class or pose them in a context of small group activities, thegeneral scenario is that only a few students become involved in thesediscussions. Allowing sufficient wait-time for students to formulate andrespond to open-ended questions orally can create a lull during which otherstudents can become off-task mentally if not physically, too. Within the timeallotted for teacher-student interaction in class, a teacher cannot usually engageevery student in a class of 25-30 students. A unique feature about impromptuwriting prompts is that each student has the opportunity to expressthemselves, in writing, to the teacher during every class in which a prompt isused. Writing also allows them the opportunity to examine their thoughts andmake changes in their statements prior to submitting their paper. Studentsanswering orally generally do not have the chance to say something, reflectupon what they said and then make changes to those statements. The teachersin this study, found that five minutes was sufficient time for students' readingand responding to a prompt, but was not disruptive to the continuation of theirother instructional practices.

Because their classes had a range of academic abilities includingstudents who were being mainstreamed from special education classes as wellas those intending to go to college, the teachers did not want to rewardstudents for writing or punish them for not writing. Some students had greatdifficulty expressing themselves in writing, particularly at the beginning of thestudy. While one might argue that students should be able to communicatetheir understanding of mathematics orally and in writing, the teachers in thisstudy did not want to handicap or embarrass students by using writingprompts as official assessment tanks. It should also be noted that this strategyof feedback is not the panacea for assessment that some look for, but aneffective additional strategy to other proven procedures. But it too has itslimitations. As noted in the results section, on a few occasions some studentsgave inadequate or ambiguous written statements which needed verbalclarification by the teacher. However the subsequent verbal discourse, guidedby the student's written response, zeroed onto particular difficulties muchmore quickly than would have been the case without this guidance.

As previously stated, this study serves as a beginning to theexamination of teacher benefits to the use of writing in mathematics classes.For the second year teacher in the study, the benefits were more pronounced


than for the experienced teacher. Both continued using writing promptsduring the second semester of the school year even though the study officiallyended with the first semester.

Teachers and mathematics education researchers should begin toinvestigate further the understudied phenomenon of benefits gained by theteachers who use writing activities in the mathematics class. Thereconfirmation and construction of pedagogical content knowledge is only onepotential benefit. Other benefits worthy of investigation are the influencereading students' writings can have on instructional practices, both short- andlong-term; and, the influence reading students writings can have on teacher-student interaction pattems. The results of this study suggest a relativelysimple, easily implemented, and fairly robust method for assessing students'understanding of mathematics at very little cost to either teachers or students.There is a need for a more controlled study measuring and documenting theteacher benefits suggested in this paper. The results could be very significantand influential to the professional development of novice and expert teachers.

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Davis, R. B., Jockusch, E., & McKnight, C. (1978). Cognitive processes inlearning algebra. Journal of Children's Mathematical Behavior, 2(1), 10-230.

Davison, D., & Pearce, D. (1990). Perspectives on writing activities in themathematics classroom. Mathematics Education Research Journal, 2 (1),15-22.

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Miller, L. D. (in press). Writing to learn mathematics. Mathematics Teacher.

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constructing pedagogical content knowledge from students’ writing in secondary mathematics

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