Basic Problems in Basic Problems in Mathematics Education Mathematics Education

ResearchResearch1, 21, 2

Leslie P. SteffeLeslie P. Steffe

University of GeorgiaUniversity of Georgia

Basic Principles of Radical Basic Principles of Radical ConstructivismConstructivism

• 1. A. Knowledge is not passively received either 1. A. Knowledge is not passively received either through the senses or by way of communication;through the senses or by way of communication;

• B. Knowledge is actively built up by the B. Knowledge is actively built up by the cognizing individual.cognizing individual.

• 2. A. Knowledge is adaptive, in the biological 2. A. Knowledge is adaptive, in the biological sense of the term, tending toward fit or viability;sense of the term, tending toward fit or viability;

• B. Cognition serves in the individual’s B. Cognition serves in the individual’s organization of the experiential world, not the organization of the experiential world, not the discovery of an objective, ontological reality discovery of an objective, ontological reality (Glasersfeld, 1995)(Glasersfeld, 1995)

A Progressive Research A Progressive Research Program.Program.

• A research program is progressive if research program is progressive if progressive problem shifts occur. (Lakatos, progressive problem shifts occur. (Lakatos, 1970).1970).

First ProblemFirst Problem

• How can researchers in mathematics How can researchers in mathematics education use mathematics teaching as a education use mathematics teaching as a method of scientific investigation?method of scientific investigation?

Main Goals of Using Teaching as a Main Goals of Using Teaching as a Method of Scientific Investigation.Method of Scientific Investigation.

• 1. To build living, experiential models of 1. To build living, experiential models of students’ mathematics. students’ mathematics.

• 2. To experience constraints when teaching 2. To experience constraints when teaching students. students.

– The constructivist is fully aware of the fact that an The constructivist is fully aware of the fact that an organism’s conceptual constructions are not fancy-organism’s conceptual constructions are not fancy-free. On the contrary, the process of constructing is free. On the contrary, the process of constructing is constantly curbed and held in check by the constantly curbed and held in check by the constraints it runs into. (Glasersfeld, 1990, p. 33)constraints it runs into. (Glasersfeld, 1990, p. 33)

Basis For Using Teaching as a Method of Basis For Using Teaching as a Method of Scientific InvestigationScientific Investigation

• The view of students as self-organizing and The view of students as self-organizing and interacting systems (the mind organizes the interacting systems (the mind organizes the world by organizing itself).world by organizing itself).

• "Cognitive processes seem, then, to be at one and the same "Cognitive processes seem, then, to be at one and the same time the outcome of organic auto-regulation and the most time the outcome of organic auto-regulation and the most highly differentiated organs of this regulation at the core of highly differentiated organs of this regulation at the core of interactions with the environment” (Piattelli-Palmarini,1980).interactions with the environment” (Piattelli-Palmarini,1980).

A Compatibility with Second-A Compatibility with Second-Order Cybernetics.Order Cybernetics.

• Approaches to inquiry have centered on Approaches to inquiry have centered on the idea of worlds being constructed by the idea of worlds being constructed by inquirers who are simultaneously inquirers who are simultaneously participants in those same worlds (Steier, participants in those same worlds (Steier, 1995). 1995).

• A science of observed systems cannot be A science of observed systems cannot be divorced from a science of observing divorced from a science of observing systems because it is we who observe.systems because it is we who observe.

Synopsis of the Researcher as TeacherSynopsis of the Researcher as Teacher

• It is critical to understand that one’s own actions and It is critical to understand that one’s own actions and interactions as a teacher are essential in student’s interactions as a teacher are essential in student’s construction of mathematics.construction of mathematics.

• More importantly, one’s own actions and interactions, More importantly, one’s own actions and interactions, when coupled with students’ actions and interactions, are when coupled with students’ actions and interactions, are essential in one’s own construction of students’ essential in one’s own construction of students’ mathematics and how it might be productively affected.mathematics and how it might be productively affected.

• A major goal of the researcher as teacher is to produce A major goal of the researcher as teacher is to produce images of the constructive possibilities of students in images of the constructive possibilities of students in

mathematics.mathematics.

