Boundary Objects Within a Replacement Unit Strategy for Mathematics Teacher Development

Abstract

We recognise that, for instructional innovations to take root in mathematics classrooms, curriculum redesign and teachers’ professional development are two necessary and mutually-reinforcing processes: a redesigned curriculum needs to be seen as an improvement in order to facilitate teachers’ buy-in—an ingredient for effective professional development; on the other hand, teachers’ professional development content needs to be directed towards actual useable classroom implements through the enterprise of collaborative curriculum redesign. In this chapter, we examine the interaction between researchers and teachers in this collaborative enterprise through the metaphor of boundary crossing. In particular, we study a basic model of how “boundary objects” located within a “Replacement Unit” strategy interact to advance the goals of professional development.

Appendices

Appendix A: A compressed version of the Practical Worksheet Used in Eastpark Secondary School

• Instructions

  • 1. You may proceed to complete the worksheet doing stages I–IV.

  • 2. If you wish, you have 15 min to solve the problem without explicitly using Polya’s model. Do your work in the space for Stage III.

    • If you are stuck after 15 min, use Polya’s model and complete all the stages I–IV.

    • If you can solve the problem, you must proceed to do stage IV—Check and Expand.

    • You may have to return to this section a few times. Number each attempt to understand the problem accordingly as Attempt 1, Attempt 2, etc.

• Stage I: Understand the Problem

• (a) Use some heuristics such as Draw a Diagram, Restate the problem, Use Suitable Numbers, etc., to help you.

• (b) I have understood the problem. (Circle your agreement below)

• Stage II & III: Devise a Plan and Carry it out

• (a) State your plan clearly, for example: (i) Use suitable Numbers and Look for Patterns; or (ii) Find the areas of all smaller triangles and work out their ratios.

• (b) Number each plan as Plan 1, Plan 2, etc.

• (c) Carry out the plan that you have stated.

• (d) Write down in the Control column, the key points where you make a decision or observation, for e.g. go back to check, try something else, look for resources or totally abandon the plan.

• Stage IV: Check and Expand

• (a) Write down how you checked your solution.

• (b) Write down a sketch of any alternative solution(s) that you can think of.

• (c) Give one or two adaptations, extensions or generalisations of the problem. Explain succinctly whether your solution structure will work on them.

Appendix B

Some of the specific problems suggested during the RU strategy

Problem name	Problem details
Choose a number	Write down an integer between 1 and 200. What is the probability/chance that there is a match?
Three children	Given a family with three children, what is the probability that the family has three boys?
Passenger seat	Mr. and Mrs. Tay and their son John goes into their family car. Mr. and Mrs. Tay can drive but John cannot. What is the probability of Mr. Tay sitting in the passenger seat? (Apart from the driver seat that must be filled, the other passengers can choose to sit in any of the remaining seats in the car.)
Loaded die	In an unbiased die, three faces are painted “1”, two faces are painted “2”, and the last face is painted “3”. Find the probability that when the die is rolled, “2” is obtained
Phoney Russian roulette	Two bullets are placed in two consecutive chambers of a 6-chamber revolver. The cylinder is then spun. Two persons play a safe version of Russian Roulette. The first points a gun at his hand phone and pulls the trigger. The shot is blank. Suppose you are the second person and it is now your turn to point the gun at your hand phone and pull the trigger. Should you pull the trigger or spin the cylinder another time before pulling the trigger?