1, 2
Leslie P. Steffe
University of Georgia

• 1. A. Knowledge is not passively received either through the senses or by way of communication;
• B. Knowledge is actively built up by the cognizing individual.
• 2. A. Knowledge is adaptive, in the biological sense of the term, tending toward fit or viability;
• B. Cognition serves in the individual’s organization of the experiential world, not the discovery of an objective, ontological reality
(Glasersfeld, 1995)

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

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

Main Goals of Using Teaching as a
Method of Scientific Investigation.
• 1. To build living, experiential models of students’ mathematics.
• 2. To experience constraints when teaching students.
– The constructivist is fully aware of the fact that an organism’s conceptual constructions are not fancyfree. On the contrary, the process of constructing is constantly curbed and held in check by the constraints it runs into. (Glasersfeld, 1990, p. 33)

Basis For Using Teaching as a Method of
Scientific Investigation
• The view of students as self-organizing and interacting systems (the mind organizes the world by organizing itself).
• "Cognitive processes seem, then, to be at one and the same time the outcome of organic auto-regulation and the most highly differentiated organs of this regulation at the core of interactions with the environment” (Piattelli-Palmarini,1980).

• Approaches to inquiry have centered on the idea of worlds being constructed by inquirers who are simultaneously participants in those same worlds (Steier,
1995).
• A science of observed systems cannot be divorced from a science of observing systems because it is we who observe.

Synopsis of the Researcher as Teacher
• It is critical to understand that one’s own actions and interactions as a teacher are essential in student’s construction of mathematics.
• More importantly, one’s own actions and interactions, when coupled with students’ actions and interactions, are essential in one’s own construction of students’ mathematics and how it might be productively affected.
• A major goal of the researcher as teacher is to produce images of the constructive possibilities of students in mathematics.

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

• First-order models are models the individual has constructed to comprehend and control his or her own experience; that is, the individual’s knowledge.
• Second-order models are models an observer constructs of the observed person’s knowledge in order to explain his or her observations.

• Conceptual analysis is involved in building explanatory models of students’ construction of mathematics.
• What mental operations (and co-ordinations thereof) can I posit in my model (the student’s mathematics) to explain the actions I have observed?
• Students’ mathematics is about the ways and means they operate mathematically.

The First Time That Conceptual
Analysis Was Used in Mathematics
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.

• Perceptual Counting Scheme.
• Figurative Counting Scheme.
• Initial Number Sequence [INS Counting
Scheme].
• Tacitly Nested Number Sequence [TNS
Counting Scheme].
• Explicitly Nested Number Sequence [ENS
Counting Scheme].
• Generalized Number Sequence.

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

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

The Epistemic Student
• Consists of the mathematical schemes of action and operation of students at the same level of construction that have been abstracted from living, experiential models of students.
• Epistemic students are dynamic organizations of schemes of action and operation in the researchers’ or teachers’ mental life. The schemes of action and operation include accommodations in the schemes.

• The epistemic students that teachers or researchers carry around in their heads are similar to what family therapists call internalized others.
• However, I think of epistemic students as interiorized others. It is the living, experiential models that are analogous to internalized others.
• Experiences of students’ mathematics are mathematical experiences.
• I consider students as rational beings and the explanations of their mathematics--the mathematics of students--as serious and important mathematics.

• The main reason for doing research on the construction of the mathematics of students:
– To construct a “school mathematics” that is based on the mathematical concepts and operations of students that are abstracted from scientific teaching.

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

• O S
• B i
• S t
• e u
• r a
• v t
• e a
• r’ t
• s i
• o
• n
Scheme’s
Situation
Scheme’s
Goal
Scheme’s
Activity
Scheme’s
Result

Trajectories of Three Students’ Counting
Schemes Who Were in the Perceptual
Stage Upon Entering Grade One.
•
•
•
FCS
PCS
Sep Oct Nov Dec Jan Feb Mar Apr May Jun Sep Oct Nov Dec Jan Feb Mar Apr May Jun
First Grade Second Grade
Brenda James Tarus

Percent of Entering First Graders in the Perceptual
Stage of the Counting Scheme Reported by
Individual School Systems in Robert Wright’s
Mathematical Recovery Program.
• In Wyoming: 61% and 12% pre-counters.
• In Wyoming: 10%
• In Arkansas: 45%
• In Maryland: 60% and 8% pre-counters.
• In Georgia*: 33%
*Data Supplied by Myself.

Tarus: January of his Second-Grade
• I presented Tarus with a cylindrical tube open at one end where a marble would just fit into the tube. After Tarus put eleven marbles into the tube, I poured three marbles out of the tube into his hand and asked him to find how many marbles were left in the tube.
• Tarus buried his head in his arms and played with a marble and then said, “ten”. In explanation, he said, “I count”.

