About:

Following her research, Stacey offers teachers some clear teaching strategies to assist in supporting students to master the difficult concepts associated with decimals (such as decimal notation) and decimal calculations without over-simplifying the mathematics.

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Professor Kaye Stacey: Hello, I’m Kaye Stacey. ‘Death by Decimal’ - this might be the title of a rather unusual detective story. But, in fact, it describes a group of accidental deaths in hospitals: deaths which are the result of a patient being given an incorrect dose of a drug because of an incorrect arithmetical calculation or incorrectly reading a numerical display. As teachers know, it is quite common to make mistakes putting the decimal point in the wrong spot after a calculation or when reading the number of zeros before or after a decimal point. It could easily be fatal for a patient to be given a drug dose too big or too small by a factor of ten or a hundred or even more. That is ‘death by decimal’. Nowadays, our hospitals have many processes in place to reduce ‘death by decimal’, so it is fortunately a very rare event. One way is to package drugs in amounts that reduce the need for calculations involving decimals. And of course, another way is to have well educated, numerate nurses and standard procedures for checking and rechecking hospital calculations and readings. However, an insidious form of mathematical ‘death by decimal’ is happening today to many students in our schools. This type of mathematical ‘death’ causes students to learn rules without reasons and to stop thinking of mathematics as a sensible subject where they can use their own powers of thinking. Fortunately, though, the cure for mathematical death by decimal is within the capacity of all teachers. The decimal system for writing numbers is very powerful and very concise and for these reasons it is difficult for some students to understand. This is often surprising to teachers. We all know that fractions are difficult, especially because it is complicated to learn the special rules we need to add, subtract and multiply and divide them. The rules for computing with decimal numbers are much simpler because they are minor variations of the rules for whole numbers, but decimals are hard in a ‘hidden’ way. They can be even more difficult to understand than fractions because the concise notation hides information: What do the digits mean? Where are the tenths? Where are the hundredths? How can one digit sometimes be both tenths and hundredths? The list goes on. To understand this difficulty, my colleague, Vicki Steinle, and I studied more than 3000 students from Years 4 to 10. We tested them twice a year over three years and we traced the changes in their understanding of the meaning of decimal numbers. This slide shows the percentage of students who had a good understanding of decimal notation, so that they understood the size of decimal numbers well. They may not know all that there is to know about decimals, but for convenience, we call them ‘experts’. The slide shows that about a quarter of students in Year 5 are experts and the percentage steadily increases through secondary school. But surprisingly, it only reaches about 70%, two thirds of the student population, even by Year 10. There are far-reaching practical consequences for those students who miss out. One of our adult students told us that she and her husband owned a breathalyser unit to avoid drink driving, but they could not use it because neither of them was sure whether a reading such as 0.12 was over or under the legal limit of 0.05. How do the students who are not experts think about decimals? There are about a dozen different yet common ways: I will show you three main types. Younger students often have what we call ‘longer-is-larger’ thinking. These students often think that a number like 3.4 means 3 plus 4 little bits or 3.5 means 3 plus 5 little bits but these little bits are of unconsidered size. They know that 3.5 comes ‘after’ 3.4, but they think that 3.42 is much bigger, because it has 42 little bits instead of 4. Here you can see that this is common thinking in primary schools, but quickly becomes less common. This type of thinking and other erroneous ideas are fostered if students never mix decimals of different lengths – if they learn about one-decimal place decimals for a year, two-decimal place decimals the next year, and so on. Throughout secondary school, about 15% of students at every year level draw too heavily on their knowledge that decimals are like fractions and they are what we call ‘shorter-is-larger’ thinkers. You can see them on this graph. They know 5.123 is less than 5.732, but they think that both of these are less than 5.68. They think of 5.732 being like a fraction with denominator 732, so it would be less than something like a fraction with denominator 68. Other shorter-is-larger thinkers consider that a number like 5.732 is made of thousandths, whereas 5.68 is made of hundredths which are bigger pieces. Although only about 15% of students are in this category at any one time, we found that about 30% of students thought like this on at least one of the occasions that we tested them during secondary school. Shorter-is-larger thinking arises from an underlying psychological association of decimals with fractions and, from there, to things that are thought of as operating in the opposite way to whole numbers, including negative numbers. That is why it becomes a little more common around Year 8. There is something else to notice on the slide. Shorter-is-larger