Kaput - three strands of algebra

BOOK REVIEW: Mathematics Educators Respond to Kaput’s "Algebra Problem": A Review of Algebra in the Early Grades
by Dan Chazan and Ann R. Edwards
JRME March 2010, Volume 41, Issue 2, Pages 203 - 204 [pdf]

In chapter 1, Kaput offers an overarching characterization of algebra and algebraic reasoning that unifies the diverse perspectives represented in this volume. His analysis presents two core aspects of algebra— algebra as the systematic symbolizing of generalizations and algebra as syntactic action in conventional symbol systems.

These core aspects are embodied in three strands of algebra:

1. algebra as the study of structures and systems abstracted/generalized from arithmetic and quantitative reasoning;
2. algebra as the study of functions (the generalization of variability); and
3. algebra as a modeling tool.

Picking up on these threads, chapter 2 (Kaput, Blanton, & Moreno) and chapter 5 (E. Smith) further explicate algebra as embodied in the linked processes of generalization and symbolization—Kaput, et al. examining symbolization in the context of equation solving and Smith focusing on the role of argumentation in the establishment of generality.

Chapter 4 (J. Smith & Thompson) takes a contrasting view, arguing that the development of algebraic reasoning can effectively be based in reasoning about relationships between physical quantities.

Korthagan & Kessels (1999) - Linking theory and practice

Korthagan, F. and Kessels, J. (1999). Linking theory and practice: Changing the pedagogy of teacher education. Educational Researcher, 28(4), 4-17.

Episteme Versus Phronesis

In teacher education there is much confusion about at least two different meanings of the word "theory." Kessels and Korthagen (1996) go back to Aristotle's concepts of episteme and phronesis to explain the difference. If a teacher educator offers epistemic knowledge, he or she uses general conceptions, applicable to a wide variety of situations; this knowledge is based on research and can be characterized as "objective" theory, theory with a big T. This is the type of knowledge that plays a central role in the traditional approach and that should certainly not be left out of teacher education programs: Now and then student teachers should be helped to see the larger picture of educational knowledge.

More often, however, they need knowledge that is situation-specific and related to the context in which they meet a problem or develop a need or concern, knowledge that brings their already existing, subjective perception of personally relevant classroom situations one step further. This type of knowledge is called phronesis. We could also call it "theory with a small t."

The character of phronesis is more perceptual than conceptual: It—often unconsciously— focuses the attention of the actor in the situation on certain characteristics of the situation, characteristics important to the question of how to act in the situation.

To put it concisely, episteme aims primarily at helping us to know more about many situations, while the emphasis of phronesis is mostly on perceiving more in a particular situation and finding a helpful course of action on the basis of strengthened awareness.

This strengthened awareness of concrete characteristics in specific situations is also the fundamental difference between phronesis and procedural knowledge (knowledge about "how to ..."). The danger of an emphasis on procedural knowledge in teacher education is that student teachers learn a lot of methods and strategies for many types of situations but do not learn how to discover, in the specific situations occurring

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Techne is craft knowledge (Kessels & Korthagan, 2001)

The craft knowledge in teaching, according to Grimmett and Mackinnon (1999), "consists of pedagogical content and pedagogical learner knowledge derived from considered experience in the practice setting" and "represents teachers judgment in apprehending the events of practice from their own perspectives as students of teaching and learning" (p.387). They argue that craft knowledge in teaching is gained primarily through experience and practice, rather than acquiring from books or lectures.

Therese Day: the craft knowledge of teaching is the professional knowledge gained by experience which teachers use everyday in their classrooms but which is rarely articulated in any conscious manner [link]

Wikipedia: Techne, or techné, as distinguished from episteme, which is often translated as craftsmanship, craft, or art. It is the rational method involved in producing an object or accomplishing a goal or objective. Techne resembles episteme in the implication of knowledge of principles, although techne differs in that its intent is making or doing, as opposed to "disinterested understanding."

As one observer has argued, techne "was not concerned with the necessity and eternal a priori truths of the cosmos, nor with the a posteriori contingencies and exigencies of ethics and politics. [...] Moreover, this was a kind of knowledge associated with people who were bound to necessity. That is, techne was chiefly operative in the domestic sphere, in farming and slavery, and not in the free realm of the Greek polis."

