Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks
Kaye Stacey and Jill Vincent
Further descriptions of modes of reasoning by topic
Area of a trapezium
Area of a trapezium was included in eight textbooks and all explanations were deduction using a general case, deriving the formula for a trapezium from the formula for a known shape: parallelogram, rectangle or triangle. The mode of reasoning is deduction using a general case because the methods build on students’ prior knowledge of calculating the area of other shapes, and use pronumerals for the side lengths and perpendicular height. Figure 4 shows the interesting variety of dissections used in the explanations.
(a) Area of a trapezium as half the area of a parallelogram (textbooks A, C, E, F and H) /(b) Trapezium dissected into two triangles (textbook I) /
(c) Dissection of trapezium into parallelogram (textbook I) /
(d) Dissection of trapezium to make a rectangle (textbook G) /
Fig. 4 Dissections of trapezia used in explanations
Area of a circle
Area of a circle was included in eight textbooks. Three modes of reasoning were identified in the explanations of the formula:
§ appeal to authority
§ experimental demonstration
§ deduction using a general case.
Only one textbook (G) reasoned by appeal to authority by stating the rule without any explanation, although this textbook explicitly rejected the method of counting squares as inadequate. Textbook E also rejected this empirical method, but like textbooks A, C, D, F, H and I, derived the formula for the area of a circle by dissecting a circle into sectors, rearranging them to form an approximate rectangle (see Figure 5), and calculating the area of the rectangle, and hence the area of the circle (approximately). We classify this mode of reasoning as deduction using a general case, even though it requires very considerable refinement related to the limit processes to become a mathematically acceptable proof. However, we judge that it functions well as a justification of the formula in this simplified version at eighth-grade level. Two other dissections and re-arrangements of the circle were also presented. The explanation in Figure 5 has both general features (e.g. the radius r) and specific features (e.g. the number of sectors), which shows that the decision of whether this is deduction using a general case or deduction using a specific case is not clear cut. In this case, we judged that the generality of the radius was predominant.
Textbook A prepared students for this explanation by preceding it with a practical version of the dissection in Figure 5, where students cut a photocopy of a circular protractor into sectors, construct the ‘rectangle’ and hence find the area of the protractor. They were asked what would happen if the protractor was cut into more, narrower sectors, thereby acknowledging the limiting processes involved in the mathematical proof. We classified this activity as an explanation in the experimental demonstration mode.
Fig. 5 Area of a circle represented as area of a rectangle of equivalent area
Textbook D presented a variety of approaches to justifying the area of a circle, approaching all through guided discovery. First, by drawing circles inside and outside squares, students find that the area of the circle must be between (so approximately ). Although this is an approximate result (and stated to be so), the mode of reasoning is still deduction using a general case. This was followed by an empirical demonstration, placing a grid over the circle and showing 316 of 400 grid squares fell inside the circle (hence obtaining an area of ). Students were then guided through two dissection explanations representing deduction using a general case.
Distributive law
Distributive law was included in nine textbooks. Five modes of reasoning were identified:
§ appeal to authority
§ qualitative analogy
§ deduction using a model, or a specific case
§ experimental demonstration
Three textbooks (E, G, I) stated the rule that , with no accompanying reasoning, thus adopting an appeal to authority approach. One textbook (A) introduced the Distributive Law with a qualitative analogy: a pictorial representation of 2 lots of 3 apples and 2 bananas as an explanation that . Fortunately this was the only one of the textbooks we investigated that fell into the trap of ‘fruit salad’ algebra, where pronumerals are used incorrectly to represent objects (apples and bananas) rather than numbers. This textbook also included a correct explanation by using an area drawn on a square grid to show that is equivalent to and also to (see Figure 6). Textbook H also used the rectangle area model for multiplication, also asking students to use the same method to show the equivalence of expressions in the exercises that followed (deduction using a model).
Fig. 6 Area model for the distributive law
Two textbooks (B, F) explained the distributive law by deduction from the link between multiplication and repeated addition. For example, it was noted that is equivalent to and that this can then be simplified to . We regard each of the explanations in textbooks B and F as deduction using a specific case. Textbook C reasoned with a ‘counters in cup’ model, which we classified as deduction using a model (see Figure 7). Each cup contained the same unknown number, x, of counters, with an extra 2 counters, so considering the total number of counters in 3 lots of x and 2 extra counters showed that . This textbook also reused the model by including similar ‘cup and counter’ problems in the exercise that followed.
Fig. 7 ‘Counters in a cup’ model for the distributive law
Textbook D used a specific example, reminding students that when working with numbers, was equivalent to and also to , and that this could be extended to pronumerals. We call this experimental demonstration.
Multiplication of powers
Multiplication of powers was included in six textbooks. Two modes of reasoning were identified:
§ appeal to authority
§ deduction using a specific case
At this eighth-grade level only positive integer powers were included. Textbook D used an appeal to authority approach: ‘when multiplying powers of x, add the indices’, . All other textbooks that included this topic calculated examples of multiplication of powers (either numbers or pronumerals) from the definition of powers as shorthand for repeated multiplication (for example, ). Four of these five textbooks then stated the rule, and one invited students to state the rule themselves. We infer that the specific examples (e.g., powers of 2 and 3) in these five textbooks are intended to be seen by students as generic, but as there was no explicit discussion that pointed to all numbers and any positive integer powers we classify these as deduction using a specific case.
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