Second ProblemSecond Problem

• How can researchers build explanatory How can researchers build explanatory models of students’ construction of models of students’ construction of mathematics in the context of using mathematics in the context of using teaching as a method of scientific teaching as a method of scientific investigation?investigation?

First- and Second-Order First- and Second-Order Models.Models.

• First-order models are models the First-order models are models the individual has constructed to comprehend individual has constructed to comprehend and control his or her own experience; that and control his or her own experience; that is, the individual’s knowledge. is, the individual’s knowledge.

• Second-order models are models an Second-order models are models an observer constructs of the observed observer constructs of the observed person’s knowledge in order to explain his person’s knowledge in order to explain his or her observations.or her observations.

Conceptual Analysis.Conceptual Analysis.• Conceptual analysis is involved in building

explanatory models of students’ construction of mathematics.

• What mental operations (and co-ordinations What mental operations (and co-ordinations thereof) can I posit in my model (the student’s thereof) can I posit in my model (the student’s mathematics) to explain the actions I have mathematics) to explain the actions I have observed?observed?

• Students’ mathematics is about the ways and Students’ mathematics is about the ways and means they operate mathematically.means they operate mathematically.

The First Time That Conceptual The First Time That Conceptual Analysis Was Used in Mathematics Analysis Was Used in Mathematics Education Research.Education Research.

• Interdisciplinary Research on Number (IRON)

• Members of the original project: Ernst von Glasersfeld, Pat Thompson, who was then a graduate student, John Richards, a philosopher of mathematics, and myself.

Children’s Counting Schemes.Children’s Counting Schemes.

• Perceptual Counting Scheme.Perceptual Counting Scheme.• Figurative Counting Scheme.Figurative Counting Scheme.• Initial Number Sequence [INS Counting Initial Number Sequence [INS Counting

Scheme].Scheme].• Tacitly Nested Number Sequence [TNS Tacitly Nested Number Sequence [TNS

Counting Scheme].Counting Scheme].• Explicitly Nested Number Sequence [ENS Explicitly Nested Number Sequence [ENS

Counting Scheme].Counting Scheme].• Generalized Number Sequence.Generalized Number Sequence.

Third Problem. Third Problem.

• What are the trajectories of epistemic students over the course of pre-college education that are abstracted from the scientific teaching of students?

Epistemic SubjectEpistemic Subject

• That which is common to all subjects at That which is common to all subjects at the same level of development, whose the same level of development, whose cognitive structures derive from the most cognitive structures derive from the most general mechanisms of the co-ordination general mechanisms of the co-ordination of actions. (Piaget, 1966, p. 308)of actions. (Piaget, 1966, p. 308)

The Epistemic StudentThe Epistemic Student

• Consists of the mathematical schemes of action Consists of the mathematical schemes of action and operation of students at the same level of and operation of students at the same level of construction that have been abstracted from construction that have been abstracted from living, experiential models of students. living, experiential models of students.

• Epistemic students are dynamic organizations of Epistemic students are dynamic organizations of schemes of action and operation in the schemes of action and operation in the researchers’ or teachers’ mental life. The researchers’ or teachers’ mental life. The schemes of action and operation include schemes of action and operation include accommodations in the schemes. accommodations in the schemes.

Mathematics of StudentsMathematics of Students• The epistemic students that teachers or researchers The epistemic students that teachers or researchers

carry around in their heads are similar to what family carry around in their heads are similar to what family therapists call internalized others. therapists call internalized others.

• However, I think of epistemic students as interiorized However, I think of epistemic students as interiorized others. It is the living, experiential models that are others. It is the living, experiential models that are analogous to internalized others.analogous to internalized others.

• Experiences of students’ mathematics are mathematical Experiences of students’ mathematics are mathematical experiences. experiences.

• I consider students as rational beings and the I consider students as rational beings and the explanations of their mathematics--the mathematics of explanations of their mathematics--the mathematics of

students--as serious and important mathematics.students--as serious and important mathematics.