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

Hypothetical Trajectory of the Three Students Who
Entered Grade One in the Perceptual Stage of the
Counting Scheme from Grade Three Through
Grade Five.
•
•
•
•
•
TNS
•
• INS
Sep May Sep May Sep May
Third Grade Fourth Grade Fifth Grade
Brenda Tarus James

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

Operations of The Explicitly Nested
Number Sequence
• The first operation: The operations that produce a unit of units.
• The second operation: The iterability of the unit of one.
• The third operation: Recursion.
• The fourth operation: The disembedding operation.

• T: (Places “27 + __ = 36” in front of Jason.)
• J: Twenty-seven—(pause of about 20 seconds).
Let me see —(another pause)—twenty-seven plus seven —it’s nine more!

Strategic Additive Reasoning
• I asked Johanna to take twelve blocks, told her that together we had nineteen, and asked her how many I had.
– After sitting silently for about 20 seconds, she said, “seven” and explained, “Well, ten plus nine is nineteen; and I take away the two-1 mean, ten plus two is twelve, and nine take away two is seven!”

Strategic Additive Reasoning as Mathematics of
Students
• By engaging in strategic additive reasoning, students construct a mathematical reality for themselves that is a product of their own ways and means of operating.
• It should be no surprise that when learning to compute using standard algorithms is emphasized, students easily lose their sense that it is they who are the agents in doing mathematics.

• An eight-year-old child, Nathan, was asked to make copies of a string of three toys and a string of four toys to make 24 toys using computer software. Rather than make the copies, the child reasoned out loud as follows:
– Three and four is seven; three sevens is 21, so three more to make 24. That’s four threes and three fours!

• The operations that produce a unit of units of units are available to students prior to activity.
• All of the operations of the explicitly nested number sequence can be carried out using composite units as well as other operations such as those demonstrated by Nathan.

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

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

• According to Thompson (1994), a quantitative scheme consists of:
• an object concept,
• a property of that object concept,
• a unit to measure the property, and
• a process by which a numerical value is assigned to the property.

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

• An extensive quantitative property of an object concept can be subdivided or partitioned.
• Number concepts can be used as templates in partitioning.
• Children’s number sequences can become measuring schemes when used in the context of extensive quantity.
• Incidentally, McLellan & Dewey (1895) commented that number arises as a need to measure things.

Construction of Object Concepts and
Quantitative Properties
• [Hypothesis] Quantitative properties of object concepts are introduced into the object concepts by the knowing subject’s actions in the construction of the object concepts.
• [Hypothesis] Children’s construction of quantitative properties are inextricably intertwined.

• How can students’ mathematical learning be constituted as a spontaneous activity in the scientific teaching of students?
• Corollary: How can students’ mathematical activity be constituted as an independent albeit a social activity ?

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

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

• I use the term to refer to:
• The non-causality of teaching actions,
• Auto-regulation of within-student interaction,
• Self-regulation of students when interacting,
• A lack of awareness of the learning process and to its unpredictability .

• If learning is placed in the context of accommodations in students’ mathematical schemes, it need not be regarded as limited to a single problem or as a limited process.
• In fact, it should not be the intention that students learn to solve a single problem, even though situations are presented to them that might be problematic.
• Rather, the interest is in understanding the students’ assimilating schemes and how students’ might make changes in their schemes as a result of their mathematical activity.

• Mathematical play is a form of cognitive play (Piaget, 1962).
• Mathematical play is involved in students' engaging in independent mathematical activity, which can be either an individual or a social activity (Steffe & Wiegel, 1994).
• As social activity, students’ independent mathematical activity comprises a self-regulating and possibly self-sustaining social system in the sense that Maturana (1978) spoke of a consensual domain of interactions.

• A consensual domain is established when the individuals of a group adjust and adapt their actions and reactions to achieve the degree of compatibility necessary for cooperation.
• This involves the use of language and the adjustments and mutual adaptations of individual meanings to allow effective interaction and cooperation. (Glasersfeld, Personal
Communication)

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

Professional Mathematics Teachers
• A professional mathematics teacher can use teaching as a method of scientific investigation.
• If a teacher establishes a relationship between two ways of thinking [the “students’” and the teacher’s], she is acting as Maturana’s (1978) secondorder observer, which is “the observer’s ability … to operate as external to the situation in which he or she is, and thus be an observer of his or her circumstance as an observer” (p. 61).

• The notion of a second-order observer opens the way for investigating learning in a way that explicitly as well as implicitly takes into account the knowledge of the teacher and the knowledge of students.
• In that the mathematics of students is precisely that knowledge that the teacher constructs, mathematics teaching and mathematics learning are recursively embedded in each other.

• Recursive models of teaching and learning are formulated from the point of view of a secondorder observer —a professional teacher’s point of view.
• As we construct trajectories of epistemic students, the principle of self-reflexivity in radical constructivism implies that we use our way of doing research in our practice as teacher educators.

• Educating mathematics teachers in such a way that their teaching practice is founded on epistemic students involves researchers constructing epistemic mathematics teacher education students whose practice is founded on epistemic students (Simon, & Tzur, 1999, Tzur, 2001).
• How this might be done is a major problem in mathematics teacher education and it involves reforming the practice of mathematics teacher education in the colleges and universities in such a way that we educate professional mathematics teachers .