Scaffolding Students' Thinking in Mathematical Investigations: Natalie McCosker and Carmel Diezmann Draw Some Pertinent Lessons about Effective Scaffolding from Four Episodes in Which the Teacher's Attempts at Scaffolding Did Not Have the Desired Effects
McCosker, Natalie, Diezmann, Carmel, Australian Primary Mathematics Classroom
Mathematical investigations are loosely-defined, engaging problem-solving tasks that allow students to ask their own questions, explore their own interests and set their own goals (Jaworski, 1994). The value of investigations for students lies in their complexity. Scaffolding plays an important role in supporting students' high-level engagement by encouraging divergent and creative thinking (Henningsen & Stein, 1997). Scaffolding is "a process that enables a child or novice to solve a problem, carry out a task or achieve a goal which would be beyond his [or her] unassisted efforts" (Wood, Bruner & Ross, 1976, p. 90). Scaffolding provides the opportunity for students to develop their independence, sense-making and self-confidence whilst working mathematically (Williams, 2008). However it is incorrect to assume that all conversations between teachers and students amount to scaffolding because not all result in high-level thinking and reasoning in students. Understanding what is ineffective and why, is one way to improve our pedagogical practice.
This article describes some of the issues that teachers might encounter when scaffolding students' thinking during mathematical investigations. It describes four episodes in which a teacher's interactions with students failed to support their mathematical thinking and explores the reasons why the scaffolding was ineffective. As a background to these episodes, we first provide an overview of the mathematical investigation. Our paper concludes with some recommendations for scaffolding during investigations.
The mathematical investigation
The mathematical investigation, undertaken by a Year 3 class of 24 students, emerged from a picture book entitled Counting on Frank (Clement, 1990). On the second reading of the book to the class, the teacher paused on a page that showed a boy surrounded by a large number of peas (Figure 1). She challenged the students to think of a way to find out how many peas had been pushed off the plate and what materials they would need for this investigation.
Small groups of students began their investigations of how they might determine the number of peas by creating a plan using drawings and text. Students were then invited to test their strategies. Upon completion, each group shared their investigatory approach and results with the rest of the class. Students then returned to their plans and recorded what their group did, using various diagrams, calculations, written explanations and flow charts.
[FIGURE 1 OMITTED]
The teacher as scaffolder
The teacher assumed the role of scaffolder many times during the peas investigation. Some of these scaffolding interactions impacted positively on the students' understanding of the task and their success with the investigation. However, in order to learn more about the characteristics of scaffolding, we examine four episodes, where the scaffolding appeared to be ineffective.
Episode 1: The need to press for meaning
In the following interaction, the teacher attempted to scaffold Marnie's thinking about the peas problem through questioning. Particularly pertinent phrases are in bold.
Teacher: I'm just having a look at the picture here that you have drawn, and I can see that you have put numbers and arrows. Can you tell me a little bit about what you have done?
Marnie: Um ...
Teacher: I can see that number one here [teacher points to the number one] is the desk, and this is a hundred block [teacher points to the drawing]. Can you tell me what you are thinking?
Marnie: Measure a hundred block on the desk and count in hundreds to see how much peas.
Teacher: Uh ... So measure ... count ... and then you're going to come up with your answer ... What an interesting idea.
Marnie's response to the first question, "Can you tell me a little bit about what you have done? …