DVORA PERETZ

ENHANCING REASONING ATTITUDES OF PROSPECTIVE

ELEMENTARY SCHOOL MATHEMATICS TEACHERS

ABSTRACT. This article presents a constructivist approach for teaching mathematics

to prospective elementary school teachers in USA. This approach employs a model of a

‘‘mathematical situation,’’ a set of physical operations and a physical language to

reason about students’ mathematical doings. One of the primary goals of this approach

is to promote a reasoning attitude toward the learning of mathematics by prospective

elementary school teachers. At the same time this approach encourages the develop-

ment of their reasoning skills. This approach provides a less rigid frame for discussing

mathematics ‘‘doings’’, [The word ‘‘doings’’ in this manuscript refers to the variety of

mathematical activities, from thinking on a mathematical problem, using manipula-

tives, graphing, solving and so forth.] which is still structured enough to allow

prospective teachers to appreciate the kinds of doings and argumentations found in

mathematics. It provides a concrete-like basis that serves to promote the understanding

of arithmetical concepts (especially fractions). In the context of teacher education, this

approach adds structure and content to the usual curriculum of basic mathematics

courses for future elementary and middle school teachers. It grants future teachers a

higher degree of flexibility in dealing with their students’ questions and learning

difficulties.

KEY WORDS: reasoning, attitudes, mathematics teachers’ education

INTRODUCTION

Most, if not all, prospective elementary school teachers in their first

semester in college would offer, if asked, various reasons for

‘‘why’’ 3� 12 ¼ 6: Among these: ‘‘because this is the way it is done,’’

‘‘my teacher said so’’ or ‘‘flip the guy and multiply.’’1 Dreyfus and

Hadas (1996) argue that proof is an answer to the question ‘‘why;’’

certainly they would not consider this ‘‘individualized’’ kind of reason-

ing a proof, though all of these responses are indeed answers to the

question ‘‘why.’’ These answers explain why the individual student

gave six as an answer and not why six is the mathematically correct

Journal of Mathematics Teacher Education (2006) 9:381–400 � Springer 2006

DOI 10.1007/s10857-006-9013-9

answer to the mathematical challenge. This line of reasoning has no

bearing on the specific situation at hand. In the above mathematical

challenge it could easily support a 2 and 16 as correct answers in the

same manner: 3� 12 ¼ 3þ1

2 ¼ 2 is true because ‘‘this is the way it is

done,’’ and 3� 12 ¼ 1

3� 12 ¼ 1

6 is true because ‘‘flip the guy and multi-

ply.’’ It is as if an external authority required the students to accept

the arguments as true.

Improving the reasoning of prospective teachers, especially in the

context of science and mathematics, is not only a question of ‘‘teach-

ing’’ reasoning in some agreed manner, but also of changing attitudes.

The latter is a much harder task and it is one of utmost importance.

There is a need to promote the urge for a reasonable reason within

our students. How do we make them start asking ‘‘why?’’ How can we

make them appreciate the need to reason about their mathematical

‘‘doings?’’

This article describes a constructivist approach Teaching–Learning–

Space (TLS) that I have employed in order to get my undergraduate

elementary school prospective teachers students to unpack their fragile

mathematical assets and to re-construct a deeper and more flexible

understanding of the basic mathematical concepts, which for the most

part they already ‘‘know.’’ The main goals of this approach are to

encourage them to understand their mathematical ‘‘doings’’ and to

promote a reasoning attitude within them, rather than just teaching

them the basic arithmetical concepts. This approach focuses on the

teaching of arithmetic as a whole, not just on the teaching of frac-

tions. Having said that, some of the applications that I found most

powerful, and a few which I cite here, do deal with fractions.

As in any teaching approach, the TLS encompasses different

aspects underlying different teaching outcomes. The modeling aspects

of the TLS are discussed in Peretz (2005); it is a formalization of the

proposed approach as an Inverse Mathematical Model. Other aspects,

such as issues of language and of teaching for understanding, are the

focus of other papers currently under preparation. In this paper, I

show how the TLS approach can be used by teachers as a tool to

advance students’ mathematical reasoning.

Thus, the conceptual framework presented is applicable to the field

of reasoning in mathematics education and supports those aspects per-

taining to issues of reasoning in the context of a constructivist teach-

ing approach. A brief overview of the approach is provided, followed

by a description of the approach vis-a-vis a class engagement. The

theoretical attributes of the proposed approach will be discussed in the

382 DVORA PERETZ

context of a few examples of its applications. The insights reported in

the concluding section are based on my observations as the teacher of

the class and offer the reader a sense of the class at the end of the

semester. The discussion highlights some of the key aspects of the TLS

approach and refers briefly to the possibility of employing it in

elementary school.