Radical Constructivism and Radical Constructivism and “School Mathematics”“School Mathematics”

• The main reason for doing research on the The main reason for doing research on the construction of the mathematics of construction of the mathematics of students:students:

– To construct a “school mathematics” that is To construct a “school mathematics” that is based on the mathematical concepts and based on the mathematical concepts and operations of students that are abstracted operations of students that are abstracted from scientific teaching. from scientific teaching.

Fourth ProblemFourth Problem

• What are the trajectories of epistemic What are the trajectories of epistemic students that are abstracted from the students that are abstracted from the scientific teaching of students who enter scientific teaching of students who enter grade one in the perceptual stage of the grade one in the perceptual stage of the counting scheme?counting scheme?

The Concept of a SchemeThe Concept of a Scheme

• O SO S• B iB i• S tS t• e ue u• r ar a• v tv t• e ae a• r’ tr’ t• s is i• oo• nn

Scheme’sActivity

Scheme’sResult

Scheme’sSituation

Scheme’sGoal

• FCSFCS

• PCSPCS

Sep Oct Nov Dec Jan Feb Mar Apr May Jun Sep Oct Nov Dec Jan Feb Mar Apr May JunSep Oct Nov Dec Jan Feb Mar Apr May Jun Sep Oct Nov Dec Jan Feb Mar Apr May Jun First Grade First Grade Second Grade Second Grade

• Brenda James TarusBrenda James Tarus

Trajectories of Three Students’ Counting Schemes Who Were in the Perceptual Stage Upon Entering Grade One.

Percent of Entering First Graders in the Perceptual Percent of Entering First Graders in the Perceptual Stage of the Counting Scheme Reported by Stage of the Counting Scheme Reported by Individual School Systems in Robert Wright’s Individual School Systems in Robert Wright’s Mathematical Recovery Program.Mathematical Recovery Program.

• In Wyoming: 61% and 12% pre-counters.In Wyoming: 61% and 12% pre-counters.

• In Wyoming: 10%In Wyoming: 10%

• In Arkansas: 45%In Arkansas: 45%

• In Maryland: 60% and 8% pre-counters. In Maryland: 60% and 8% pre-counters.

• In Georgia*: 33%In Georgia*: 33%

*Data Supplied by Myself. *Data Supplied by Myself.

Tarus: January of his Second-GradeTarus: January of his Second-Grade

• I presented Tarus with a cylindrical tube open I presented Tarus with a cylindrical tube open at one end where a marble would just fit into the at one end where a marble would just fit into the tube. After Tarus put eleven marbles into the tube. After Tarus put eleven marbles into the tube, I poured three marbles out of the tube into tube, I poured three marbles out of the tube into his hand and asked him to find how many his hand and asked him to find how many marbles were left in the tube. marbles were left in the tube.

• Tarus buried his head in his arms and played Tarus buried his head in his arms and played with a marble and then said, “ten”. In with a marble and then said, “ten”. In explanation, he said, “I count”. explanation, he said, “I count”.

Fifth Problem Fifth Problem

• What are the trajectories of epistemic What are the trajectories of epistemic students that are abstracted from the students that are abstracted from the scientific teaching of students who enter scientific teaching of students who enter grade three in the INS stage of the grade three in the INS stage of the counting scheme? counting scheme?

Hypothetical Trajectory of the Three Students Who Hypothetical Trajectory of the Three Students Who Entered Grade One in the Perceptual Stage of the Entered Grade One in the Perceptual Stage of the Counting Scheme from Grade Three Through Counting Scheme from Grade Three Through Grade Five.Grade Five.

TNSTNS

• • INSINS

• Sep May Sep May Sep May Sep May Sep May Sep May •

• Third Grade Fourth Grade Fifth GradeThird Grade Fourth Grade Fifth Grade

•

• Brenda Tarus JamesBrenda Tarus James

Sixth ProblemSixth Problem

• What are the trajectories of epistemic students What are the trajectories of epistemic students concerning the construction of strategic additive concerning the construction of strategic additive and multiplicative reasoning that are abstracted and multiplicative reasoning that are abstracted from the scientific teaching of students who from the scientific teaching of students who enter their first grade in the ENS stage of the enter their first grade in the ENS stage of the counting schemecounting scheme?