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Kessels & Korthagan, 2001, Ch.2 of Linking practice and theory: the pedagogy of realistic teacher education By F. A. J. Korthagen, Jos Kessels [link]

Shulman (1987) - Knowledge and teaching (PCK)

Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.

Categories of the Knowledge Base
If teacher knowledge were to be organized into a handbook, an encyclopedia, or some other format for arraying knowledge, what would the category headings look like? At minimum, they would include:

• content knowledge;
• general pedagogical knowledge, with special reference to those broad principles and strategies of classroom management and organization that appear to transcend subject matter;
• curriculum knowledge, with particular grasp of the materials and programs that serve as "tools of the trade" for teachers;
• pedagogical content knowledge, that special amalgam of content and pedagogy that is uniquely the province of teachers, their own special form of professional understanding;
• knowledge of learners and their characteristics;
• knowledge of educational contexts, ranging from the workings of the group or classroom, the governance and financing of school districts, to the character of communities and cultures; and
• knowledge of educational ends, purposes, and values, and their philosophical and historical grounds.

Among those categories, pedagogical content knowledge is of special interest because it identifies the distinctive bodies of knowledge for teaching. It represents the blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction. Pedagogical content knowledge is the category most likely to distinguish the understanding of the content specialist from that of the pedagogue. While far more can be said regarding the categories of a knowledge base for teaching, elucidation of them is not a central purpose of this paper.

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Excerpt from Shulman, L. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, Vol. 15, No. 2 (Feb., 1986), pp. 4-14

p9:
Pedagogical Content Knowledge. A second kind of content knowledge is pedagogical knowledge, which goes beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teaching. I still speak of content knowledge here, but of the particular form of content knowledge that embodies the aspects of content most germane to its teachability.

Within the category of pedagogical content knowledge I include, for the most regularly taught topics in one's subject area, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations- in a word, the ways of representing and formulating the subject that make it comprehensible to others. Since there are no single most powerful forms of representation, the teacher must have at hand a veritablea rmamentariumo f alternative forms of representation, some of which derive from research whereas others originate in the wisdom of practice.

Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons.

Leinhardt & Greeno (1986) - The Cognitive Skill of Teaching

Leinhardt, G. and Greeno, J. (1986). The cognitive skill of teaching. Journal of Educational Psychology, 78(2), 75-95.

L&G characterize teaching as a complex cognitive skill amenable to analysis in a manner similar to other skills described by cognitive psychology. According to L&G, teaching skill rests on two fundamental knowledge systems: lesson structure and subject matter.

L&G propose that a skilled teacher has a complex knowledge structure composed of interrelated sets of organized actions. L&G refer to these organized actions as schemata. They are applied flexibly and with little cognitive effort in circumstances that arise in the classroom. The main feature of the skilled teacher's knowledge structure is a set of schemata for teaching activities. These schemata include structures at differing levels of generality, with some schemata for quite global activities such as checking homework and some for smaller units of activity such as distributing paper to the class.

A characteristic of skilled performance is that many component actions are performed with little effort because they have become automatic through practice. L&G conclude that skilled teachers have a large repertoire of activities that they perform fluently. L&G refer to these activities as routines (Leinhardt, Weidman, & Hammond, in press). For routines to be effective, the students as well as the teacher must have developed an organization of actions or schemata for the actions that are performed. Routines play an important role in skilled performances because they allow relatively lowlevel activities to be carried out efficiently, without diverting significant mental resources from the more general and substantive activities and goals of teaching. Thus, routines reduce cognitive load and expand the teacher's facility to deal with unpredictable elements of a task.

Skilled teaching requires decisions about whether to proceed with the next component of a lesson, based on students' readiness for new material and the likelihood that students will succeed in solving instructional problems, or involving selection of students to ask questions or give special help.

The article describes a homework check activity of an expert math teacher and a novice one.

Expert teachers constructed their mathematics lessons around a core of activities. The expert teachers had, with the class, a large repertoire of routines, usually with several forms of each one. The expert's lesson can be characterized as an action agenda consisting of a list of action segments.

