Sunny with a chance ... of tenths! Using the familiar context of temperature to support Belinda Beaman teaching decimals St James the Apostle Primary School, Vic. <[email protected]> Introduction As teachers we are encouraged A collective sigh filled the room when I announced that we would be investigating to contextualize the mathematics decimals. Mumbles about “the numbers with the full stops” accompanied the folded arms, that we teach. In this article, and before I knew it, 27 grade 5 students were disengaged before we had even begun. Belinda Beaman explains how Have you been there? Decimals are grouped with ratio, she used the weather as a percentages and fractions under the ‘rational number’ umbrella, and this group of topics is context for developing decimal not generally favoured by teachers or students alike. Faced with this dilemma, I found a way understanding. We particularly to teach decimals to tenths that sneaks up on students before they even realise they are enjoyed reading how the students learning—and all teachers like that. were involved in estimating. Why the weather? The term ‘context’ might be described as the “non-mathematical meanings present in the problem” (Huang, 2004, p. 278). A connection, therefore, can form between mathematical content and application, based on the familiarity of the context to the problem solver (Huang, 2004). The teaching of decimals needs to be anchored to where the student is ‘comfortable’ and requires the provision of a context for teaching with which a student can relate (Chinn, 2008). Showing students that decimals are part of their lives will ground the development of decimal sense in “meaningful experience” (Caswell, 2006, p. 28). In order to support students in their out-of-school learning 26 APMC 18 (1) 2013 Sunny with a chance of tenths! Using the familiar context of temperature to support teaching decimals of mathematics, familiar contexts need to be consulted the Bureau of Meteorology recreated within the classroom so that students website, which is refreshed several times can begin to connect their classroom tasks to each hour, and obtained the temperature everyday experiences. for 9.00 am, 10.00 am and 11.00 am, as In summary, I needed to find a context that our mathematics sessions most days took would engage students and show them how place after this time. I asked my students decimals are used in real life. I wondered how to guess what the temperature was at 9 using the weather might work. This article o’clock. I played the ‘higher/lower’ game, describes sessions that were undertaken in my and almost all 27 hands were in the air, eager grade 5 classroom. Students observed hourly to ‘get it’! However, many hands went down, changes in temperature each school day for and excitement was replaced with confusion five days and plotted their findings on a graph. when it became clear that the temperature was above 15 degrees but below 16 degrees. How could that be? I allowed the quiet chatter that followed, as students speculated if I had made a mistake. It took only seconds before one student waved his hand in the air with gusto. “It must have tenths!” he yelled. All eyes were on him, and a few hesitant hands punctuated the air. Within seconds, hands were flying, guesses were forthcoming, and I was ready to launch into teaching tenths. Prior to this, we had looked at decimals using Linear Arithmetic Blocks (LAB; see Victorian Government Department of Education and Early Childhood Development, 2010). However, no contextual Figure 1. Student graph. link had been made prior to this. The weather lady Organising the graph We would start every mathematics session the Each student received a sheet of grid paper same way, plotting temperatures for that day and set up their graph according to the and catching up on temperatures from the teacher’s displayed large copy. The hours previous afternoon. Students would open up of the school day ran across the horizontal their graphs and chat about their estimations axis, and temperatures in degrees lined the with the person next to them, and I would vertical. We started our graph at 14 degrees. be the weather lady, ready with my hourly It was November and temperatures were temperatures. unlikely to creep below this, even at 9.00 am. I would begin taking estimations. Each This decision was based on quite a discussion guess had to be accompanied by a brief to ensure students understood that if we did justification, to which I would share some our graph in winter then we would need to praise, then inevitably announce, “Higher!” include much lower temperatures. By this or “Lower!” which would always be met with stage, students were unaware that this activity an edge-of-the-seat reaction of excitement or had anything to do with decimals; I had not disappointment. Perhaps their estimation was even mentioned them yet. out! However, while other hands were in the air, there was still time to think of another one. Students actively developed their Decimals sessions heat up understanding of tenths, as can be evidenced With graphs prepared and interest captured, by the scale shown in Figure 2. Here, the we began to plot the temperature. I had student has marked in her own tenths APMC 18 (1) 2013 27 Beaman without teacher prompting. It is even more interesting to note that she made the fifth tenth more prominent each time. I had continued to reinforce the idea that the fifth tenth was halfway between one temperature and the next, and it would appear that this student had consolidated this concept here, for herself. The graph sheets had been deliberately created so that the scale was not connected to centimetres. Even if this student made a connection with the visual of a ruler, she had brought her own prior knowledge to this task, as I had not made this explicit link for the students. Figure 3. Wall display of graph and charts. Sixteen point ten The development of addition and subtraction began with recording the increase or decrease of the temperature. When the temperature