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APA style, Times New Roman:12 POINT. Double spacing.
Consider in detail the two activities (Turnip field, Downton and Wright
and The marvellous multiplier, Hurst and Hurrell) to provide a conceptual understanding of place value.
Write a 500 word synopsis of each article and provide links to the NSW NESA syllabus outcome(s), the appropriate working mathematically outcome(s) and the Australian curriculum code(s).
Write a 400 word reflection on the how the activities mentioned are superior to a textbook activity and link the reflection to at least two learning theories or theorists.
Opening Up Mathematics Education Research Sliding into Multiplicative Thinking: The Power of the ‘Marvellous Multiplier’ Chris Hurst Curtin University [email protected] Derek Hurrell University of Notre Dame Australia [email protected] Multiplicative thinking is a critical stage in mathematical learning and underpins much of the mathematics learned beyond middle primary years. Its components are complex and an inability to understand them conceptually is likely to undermine students’ capacity to develop beyond additive thinking. Of particular importance are the ten times relationship between places in the number system and what happens when numbers are multiplied or divided by powers of ten. Evidence from the research project discussed here suggests that many students have a procedural view of these ideas, and that a conceptual understanding needs to be developed. It is suggested that this may be possible through the use of a device called ‘The Marvellous Multiplier’. Background Multiplicative thinking is rightly considered to be a ‘big idea’ of mathematics as it underpins important mathematical ideas such as multiplicative partitioning, proportional reasoning, and algebraic generalisations (Hurst & Hurrell, 2014; Siemon, Bleckley & Neal, 2012). It is well documented that students who do not develop into adequate multiplicative thinkers are likely to struggle with mathematics beyond primary school, or even in the later years of it, yet a large number of students leave primary school without that necessary understanding (Clark & Kamii, 1996; Siemon, Breed, Dole, Izard & Virgona, 2006). Part of the problem may be that students develop a procedural view of mathematics in general and multiplicative thinking in particular and perhaps this is, at least in part, due to the way in which mathematics is taught. Thanheiser, Philipp, Fasteen, Strand & Mills (2013) recently interviewed pre-service teachers about their level of conceptual understanding and uncovered a prevalent and rather confronting attitude that knowing procedures for doing mathematics was all that was required. They developed three principles for pre-service teacher knowledge which could well be applied to teachers in general: Underlying concepts serve as the foundation for mathematical procedures, knowing the foundations for the procedures has value including knowing why each procedure yields correct answers, and, until they themselves learn to make sense of mathematics, pre-service teachers (PSTs) will be unprepared to support their future students beyond learning procedures. (Thanheiser et al., 2013, p. 138) The on-going research project on which this paper is based has shown that there are indeed many primary aged children with a procedural view of aspects of multiplicative thinking. To date over 400 students have participated in the study and whilst the great majority of them can identify the commutative and distributive properties of multiplication, the inverse relationship between multiplication and division, and the extension of basic multiplication and division facts by powers of ten, very few can explain conceptually why those properties and relationships work. Most explanations are along the lines of ‘swapping the numbers around’ (commutative), ‘splitting the numbers’ (distributive), ‘they’re the same family’ (inverse), and ‘adding or taking a zero’ (extension). Similarly, many of the 2016. In White, B., Chinnappan, M. & Trenholm, S. (Eds.). Opening up mathematics education research (Proceedings of the 39th annual conference of the Mathematics Education Research Group of Australasia), pp. 328–335. Adelaide: MERGA. 328 mailto:[email protected] mailto:[email protected] students know how to quickly give an answer to an exercise like 74x10 or 74x100 and almost all described and explained the process as a result of ‘adding a zero’. Methodology The research is built on two instruments – a semi-structured diagnostic one-on-one interview and a written version of the interview (Multiplicative Thinking Quiz – MTQ). The latter was developed in order to gather a large amount of data from a comparatively large sample in a short time. Each interview takes approximately 35-40 minutes whereas the MTQ can be administered to a whole class group in the same time. In general, the MTQ is administered and students are identified for later interviewing in order to probe their understanding. This paper reports only on the use of the interview with a group of 16 Year Five students. The particular sample was chosen as the basis for this discussion because the students’ responses to the MTQ indicated that they had or were developing a measure of conceptual understanding of multiplicative thinking. The interviews were audio recorded and later transcribed. The theme considered in this paper encompasses the relationships between the concepts of ‘times bigger’, extended number facts, and multiplying and dividing by powers