Operations of The Explicitly Nested Operations of The Explicitly Nested Number SequenceNumber Sequence

• The first operationThe first operation: : The operations that The operations that produce a unit of units.produce a unit of units.

• The second operationThe second operation: : The iterability of The iterability of the unit of one. the unit of one.

• The third operationThe third operation: : Recursion.Recursion.

• The fourth operationThe fourth operation: : The disembedding The disembedding operation. operation.

Jason solving “27 + __ = 36” in Jason solving “27 + __ = 36” in March of his second gradeMarch of his second grade• T: (Places “27 + __ = 36” in front of Jason.)T: (Places “27 + __ = 36” in front of Jason.)• J: Twenty-seven—(pause of about 20 seconds). J: Twenty-seven—(pause of about 20 seconds).

Let me see—(another pause)—twenty-seven Let me see—(another pause)—twenty-seven plus seven—it’s nine more! plus seven—it’s nine more!

Strategic Additive ReasoningStrategic Additive Reasoning

• I asked Johanna to take twelve blocks, I asked Johanna to take twelve blocks, told her that together we had nineteen, told her that together we had nineteen, and asked her how many I had. and asked her how many I had.

– After sitting silently for about 20 seconds, she After sitting silently for about 20 seconds, she said, “seven” and explained, “Well, ten plus said, “seven” and explained, “Well, ten plus nine is nineteen; and I take away the two-1 nine is nineteen; and I take away the two-1 mean, ten plus two is twelvemean, ten plus two is twelve, and nine take away two is seven!”

Strategic Additive Reasoning as Mathematics of Strategic Additive Reasoning as Mathematics of

StudentsStudents

• By engaging in strategic additive reasoning, By engaging in strategic additive reasoning, students construct a mathematical reality for students construct a mathematical reality for themselves that is a product of their own ways themselves that is a product of their own ways and means of operating. and means of operating.

• It should be no surprise that when learning to It should be no surprise that when learning to compute using standard algorithms is compute using standard algorithms is emphasized, students easily lose their sense emphasized, students easily lose their sense that it is they who are the agents in doing that it is they who are the agents in doing mathematics.mathematics.

Strategic Multiplicative Strategic Multiplicative ReasoningReasoning

• An eight-year-old child, Nathan, was asked to An eight-year-old child, Nathan, was asked to make make copies of a string of three toys and a string copies of a string of three toys and a string of four toys to make 24 toys using computer of four toys to make 24 toys using computer software. Rather than make the copies, the software. Rather than make the copies, the child reasoned out loud as follows:child reasoned out loud as follows:

– Three and four Three and four is seven; three sevens is 21, so three is seven; three sevens is 21, so three more to make 24. That’s four threes and three fours!more to make 24. That’s four threes and three fours!

Generalized Number SequenceGeneralized Number Sequence• The operations that produce a unit of units of The operations that produce a unit of units of

units are available to students prior to activity. units are available to students prior to activity.

• All of the operations of the explicitly nested All of the operations of the explicitly nested number sequence can be carried out using number sequence can be carried out using composite units as well as other operations such composite units as well as other operations such as those demonstrated by Nathan.as those demonstrated by Nathan.

Seventh ProblemSeventh Problem

• How do students use the unitizing How do students use the unitizing operation throughout their mathematics operation throughout their mathematics education in the construction of operations education in the construction of operations that produce systems of units in the that produce systems of units in the context of the scientific teaching of context of the scientific teaching of students?students?

Eighth ProblemEighth Problem

• What are the trajectories of epistemic What are the trajectories of epistemic students concerning the construction students concerning the construction of quantitative schemes that are of quantitative schemes that are abstracted from the scientific teaching abstracted from the scientific teaching of students?of students?

Quantitative SchemesQuantitative Schemes

• According to Thompson (1994), a According to Thompson (1994), a quantitative scheme consists of: quantitative scheme consists of:

• an object concept, an object concept,

• a property of that object concept, a property of that object concept,

• a unit to measure the property, and a unit to measure the property, and

• a process by which a numerical value is a process by which a numerical value is assigned to the property.assigned to the property.

