previous table of contents appendix B table of contents
3.17 Ira: "Wait, wait. We're only supposed to make it up to five. . . Now what was it again Jared?"
3.18 From the audio tape it appears that Ira and Jared removed some blocks. Then they recounted and came up with the answer 26.
3.19 Both Ira and Jared try to get the teacher's attention. They are shouting out, "We got 26!" The teacher goes over to them and asks how they came up with their answer. They explained that they counted the blocks. (3:45) EHT Simple Deduction
3.20 The teacher asks if there would be a quick way to count the number of blocks and suggests counting by fives.
3.21 Jared counts by fives and gets 25. Ira then counts and gets 25 as well.
3.22 Jared recounts the blocks again by counting each block.
4.0 The teacher helps Veronica. She is having difficulty recreating the square pattern. She has made square 1 and 2 of the pattern but stumbles on the third. She has created a 2x3 rectangle for the third "square". (3:35)
4.1 Veronica: "That's not a square."
4.2 Teacher: "What are you going to do to make it a square. . . look at the picture, how did they make it?"
4.3 Veronica: "They put three all over the place." Veronica creates the third square EHT Induction
5.0 The teacher asks Cynthia if she can predict how many blocks would be in square 6 of the pattern after she has correctly created and figured out the number of blocks for square 5. (3:36)
5.1 Cynthia: "1, 2, 3, ...30. (Cynthia is busy counting each block in the fifth square, then counts an extra 5 at the top of it.) I think there is going to be thirty in the next one."
5.2 Teacher: "Why do you think there is going to be thirty."
5.3 Cynthia: "Cause I counted one extra row." EWT Simple Deduction
6.0 Cynthia begins to work on the triangle pattern. (3:44)
6.1 Cynthia: "I do not need triangles. I could use these. " Cynthia shows the teacher the diamonds. She begins to replicate the pattern shown on her sheet. " This will make it faster too." (Cynthia does not build her pattern at this time.)
EWP Simple Deduction
7.0 Leonard begins to make his own pattern (3:48)
7.1 Leonard starts his pattern off with 1 diamond. Then he makes number 2 of his pattern using 1 isosceles triangle on top, with 2 diamonds on the bottom. (3:50)
7.2 Leonard continues to make this pattern grow by continually adding on two more diamonds each time. He continues until he has 3 parts of a pattern created.
7.3 Leonard is getting frustrated with illustrating his pattern. He explains that he can make the pattern grow using the blocks but it is too tricky to draw. The teacher asks Leonard to predict how many pattern blocks he would use in number 4 of his pattern. (3:55)
7.4 Leonard: " Seven." Leonard makes the next pattern to verify his guess.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-145
To Beginning of Section | To End of Section
7.5 The teacher then asks Leonard how many blocks will be needed for the next one. Leonard quickly states nine. "Seven, eight, nine . . . it's going on by twos cause I added on two at the bottom. " (3:57) EWT Induction
7.6 Leonard finishes his pattern, explaining that there is five items in it. The teacher asks Leonard to explain how he created his pattern. (3:58)
7.7 He tells the teacher that he used triangles and diamonds. When asked how he made the pattern grow he says he added on two diamonds each time. He looks at his pattern that he has created which is: #1 - a diamond, #2 - a triangle and 2 diamonds, #3 - a triangle and 4 diamonds, #4 - a triangle and 6 diamonds, #5 - a triangle and 8 diamonds. Induction
7.8 Leonard: "In the second one, I added this green thing (triangle). No this ain't right here. (Leonard takes away the diamond from #1 and replaces it with a triangle.)
7.9 The teacher asks why he did that.
7.10 Leonard: "Cause I changed it. . . cause if that (the triangle) was there (#1), then that would be adding on two." EHT Simple Deduction
8.0 Ira is busy building the triangle patterns. (3:49)
8.1 Ira says to the teacher without any prompting, "I know the next one will be seven." Ira is referring to the number of blocks that will be in the bottom row of triangle number four.
8.2 The teacher asks him why he thinks the number will be seven.
8.3 Ira: "Because I counted one more line onto this one." EHT Induction
8.4 The teacher asks Ira how many blocks he thinks will be in triangle number 4.
8.5 Ira has finished his fifth triangle and counts the blocks he used. He states that he needs 25. (4:00)
8.6 Ira realizes the similarity between the previous chart he just completed for the squares and the triangles and gets the teacher's attention.
8.7 Ira: "Hey! It's the same. Look, 25 for the squares and 25 for the triangles." Induction
9.0 Veronica and Cynthia work together to create the fourth triangle. (3:58)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-146
To Beginning of Section | To End of Section
Activity:
Students were given two sheets to do containing examples of "growing patterns." The two patterns that they were asked to make and observe using pattern blocks were the square and the triangle. For both patterns the first three shapes of each pattern were depicted, labelled as 1, 2 and 3. The students were then required to come up with the 4th shape in each pattern on their own and eventually make predictions about the number of pattern blocks needed to make the 5th shape in each pattern.
Two charts were given to be completed, one for each shape pattern which resembled the following:
|
Number |
Number of blocks in bottom row |
Number of blocks in square/triangle |
|
1 |
|
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
Upon completion of these two sheets, students were given the opportunity to create their own growing patterns. Many students did not reach this stage.
Students:
Notes:
Only the last half of this session was videotaped - sorry but it was a VERY OFF day for me. I've tried to make the best of what we do have from the audio and available video.
You may get a thing or two from this but there's not much here...
-------------------------------------------------------------------------------------------------------------
1.0 Seren has completed the first four squares of the pattern. The teacher asks her to predict how many blocks she would need to make the fifth square of the pattern.