of ten, as well as the use of the ‘Marvellous Multiplier’, a sliding strip device designed to enhance the development of understanding of those concepts. This is explored by considering students’ responses to a set of questions from the interview and comparing their thinking before and during the use of the Marvellous Multiplier. The ‘Marvellous Multiplier’ (M/M) (Figure 1) is a piece of laminated card showing whole number place value columns into the millions. There is a corresponding row of empty columns where numbers can be written. A second laminated strip showing a single digit number is inserted into two slits at the left and right ends of the place value columns. The M/M is operated by sliding the numbered strip to the left or right and a zero can be written to fill the empty place, or removed, if sliding to the right. Figure 1: Marvellous Multiplier showing sliding strip at original position and after being slid one place 329 Figure 2: Decimal Marvellous Multiplier showing sliding strip in original and after being slid one place Figures 1 and 2 shows the M/M with the sliding strip in its initial position, and when the strip had been slid by one place. The purpose of the M/M was to assist students to understand that when numbers are multiplied or divided by a power of ten, all of the digits move one place to the left (for multiplication) or one place to the right (for division) for each power of ten. This equates to a measure of conceptual understanding as opposed to the explanation of ‘adding a zero’ which is deemed to be procedural in nature. The research team wanted to see if the language used by the students changed when the M/M was introduced and whether or not the students’ understanding shifted or was clarified. Results and Discussion Theme 1 – ‘Times bigger’, extended number facts, and powers of ten The following interview questions were asked in order to generate data about the theme: _ How many times bigger is 40 compared to 4, 400 compared to 40, 4000 compared to 400, and 400 compared to 4 (These questions were asked separately). _ My friend says that if you that 17x6=102, then you must know the answer to 170x6. Is he right? How do you know? _ Write as many other number sentences as you can like 170X6 (with their answers). _ What happens to a number when you multiply by ten, like 74x10? Please explain. _ What happens to a number when you divide it by 10, like 160÷10? Please explain. The M/M was introduced as and if needed to most students in combination with the fourth and fifth questions, depending on each student’s response/s. For some students it was used on several occasions to further probe a point or clarify some point of understanding. Data that were generated from these questions were analysed to see what connections might exist between the embedded ideas and to see whether or not the use of the ‘Marvellous 330 Multiplier’ would have any effect on the student’s apparent understanding. Table 1 contains a summary of responses to the questions. Table 1 Responses of the 16 Year Five students to the Theme One questions Question Correct response Partially correct Correctly identifies ‘how many times bigger’ is one number than another. 13 2 Explains conceptually how the answer is obtained in extended number facts 4 4 Gives range of extended multiplication facts based on 170X6 or 102÷6 5 7 Gives range of extended division facts based on 170X6 or 102÷6 5 4 Explains conceptually what occurs when a number is multiplied by ten (digits move) 3 4 Explains conceptually what occurs when a number is multiplied by ten (Working with Marvellous Multiplier) 14 2 Explains conceptually what occurs when a number is divided by ten (digits move) 1 8 Explains conceptually what occurs when a number is divided by ten (Working with Marvellous Multiplier) 14 2 It is evident from Table 1 that most students correctly identified the multiplicative relationship between pairs of numbers which seems to indicate an understanding of the notion of 'times bigger’. However, whilst most of them were able to provide a range of extended number facts, only half were able to conceptually explain (some to a limited extent) what happened when extending number facts. This entailed them expressing that multiplying by ten made the number ten times bigger and for whole numbers, a zero was added. Conversely, some explained that ‘adding a zero’ means making the number ‘ten times bigger’. The four students who did explain the situation well did so in a couple of ways. Three of them used the example of 170x6 and the distributive property to show that (17x10) x6 was the same as (17x6) x10. One other student used the same example (170x6) and explained that “It makes the number a different place value into the thousands” (student Craig). However, only three students were able to say initially that digits in a number moved one place value column for each power of ten by which the number was multiplied, and one student could do that for division. The following excerpts from interview transcripts provide some insight into the extent to which the M/M helped to clarify students’ thinking. The first two students had demonstrated some measure of understanding but had had been limited to the type of explanation in the previous paragraph. With students Jacob and Pete, both the whole number and decimals versions of the M/M were introduced several times for the various examples indicated. 331 INT: You said that you add a zero when you multiply by ten. What happens if I move this number (4) to here (tens column)? What’s it worth now?