Using Designed Instructional Activities to Enable Novices to Manage Ambitious Mathematics Teaching

Lampert, M., Beasley, H., Ghousseini, H., Kazemi, E., & Franke, M. (2010). Using Designed Instructional Activities to Enable Novices to Manage Ambitious Mathematics Teaching. In M. K. Stein & L. Kucan (Eds.), Instructional Explanations in the Disciplines (pp. 129–141). Boston, MA: Springer.

Instructional Activities Using Routines as Tools for Teacher Education
• Choral counting: The teacher leads the class in a count, teaching different concepts and skills by deciding what number to start with, what to count by (e.g., by 10s, by 19s, by 3/4s), whether to count forward or backward, and when to stop. The teacher publicly records the count on the board, stopping to elicit children’s ideas for figuring out the next number, and to co-construct an explanation of the mathematics that arises in patterns.
• Strategy sharing: The teacher poses a computational problem and elicits multiple ways of solving the problem. Careful use of representations and targeted questioning of students are designed to help the class learn the general logic underlying the strategies, identify mathematical connections, and evaluate strategies in terms of efficiency and generalizability.
• Strings: The teacher poses several related computational problems, one at a time, in order to scaffold students’ ability to make connections across problems and use what they know to solve a more difficult computational problem. This activity is used to target a particular strategy (as compared to eliciting a range of strategies). For example, posing 4 × 4, then 4 × 40, and then 4 × 39 is designed to help students consider how to use 4 × 40 to solve 4 × 39, developing their knowledge of compensating strategies in multiplication (Fosnot & Dolk, 2001).
• Solving word problems: The teacher first launches a word problem to support students in making sense of the problem situation, then monitors while students are working to determine how students are solving the problem, gauges which student strategies are best suited for meeting the instructional goal of an upcoming mathematical discussion, and makes judgments about how to orchestrate the discussion to meet those goals.

[a fifth IA in recent articles is Quick Images: The goal of this activity is to build students' ability to visualize a quantity.]
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A Focus on Instructional Dialogue
A relatively recent focus of Leinhardt’s work on teaching routines has been how they are used in “instructional dialogue” (Leinhardt & Steele, 2005), a practice we would consider to be the centerpiece of ambitious mathematics teaching. In this kind of teaching, an explanation is co-constructed by the teacher and students in the class during an instructional conversation. Maintaining a coherent mathematical learning agenda while encouraging student talk about mathematics is perhaps the most challenging aspect of ambitious teaching. In their study of teaching through instructional dialogues, Leinhardt and Steele (2005) demonstrated the use of eight kinds of “exchange” routines in this kind of teaching to accomplish explanatory work, including maintaining mathematical clarity in the face of student inarticulateness, fixing the agenda of the class on a single student’s idea, making it safe for students to revise incorrect contributions, and honing students’ contributions toward mathematical accuracy and precision. The exchange routines that Leinhardt and Steele (pp. 143–144) identified include the following:
• The call-on routine, which is initiated by a rather open invitation to discussion and has two separate components: the initial identification of a problem and the speaker who responds, followed by a second part in which the class is prompted to analyze, justify, or critique the statement given by the first speaker or another speaker in the discussion.
• The related revise routine in which students were asked to rethink their assertions and publicly explain a new way of thinking about their solutions.
• The clarification routine “which was invoked when a confusion arose regarding an idea or conjecture volunteered into the public space, which in turn involved understanding the source of confusion.”

Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis

Slavin, R. E., Lake, C., and Groff, C. (2009). Effective Programs in Middle and High School Mathematics: A Best-Evidence Synthesis. Review of Educational Research, 79(2):839-911. (abstract)

This review examined 100 studies of three types of programs designed to improve achievement in mathematics (Slavin, Lake, & Groff, 2009). In this review, 40 studies of mathematics curricula found very small effects (ES = +0.03); 38 studies of computer-assisted instruction found small effects (ES = +0.10); and 22 studies of instructional process programs found small effects (ES = +0.18); although the effects of specific programs varied widely, with studies of two forms of cooperative learning having medium effects (ES = +0.48).