1.1 Seren: " I don't know, I'll check."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-147
To Beginning of Section | To End of Section
1.2 Preston begins counting blocks and shouts out sixteen. (I assume from the audio tape that he is counting the number of blocks for square 4 and not square 5 like Seren.)
1.3 Seren: "Twenty!"
1.4 Preston: "No sixteen."
1.5 The teacher asks Seren why she thinks the answer will be twenty.
1.6 Seren: "Because I counted on. . . there will be sixteen on that one. . .there will be five on the bottom . . . EHT Simple Deduction
1.7 Preston and Seren both talk out loud trying to get the answer.
1.8 Preston: "First then there's 1, then there's 2, . . . then there's seven (blocks on the bottom row.)"
1.9 The teacher asks once again how many blocks would be in the fifth square.
1.10 Preston: "Five."
1.11 The teacher agrees that the fifth one will have five blocks on the bottom row and then asks again how many blocks in all will the square contain.
1.12 Preston counts on from 16 to 20 (I assume he is simply counting on an extra "level" to his fourth square as Seren did previously.): "On the fifth one there will be twenty."
1.13 Seren: "Because 5, and 5 and 5 and 5 is . . ." Seren continues to add up fives. She reaches a total of thirty.
1.14 The teacher asks Seren how many fives would be in the fifth square.
1.15 Seren: "Five rows of five." Seren begins to try to add up the fives again. She still is unable to come up with the answer. EWT Induction
1.16 Preston counts blocks by ones. (I assume he has created square 5 at this point.) "In the next one twenty-five. "
2.0 Shelley informs the teacher that she thinks square 5 will have 25.
2.1 Shelley tells the teacher without being asked for an explanation, "I counted like this, 5, 10, 15, 20, 25." EHN Empirical test
3.0 Joline has the first four squares completed. She is asked how many square pattern blocks will be used for the fifth square. She says thirty, to arrive at this answer she counts all the blocks she has used for the first four squares which totals 30. Joline is directed to build a fifth square to check her guess.
3.1 Joline shouts out "55." She must have created the fifth square that has a total of 25 blocks and then counted all of the blocks she used for the entire pattern.
Videotape available from here!
4.0 Shelley and Seth have both created the first three triangles and are busy working at the fourth triangle. (3:41)
4.1 Shelley tells the teacher that she is unable to make the fourth triangle. The teacher suggests that she start by beginning at the bottom instead of the top this time.
4.2 Shelley: " I can't start at the bottom. It's too hard and I don't know how to make it."
4.3 The teacher works with Shelley examining how many blocks were in the bottom row of each of the first three triangles that she created.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-148
To Beginning of Section | To End of Section
4.4 Shelley identifies that the number of blocks that are in the existing triangles first rows are 1, 3, and 5 and predicts that 7 will be in the next triangle. (3:43)
4.5 Shelley tries to make the fourth triangle. She makes the bottom row. The teacher then has her look at the number of blocks in each of the second rows (0, 1, 3). She predicts that the second row of the triangle 4 will have 5 blocks and begins to construct the second row. Induction
4.6 Shelley completes the fourth triangle. (3:45)
5.0 Seren: "I did it! Same as the last page." (3:43)
6.0 Seth shouts out excitedly, "Hey, I made it to the fourth one (triangle)!" (3:45)
6.1 The teacher asks Seth to explain how he figured out how to make triangle 4.
6.2
Seth: "It's kinda like the same thing as the square ones. Put four on the
bottom." He means that he used 4 triangles that are sitting flat on the
bottom row of the fourth triangle.
EWT Analogy, Induction
6.3 Seth turns over his sheet and looks at the information that he wrote in the first chart and copies it onto the chart for the triangles. (Note that he is sitting right across from Seren who made the observation that the two charts were the same.)
7.0 Preston is trying to make a growing pattern of his own. (3:47)
7.1 Preston experiments with several different shapes.
8.0 Seren is working on her own pattern. (3:47)
8.1 Seren has created the following: #1 - 1 square, # 2 - a 3x 2 rectangle created by using 6 squares.
8.2 The teacher asks Seren about her pattern - she asks how she is making it grow.
8.3 Seren: "1, 4 - - by twos, - - by threes." EHT Induction
8.4 Seren continues to work on her pattern.
9.0 Seth is busy making a pattern using the pattern blocks. He has created a design rather than repeating a specific shape to make it grow as the first two sheets guided him to do. (3:48)
9.1 Teacher: "You have to start off with a shape and make it get bigger."
9.2 Seth: "I did! " Seth explains that he started with a triangle first and then added on blocks to it to make it bigger. (He has a design created with a green triangle in the centre that is surrounded by a number of yellow trapezoids.)
9.3 The teacher provides some additional examples of shapes that are growing which she has drawn on a sheet.
9.4 Seth continues to add blocks onto his design.
10.0 Shelley has created a pattern using trapezoids. #1 - 1 hexagons, #2- four hexagons formed into a "square-like figure", #3 - nine hexagons formed in a "square-like figure". (3:54) The teacher asks Shelley how many hexagons she would need for the fourth shape in her pattern.
10.1 Shelley: "I'm just going to do three."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-149
To Beginning of Section | To End of Section
10.2 The teacher asks her to predict how many would be in the next one if she were going to make it.
10.3 Shelley: "Eleven."
10.4 Teacher: "Why do you think it will be eleven?" Induction
10.5 Shelley: "Cause you skip one, 9, 10, 11." EWT
10.6 The teacher asks Shelley if she skipped a number each time in creating the first three.
10.7 Shelley: "No."
10.8 Shelley looks at her pattern but she does not attempt to verify or check her guess.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-150
To Beginning of Section | To End of Section
The students were given a sheet of paper and a hole punch. They were asked to make a given number of folds in the paper and a specified number of holes and then asked to predict what the results would be when the paper was unfolded.