was finally obtained, we would count, by tenths, from the temperature of the previous hour. This way we could find the difference, or the change, in temperature. If we counted 18 tenths, we knew that this was one whole degree plus 8 tenths, which could be written as 1.8. Counting each time was necessary, particularly when the skill of bridging was required. The volume of students’ responses would get conspicuously softer as students approached, for example, “…16.8, 16.9, 17…”. Figure 2. Tenths. I am sure some hoped there was a number we could call ‘sixteen point ten’, but the practice As the days went on, our estimations of continually counting allowed these students became more informed. We kept charts to consolidate the bridging concept. alongside our graph on the wall, which listed Some students did not need to count up each day, each hour, and each temperature. by tenths when they discovered the addition These charts also had a space to record the and subtraction that could be carried out. The change in temperature from one hour to strategy of reaching the next whole was the the next, which was used during the week most frequent. For example, a student might to develop mental addition and subtraction say, “To get from 15.8 to 16.2, I know it’s four of decimal numbers. Students were able to tenths because 15.8 plus two more tenths is base an estimation on the temperature at the 16, then another two tenths makes 16.2.” This same time on previous days, or based on the was impressive, considering it had not been change in temperature experienced at this not modelled for students immediately. Some time on other days. I gave students time to students, who were looking for a quicker way develop a justification for their estimation of calculating the difference, followed these and talk about it with the person beside examples and copied quite keenly. After all, it them. That also freed me up to approach was a strategy that worked which the teacher individuals needing support, or who simply had not even shared yet! This sharing among wanted to talk to me about their thinking. peers was most effective. 28 APMC 18 (1) 2013 Sunny with a chance of tenths! Using the familiar context of temperature to support teaching decimals Bring a jacket! The forecast My favourite conversations to overhear were Students in my grade 5 classroom the ones regarding memory of the actual completed pre- and post-testing of their weather. “No,” a student might say, “I’m sure decimals learning and the final results were that this morning it was colder than yesterday impressive, but, from the learning I had morning because I wasn’t wearing a jumper witnessed, this testing was only proof of what yesterday.” Students were not even thinking, I knew. Using the temperature, which is a here, in terms of numbers first. They truly strong and importantly, familiar, contextual were relying on real-world experience to guide examples of decimals in operation proved to their own estimation of the temperature. Some be a most effective way to provide students would even say, “It was colder by more than a with opportunities to practise and refine tenth because that would hardly even be any their understanding of tenths, especially different.” Even students who usually required when combined with LAB as concrete support in mathematics were grasping the representation. This combination of ideas size of tenths by relating them to whole worked in my room, where dialogue was degrees, and then, to their own experience unprompted, unscripted and never boring, of the weather. They would even quite keenly rather like Melbourne’s weather. In fact, it add this reasoning to their justifications. And appeared as if Melbourne’s weather had, for why not? Students were blurring the line a change, been quite reliable indeed. between mathematics and real life experience smoothly and with conviction. Cool feedback References Contextual examples of decimals in operation Australian Government. (2011). Bureau of Meteorology website. Accessed at http://www.bom.gov.au. have their limitations (see Steinle, 2004; Caswell, R. (2006). Developing decimal sense. Australian Irwin, 2001), and this present example is not Primary Mathematics Classroom, 11(4), 25–28. excluded. It deals exclusively with wholes and Chinn, S. (2008). The decimal point and the ths. tenths, and the use of the word ‘degrees,’ Mathematics Teaching Incorporating Micromath, 208, 19. Huang, H. (2004). The impact of context on children’s which is also the unit of measurement for performance in solving everyday mathematical angles, could prompt some criticism. My problems with real-world settings. Journal of Research students were not disadvantaged by these in Childhood Education, 18(4), 278–289. Irwin, K. C. (2001). Using everyday knowledge of limitations. I did teach decimals beyond decimals to enhance understanding. Journal for tenths using the LAB. Thanks to our Research in Mathematics Education, 32(4), 399–420. temperature observations, students had a Steinle, V. (2004). Detection and remediation of lively and developing understanding of tenths, decimal misconceptions. In B. Tadich, S. Tobias, C. Brew, B. Beatty & P. Sullivan (Eds.), Towards including simple operations, that they were excellence in mathematics. (Proceedings of the MAV able to apply to their new discoveries about Annual Conference, pp. 460-478). Melbourne, hundredths and thousandths. Regarding the Australia: MERGA. use of ‘degrees’, I pointed out very early to my Victorian Government Department of Education and Early Childhood Development (2010). Linear students that angles are measured in degrees Arithmetic Blocks, http://www.education.vic.gov. too, but a different kind of degree. We had au/studentlearning/teachingresources/maths/ investigated angles two terms earlier, and mathscontinuum/number/lab.htm students were satisfied with a brief discussion to clarify this sharing of terms. It did not prove to be an issue, and was not raised by any student throughout the sessions. APMC 18 (1) 2013 29