In the last lecture, we looked at several examples of students ideas in science, now, I'd like to look at some examples of students ideas in Mathematics. Stella Bosnia du. Here, will talk about some students ideas involving fractions. One of these areas is rational number. So the research that we need is mostly with fractions. Because understanding fraction violates many of the basic modes of whole numbers. For example, let's take the symbolic notation of a fraction. Fraction consists of two numbers and an immediate there is a line. Now, this is different from 1, 2, 3, 4, and 5, where you have one number. So in the case of fractions, you're dealing with a different symbolic representation. So we figured that may be children will have difficulty understanding the symbolic notation. What really does fraction mean? Yes, they do. So for example, if you ask children, you give them two fractions to compare, they will tell you that the fraction with a bigger numbers as denominators numerators is the biggest fraction. So actually, this is known in the research community is a whole number bias. But then, bias express itself in many, many mistakes that students make. Fractions are really difficult to understand. If you look at the research, many children as much as 80 percent in some studies cannot add fractions. They do not know how to add fractions, they add the numerators and denominators. They don't know how to multiply with fractions. Multiplication with fraction is a very difficult thing because we have seen that multiplication with whole number makes bigger with multiplication with fractions, that's not make bigger. This is a very counter interactive idea. One idea that cuts across science and mathematics and it's important in both is the idea of volume. Here, Professor Carol Smith talks about some students difficulties with volume which might be surprising. Students have lots of difficultly with volume not just weight, volume is very complicated like maybe teachers recognize more the difficult with volume, but I can tell you that when you do interviews with kids and you ask them to try to figure out the volume of an object, and you give them little cubes and you ask them can you use these tubes to figure out how much space this is taking up? What's the volume of the object? They do all sorts of things with those cubes other than find the volume of the object and teacher might not realize how many possibilities are out there for them? When you're thinking about space, there's so many different dimensions of space, or aspects of space. So some of the things they typically do is they might just put those cubes on top of the object on the surface of the object and really are finding the surface area of the, not the whole surface area but just the area of the top face of the object. Here's another very creative thing they do, they use a little cubes and they increase the object, they surround it rather than build a little replica of the object that recreates the object in terms of a decomposition into cubes. What they're really doing is trying to cover the object with the cube in some way tangibly and figure up. Finally, I'd like to talk about student understanding of the equal sign, and this is from a paper by Eric Knuth and others, asking does understanding the equals sign matter? So let's look at this question that they asked, 3 plus 4 equals 7. What does that symbol means that the arrow is pointing to? So if we think about it, of course, it means that what's on the left-hand side of the equal sign is equal in quantity to what's on the right-hand side. That's just what an equal sign means. A number of students express this idea in their own words. It means that what's left and right of the equal sign mean the same thing, etc. However, a number of students, in fact, a majority of students expressed a different view of what the equal sign means. It means something to the effect of and answer is. So they would say things like a sign connecting the answer to the problem. What's the problems answer is, the total how much the numbers added? So they seem to be thinking of the equal sign as something that makes something happens, like when you press the equal sign on a calculator, you get the answer to whatever numbers you are entering earlier. So this view of what an equals sign is, we can see that a larger number of students have this operational view of the equal sign then had the relational view of the equal sign. What are the implications or the consequences of this? This is a further part of the paper, they went on to ask question such as what value of m would make the following sentence true? So looking at 4m plus 10 equals 70. Many students try to guess and check method. Okay well, let me try five. Okay, so if I put five in there then four times five is 20 plus 10 is 30. Okay, that's less than 70. So I guess I m needs to be bigger. So they would keep trying a different value of m until they got the right answer. Other students would use them more algebraic approach,4m plus 10 equals 70. So let's subtract 10 from each side. So now, 4m equals 60, m equals 60 divided by four which is 15. So that's an algebraic a much more sophisticated way of arriving at the answer through algebraic means. What they found was that students who had a relational view of the equal sign were far more likely to use this algebraic method of solving these kinds of problems than students who did not express that kind of a relational view of the equal sign. So thinking back over these examples of students ideas in science and mathematics, I think we can see some commonalities, some overarching ideas. One is that these ideas are often surprising to teachers and researchers. Unless you ask students very carefully how they're thinking about things and then listen to the students, you can often miss their ideas. So for example, Jim Minstrel was very surprised when he said let's just remove air from the room that students thought that then things would float off. These ideas are often very reasonable to students who express them. So for example, the students talking about air pressure causing gravity talked about well if an astronaut is out in space in their space suit walking near their spaceship, they are floating around because there is no air. Therefore, air must cause gravity. Finally, these ideas are consequential. Obviously, if you think that air pressure causes gravity, that's going to cause difficulties in thinking about gravity and solving problems involving gravity. As we saw with the algebra, if students think of the equal sign as operational rather than relational, then they're far less likely to be able to make sense of the algebraic manipulations involved in solving equations using those kinds of manipulations. In the upcoming lecture, we'll zoom back and take an overview look at students ideas, and try to make sense of what are students ideas? How can we make sense of students ideas? How can we use those insights than in instruction?