Students:
Notes:
The classroom teacher had exposed the students to this activity on the morning of this activity and thus, some students based their "guesses" on things that they remembered having seen that morning. As the activity progressed, the teacher involved began by having students make three folds in the paper that was not done by the classroom teacher.
In the notes, I have made reference to diagrams. These refer to the hard copies of the work that the students did which I have kept in a file. The sheets with the holes punched in them are also labelled from 1 - 6.
1.0 David is instructed to fold his paper (#1) twice and punch one hole in it. (3:22)
1.1 David states that he thinks there will be eight holes in the paper and draws where he believes they will be. The teacher asks him why he drew the holes in the position he did and he responds, "because I saw how `Miss' done it today." (Diagram 1-a) EWT Simple Deduction
1.2 He opens up the paper and discovers that there are only four holes in the paper. (3:23)
2.0 David refolds the same piece of paper and punches a second hole in it. (3:23)
2.1 The teacher asks David how many holes does he think will be in the paper now and he responds, "It'll be the same as that (he points to the diagram he just drew for the first guess). It's gonna be eight!" EWT Simple Deduction
3.0 The teacher begins to instruct David to refold the paper again and punch a third hole and David shouts out, "twelve". (3:24)
3.1 The teacher asks why he thinks the number of holes will be twelve.
3.2 David: "Because it's gonna add on four more." EWT Induction
3.3 David opens the paper to find that he indeed has created 12 holes.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-151
To Beginning of Section | To End of Section
4.0 David correctly predicts that when there are four holes punched in the paper, there will be sixteen holes but is unable to draw the result (see diagram 1-b). (3:24)
5.0 David is asked to fold a new piece of paper (#2) three times. He then punches 1 hole in the piece of paper and is asked to predict how many holes will be in the paper and draw his prediction (see diagram 2-a). (3:26)
5.1 David makes the prediction that there will be sixteen holes in the paper and offered the following explanation when asked, "because it's gonna be - - no eighteen! It's a guess."
5.2 David draws two vertical rows of nine on his paper as his prediction. (3:27)
5.3 David unfolds the paper and quickly counts to discover that he has created 8 holes.
6.0 David punches a second hole in his paper and makes the prediction that there will be 12 holes in his paper. (3:28)
6.1 Teacher: "Why do you say twelve?"
6.2 David: " - - Sixteen!!"
6.3 Teacher: "Why did you change your mind?"
6.4 David: "Because there was eight first and now it's gonna add on another eight so eight plus eight is sixteen." EWT Induction
6.5 The teacher asks David to draw his prediction. This time he draws two vertical rows of eight. (See diagram 2-b).
6.6 David opens up his sheet. The teacher asks him if he expected the pattern to be as it turned out. "No! It's x's."
6.7 David pauses for a while and then adds, " because that time I done it diagonally." EWX Simple Deduction
7.0 David refolds the paper and punches a third hole in his paper. (3:29)
7.1 David punches his third hole and says with a grin, "Now I knows what it's gonna look like!"
7.2 Teacher: "What is it going to look like?"
7.3 David: " . . .It's gonna be add on one (to each part of the x's)." EWT Induction
7.4 David draws two x's; each constructed using 9 holes.
7.5 The teacher asks David how many holes will be created and David correctly states 24 yet he does not alter his diagram to reflect this.
7.6 David opens up the paper and says, "24 see!" before he even counts them. Then he checks. (3:30)
8.0 Maurice is instructed to fold a piece of paper (#3) three times anyway he wishes to and punch one hole in it. (3:32)
8.1 Maurice is asked how many holes he thinks will be in the paper when it is unfolded and then asked why he thinks so.
8.2 Maurice: "Nine. Because three plus three is six. And six plus nine is - - (he shakes his head) six plus three is nine."
8.3 Teacher: "Why are you adding threes?"
8.4 Maurice: "Because I folded it three ways." (3:33) EWT analogy
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-152
To Beginning of Section | To End of Section
8.5 Maurice draws his prediction on a sheet of paper. (See diagram 3-a)
8.6 Maurice opens up the paper and discovers that he created eight holes.
9.0 Maurice refolds his paper and punches a second hole in it. (3:35)
9.1 The teacher asks Maurice how many holes does he think the paper will have in it now and he quickly states sixteen. The teacher asks him why.
9.2 Maurice: "Because eight plus eight is sixteen." EWT Simple Deduction
9.3 Maurice draws his prediction (diagram 3-b). He draws five vertical rows, all containing three with the exception of one that contains four. (3:36)
9.4 Maurice checks his prediction.
9.5 Maurice: "It's four groups of four."
10.0 Maurice refolds the paper and punches a third hole in it. (3:38)
10.1 Teacher: "How many holes will be in it this time?"
10.2 Maurice pauses for a while and then says eighteen. The teacher asks him why. Maurice pauses and rests his head on one hand and appears to be thinking. He is moving his fingers as if he is counting. Finally, he states 25.
10.3 Teacher: "What did you add up to get 25?"
10.4 Maurice: "I added up 8 and - - 16."
10.4 Teacher: "Where did you get the 16 to?"
10.5 Maurice smiles and then says, "how many were there before?" and begins to open up the paper. The teacher asks him to draw his guess before opening the paper. (3:40) EWT Induction
10.6 Maurice draws a diagram of the pattern created after punching two holes and remarks that there were 16 holes in it. He then adds 9 holes to the diagram by "guessing" where they would be. (Diagram 3-c)
10.7 Maurice opens up the paper and comments that there are rows of three.
11.0 Alicia is asked to fold a piece of paper (#4) three times. (3:42)
11.1 The teacher then asks Alicia to guess how many sections will she have created in the paper when she unfolds it.
11.2 Alicia makes a guess of 6 and then opens the paper to check. She discovers that she has created 8 sections in the paper.