Reorganization HypothesisReorganization Hypothesis

• Operative quantitative schemes can be Operative quantitative schemes can be constructed as reorganizations of number constructed as reorganizations of number sequences.sequences.

Extensive Quantitative Extensive Quantitative PropertiesProperties

• An extensive quantitative property of an object An extensive quantitative property of an object concept can be subdivided or partitioned.concept can be subdivided or partitioned.

• Number concepts can be used as templates in Number concepts can be used as templates in partitioning.partitioning.

• Children’s number sequences can become Children’s number sequences can become measuring schemes when used in the context of measuring schemes when used in the context of extensive quantity. extensive quantity.

• Incidentally, McLellan & Dewey (1895) Incidentally, McLellan & Dewey (1895) commented that number arises as a need to commented that number arises as a need to measure things.measure things.

Construction of Object Concepts and Construction of Object Concepts and Quantitative Properties Quantitative Properties

• [Hypothesis] Quantitative properties of object [Hypothesis] Quantitative properties of object concepts are introduced into the object concepts concepts are introduced into the object concepts by the knowing subject’s actions in the by the knowing subject’s actions in the construction of the object concepts.construction of the object concepts.

• [Hypothesis] Children’s construction of [Hypothesis] Children’s construction of quantitative properties are inextricably quantitative properties are inextricably intertwined. intertwined.

Ninth ProblemNinth Problem

• How can students’ mathematical learning be How can students’ mathematical learning be constituted as a spontaneous activity in the constituted as a spontaneous activity in the scientific teaching of students?scientific teaching of students?

• Corollary:Corollary: How can students’ mathematical How can students’ mathematical activity be constituted as an independent albeit a activity be constituted as an independent albeit a social activitysocial activity??

On the Spontaneity of On the Spontaneity of Spontaneous DevelopmentSpontaneous Development

• It appears to be extremely difficult to define It appears to be extremely difficult to define “mathematical contexts,” especially with “mathematical contexts,” especially with reference to young children. Given the very reference to young children. Given the very general basis for construction of logical-general basis for construction of logical-mathematical operations … almost any situation mathematical operations … almost any situation that can be commented on, asked about, that can be commented on, asked about, indicated as desirable, etc., can lead to actions, indicated as desirable, etc., can lead to actions, utterances, gestures, or other communicative utterances, gestures, or other communicative acts that have something to do with logic or acts that have something to do with logic or mathematics (Sinclair, 1990, p. 25). mathematics (Sinclair, 1990, p. 25).

The Use of “Spontaneous” in The Use of “Spontaneous” in the Context of Learningthe Context of Learning

• I do not use “spontaneous” in the I do not use “spontaneous” in the context of learning to indicate the context of learning to indicate the absence of elements with which absence of elements with which students interact.students interact.

The Use of “Spontaneous” in The Use of “Spontaneous” in the Context of Learningthe Context of Learning

• I use the term to refer to:I use the term to refer to:

• The non-causality of teaching actions,The non-causality of teaching actions,• Auto-regulation of within-student interaction,Auto-regulation of within-student interaction,• Self-regulation of students when interacting, Self-regulation of students when interacting,

• A lack of awareness of the learning processA lack of awareness of the learning process and to and to

its unpredictabilityits unpredictability. .

Learning as Accommodations in Learning as Accommodations in SchemesSchemes• If learning is placed in the context of accommodations in learning is placed in the context of accommodations in

students’ mathematical schemes, it need not be regarded students’ mathematical schemes, it need not be regarded as limited to a single problem or as a limited process. as limited to a single problem or as a limited process.

• In fact, it should not be the intention that students learn to In fact, it should not be the intention that students learn to solve a single problem, even though situations are solve a single problem, even though situations are presented to them that might be problematic. presented to them that might be problematic.

• Rather, the interest is in understanding the students’ Rather, the interest is in understanding the students’ assimilating schemes and how students’ might make assimilating schemes and how students’ might make changes in their schemes as a result of their changes in their schemes as a result of their mathematical activity. mathematical activity.

Independent Mathematical Independent Mathematical ActivityActivity

• Mathematical play is a form of cognitive play (Piaget, 1962). Mathematical play is a form of cognitive play (Piaget, 1962).

