Chapter 6: Teacher Development and Mathematical Challenge
by
Derek Holton
Kwok-Cheung Cheung
Sesutho Kesianye
Maria de Losada
Roza Leikin
Gregory Makrides
Hartwig Meissner
Linda Sheffield
Ban Har Yeap
In this chapter we look at the issues around teacher professional
development as they relate to teaching using mathematical challenges. We
look at what mathematics is; discuss why challenging mathematics
problems are important in school classrooms; give some examples of
problems that can provide a challenge in a classroom situation; and suggest
some barriers that might inhibit the use of challenging problems. We then
look at the mathematics education research that is relevant to the theme of
the chapter. This is followed by effective pedagogy and teacher
preparation, which includes both theoretical and practical aspects and
suggestions for someone who is about to develop a professional
development project using challenging mathematics. The overall structure
of this chapter is presented below.
I. Introduction
1. What is mathematics and what are mathematical challenges?
2. Why are challenging mathematics problems important in school?
3. What do challenging mathematical problems for school classrooms look like?
4. What barriers might prevent teachers from using challenging problems?
II. Research
1. What do we know about the effect of teachers’ knowledge and beliefs on the teaching
and learning of challenging mathematics?
2. What other factors are important to teaching and learning challenging mathematics?
3. What are other needed areas of research?
III. Effective Pedagogy
1. What is the role of the teacher in a class where challenging problems are used?
2. What is effective pedagogy for classrooms using challenging mathematical problems?
IV. Teacher Preparation
1. What is the role of professional development in encouraging classes with challenging
mathematical problems?
2. Some in-service and pre-service programs
V. Summary
1. Summary
Excerpt from Chapter 6:
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(the book is available at Springer)
3.2 Decimal Grid Task: In the grid of Figure 2, select a path from A to B. Change the
direction after each numbered step. Multiply together (with a calculator) the decimal
numbers1 on each step you go along. Find the path with the smallest product.
You have 4 trials. What is your answer? What is your smallest product?
The problem solvers start working, individually or in small groups. The reader should do the
same; please answer the questions below before you continue reading. Be warned, there is
more to this problem than you might think.
Figure 2
After about 10 to 15 minutes the class might be asked for solutions. Who got 62.121... as their
smallest product? Who got 1.89? Who got 0.3564? Who else got a product between 1 and
0.1? Who got a product between 0.1 and 0.01; who got one between 0.01 and 0.001? Is there a
product smaller than 0.001?
Then the challenge continues: Find a better path than you had. Try to explain the paths to
others. Are there rules to find better paths? Is there a chance to find the best path?
1
In Germany a comma is used instead of a decimal point
Before you continue reading, please make three or four notes about possible strategies or
arguments which could come up in the discussion.
Now we will summarize strategies and experiences that have been reported in the past.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
I do not remember why I chose a specific path;
I always took the smallest number at each point (this strategy gives 62.121...);
I took as few steps as possible (2.5 × 8.4 × 0.3 × 0.3 = 1.89);
I looked for as many “zeros” as I could (13.2 × 0.5 × 0.6 × 0.3 × 0.3 = 0.3564);
after 5.9 (second step) we can continue with smaller factors (0.7 × 5.9 × 0.5 × 0.6 × 0.3 ×
0.3 = 0.11151);
I was looking for small “detours” (instead of 13.2 go 0.7 × 5.9; instead of 5.4 go 0.5 × 0.6;
instead 8.4 go 5.9 × 0.7; ...).
I discovered a “cycle” (... × 0.6 × 0.8 × 0.3 × ...);
I tried to use a “small path” more than once (“small path” means decimal less than 1);
I ran through the cycle (see (g)) many times.
Analyzing the different paths we also find some assumptions that were made but that were not
mentioned:
(j) multiplication produces bigger numbers;
(k) make no “detours”;
(l) reach the goal B as quickly as possible;
(m) do not go “backwards”.
In this problem several cognitive jumps can occur. For example:
(n) an unconsciously existing notion, “Multiplication makes bigger”, may be destroyed;
(o) a new experience can become conscious: “Multiplication does not always give a bigger
result”;
(p) a more concrete realisation is “for multiplication, decimals bigger than 1 are quite
different from decimals less than 1”;
(q) another new experience might be “More factors may lead to a smaller product”;
(r) repeating the “cycle” (... x 0.6 x 0.8 x 0.3 x ...) several times makes the product smaller
and smaller;
(s) repeating the “cycle” (... x 0.6 x 0.8 x 0.3 x ...) again and again, we might reach zero.
Discussion is very important in the completion of problems like the above because there often
are conflicts between intuitive and analytical Vorstellungen. Speaking about their experiences
makes unconscious Vorstellungen conscious and leads to revelations, as, for example, in (n)
to (s).