11.3 Alicia refolds the paper and the teacher asks Alicia to punch 1 hole in the paper.
11.4 Teacher: "How many holes are going to be in here?" (3:45)
11.5 Alicia: "Eight."
11.6 Teacher: "Why do you say eight?"
11.7 Alicia: "Because it was eight blocks." EWT Induction
11.8 Alicia is asked to draw her prediction (diagram 4-a).
11.9 Alicia opens the paper to check her guess.
12.0 Alicia refolds the paper and punches a second hole in it. (3:46)
12.1 Teacher: "How many holes do you think are going to be in it now?"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-153
To Beginning of Section | To End of Section
12.2 Alicia pauses for a moment and then states, "Seventeen."
12.3 The teacher asks Alicia why she thinks the number of holes will be 17.
12.4 Alicia explains, ". . .I think that's what Ms. Hudson did today. She was doing kind of this experiment." EWT Simple Deduction
12.5 Teacher: "Are there any numbers we could use to figure it out?"
12.6 Alicia shakes her head and states "no."
12.7 Alicia draws a diagram of her prediction (diagram 4-b). It is a rectangular arrangement of circles, 3 wide by 4 high.
12.8 The teacher asks her about the number of holes she has drawn. Alicia realizes that she has only drawn 12 so alters her diagram. First she says it should be 4 more which would make sixteen and begins to add in another column of four but then she changes her mind and erases it. Then she says it should be fifteen, so she adds on a row of three.
12.9 Alicia checks her guess by unfolding her paper. (3:48)
12.10 The teacher asks her to count the holes. (There are actually sixteen). She counts and gets 15. The teacher asks her to double-check her answer. On the second count she gets 16.
12.11 Teacher: "I wonder which one is it?"
12.12 Alicia: "What's 8 plus 8?"
12.13 Teacher: "Would 8 plus 8 be an odd answer or an even answer?"
12.14 Alicia: "An even answer because there's eight on each side."
12.15 Teacher: "So would the answer be 15 or 16?"
12.16 Alicia: "Um - - (long pause!) 16." EWT Simple Deduction
13.0 Alicia refolds the paper and punches a third hole in the paper. (3:50)
13.1 The teacher asks Alicia how many holes will be in the paper now.
13.2 Alicia: "17. Not 17! The last time was 16. - - 21? "
13.3 Teacher: "Why do you say 21?"
13.4 Alicia: "Um, 22!"
13.5 Teacher: "Why do you say 22?"
13.6 Alicia: "I'm guessing."
13.7 Alicia opens up the paper and checks her guess.
14.0 Seren is asked to fold a piece of paper (#5) three times. Note: She folds the paper unevenly, creating two different "thicknesses". (3:51)
14.1 The teacher asks Seren to punch one hole in the paper and predict how many holes will be in the paper when she opens it up.
14.2 Seren: "Three."
14.3 Teacher: "Why do you say three holes?"
14.4 Seren: "Because I folded up the paper three times." EWT Simple Deduction
14.5 She opens the paper and discovers that she has created four holes. She wonders if she folded the paper four times by mistake and checks by refolding the paper.
15.0 Seren punches a second hole in the paper. This time Seren has punched a hole in the section of the paper that is "thinner". (3:53)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-154
To Beginning of Section | To End of Section
15.1 The teacher asks Seren how many holes does she think she will have now.
15.2 Seren responds by saying three.
15.3 Teacher: "Three all together?"
15.4 Seren: "Ah! Eight!"
15.5 Seren draws a diagram of her prediction. She has drawn seven holes. (3:54)
15.6 The teacher observes that she has drawn only seven holes (diagram 5-a) and questions Seren about it. Seren says she thinks there will be seven holes this time.
15.7 Seren opens up the paper to discover that she has created seven holes.
15.8 The teacher asks Seren why she guessed seven holes and she states that she doesn't know.
16.0 Seren refolds the paper and punches a third hole in the paper. This hole in punched in the "thinner" section once again. (3:54)
16.1 The teacher goes over in order the holes that Seren has punched questioning her about the number of holes each created. When the teacher gets to the third hole Seren says, "I think there's going to be 10 altogether."
16.2 Teacher: "Why do you think there's going to be ten?”
16.3 Seren: "Because I punched this one (hole 2) and it's sort of the same place as that one and they have the same amount of paper beneath them. And seven plus three equals ten." EWT Simple Deduction
16.4 Seren draws her prediction (diagram 5-b).
16.5 Seren checks her prediction by unfolding the paper.
17.0 Seren refolds the paper and punches a fourth hole in the paper. This time she places it on the "thick" side. (3:56)
17.1 Seren says, "It's going to be fourteen holes" without being asked by the teacher.
17.2 The teacher asks her why she has thinks the number of holes will be fourteen.
17.3 Seren: "Because before there (hole 1) there was four holes. Probably the other one will have four holes." EWT Induction
17.4 Seren unfolds the paper to see that she was correct.
18.0 Seren refolds the paper. The teacher asks Seren to punch a hole so that the paper will have 17 holes in it. (3:57)
18.1 Seren quickly punches a hole in the thin side and discovers that she is correct.
18.2 The teacher questions Seren why one side seems to make more holes than the other side.
18.3 Seren: " Because I folded it once more." EWT Simple Deduction
19.0 Seren folds another sheet of paper (#6) two times and punches 1 hole in it. (3:58)
19.1 She is asked to predict how many holes will be in the paper. After a brief pause she says four.
19.2 The teacher asks her why she thinks there will be four holes in the paper.
19.3 Seren explains that the paper is folded something like the last one (I think, I can't hear what she says clearly)).
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-155
To Beginning of Section | To End of Section
19.4 Seren opens the paper and discovers that she is correct.
19.5 Seren continues to punch holes and correctly predict how many will be created.
19.6 The teacher asks Seren if she could punch a hole so that the number of holes would not go up by fours.
19.7 Seren quickly states no and offers this explanation when asked, "I folded it all the same." EWT Simple Deduction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-156
To Beginning of Section | To End of Section
Activity: Students were given paper to fold which they punched holes into and then were asked to predict how many holes would be created and what the pattern of holes would look like.