CONCEPTUAL FRAMEWORK

Promoting reasoning skills of students is treated extensively in the

literature. This article contends that students first need to feel the

‘‘need’’ to reason; they need to develop a reasoning attitude, or a rea-

soning habit of mind. Otherwise, teaching them reasoning skills, the

‘‘how’’ to reason, is algorithmic, or as Selden and Selden (1995) say,

ritualistic. They call for building an early experiential base for the

concept of proof as they claim that:

Both weak validation skills and viewing proofs as ritualistic, and unrelated to

common sense reasoning, may be partially traceable to the absence of arguments,especially student-produced arguments, in school mathematics (p. 141).

The kind of reasoning to employ depends on the specific field-context

and on the specific level of formality required (e.g., the traditional two-

columns (claim–reason) vs. sketches in geometry or epsilon–delta in

calculus). Offering prospective teachers many opportunities to exercise

self-produced argumentation in contexts which differ in their level of for-

mality might serve not only to build an experiential base for the concept

of proof within the prospective teachers themselves, but also to suggest

ways in which they might encourage their future students to build an ear-

ly experiential base for the concept of proof as Selden and Selden (1995)

recommend.

Fischbein and Kedem (1982) describe the confusion of students in

the context of proof and evidence. It seems that some students are

willing to accept proofs as valid, but still prefer to strengthen their

conviction by additional empirical evidence. This might be linked to

Hersh’s (1993) observation:

mathematical proof can convince, and it can explain. In mathematics research, its

primary role is convincing. At the high-school or undergraduate level, its primaryrole is explaining (p.398).

Strengthening the links between ‘‘empirical’’ evidence and

‘‘accepted’’ proof might not only carry explanatory power for the

383ENHANCING REASONING ATTITUDE

student, but might also clear up some of the confusion that Fischbein

and Kedem (1982) discuss.

In Show How You Know: A Visual Medium for Demonstrative

Discourse, the authors Moore and Schwartz (1994) describe an effort

to create a mathematics learning environment in which the social

nature of the classroom facilitates conceptual conjecture and justifica-

tion. A visual representation is designed which students can manipu-

late to obtain demonstrative proofs. They describe how students began

to create a culture in which demonstration became a social phenome-

non. Such a social climate in a class of young students or of prospec-

tive teachers might contribute to the development of a sense of the

need to reason about any given mathematical situation, which in turn

might promote a reasoning attitude.

Moore and Schwartz contend that ‘‘to prove’’ is to ‘‘show’’ that

your statement is ‘‘true.’’ One way of doing this is to bring sufficient

‘‘adequate’’ reasons to convince everyone in your community. Another

way is simply to ‘‘show’’ that it works. For example, to prove the

Graham–Schmidt theorem, which asserts the existence of an orthogo-

nal basis in a finite dimension inner product space,2 one ‘‘builds’’ an

ortho-normal basis from scratch – it is as if one just provides the

required basis, which not only proves its existence, but also suggests a

constructive algorithm for finding such a basis in any given situation.

From here, we see that answering the right ‘‘whys,’’ i.e., why is a

mathematical statement true/not true constitutes a proof. Mathemati-

cal proofs or mathematical lines of reasoning differ from each other in

the context in which they are performed and in the level of their for-

mality. To prove that 2 + 3 = 5 one might use concrete blocks or

one could turn to set theory, use sets of appropriate sizes and refer to

the definition of binary operations on sets, such as union. We treat

both proofs as appropriate in different contexts: set theory might be

appropriate for an undergraduate while concrete blocks might be

appropriate for elementary school students. I would not expect pro-

spective elementary teachers to use set theory rigorously, yet I believe

they would benefit from knowing the relevant concepts from set

theory and the ways in which they are related to the arithmetic

operations, as it is done in the TLS approach.

What might a constructivist perspective imply for teaching mathe-

matics to prospective elementary school teachers? Guided inquiries,

explorations, evaluative reflections promote the construction of under-

standing in both individual and social contexts. Interpretative dis-

course allows variety of interpretation and leads to negotiation and

384 DVORA PERETZ

agreement rather than to the ‘‘right solution’’ or the ‘‘right method,’’

and ‘‘The pedagogical structure for learning [...] is shared by the

learner and the teacher’’ (Yore, 2001).