Wells (2002) - Spiral of Knowing

Wells, G. (2002). Learning and teaching for understanding: the key role of collaborative knowledge building. In Social Constructivist Teaching, Volume 9, pages 1–41. Elsevier Science Ltd. PDF

In this chapter, Wells explored the role of language – and of meaning-making practices more generally – in promoting students’ learning in all areas of the curriculum. Wells gives some attention to reading, broadly conceived, since acquiring information from books, maps, diagrams, and texts of all kinds, plays an increasingly important role in education as students increase in age (Kress, 1997; Lemke, 2002). Wells also devotes some attention to writing – in non-narrative as well as narrative genres – as, with Langer and Applebee (1987) Wells believes that it is in the writer’s dialogue with his or her emerging text that an individual’s understanding of an issue or topic is most effectively developed and refined.

Brophy & Good (1974) - Teacher-student relationships - Expectation effects

Brophy, J., & Good, T. (1974). Teacher-student relationships: Causes and consequences. New York: Holt, Rinehart and Winston. Ch. 10 Classroom research: Some suggestions for the future

Expectation effects

• The book documents specific ways in which teachers have been found to communicate low expectations of student, and what the effects are
• Teacher expectations can influence students' achievement and affect (attitudes) toward school, learning, and subject matter.
• Teachers often rank their students in order of achievement after only a few days of class, and these rankings are stable over time (p302)
• Proactive teachers appear to be undeterred by their expectations for low achieving students, so they spend more time interacting with lows than highs
• Reactive teachers tend to "let nature takes it's course" and tend to allow high students to dominate classroom life because of their ability and initiative
• A third type of teacher overreacts to student differences by overtly favoring high students, which then tends to magnify the difference with lows
• Global expectations of an entire class have been related to student gain. Teachers who overestimate the ability of students get better results than teachers who underestimate.
• Some teachers seem to pick certain students in order to cue their teaching tactics, especially when to move on to a new topic. These students then determine the tempo and pacing of instruction.

Matching Students and Teachers

• Teachers have often been found to state strong preferences for certain types of learners (e.g., over- or underachievers, passive or aggressive, achievement oriented, similar backgrounds, etc.)
• Students who receive more personalized and positive attention from teachers seem to do better

Peer expectations can also affect student performance

1) How do teachers’s expectations of students change over time and experience?
2) Do the expectations of teachers and ways in which they communicate these expectation vary with experience, local context and other factors?

Rowan 2002 - Alternative views of the task of teaching: task variety & uncertainty

Rowan, B. (2002). Teachers’ work and instructional management, part I: Alternative views of the task of teaching. In W.K. Hoy and C.G. Miskel (eds.), Theory and Research in Educational Administration, Volume 1, pp.129-149. Greenwich, CT: Information Age Publishing, 2002.

Task variety refers to the range of instructional activities that teachers engage in as they teach in classrooms. Task uncertainty refers to the extent to which teachers have access to a systematic body of knowledge to guide this work.

Organizing Schools for Improvement: Lessons from Chicago

Bryk, A.S., Sebring, P.B., Allensworth, E., Luppescu S., & Easton, J.Q. (2010). Organizing Schools for Improvement: Lessons from Chicago. Chicago, IL: University of Chicago Press

[Excerpt from a book review written by Nathan Meyer in the Journal of Education for Students Placed at Risk, 15: 329–331, 2010]

Bryk et al. (2010) examined a total of 390 neighborhood elementary schools in Chicago for gains in math, reading, and attendance in grades 2–8. The authors used seven years of data from the Iowa Test of Basic Skills as their platform in identifying student improvement. Bryk and colleagues identified schools with similar demographics that remained academically stagnant in the lower quartile against schools that moved in the top quartile of performance. The authors found five intertwining essential supports for all schools:

1. school leadership
2. parent-community ties
3. professional capacity
4. student-centered learning climate
5. instructional guidance

A school having strong support on any of the five supports was four to five times more likely to demonstrate substantial improvement in reading and math than those in the bottom quartile of the same indicator (p. 84). The authors use the metaphor that missing an essential support is comparable to baking a cake: "if one of the ingredients is absent, it is just not a cake’’ (p. 66). Bryk and colleagues found that if any one of the five essential supports were lacking, schools did not improve.