Notes: This activity was recorded on audiotape only. There was another group on the same day being recorded at the same time by video only.
1.0 Ira is given a piece of paper and instructed to fold it three times. He folds the paper and then punches one hole in it. The teacher asks him how many holes will be in the paper and he responds by saying six. The teacher asks Ira why he thinks it will be six. He says, “because three plus three is six.” EWT Simple Deduction
1.1 The teacher asks Ira to draw the pattern he predicts will be on the paper when he unfolds it. He draws diagram I1-a.
1.2 Ira unfolds the paper and discovers that there are eight holes in his paper.
1.3 The teacher asks Ira to refold his paper and punch another hole. He is asked how many holes the paper will have when it is unfolded.
1.4 Ira: “Seventeen.”
1.5 Teacher: “Why do you say seventeen?”
1.6 Ira: “Because I done this in my math book.”
1.7 The teacher asks Ira what he means by, `he did it in his math book.’ Ira clarifies, with teacher questioning, that he added up eight plus eight before in his math book. He says eight plus eight is seventeen. EWT Simple Deduction
1.8 Ira is asked to draw a diagram of his prediction (diagram I1-b). He only draws seven holes on his sheet of paper.
1.9 Ira unfolds his paper to discover that there are now 16 holes.
1.10 The teacher tells Ira to look carefully at the pattern of holes that have been created before he refolds the paper. Ira refolds the paper and puts a third hole in the paper.
1.11 The teacher asks Ira how many holes are going to be in the paper. He does not answer so the teacher asks him to recall how many were in the paper after the first hole was punched but he is unable to remember.
1.12 Ira makes a guess of 27 holes.
1.13 The teacher asks him to draw what he thinks the pattern of holes will look like.
1.14 Ira draws a diagram (diagram I1-c) closely resembling the actual pattern created. It contains 24 holes (which is the actual number created by punching 3 holes.)
1.15 Ira unfolds the paper and discovers that he has correctly predicted how many holes were in the paper.
1.16 The teacher asks Ira if he has any idea how many holes would be created if he punched another hole.
1.17 Ira guesses 26 without giving an explanation why. Ira punches a fourth hole.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-157
To Beginning of Section | To End of Section
1.18 Ira then draws his diagram by simply adding on two more circles to each group of six in his previous diagram (diagram I1-c). By doing this he counts to find out that he has drawn 32 holes on his diagram.
1.19 Ira opens his paper and discovers that there are indeed 32 holes in the paper.
1.20 The teacher asks Ira why he added on two circles to each group of six from his previous diagram when he trying to predict the pattern for 4 punched holes.
1.21 Ira: “Because they’re adding up two more on every square.” EWT Induction
2.0 Kirsten is instructed to fold a piece of paper three times. She punches one hole in the paper and is asked by the teacher how many holes the paper will have in it when she unfolds it. Kirsten makes a prediction of ten. #K1
2.1 Kirsten says that ten is just a guess.
2.2 Kirsten unfolds the paper and quickly discovers that there are eight holes in the paper.
2.3 The teacher asks Kirsten to carefully examine the pattern of holes that were created in the paper because she will be asked to draw a diagram of the holes after she punches a new hole.
2.4 Kirsten refolds the paper and punches a second hole.
2.5 Kirsten guesses that there will be nine holes in the paper and draws a diagram (K1-a) of her guess. She draws a horizontal line containing thirteen circles yet the first pattern of holes that was created were scattered over the page.
2.6 The teacher questions Kirsten about the number of holes she has drawn. Kirsten says that she thinks the paper will have thirteen holes now.
2.7 Kirsten unfolds the paper and sees that there are sixteen holes created. The teacher asks her how many additional holes were created by punching the second hole but she does not give an answer.
3.0 Kirsten is given another sheet of paper and asked to fold it three times but differently than the last time. #K2
3.1 Kirsten punches a hole in the paper. The teacher asks her to predict how many holes would be in the paper when it is unfolded. Kirsten states six without offering an explanation.
3.2 The teacher asks Kirsten to recall how many holes were created by punching one hole in the first sheet of paper. Kirsten says, “16.”
3.3 Kirsten unfolds the paper and discovers that there are 6 holes in the paper.
3.4 Kirsten refolds the paper and punches a second hole in the paper. When asked to predict how many holes will be in the paper, she guesses 12. The teacher asks her why she says twelve.
3.5 Kirsten says there is no reason why she guessed twelve.
3.6 Kirsten unfolds the paper and discovers that there are 12 holes in the paper.
3.7 The teacher points out that first there were 6 holes, then there were 12 holes.
3.8 Kirsten refolds the paper and punches a third hole in the paper.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-158
To Beginning of Section | To End of Section
3.9 Kirsten predicts that there will be 19 holes in the paper when asked to make a prediction.
3.10 Kirsten shrugs her shoulders when questioned why she said 19.
3.11 Kirsten unfolds the paper and counts 14 holes (There are actually.
3.12 The teacher asks Kirsten how many extra holes were created. She says three.
3.13 Teacher: “Why do you say three? How many were in there when you just punched two?”
3.14 Kirsten: “Three.”
4.0 Preston is asked to fold a piece of paper three times. (#P1) As he makes each fold, he counts the layers of paper he is creating.