An earlier review examined 33 studies of four types of programs designed to improve achievement in reading (Slavin, Cheung, Groff, & Lake, 2008); Regarding these programs, no studies of secondary reading curricula met the criteria to be included in the review; eight studies of computer-assisted instruction found small effects (ES = +0.10); 16 studies of instructional-process programs had small effects (ES = +0.21); and nine studies of two mixed-method models that combined large-group, small-group, and computer-assisted, individualized instruction had small effects (ES = +0.23). The third review was conducted by the What Works Clearinghouse based on three studies of a computer-based adolescent literacy program that supplements regular classroom reading instruction in grades K-8. The review found that the program had small effects on reading comprehension (ES = .27) and literacy achievement (ES = .28).


Abstract
This article reviews research on the achievement outcomes of mathematics programs for middle and high schools. Study inclusion requirements include use of a randomized or matched control group, a study duration of at least 12 weeks, and equality at pretest. There were 100 qualifying studies, 26 of which used random assignment to treatments. Effect sizes were very small for mathematics curricula and for computer-assisted instruction. Positive effects were found for two cooperative learning programs. Outcomes were similar for disadvantaged and nondisadvantaged students and for students of different ethnicities. Consistent with an earlier review of elementary programs, this article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

Mind in action: A functional approach to thinking - Scribner (1983)

Scribner, S. (1983) Mind in action: A functional approach to thinking. In M. Cole, Y. Engestrom. And O. Vasquez (Eds.) Mind, Culture and Activity: Seminal papers from the Laboratory of Comparative Human Cognition, (pp. 354-368). NY, NY: Cambridge University Press.

This "classic" and widely cited article is about the use of mathematics knowledge by dairy workers who assemble and price orders and take inventory in the warehouse.

The first thing we learned from our systematic observations is that the preloaders had a large repertoire of solution strategies for what looked like the "same problem."

We postulated a "law of mental effort": "In product assembly, mental work will be expended to save physical work."

By comparing various modes of solution in terms of the number of moves they required, we could determine which strategy represented a "least-physical-effort solution" under a given set of circumstances. We refer to these as optimal solutions.

Pricing delivery tickets is all symbolic work. Speed and accuracy count.

A problem by problem analysis of solution strategies showed that the case price technique functioned as an effort saver in a manner analogous to the nonliteral optimal solutions in the product assembly task - with an important difference. The effort saved here was mental, not physical, Use of case price either eliminated computation altogether or simplified it.

Practice makes for difference - the problem-solving process is restructured by the knowledge and strategy repertoire available to the expert in comparison to the novice.

One feature of skilled problem-solving is the dependency of problem solving strategies on knowledge about the workplace. Skill in the dairy was not content-free.

Variability was an outstanding feature of skilled performance on all tasks.

Skilled practical thinking at work is goal-directed and varies adaptively with the changing properties of problems and changing conditions in the task environment.

In contrast to the conventional psychological model of learning which assumes a progression from the particular and concrete to the general and abstract, skill acquisition at work seems to move in the direction of mastery of the concrete.

Work activities have certain peculiarities and cannot be considered representative of all practical thinking in action.

At the end of one interview, a seasoned delivery driver described to me the public's image of a milkman. He said , "Most people believe you only need a strong back to be a milk man. But, come to think of it, there is a lot of brain work involved." I think he is right.

What’s all the fuss about metacognition?

Schoenfeld, A. (1987). What’s all the fuss about metacognition? In A. Schoenfeld (Ed.), Cognitive Science and Mathematics Education, pages 189-215. Hillsdale, NJ: Erlbaum.

It's about how teaching metacognitive skills can increase students'
understanding and help them become better problem solvers. I liked that this
article discusses strategies and examples of how to teach metacognitive skills.

What is it?
1. How accurate are you at describing your own thinking?
2. Control or self-regulation
3. Beliefs and intuitions

Why is it important?
1. Students need good study skills, using what you know efficiently, managing time
2. Students with metacognitive skills will learn more, have greater & deeper conceptual understanding, are generally better problem solvers, and will likely enjoy learning more.

What to do about? How do you teach metacognitive skills?
1. Use video tapes
2. Teacher as role model for metacognitive behavior
3. Whole class discussions of problems with teacher serving as "control"
4. Problem solving in small groups

Questions that he used to prompt student thinking and metacognition:
What exactly are you doing?
Why are you doing it?
How does it help you?
Is this likely to be productive?