Scenario: Christiana’s teaching multiplication by powers of ten
Grade (student age): Year 4 (age 9-10)
Contributed by: Marilena Petrou
Context – national, curricular, professional, other
In Cyprus textbooks are used on a daily basis in the classroom, and are produced by policy makers. The mathematics textbooks in Cyprus are year-based, and all students at a particular grade are taught the same mathematics for the same length of time from the same textbook. These textbooks are consistent with fundamental principles of constructivist views of learning and teaching mathematics. Activities are designed with the purpose of helping students to construct their own understanding of different topics, while teachers are seen as facilitators in the process of learning, whose aim is to create a teaching environment that supports exploration of different ideas.
Christiana, a final year student in a teacher preparation programme, was teaching multiplication by powers of ten with a Year 4 (pupils age 9-10). The lesson took place in a school-based placement towards the end of her four-year teacher preparation.
She considered the textbook to be the main resource both for her planning and teaching, but, she tended to elaborate upon it in ways that she believed would make the lesson more meaningful for her students.
Below I provide an example.
Here Christiana designed an investigative activity that encouraged children to develop their own informal method of mental calculations when multiplying by powers of ten. The teachers’ guide suggests starting the lesson by asking students to use their calculator to complete the table shown in Figure 1
Figure 1: Multiplication by powers of ten (Grade 4, Teachers’ Guide, p.99)
Christiana elaborated on what was suggested by techer’s guide and and developed the worksheet shown in Figure 2
Figure 2 Multiplication by powers of ten - Christiana’s worksheet
Knowledge Quartet Coding Commentary
Contributed by: Marilena Petrou
Knowledge Quartet Dimension: Transformation
Knowledge Quartet Code: Use of instructional material (UIM)
Scenario: Christiana teaching multiplication by powers of ten
The textbooks had been developed following a mainly constructivist view of learning, with the emphasis on promoting students’ conceptual understanding. Christiana seemed to understand the philosophy of the proposed activity, using the textbook suggestion and developing the worksheet above. Her transformational knowledge was demonstrated in her additions to the textbook material which focus on promoting coneptual undertanding and therefore this is offerred as an example of UIM instead of ATB
The questions included in the activity were designed to present pupils with increasing challenge as they progressed through them, moving from easy questions to harder ones. The students discovered the rule of ‘adding zero’ at the end when multiplying by powers of ten. However, Christiana wanted to help her students develop understanding that went beyond simply knowing how, to include knowing why. She challenged her students to understand the connection between their ‘rule’ and place value concepts. Here is an indicative extract from her lesson:
Christiana: Let’s take the number 542. What is the value of 5 in this number?
Victoria: It is five hundred.
Costas: When multiplied by 1000 what is the value of the digit 5?
Victoria: Now it shows 5 hundred thousands.
Christiana: Thus the hundreds digit became hundred thousands.
[The same discission continued with the digits 2 and 4]
Christiana: Therefore, notice that the value of each digit in the number changes respectively. If, for example, we multiply by ten, then the value of each digit becomes ten times bigger. Thus, by adding zero at the end we indicate the change of value of each digit.
In the example above there was evidence of conceptual understanding of multiplication and place value both in Christiana’s questions and explanations.
Research suggests that teachers interpret textbook messages about the aims of instruction and about what is good mathematics teaching in different ways, and that their interpretation influences the ways they used textbooks in their teaching (Ball and Cohen, 1996). Adding to this Nicol and Crespo (2006) claim that factors influencing how they use textbooks include the classroom context, and the teachers’ understanding of mathematics. Christiana seemed able to draw on her own understanding to construct an appropriate activity to elaborate on the textbook activity.
Ball, D. L. and Cohen, D.K (1996) Reform by the book: What is: Or might be: The role of curriculum materials in teacher learning and instructional reform?, Educational Researcher, 25(9), 6-8,14.
Nicol, C.C. and Crespo, S.M. (2006) Learning to teach with mathematics textbooks: How preservice teachers interpret and use curriculum materials, Educational Studies in Mathematics, 62(3), 331-355.