The challenge of self-regulated learning in mathematics teachers' professional training

Abstract

This study investigated mathematics teachers' professional knowledge among elementary school teachers exposed to a professional training program that either supported self-regulated learning (SRL) or offered no SRL support (no-SRL). The SRL support was based on the IMPROVE metacognitive self-questioning method that directs students' attention to understanding when, why, and how to solve problems (Kramarski and Mevarech, Am Educ Res J 40:281–310, 2003). Sixty-four Israeli elementary teachers participated in a month-long professional development program to enhance mathematical and pedagogical knowledge. The course was part of a 3-year professional development program sponsored by the Israeli Ministry of Education. This mixed-method study included quantitative assessments of teachers' professional knowledge in mathematical problem solving for an authentic task based on Program for International Student Assessment's framework (Program for International Student Assessment, 2003) and in lesson planning, as well as qualitative interviews and videotaped observations of two teachers. Results indicated that teachers in the SRL program outperformed those in the no-SRL program on various problem solving skills (e.g., reflection and conceptual mathematical explanations) and lesson planning (e.g., task demands and teaching approach). Videotaped observations of actual teaching indicated that the SRL-trained teacher demonstrated more teaching practices that aimed to promote students' understanding and better supported students' regulation of their own learning, compared to the no-SRL-trained teacher. We discuss educational and practical implications.

Appendices

Appendix 1

Sample authentic task from the in-service teachers' training workshops

The parking lot task

Dan and Dina need to park their cars in a public parking lot. There are two parking lots nearby. Read the signs at the entrance to the parking lots:

Tel Aviv Parking Lot—every hour costs 6 shekels.

My Parking Lot—entrance costs 20 shekels + 2 shekels per hour.

• Assume Dan needs to park for 4 h and Dina needs to park for 7 h. Which parking lot should each of them pick?

• Explain the factors that guide Dan and Dina in their choices.

• Find an algebraic expression that represents the total cost for each parking lot. Explain in detail.

• List guidelines for choosing a parking lot so that it costs the least amount possible. Explain.

Appendix 2

Mathematical explanation quality: sample teacher arguments for the response “Mayan's plan was more profitable” (on the reflective open opinion item “Which savings plan is more profitable?”)

Examples of no argumentation (score = 0):

• We can see from the figure and from the number values.

• She saves more money each day.

Examples of procedural argumentation (score = 1):

• After the fourth day, Mayan's money increases more than Shai's money. On the fifth day, Mayan's money increases by 1 NIS; on the sixth day, the amount increases by 2 NIS; on the seventh day, the amount increases by 3 NIS; and on the eighth day, it increases by 4 NIS, etc.

• From the table below, we can see differences in the amounts of money that increase each day: Mayan's money increases by 2 NIS and Shai's by 1 NIS.

Money saved	Day
	1	2	3	4	5	6	7
Mayan	2	4	6	8	10	12	14
Shai	5	6	7	8	9	10	11

Examples of conceptual argumentation (score = 2):

• We can see in the figure that the slope in Mayan’s line is steeper than Shai’s.

• The change rate of Mayan’s savings plan is greater. From the fifth day, Mayan’s savings plan money is larger than Shai’s in spite of the advantage that she had—money on hand at the beginning of the savings program.

Appendix 3

Sample items from the two tests: (a) pedagogical prior knowledge test and (b) the annual Israeli Ministry Education mathematical and pedagogical knowledge test

• 1. Pedagogical prior knowledge test (using representations for explanation purposes)

When 23 is divided by 4, three possible answers are given

• (a) 5.75

• (b) $5\frac{3}{4}$

• (c) 5 with remainder 3

For each of them, write one story problem for which that answer is most appropriate.

• 2. The annual mathematical knowledge test:

• Task 1: Argument: The sum of a two-digit number and a number with the same digits written backwards is divisible by 11.

• (a) Provide an example in support of this argument

• (b) Prove the argument

• (c) Is this argument true for the sum of a three-digit number and a number with the same digits written backwards? If so, prove this. If not, supply a counter-example

• Task 2: a − 1 = b + 2 = c − 5 = d + 10 (a, b, c, d ≠ 0)

Which of the parameters a, b, c, and d is the biggest one? Explain.

• Task 3: A student received the following exercise: $\frac{1}{2} + \frac{1}{5} = K$ and answered $\frac{{3\frac{1}{2}}}{5} = K$

Is the answer correct?

• If the student was right, how could you explain his answer to the rest of the class?

• If the student was wrong, how should you explain his mistake to him?

Appendix 4

Sample lesson plan for the money-saving task (with scoring)

• 1. Task demands

Goal of the lesson: Grade 5 students should learn to generalize mathematical properties by using different representations like tables and graphs.

Prior knowledge: The lesson is based on students' knowledge about organizing data in a table and presenting data in a graph. In addition, I think that students should be able to draw conclusions from data presented in a table and graph and to explain their reasoning with conceptual terms.

Learning difficulties: I expect some difficulties in administering the task in grade 5, like reading/comparing data from two lines on the same graph, relating to each one on the same axes, obtaining conclusions, and finding generalizations.

Scoring: This teacher received a score of 3 for this category of the plan: The planning referred to all three elements of the task's demands: clear learning goals, prior knowledge, skills, and expected difficulties with regard to interpreting the two lines on the figure.

• 2. Task design

The task: Shai and Mayan received pocket money from their parents. Each day, Shai received 1 NIS and Mayan received 2 NIS. Before the parents started giving an allowance to the girls, Shai had saved 4 NIS and Mayan did not have any money. The following is the graph that represents the money savings in two colors (attaches graph with a red line depicting girl I and a blue line depicting girl II).

• 1. Which graph represents Shai’s savings? Explain.

• 2. Which graph represents Mayan’s savings? Explain.

• 3. From the graph, is there a day that both girls save the same amount of money?

  • If the answer is yes, indicate which day and the amount of money?

  • If the answer is no, explain the reason?

• 4. Write the data in a table.

• 5. Calculate the amount of money that each girl will receive after 3 days.

• 6. Find a pattern that calculates the money each girl saved (represents the amount of days).

• 7. Write an exercise for Shai’s savings plan after 3 days. Explain.

• 8. Write an exercise for Mayan’s savings plan after 3 days. Explain.

• 9. How much money will each girl receive after 10 days? Show your work.

• 10. Which savings plan is more profitable? Explain.

Scoring: This teacher received a score of 3 for this category of the plan: The planning referred to all three elements of the task's design: using diverse representations (tables, graphs, and a pattern expression), various problem solving skills (e.g., reproduction, connection, and reflection), and didactical considerations for presenting the task, such as using different colors for the figure.

• 3. Teaching approach

The lesson aims to foster all students' understanding by integrating questions with different levels of difficulty (e.g., item 3 vs. 10) and by asking students to draw conclusions (e.g., item 10). Special emphasis is given to the requirement for explaining students' mathematical reasoning (five out of ten items).

Scoring: This teacher received a score of 2 for this category of the plan because she referred to engagement of students in understanding and her questions required students to draw conclusions and to explain their reasoning. However, no specific emphasis was placed on the third element of identifying different paths to the solution (e.g., show your work in different ways).