Richards (1991) describes how a teacher might go about thinking

about a constructivist teaching approach that is relevant for the

proposed TLS approach:

It is necessary to provide a structure and a set of plans that support the develop-

ment of informed exploration and reflective inquiry without taking initiative orcontrol away from the student. The teacher must design tasks and projects thatstimulate students to ask questions, pose problems, and set goals. Students will

not become active learners by accident, but by design, through the use of the plansthat we structure to guide exploration and inquiry (p. 38).

Next, I present the Teaching–Learning–Space (TLS) approach,

which promotes the urge to reason by: (1) Making it a habit to ask

the right ‘‘why;’’ (2) Having the students reason in different contexts,

(3) Involving the students in a mathematical-like discourse that is less

formal than the traditional mathematical discourse, yet maintains a

certain level of rigor, and (4) Reconstructing an intuitive basis for the

learning of mathematics.

THE TEACHING–LEARNING–SPACE APPROACH

The Teaching–Learning–Space (TLS) is a teaching model for teaching

arithmetic-based mathematics. I developed this model when I was in

Michigan State University, teaching prospective elementary school

teachers their first mathematics course. The course aimed at promoting

number sense at an elementary level. Given the limited ability of the

students to explain their mathematical ‘‘doings,’’ I found myself look-

ing for tools not only to help the students’ reason about a mathemati-

cal situation, but first and foremost to make them aware of the need

for this reasoning.

The core of the TLS in which the teaching–learning experience takes

place is perceived as consisting of three sub-spaces: the contextual, the

abstract, and the physical-abstract. These three sub-spaces differ in the

kinds of reasoning employed in each, in the language used in each and

in the kinds of activity performed in them. The transition among the

three sub-spaces is effected by a goal-driven means of re-phrasing in the

language used in the new sub-space. The TLS model provides a frame

for relating various concepts across different representations and across

different situations and facilitates a critical discourse. A comprehensive

385ENHANCING REASONING ATTITUDE

support system surrounds the core of the TLS to address motivational

issues (see Figure 1).

There are many paths through the different sub-spaces of the TLS,

and many ways of using the model to promote reasoning and under-

standing. The main continuum provides tools for studying one specific

mathematical situation: mathematical situation – contextual situation –

abstract set model – physical abstract situation – contextual situation

– mathematical situation. Other continuums follow chronological pro-

cesses: whole number to fractions or additive to multiplicative continu-

ums. The latter two continuums are interrelated and embedded in each

other and in each of the applications of the main continuum.3 Next, I

describe the different sub-spaces vis-a-vis a description of an actual

class engagement aimed at adding meaning to the mathematical algo-

rithm 3 14� 2 1

5 ¼ 134 � 11

5 ¼ 134 � 5

11 ¼ 6544 ¼ 1 21

44 and prove it (reason

about it). Some of the activities in the class were based on a whole

class forum and others were based on individual student work.

The Contextual Sub-Space

By the contextual sub-space I refer to the level of the story problem.

Thus, the language here is real and is related to the ‘‘story,’’ and so

are the activities, as well as the reasoning. The transition from the

Figure 1. The Teaching–Learning–Space model.

386 DVORA PERETZ

contextual to the abstract space is motivated by the ‘‘problem,’’ which

is first stated in the ‘‘contextual language.’’

In Class

We4 begin by constructing a real problem for the mathematical prob-

lem: Since we are dealing with division, we think about whole number

division, for example 6� 2: We try to come up with a real situation of

sharing 6 between 2, or ‘‘measuring’’ how many 2s are in 6. We look

at real problems that also make sense in the case of fractions. (thus,

we have x pizzas to share among y people’ is inappropriate). Eventu-

ally we agree on something like: A gallon of ice cream is made with 2 15

cups of sugar. Liz has 3 14 cups of sugar. How much ice cream, in gal-

lons, can she make?5 This is followed by a discussion: Will she make

at least 1 gallon of ice cream? 2 gallons? More? Why? The reasoning

here is based on ‘‘she has more than 3 but fewer than 4 cups of sugar

– so she’ll make more than 1 gallon but fewer than 2 gallons of ice

cream.’’ This discussion takes place without any actual writing, draw-

ing or calculating. We conclude the discussion with a real strategy to

find out ‘‘how many gallons’’ (whole number language) of ice cream

she can make. For example, put the amount of sugar that is required

for each gallon of ice cream in a separate heap, until there is no more

sugar left, then count the ‘‘number of heaps’’ (whole number

language). Thus, we first build the ‘‘story’’ – the modeling in the next

phase will take us back to the division of fractions.