• Mathematical play is involved in students' engaging in Mathematical play is involved in students' engaging in independent mathematical activity, which can be either an independent mathematical activity, which can be either an individual or a social activity (Steffe & Wiegel, 1994). individual or a social activity (Steffe & Wiegel, 1994).

• As social activity, students’ independent mathematical activity As social activity, students’ independent mathematical activity comprises a self-regulating and possibly self-sustaining social comprises a self-regulating and possibly self-sustaining social system in the sense that Maturana (1978) spoke of a system in the sense that Maturana (1978) spoke of a

consensual domain of interactions.consensual domain of interactions.

Consensual Domain.Consensual Domain.

• A consensual domain is established when the A consensual domain is established when the individuals of a group adjust and adapt their individuals of a group adjust and adapt their actions and reactions to achieve the degree of actions and reactions to achieve the degree of compatibility necessary for cooperation. compatibility necessary for cooperation.

• This involves the use of language and the This involves the use of language and the adjustments and mutual adaptations of individual adjustments and mutual adaptations of individual meanings to allow effective interaction and meanings to allow effective interaction and cooperation. (Glasersfeld, Personal cooperation. (Glasersfeld, Personal Communication)Communication)

Tenth ProblemTenth Problem

• How can mathematics teacher education be reformed so we can educate professional mathematics teachers?

Professional Mathematics TeachersProfessional Mathematics Teachers

• A professional mathematics teacher can use A professional mathematics teacher can use teaching as a method of scientific investigation. teaching as a method of scientific investigation.

• If a teacher establishes a relationship between If a teacher establishes a relationship between two ways of thinking [the “students’” and the two ways of thinking [the “students’” and the teacher’s], she is acting as Maturana’s (1978) teacher’s], she is acting as Maturana’s (1978) second-order observer, which is “the observer’s second-order observer, which is “the observer’s ability … to operate as external to the situation in ability … to operate as external to the situation in which he or she is, and thus be an observer of which he or she is, and thus be an observer of his or her circumstance as an observer” (p. 61).his or her circumstance as an observer” (p. 61).

A Professional Mathematics A Professional Mathematics TeacherTeacher

• The notion of a second-order observer opens the way for The notion of a second-order observer opens the way for investigating learning in a way that explicitly as well as investigating learning in a way that explicitly as well as implicitly takes into account the knowledge of the teacher implicitly takes into account the knowledge of the teacher and the knowledge of students. and the knowledge of students.

• In that the mathematics of students is precisely that In that the mathematics of students is precisely that knowledge that the teacher constructs, mathematics knowledge that the teacher constructs, mathematics teaching and mathematics learning are recursively teaching and mathematics learning are recursively embedded in each other. embedded in each other.

A Professional Mathematics A Professional Mathematics TeacherTeacher

• Recursive models of teaching and learning are Recursive models of teaching and learning are formulated from the point of view of a second-formulated from the point of view of a second-order observer—a professional teacher’s point of order observer—a professional teacher’s point of view.view.

• As we construct trajectories of epistemic As we construct trajectories of epistemic students, the principle of self-reflexivity in radical students, the principle of self-reflexivity in radical constructivism implies that constructivism implies that we use our way of we use our way of doing research in our practice as teacher doing research in our practice as teacher educators.educators.

Reforming the Practice of Reforming the Practice of Mathematics Teacher EducationMathematics Teacher Education

• Educating mathematics teachers in such a way that their Educating mathematics teachers in such a way that their teaching practice is founded on epistemic students involves teaching practice is founded on epistemic students involves researchers constructing epistemic mathematics teacher researchers constructing epistemic mathematics teacher education students whose practice is founded on epistemic education students whose practice is founded on epistemic students (Simon, & Tzur, 1999, Tzur, 2001). students (Simon, & Tzur, 1999, Tzur, 2001).

• How this might be done is a major problem in mathematics How this might be done is a major problem in mathematics teacher education and it involves reforming the practice of teacher education and it involves reforming the practice of mathematics teacher education in the colleges and universities mathematics teacher education in the colleges and universities in such a way that we educate in such a way that we educate professional mathematics professional mathematics teachersteachers. .