What About Correctness? It was already clear to some thinkers that a conception of knowledge that required correspondence to a real world was illusory, because there was no way of checking any such correspondence. In order to judge the goodness of a representation that is supposed to depict something else, one would have to compare it to what it is supposed to represent. In the case of knowledge this would be impossible, because we have no access to the real world except through experience and yet another act of knowing and this, by definition, would simply yield another representation. It is logically impossible, however, to compare a representation with something it is supposed to depict, if that something is supposed to exist in a real world that lies beyond our experiential interface. (von Glasersfeld, 1995) Because it is so often taken for granted that mathematical expressions can be understood without carrying out the operations they symbolize, formalist mathematicians are sometimes carried away and declare that the manipulation of symbols constitutes mathematics. Without the mental operations they indicate, however, symbols are reduced to meaningless marks. (von Glasersfeld, 1995) Maya is a 2 nd grade student LS:If you start from there and take 3 cards at a time to make a pile, I wonder how many piles of 3 you could make? M:(Sits silently in deep concentration for about 2 minutes). 7! LS:When you counted, what did you say? M:21, 20, 19 that would be one. 18, 17, 16 that would be two. This method was not suggested to Maya and differed substantially from the algorithm her teacher had tried to teach her. LS: Can you give me a multiplication problem for that? M:21 times 3? LS:What does 21 times 3 mean? M:21, and take 3 out of 21. LS:Is that 21 times 3 or 21 divided by 3? M:21 divided by 3! LS: Can you give me a multiplication problem? M:(Sits silently for over a minute). LS:What are you doing? M:I am figuring out how many 3s equal 7. LS: Put 18 blocks into that container. You can count by 3s if you want. M:(Takes 3 at a time and places them in the container) LS:Give me a multiplication problem for that. M:(Long pause) 3 times 6 equals 18! LS: Put 15 blocks into that other container. M:(Puts them in by 3s, and this time she keeps track and says that she has 5 groups of 3 in the container). LS:(Pours the contents of the two containers together.) Can you find how many blocks there are using 3s? M:(Thinks a long time) times 6 is 30. LS: This is 36 inches long. How many 12-inch rulers is it going to take to measure it? M:3. LS:How did you know? Did you know that 3 times 12 is 36? M:(Nods yes). LS:(Gives Maya a 10-inch ruler and a 2-inch ruler that make up a 12-inch ruler). LS: Now, if you put the tens down first, how many of those would it take and how many of the twos? M:It would take 3 of these (the tens) and 3 of these (the twos). LS:Tell me why. M:Because 3 tens is times 3 is 6 and 30 plus 6 would be 36. LS: Lets see if you can do this one 84 inches. Say you were to use the 10-inch one first and the 2-inch one second. They have to be equal like they were before. How many of each kind would it take? M: (Long pause). Do they have to be equal? LS:Tell me one that isnt. M:I dont have one. LS:If you find one, tell me. M:You could use 8 of these (10 inch) and 2 of these (2 inch). LS:Can you find two that are equal? M: (Long pause). I cant find any! LS encouraged Maya to keep experimenting. She tried 9 upon his suggestion and found that it didnt work because 9 times 10 is 90. She then tried 7. After finding the two sub-products and adding their results, LS asked her to tell him the result of multiplying 7 and 12: LS:By the way, what is 7 times 12? M:84. LS:How did you know that? Did you just know it or did you use this to help you? M:No. LS:Could you use this to help you? Suppose you did not know 7 times 12, how could you use it? M:Seven 12s, you could put those two together (the 10 inch and the 2 inch rulers). Carissa (a 9 th -grade student) came upon the value of 10 cm in 8 seconds, and the teacher asked her why that worked: T:Why is 20 cm in 8 sec the same as 10 cm in 4 sec? C:I think, because 10 cm in 4 sec is half of 20 and 8. T:Okay. C:So its going the same speed. They are going the same way but its just that they have a double to do. T:A double to dowhat do you mean? C:Once he hits 10, he has another 10 to do. T:Why dont you show me with a picture? C:Like, theyre both going to 10 right here and they stop at the same time. Then the other one has to do exactly the same thing to 20. C:So its like theyre going the same way for both of them. I dont know how to explain it. The frog is going the same speed until 10 and then he stops at 10. T:What does the 10 and the 20 mean there? C:Centimeters, the distance. T:I dont see any time in your drawing. Where does time come in? C:Oh (quickly adds 4 and 4 to the drawing). C:It takes both of them 4 seconds to go to here and the rest it takes 4. Clown walked 7 cm in 2 seconds and Frog walked 9 cm in 3 seconds. Who walked faster? Carissa would divide 7 by 2 and 2 by 2 to get 3.5 cm in 1 second, and she would divide 9 by 3 and 3 by 3 to get 3 cm in 1 second, and conclude that Clown walked faster. Her strategy seemed very procedural, so her teacher developed a problem on the spot to assess her understanding: The teacher wanted to know if Carissa would just divide 16 by 4 because those were the two numbers provided. C:(Divides 16 / 4 = 4 and 4 / 4 = 1 and wrote 4 cm in 1 sec near the first tick mark in the drawing) T:Why dont you continue that through the problem to check your work? 4 cm 1 s2 s 3 s 4 s 8 cm12 cm 16 cm C:Uh oh. T:Whats wrong? C:Somethings wrong. Im not supposed to get that until the end. I dont understand. T:What you did (dividing by 4) looks like this. Whats different about this drawing and your drawing? C:(Unintelligible mumble) T:When youre dividing by 4 over here, thats splitting up into 4 equal parts. But this (Carissas drawing) is split into how many equal parts? C:Eight. T:So if you were going to figure out how far you went in each one of these little things, what would you need to divide each number by? C:(Thinks a long time)8? T:Yes! C:I divide the 4 by the 8 too? T:What do you think? C:I think maybe I should. T:Why dont you try it and see what happens? C:(Divides and gets 2 cm in 0.5 seconds) T:Does tat make sense? C:Yeah. T:Why? C:Cause its going 2 cm in half a second. C:And then 4 in 1 second. And if I keep going C:You have to divide by how many equal parts you want it to be. Composite Unit: The row is a composite unit (a unit made out of units) Composite Unit: The column is a composite unit (a unit made out of units) A row or a column is a construction. It is not an entity or an object. Absent the learners construction of it as a unit, it doesnt exist. A major methodological shift that came with RC is that researchers started asking what students perceive, construct, and understand. They abandoned the assumption that students mathematical worlds are the same as (or poor copies of) ours.