The Abstract Sub-Space and the Abstract-Set-Model

By modeling the contextual situation we move to the abstract sub-

space and use abstract sets-language. Complex situations involve

staging or breaking down procedures and using the model iteratively;

but the Basic Model refers only to the basic binary operations and

has three components, and a theoretical goal associated with it. The

model components are: The Number of Disjoint Sets that are in-

volved in the situation (first); the Number of Elements in each such

whole set (second); the Number of Elements in the Union Set (third).

The theoretical goal stems from the desire to ‘‘complete’’ the model

(i.e., to find the missing component), or to ‘‘describe’’ the relation-

ship between two of the model’s components. In the latter case, we

would refer to it as a relational-theoretical goal.

387ENHANCING REASONING ATTITUDE

Hence, we define two basic types of the Abstract Set Model

(Figure 2): The additive model and the multiplicative model. While a

constant number (2) of disjoint sets that are involved in the situation

characterizes the additive model, it is the equal size of the disjoint sets

that characterizes the multiplicative model. In both models, the third

component is the number of elements of the union set of all the

disjoint sets. The contextual space determines the theoretical goal for

the abstract space, which is therefore expressed in sets-language. If the

theoretical goal is to expose the third component (i.e., number of ele-

ments in the union set), then the model represents an addition (additive

model) or multiplication (multiplicative model) situation, whereas if the

theoretical goal is to expose the second (i.e., number of elements in

each whole set), then the model represents a subtraction (additive mod-

el) or a division (multiplicative model) situation. The latter is tradition-

ally referred to as the partitive approach to division. If the theoretical

goal is to expose the first component (number of sets), which is rele-

vant only in a multiplicative model, then the model describes what is

usually referred to as the measurement division approach. Relational-

theoretical goals in the additive model could describe either additive

relations (bigger–smaller) that are basically subtraction situations, or

multiplicative relations, which are ratio (or proportion) situations.

In Class

Since we do not have ‘‘exactly’’ two sets (heaps) and since all of our

sets (heaps) must have the same number of elements (same number of

cups of sugar), the modeling of the real strategy leads to a multiplica-

tive model; where the number of sets is unknown, there are 2 15 elements

in each whole set and 3 14 elements in the union set; thus the theoretical

Figure 2. The abstract set model.

388 DVORA PERETZ

goal is to ‘‘expose’’ the number of sets. The reasoning here will revolve

around the appropriateness of the other models, too. For example,

could it be that 2 15 is the number of sets and 3 1

4 is the number of

elements in each whole set? That would suggest that we have more

than 2 sets each with more than 3 elements, so in total we need to

have more than 6, which we do not. What would this imply in the real

situation?

The theoretical goal leads to a practical goal: To do ‘‘physically’’ in

order to achieve the theoretical goal (i.e., to join sets, to take-away

elements, to put-equally into sets, etc.).

In Class

We set the practical goal: to use all of our resources (3 14 elements) to

make as many sets as possible, each with 2 15 elements (whole number

language), which makes the transition to where we actually do the

mathematics.

The Physical-Abstract Sub-Space

The most significant feature of this space is the (mental) ‘‘physical’’

doings and language that are applied to the abstract objects (sets and

elements). These physical doings also serve to reason ‘‘physically’’

about the situation. All the doings here are accompanied by drawings

to show how, and by words to explain why.

In Class

The doings of a student are described in Figure 3. Here we see that

the student starts making sets of the required size. He first cuts up one

element into five equal pieces and then he puts 2 whole elements and

one fifth of an element together to make up one set. Since there are

not enough elements to make one more whole set, the student puts all

the leftovers together to make as ‘‘many’’ (whole number language)

sets as possible. Now, the student needs to count the number of sets

he made. To do this, he rephrases the ‘‘names’’ of the elements (uses

different units, as the original unit of 1 element is not practical any-

more), by cutting up each fourth into 5 equal pieces and each fifth

into 4 equal pieces to get many ‘‘new’’ elements ð 120Þ:

We also suggest and discuss alternative ways. For example, chang-

ing the unit of reference in the beginning by cutting each 14 of an

element into 5 equal pieces and each 1/5 into 4 equal pieces we move

389ENHANCING REASONING ATTITUDE

to work with 120 of an element instead of one whole element. Thus, we

might rephrase our problem as the whole number problem of: 65� 44:Finally, we take our ‘‘products’’ back to.

The Contextual Realm

To insert/relate our products/results into the real situation; 1 2144 what?