4.1 The teacher asks Preston to punch one hole in his paper. He thinks “it might” have six holes in it since he thinks there are six layers of paper. EWT Simple Deduction
4.2 Preston comments on how difficult it is to punch a hole in the paper, “I think the more you fold, the harder it is to punch it.” EWP Induction
4.3 The teacher asks Preston if he can draw a diagram of how he thinks the pattern of holes will look when he unfolds the paper.
4.4 Preston: “It’s pretty hard!” Preston draws diagram P1-a. “That might be it but I’m not sure.”
4.5 Preston unfolds the paper and discovers 8 holes. He immediately states, “It’s kinda hard - - let me see how many times I folded it.” Preston counts the sections created by the folds and concludes, “I folded it eight times.”
4.6 The teacher asks Preston to refold the paper and check how many times it was folded. With teacher assistance, the paper is refolded. It is indeed folded three times. Preston comments, “Oh yeah.”
4.7 Preston is instructed to punch another hole. Without being asked, Preston states, “So if I do this one, I’ll have sixteen holes.”
4.8 Teacher: “What made you come up with that guess?”
4.9 Preston: “I’m not sure. I just guessed.”
4.10 The teacher comments that she heard Preston counting some numbers before he made his guess. She asks him what he was doing.
4.11 Preston: “I counted on.”
4.12 Teacher: “How many more did you count on?”
4.13 Preston: “Eight”
4.14 Teacher: “Why did you count on eight more?”
4.15 Preston: “There might end up being eight more holes where I’m - - it’s kinda hard.” EWT Simple Deduction
4.16 Preston draws diagram P1-b. Preston unfolds his paper and discovers that there are sixteen holes in the paper.
4.17 Preston refolds the paper and punches a third hole in the paper. “I think there’ll be eight more along here.” Preston adds another vertical row of eight to diagram P1-b.
4.18 Preston opens up his paper after he punches the third hole. He asks, “How did that happen?”
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-159
To Beginning of Section | To End of Section
4.19 Teacher: “How did what happen?”
4.20 Preston: “ I think I know what’s happening. It’s going in all the spaces. See where I’m going like that . . .he’s going up the side of the paper. But if I go like - -.” Preston tries to punch a hole further away from the edge of the paper. EWT Induction
4.21 The teacher asks where the holes will be.
4.22 Preston: “Up in the middle.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-160
To Beginning of Section | To End of Section
Activity: Paper Folding Activity
Notes:
On this day, one group was being videotaped (this group), while another tape was being audiotaped. Thus, there is no audio available for this activity.
Also, while the transcripts were being typed, the work samples of the students were unavailable, thus there is no reference to any specific work samples.
1.0 Leonard is instructed to fold a piece of paper three times and to punch two holes in it when it has been folded. (3:18)
1.1 Leonard is asked if he can draw the pattern of holes he expects to see when he opens up the paper.
1.2 Leonard states that he needs to open up the paper.
1.3 The teacher asks him to make a quick guess before he opens up the paper.
1.4 Leonard makes a guess of seven and unfolds the paper twice to see eight holes. He asks if he can make another guess. He looks at the paper and says, "Eight plus eight is sixteen. 16!" EWX Simple Deduction
1.5 Leonard unfolds the paper completely and shows the teacher that it does have 16 holes.
1.6 Leonard is instructed to refold the paper and punch another hole in it. (3:21)
1.7 Leonard is asked to make a chart of the holes, so he unfolds the paper before putting a third hole in it and draws the pattern created by punching two holes onto a sheet of paper.
1.8 Leonard refolds the paper. As he is trying to punch a third hole in the paper, he states, "It's going to be 16 + 8 - - no, 16 + 16."
1.9 Leonard indicates that the answer will be thirty-two. The teacher asks him how he got thirty-two. "Cause 15 each, and that makes thirty. There's one more, so add two. So I guess it's thirty-two."
1.10 After Leonard gets one hole punched into the paper, he looks at it and says that it will be eight more. He asks the teacher, "what's 8 + 16?" Leonard counts on from 16 to arrive at the answer of 24. He states that there will be 24 holes in the paper. EWT Induction
1.11 Leonard unfolds the paper - there are twenty-four holes in it. Leonard looks at the paper and says, "It's the same thing! Look!"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-161
To Beginning of Section | To End of Section
1.12 The teacher asks him to count the holes. Leonard counts the holes. He begins by counting groups of sixes, "6, 12 " then continues by counting the remaining holes by ones. He discovers that there are 24 holes. He raises his arms in the air triumphantly and says, "I did it!" (3:24)
1.13 The teacher asks Leonard to punch another hole. He quickly states, "It's going to be 32!"
1.14 The teacher asks Leonard why he says that the answer will be 32.
1.15 Leonard: "Cause I remember saying it from the last one." Leonard then proceeds to punch a fourth hole in the paper. (3:24) EWT Induction
1.16 Leonard unfolds the paper, one fold at a time, noting how many holes he can see. " Eight holes. (He unfolds one more time.) There's eight plus eight, sixteen. (He finishes unfolding the paper.) I think it's 32." EWT Induction
1.17 The teacher asks Leonard if he is sure so he counts the holes by ones. EWX
1.18 Leonard is asked to refold the paper and punch a fifth hole in it. Leonard responds by saying, "One more hole and that's it!"