thinkers say correctly that 5.123 is less than 5.68, but they do it for the wrong reasons. Most misconceptions, including this one, lead to some correct answers and some incorrect answers. Students and teachers think they just make a few little mistakes, not realising that their entire conception may be incorrect. The next slide shows an unexpected group of students – about 10% of secondary school students. These students puzzled us. We expected that all students would know from whole number place value that every digit adds something to the value of a number. For example, that 4.78 is 4 + 7 tenths + 8 hundredths, and 4.783 is the same plus a little bit more. However, these ‘money thinking’ students often think these numbers are equal or are unsure which is the larger. This problem can be caused by too much reliance on money in teaching decimals and too frequent rounding to two decimal places in school calculations. Some of these students have explained to us that the later digits have no meaning at all, and should be thrown away. With money, students see them representing a part of a cent that cannot be paid or with numbers, just like the superfluous digits that a calculator sometimes shows after a calculation. As you can see on the slide, this ‘money-thinking’ phenomenon shows no sign of decreasing throughout secondary school. However, we think it is easy to fix with a little attention. Faced with evidence of students’ difficulties, some teachers conclude that mathematics is very hard so it needs to be simplified (for example, to consider only two-decimal place numbers) and that improving understanding requires years of slow progress. However, we have found that relatively little, well targeted instruction can address fundamental misunderstandings and change children’s ideas rapidly. Let me tell you about a study that we did several years ago, in an ordinary suburban primary school with four Grade 6 classes. This school had provided data for our longitudinal study of decimal understandings and their results showed clear room for improvement. We went to the school for one lunch hour in August and demonstrated to teachers a variety of teaching activities to address the problems. We showed them how to use concrete materials (what we call ‘linear arithmetic blocks’, which we made ourselves) for explaining the meaning of decimal numbers and games which focused on mixing decimals of different lengths. We asked the teachers to use the activities to target the meaning of decimal notation, in a short intervention which they designed themselves. Year 6 had ‘done’ decimals early in the year, so we were asking them to revisit a topic the teachers felt they had completed satisfactorily. In October, two months after the decimal activities, we returned to the school and retested the students, looking for long-term learning. This slide shows which students understood decimal notation well (that is, ‘the experts’) in each of the classes: about one third of Class 1, about three quarters of Class 2 and half of Classes 3 and 4. Let’s now see what the students knew two months later. All of Class 2 now understood decimal notation, after one targeted lesson. And about two thirds of Class 2 were now experts, instead of one third. Classes 3 and 4 show a different outcome. For various reasons, their teachers did not do any decimal activities. And we see that there was no change in any of their students’ understanding. What do we learn from this intervention? Firstly, we learn that this is not a difficult issue to address. Instruction works well when it targets the place value and base ten properties of decimal numbers, through concrete materials, games and accompanying direct explanation. Moreover, this learning can be retained. Remember that we tested about two months later. In Class 1, seven of the 16 non-expert students were experts two months later: the short intervention was enough for them. However, it was not sufficient for the nine others: they needed more than this. And Classes 3 and 4 remind us that, without attention, students’ misconceptions persist. We know from our research that they sometimes persist for years, even into adulthood. What happens when students do not understand the meaning of decimal notation? There are far-reaching consequences. For example, the rules for rounding decimals make no real sense to students with these misconceptions, so they just have to be learned parrot fashion. And probably then forgotten. Mathematics stops being something where you use your brains. Students begin to approach maths learning as a task of accumulating knowledge of hard-to-remember isolated facts, rather than a sensible and connected whole. With difficulties interpreting decimals, students will have problems placing numbers on a number line and so plotting points on coordinate axes for graphing. Students cannot check whether the answers that they get by calculator or by-hand calculations are reasonable. And as adults, they are disadvantaged when they cannot select the lowest interest rate or interpret the breathalyser output. Targeted teaching, revisited as needed over the school years; emphasising the place value and base ten principles; and using a good concrete model and confronting decimals with many digits: these are the ingredients of the cure for mathematical death by decimal. If you are interested in trying some of the teaching activities with your class, look at the ‘Teaching and learning about decimals’ CD ROM. There is information and a sample on our website. Music .

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Death by Decimal