Does it make sense? Did we get a negative number for the number of

Figure 3. The physical doing of 3 14� 2 1

5 :

390 DVORA PERETZ

gallons? Did we get more than 1 gallon and fewer than 2 gallons, as

we estimated in the beginning? We also discuss our results in terms of

the measurement approach. For example, we ‘‘measure’’ how much

sugar Liz has by the ‘‘number of gallons of ice cream it can make’’ (not

by cups) or by a ‘‘special measuring cup’’ equal to the size of 2 15 cups;

Then back to.

The Mathematical Realm

To discuss the mathematical principles that underline our ‘‘doings,’’

such as issues of commutative, associative, distributive, multiplica-

tive, and additive inverses. What are the ‘‘physical’’ sources or the

‘‘physical’’ indications for them in our physical doings? For exam-

ple, what does the ‘‘flipping’’ mean? Why does it yield the ‘‘true’’

answer to our mathematical challenge? Looking at both the mathe-

matical algorithm and at our physical-abstract doings (Figure 3),

we try to identify the roles of each number in both procedures.

Here ‘‘we’’ means teacher and students – I am describing–summa-

rizing what we did in class: usually it is the students who do most

of the talking. We see that 120 plays a role in the physical abstract

realm but not in the mathematical. How is this? A closer observa-

tion reveals that, in order to be able to describe the number of

elements in all the sets in the situation, we need a common unit

of reference to describe the fourths and the fifths of an element.

Thus, we cut each fourth into five equal pieces and each fifth into

four equal pieces to get our ‘‘new’’ elements 120. Using 1

20 of an ele-

ment as the new unit of reference to describe our sets re-phrases

our practical goal: to make sets of 44 120-elements when we have a

total of65 120-elements. We can link these facts to the original prac-

tical goal of making sets of 115 elements when we have a total of 13

5

elements to explains the ‘‘cross’’ multiplication of the numerator of

the first fraction by the denominator of the second (13� 5 = 65),

and of the numerator of the second fraction by the denominator

of the first (11� 4 = 44). Now since our theoretical goal was to

discover how many sets we have, not only do we see that the

‘‘new’’ units of reference ( 120-elements in this case) are not relevant

in the final product, but we also see that the question of how

much of the second set we made calls for a fraction notation

where the 44 is the denominator and the 65 is the numerator.

This results in the ‘‘flipping’’ effect.

391ENHANCING REASONING ATTITUDE

THE MAIN PRINCIPLES OF THE TLS APPROACH

The TLS approach reverts to natural–physical doings in trying to pro-

mote a natural or intuitive understanding of the basic mathematical

concepts as a well-established, firm basis for the understanding of the

more abstract, formal concepts. The physical doings serve to reason in

a ‘‘practical’’ manner about the situation on hand. For example:

‘‘3� 12 ¼ 6’’ since I have physically made 6 sets, each having 1

2 ele-

ments until my resource set of 3 elements is exhausted. Alternatively, I

put all my 3 elements equally into ‘‘all’’ my 12 empty sets. Since each

whole set in the situation is made up of two such ‘‘12 sets’’ and since

‘‘equally’’ means that there are the same number of elements in both

halves of a set, i.e., 3, then there are 6 elements in one whole set. In

both cases we support our arguments by presenting ‘‘physical’’ evi-

dence by drawing all the stages of our ‘‘doings.’’

Instead of concentrating on re-teaching the basic mathematical

arithmetic concepts, it employs a mathematical-like critical discourse

about these mathematical concepts using non-mathematical language.

The mathematical-like discourse is a ‘‘formalization’’ of informal

methods: it uses pre-defined structures to lead the otherwise ‘‘infor-

mal’’ class discussions, as was illustrated in the third section. This dis-

course emphasizes the asking of the right ‘‘whys’’ and leads the

students through finding appropriate lines of reasoning to answer

these ‘‘whys.’’ From my experience, this critical approach also helps

raise the students’ attention level, which is rather low when they study

topics that they regard as already known.

The ‘‘doings’’ in class, the mathematical-like discourse, and the

rigid use of our model, are tied explicitly to the kinds of doing and the

rigid argumentation discourse used in mathematics. For example, before

‘‘doing physically’’ to ‘‘prove’’ the distributive law with our model and

our language, we discuss the way a mathematician would go about

proving it, tying mathematical definitions and operations to our basic

‘‘doings’’ (break-down, join, stage etc.) and mathematical arguments

to our ‘‘physical’’ proofs.