1.19 Leonard states that there will be forty holes in the paper. Then he says, "let's see if I'm right." When he opens up the paper, he notices that the holes are in groups of tens. "10 + 10 + 10 + 10. Four tens." He sits back in his chair with his arms crossed with a smile on his face. Induction
2.0 Veronica is instructed to fold a piece of paper two times. The paper comes unfolded and she has difficulty refolding it so she starts a new one. (3:28)
2.1 Veronica punches a hole in the paper. She is asked to guess how many holes are in the paper. She says four. She then draws a diagram on a sheet of paper to show where she thinks the holes will be. (Note: I'm not sure if she completes her diagram or not - it is difficult to tell from the tape.)
2.2 Veronica unfolds the paper and sees that it has four holes in it. She refolds the paper and punches a second hole.
2.3 The teacher asks her how many holes the paper will have when it is unfolded. Veronica says eight.
2.4 Veronica opens the paper and counts the holes. She counts seven. The teacher asks her why she thinks she got seven and not eight. She examines the paper and says, "There is eight! This one never got out." She removes a circle from the paper that was not completely punched.
2.5 Veronica tries to refold the paper. She notices that the "outside" piece has three holes in it. The teacher tells her that she didn't fold the paper up the same way but that is okay. She punches another hole in the paper.
2.6 The teacher asks her how many holes she thinks will be in the paper. Veronica says 13. She explains that she got thirteen by guessing.
2.7 Veronica tries to fold her paper up and once again has difficulty.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-162
To Beginning of Section | To End of Section
3.0 The teacher folds a piece of paper three times for Veronica. (3:34)
3.1 A hole is punched in the paper and Veronica guesses that there will be five holes in it. She draws a diagram of where she thinks the holes will be.
3.2 Veronica unfolds the paper and discovers that there are eight holes in the paper. The teacher refolds the paper for her. She punches a second hole in the paper. (3:36)
3.3 Veronica is asked how many holes will be in the paper. She does not respond so the teacher questions her about how many holes were in the paper after the first hole. At first she says five. The teacher informs her that five was what she predicted would be in the paper. Veronica then remembers that there were eight holes in the paper.
3.4 Veronica is then asked again how many holes she feels will be in the paper now. This time she pauses for a moment then responds by saying 16.
3.5 The teacher asks her how she got 16. Veronica says, "(Note: It is difficult to know exactly what she says here.) I can make four again."
3.6 Veronica opens up the paper and counts 15 holes in it.
3.7 The teacher takes the sheet of paper and asks Veronica how many holes are on each side. She says eight. The teacher asks her what 8 + 8 is. After a moment, she says 16.
3.8 The teacher refolds the paper and has Veronica punch a third hole in the paper. (3:38)
3.9 Before Veronica punches the hole she says that there will be twenty holes this time "cause there is now more." EWT Induction
3.10 Veronica opens the paper.
4.0 Jared is asked to fold a piece of two times and punch a hole in it. (3:41)
4.1 The teacher asks Jared to draw a picture of what he thinks the paper will look like when he unfolds it.
4.2 Jared unfolds the paper and discovers it has four holes. He quickly refolds the paper and punches a second hole in it. He says that there will be eight holes in the paper when he is asked by the teacher.
4.3 Jared opens the paper and discovers that there are eight holes in the paper. (3:43)
4.4 Jared punches a third hole in the paper and is asked by the teacher how many holes will be in the paper. He says 13 and draws his guess on a sheet of paper.
4.5 Jared opens the paper and finds that there are twelve holes.
4.6 Jared refolds the paper. The teacher asks, "If you punch another hole, how many will there be?" (3:45)
4.7 After thinking for a moment, Jared says there will be 16 holes. He adds circles to his drawing and then punches the fourth hole.
4.8 Jared opens the paper and says, "I'm right."
5.0 Jared folds a new piece of paper three times. The paper is doubled with each fold. (3:46)
5.1 Jared punches a hole in the paper and then draws a picture of how he thinks the paper will look.
5.2 The teacher asks Jared why he has drawn his diagram the way he has. Jared explains, "This time it started off with four and I added four more on and it equalled eight." EWT Induction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-163
To Beginning of Section | To End of Section
5.3 The teacher asks Jared if adding on an extra fold means he adds on four more. Jared says yes but is unable to offer an explanation when asked why.
5.4 Jared unfolds the paper. He quickly says, "I'm right!" (3:47)
5.5 Jared refolds the paper and punches a second hole in his paper. He adds to his diagram the number of holes he thinks will be on the piece of paper.
5.6 Jared unfolds the paper and counts sixteen holes. He labels the holes. (3:50)
5.7 Jared refolds the paper and punches a third hole in it. He adds to his diagram to make a total of 24 holes.
5.8 The teacher moves away for a moment so Jared quickly opens up the paper. He does not get a chance to count the holes before the teacher returns. When the teacher returns he counts the holes drawn on his diagram then the sheet.
5.9 Jared refolds the paper and adds onto his diagram to figure out how many holes will be in the paper after he punches a fourth hole in it.
5.10 Jared says there will be 28 holes after drawing his diagram.
5.11 The teacher notes that he thinks four more will be added on to the number of holes. Jared says yes, four more will be added on.
5.12 Jared punches the fourth hole and unfolds the paper. He counts 33 holes.
5.13 Jared labels the new holes.
5.14 Jared realizes that there are 32 holes in the paper. (3:57)
5.15 Jared looks back at his diagram and questions, "How many is here?" He discovers that he had 27 circles drawn.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-164
To Beginning of Section | To End of Section
Activity: Geoboards. The students were asked to create squares on their geoboards and then asked questions about area related to the squares. Later some students were asked to figure out the area in diamonds.
Notes: I found it very difficult to describe the events of 4.0 - 4.9 for several reasons. You may want to look at it again with the geoboard next to you as a visual aid. - - I kinda' got lost!