Building on a deep understanding of simple whole number situations as a

basis for all further learning we discuss mathematical situations such

as 12 16� 1 3

4 using whole number language: ‘‘we have 1 34 groups ...’’ rather

than ‘‘we have one group and three quarters of a group.’’ The use of whole

number language motivates the use of whole number (‘‘natural’’) models to

deal with ‘‘new’’ non-natural kinds of numbers: a division like this: 12 16� 1 3

4 ;

is much the same as a division like this: 6� 5, so, we need to do the same!

392 DVORA PERETZ

Using the abstract model, which conceptualizes all basic mathemati-

cal situations as the same, to channel and to mold reasoning – ‘‘filling

in’’ the abstract set model and setting the theoretical and practical

goal accordingly: 1 34 sets each having an unknown number of elements

and the union-set has 12 16 elements; A Multiplicative Model; Theoreti-

cal Goal: To reveal the number of elements in each set; The practical

goal is to put equally all the 12 16 elements in all the ‘‘empty’’ 1 3

4 sets,

until we exhaust our resource set. Figure 4 presents a student’s ‘‘physi-

cal’’ doings.

There are various ways in which the TLS approach promotes a

reasoning attitude: Any statement is accompanied by answers to: Why

is it true/false? Why is it important/not important? Each situation is

explored thoroughly rather than employing a ‘‘task-oriented’’ explora-

tion. The students carry out massive (i.e., lots of) reasoning tasks such

as the above and others, and massive investigations (trying to reason)

of other students’ understandings and solutions.

The TLS approach also contributes to developing awareness of the

meta-cognitive processes. This is done by using the abstract set model

to compare the same/different situations, reasoning, and doings; by

using the same language/doings in comparable situations; by reasoning

about the thinking: ‘‘Why this procedure and not another,’’ etc. and

by employing a holistic approach working up from the mathematical

situation through the contextual to the abstract and then back to the

‘‘physical-abstract’’ realm and again for a closure to the contextual

and to the mathematical strata.

SCAFFOLDING AND CHANNELING REASONING

There are many applications of the TLS that make mathematics

more meaningful and more accessible to the students. One of its

most significant applications is the scaffolding of the understanding

of others. To illustrate, I describe how students make sense of an

unusual partial solution for a long division problem (Figure 5). The

students were asked to justify, to reason and to ‘‘finish’’ Jack’s

work. Only those who wrote down both options for the multiplica-

tive model (see Figure 6) and searched for ‘‘clues’’ in Jack’s work

to try and understand which model underlines his doings,

succeeded.

Each model ‘‘leads’’ to a different practical goal and setting (see

Figure 7): Model A leads to ‘‘put equally all 3674 elements into 87

393ENHANCING REASONING ATTITUDE

empty sets’’ and Model B leads to ‘‘make as many sets as possible, 87

elements each, with all your 3674 elements.’’

If Jack was thinking about Model A, we need ‘‘to see’’ 87 sets (or

something close to that) filling up. Since nowhere in Jack’s work do

Figure 4. Physical doings of 12 16� 1 3

4 :

394 DVORA PERETZ

we see 87 ‘‘empty sets’’ or anything close, nor do we see anything that

might be understood as ‘‘putting equally,’’ we cannot really convince

ourselves that he has opted for Model A. That leaves us with Model

Figure 5. Jack’s partial doings of3674� 87:

Figure 6. The two models.

Figure 7. Physical settings of models A and B.

395ENHANCING REASONING ATTITUDE

B. Here he is supposed to start only with his resource set, and we

need to see some sort of ‘‘making of sets,’’ each with 87 elements.

Using these glasses to look at Jack’s work, we see that he says ‘‘30

100s’’ which are 30 sets of a ‘‘fixed’’ number of elements. We can

now relate this to the practical goal in Model B ‘‘make sets of size

x.’’ With this in mind, we can follow his work when he makes sets

of hundreds and then ‘‘corrects’’ them to the desired size. This also

explains the 13s that we see in his work, about whose derivation few

students at the outset had any clue. Now we can finish his work in

the same way, see Figure 8.

DISCUSSION

There are several aspects of the TLS approach that are relevant to our

discussion about reasoning: asking the ‘‘whys,’’ reasoning ‘‘physically’’

Figure 8. The physical reasoning of model B.

396 DVORA PERETZ

about mathematics, using the abstract set model to channel the

reasoning and the holistic path that connects the formal to the infor-

mal and to the semi-formal. But perhaps the most important question

is whether this approach actually changes a student’s attitudes towards

reasoning. One could not expect a drastic change in attitudes after a

period of one semester. Yet this approach makes the asking of the

right ‘‘why’’ and supporting one’s answer with a relevant ‘‘proof’’ an

integral part of the discourse. Asking ‘‘why’’ creates the need for proof

(Dreyfus & Hadas, 1996). The students were constantly engaged in

explaining each step of their contextual, abstract, and physical

‘‘doings,’’ and their motivation for doing so at different levels and in

the different ‘‘languages.’’

Asking ‘‘why’’ promotes the urge to reason. The students were

exposed to the ‘‘right’’ ‘‘whys’’ to ask and they were confronted by

‘‘whys’’ to which they had no immediate answer. By the end of the

semester, not only were most of the students able to ask ‘‘good’’ ques-

tions and offer ways to find answers to these questions, but they also

understood the need to ask these questions. This could be seen in the

students’ weekly reasoning assignments, as well as in the class discus-

sions, as illustrated in the examples presented in the article. This

agrees with Richards (1991, p. 68) portrayal of a constructivist teach-

ing approach: ‘‘The teacher must design tasks and projects that

stimulate students to ask questions, pose problems, and set goals.’’

The TLS approach, similar to that of Moore and Schwartz (1994),

facilitates conceptual conjecture and justification and creates a culture

in which reasoning and demonstration became a social phenomenon.

Doing mathematics ‘‘physically’’ offers informal ‘‘proofs’’ and justifica-

tions; it serves in place of formal theorems and logic, which are used

by mathematicians to prove/understand mathematical theorems. By

‘‘physically’’ tracing each step of the ‘‘statement’’ (solution algorithm,

commutative rule, etc.), we prove it to be true or false. Moreover, the

‘‘physical-abstract-doing’’ serves what mathematicians refer to as

insightful proof, a proof that offers a ‘‘deep’’ understanding of the

situation at hand.

The abstract set model provides a channeling device to scaffold the

reasoning process. It helps the students to analyze the abstract mathe-

matical situation and leads them to construct a practical strategy to

solve it. Many students have difficulties deciding ‘‘what is next’’ in

order to construct a valid line of reasoning; the abstract set model and

the associated theoretical goals with their ‘‘fill in’’ structure lead the

students from one stage to another and help them decide ‘‘what is

397ENHANCING REASONING ATTITUDE

next’’ (see Richards’, 1991, quote in the conceptual framework

section).

The use of the Abstract Set model together with the ‘‘doings’’ in

the physical-abstract realm accentuates the differences between addi-

tive and multiplicative structures. It channels the students’ reasoning

in distinguishing between these structures – an area that is long known

to be one of great difficulty for students. Making the different charac-

teristics of the two models (the additive and the multiplicative) explicit

in the many different situations that we explore in class helps the stu-

dents to see the similarities and the differences between additive and

multiplicative situations; this, in turn, helps them to make the distinc-

tion when necessary. The students were constantly required to relate

various representations (i.e., contextual, formal-mathematical,

abstract-physical) and ‘‘doings’’ across situations and across concepts.

This holistic approach grants the students a higher degree of flexi-

bility in dealing with mathematical problems. Like many of the con-

structivist approaches to the teaching and learning of mathematics, the

TLS approach also emphasizes the importance of both aspects of the

mathematical experience: the concrete and the abstract, and the inter-

play between them. The physical-abstract realm offers a mediator con-

text in which students might understand the links between the concrete

situation and the abstract mathematical situation.

The prospective elementary school teachers students were encour-

aged to construct their own individual ‘‘doings;’’ the more advanced

students were challenged to also try ‘‘awkward’’ procedures, and not

only the most ‘‘efficient’’ one. For example: ‘‘...try to put 12 elements

into each set first, even if it makes the ’leftovers’ in the resource set an

’ugly’ number....’’ The use of ‘‘ugly’’ numbers veils intuition and there-

fore forces students to turn to logical arguments. Here, students could

test their understanding of the mathematical situation as well as their

reasoning skills.

In the context of abstract mathematics, the ‘‘physical’’ ‘‘doings’’

help strengthen the students’ conviction (Fischbein and Kedem, 1982)

by calling on their intuition (Fischbein, 1987) for certitude. The differ-

ent ‘‘doings’’ and reasoning in the TLS offer proofs that explain

(Hersh, 1993), as well as validation and demonstration (Selden &

Selden, 1995). Many students appreciate the concise view of the situa-

tion that the model affords, as well as the rigid frame it provides to

steer their ‘‘doings’’ and reasoning in a new situation. Also, they seem

to enjoy the flexibility that using the non-formal mathematical

language permits. By the end of the semester it was evident that the

398 DVORA PERETZ

quality of the class discussions had changed for the better. By that

time students could discuss the whole mathematical situation – from

constructing a ‘‘good’’ problem-story, to the ‘‘doing’’ to solve it, and

could reason about all its different aspects. Judging from the flexibility

students exhibited by the end of the semester in their response to the

many reasoning tasks, one could expect that the students’ ability to re-

spond to their future students’ questions was likewise enhanced.

Some readers might feel, as did a few of my students, that this

approach makes things more complicated. However, one needs to

remember that the TLS approach is not designed for teaching the

arithmetic mathematical concepts. Rather, its main goal is to promote

within the students reasoning skills and reasoning attitudes in the con-

text of arithmetic; indeed this is a much harder task. When things are

simple, students treat them as trivial and find no need to justify them

or to reason about them. The critical approach puts the ‘‘already

known’’ concepts into a new, more complex context. When students

try to accommodate the mathematical situation into the three-faceted

model they are forced to examine it from different viewpoints. The

requirement to reason about each of these steps not only helps them

develop appropriate lines of reasoning, but also makes reasoning an

integral part of any mathematical doing. This I believe is a step to-

wards changing prospective elementary school teachers’ attitudes

towards reasoning in the context of mathematics; prospective teachers

might also take with them ideas about how to create a teaching envi-

ronment which promotes reasoning attitudes and skills within their

elementary school’s students.

NOTES

1 We would usually consider ‘‘flip the guy and multiply’’ as an image, a metaphor

or a remembered procedure, however, here as an answer to the question, Why, it as-

sumes the role of a ‘‘line of reasoning’’ or of an ‘‘individualized proof.’’2 This can be found in any basic linear algebra textbook, such as Linear algebra with

application by Steven J. Leon (1999), 5th edition. Upper Saddle River, NJ: Prentice

Hall.3 Usually, we begin by exploring addition situations which gradually evolve into

multiplication situations; when discussing addition we explore whole number con-

texts and move on to contexts which involve fractions (The basic assumption is that

students already know fractions). When getting into multiplication situations, we

again ‘‘meet’’ fractions, from yet another aspect, i.e., their ‘‘construction.’’ Here the

discussion ‘‘starts over:’’ we re-examine characteristics of the ‘‘new’’ constructs, ‘‘re-

define’’ operations on them and explore attributes of the ‘‘new’’ operations. In such

399ENHANCING REASONING ATTITUDE

ways our critical discourse is evolving in overlapping circles.4 We here relates to myself as teacher and the students in my class.5 I recognize that this problem is not very realistic, but it demonstrates in a real way

the need for division of fractions.

REFERENCES

Dreyfus, T. & Hadas, N. (1996). Proof as answer to the question why. InternationalReviews on Mathematical Education, 28(1), 1–5.

Hersh, R. (1993). Proving is convincing and explaining. Educational Studies inMathematics, 24(4), 389–399.

Fischbein, E. & Kedem, I. (1982). Proof and certitude in the development of

mathematical thinking. In A. Vermandel (Ed.), Proceedings of the Sixth Interna-tional Conference for the Psychology of Mathematical Education (pp. 128–131).Antwerp.

Fischbein, E. (1987). Intuition in science and mathematics: An educational approach.

Dordrecht: Reidel.Moore, J. L. & Schwartz, D. L. (1994). Show how you know: A visual medium fordemonstrative discourse. Paper presented at the Annual Meeting of the American.

Educational Research Association (New Orleans, LA, April 4–8, 1994).Peretz, D. (2005). Inverse mathematical model – yet another aspect of applicationsand modeling in undergraduate mathematics for prospective teachers. Teaching

Mathematics and its Applications Journal, 24(2–3), 134–142.Richards, J. (1991). Mathematical discussions. In E. Glasersfeldvon (Ed.), Radicalconstructivism in mathematics education (pp. 13–51). Dordredcht, The Netherlands:

Kluwer.Selden, J. & Selden, A. (1995). Unpacking the logic of mathematical statements.Educational Studies in Mathematics, 29(2), 123–151.

Yore, L. D. (2001). What is meant by constructivist science teaching and will the

science education community stay the course for meaningful reform? ElectronicJournal of Science Education 5(4).

Center for Advanced Studies in Mathematics

Ben Gurion University

P.O. Box 653

Beer Sheva, 84105

Israel

E-mail: pere@cs.bgu.ac.il

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