1.0 David is asked to make a square on his geoboard. He quickly makes one, using the entire geoboard. (3:26)
1.1 The teacher asks David how many little squares are inside the big square he has created. He immediately starts figuring it out by making little squares within the large square.
1.2 After he has 6 created, the teacher asks him how many will there be. David quickly counts on the geoboard to get 16.
1.3 The teacher asks David if he could construct a square that would have 9 squares in it.
1.4 David counts the blocks from the top of the big square until he gets nine: "No! You can't do that!"
1.5 The teacher asks David to make a smaller square and count how many small squares are in it.
1.6 David quickly makes another square and says, "We made a square with nine in it!" David counts the pegs (which total nine.)
1.7 The teacher points out that there are only four little squares in his square and says that he would like to have a square made with 9 little squares in it. (3:28)
1.8 David says, "I can't do it. - - the only thing I can do is a thing like that." David constructs a 2x4 rectangle with a small square joined onto it. David counts the nine small squares contained in the figure. "It's a rectangle with a little square." EWT Simple Deduction
2.0 The teacher asks David if he can make the rectangle look more like a square. (3:29)
2.1 Teacher: "So we can make a square with 16 little squares and four little squares but there's nothing in between?"
2.2 David: "Nope!"
2.3 The teacher creates a 3 x 3 square and asks David to count the number of small squares in it. David counts nine and says, "he done it! He's a genius!"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-165
To Beginning of Section | To End of Section
3.0 The teacher makes a shape on the geoboard and asks David what it is. David immediately says it's a diamond. The teacher turns the diamond and asks if it is a square now and David agrees, "yeah, but it's on an angle." (3:30)
3.1 The teacher asks David to make a big diamond. David makes a diamond on the geoboard.
3.2 The teacher asks David how many squares are in the diamond. David starts to place an elastic band within the diamond. He creates a triangle at the top of the diamond. "I don't know. That's (the triangle) that big. It's half of a diamond." EWT Simple Deduction
3.3 David is guided to make a small square "around" the triangle. The teacher asks if the triangle he has made is half of a square and David nods in agreement.
3.4 The teacher makes another triangle in the adjacent section of the diamond. David notes that it is the other half of a square. The teacher asks what they would be if they were put together and David says it would be a triangle.
3.5 The teacher makes a small square and asks if it is bigger or smaller than the triangle created with the two half squares.
3.6 David says it is the same size. When asked why, David explains, "That and that (He places an elastic band over each small triangle as he speaks.) - - it's the same size." EWT Simple Deduction
3.7 The teacher asks David how many small squares are in the diamond if he adds up all the small squares and small triangles (half squares). (3:34)
3.8 David says one. The teacher asks, "in the whole big diamond?"
3.9 David begins to use small elastic bands to create small triangles and small squares. When he is finished, he says eight. The teacher asks him how many small squares, he says four - he counts the triangles two at a time as one.
3.10 The teacher asks how many would fit in if the small squares were cut in half. David quickly says eight.
4.0 The teacher makes another figure on the geoboard and asks David if it is a square or a diamond. (3:36)
4.1 David: "That's a diamond on an angle - - that's a square on an angle."
4.2 The teacher asks how big the shape is. David responds, "that's probably the same size as the (previous) square." (Note: It is the same size.) EWT Induction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-166
To Beginning of Section | To End of Section
4.3 The teacher asks how many little squares are in the square.
4.4 David sizes up the square and says one. When asked if there are any more, he says no.
4.5 The teacher makes a shape on the geoboard within the square. (Note: I am not sure from the tape but I believe a triangle is created.)
4.6 David makes several triangles within the square. He then becomes frustrated and states, "I can't do it." (3:39)
4.7 The teacher asks David if he would like to get some help from a friend. He goes to get Saul. Saul comes over and David asks him if he can create another triangle like the one created by the teacher. (3:41)
5.0 The teacher points out a diamond and asks Saul how many little squares could fit into it. (3:49)
5.1 Saul examines the figure and says, "it would be triangles."
5.2 The teacher agrees. Then Saul questions, "how many squares?"
5.3 The teacher points out that he could put triangles together to form squares.
5.4 Saul: "It's easy to put a square into a triangle. All you have to do is do a square. . ." Saul goes on to describe that he could cut it by using elastic bands.
6.0 The teacher asks Saul to invite someone over to show him or her what he has been doing. Saul goes and gets Maurice. (3:52)
6.1 Saul: "Three things. Can you fit a triangle in here? (Saul points to a small square.)"
6.2 Maurice starts to manipulate the elastic band to make a triangle and Saul reassures him that there's a way to do it.
6.3 Maurice makes a triangle in the small square.
6.4 Saul directs Maurice to make a second triangle within the small square. Maurice does so easily.
6.5 Then Saul asks Maurice if he can fit a square into a larger triangle that is on the geoboard.
6.6 The teacher tells Maurice that he may need to cut it up. (3:53)
6.7 Maurice makes a square but the teacher points out that he cut parts of it off.
6.8 Maurice continues to fit a square within the triangle.
6.9 Saul asks Maurice to figure out how many squares can be fit into the diamond.
6.10 Maurice places elastic bands on the geoboard but is unable to figure it out.
6.11 The teacher removes the elastics from the geoboard.
7.0 The teacher asks Maurice if he can make a square bigger than the unit square left on the geoboard. (3:56)
7.1 Maurice quickly makes a 4x4 square. The teacher asks how many little squares can fit into his big square.
7.2 Maurice uses elastic bands to start creating small squares within the big square. Maurice figures out that there are 16 small squares in the large square. (3:59)
7.3 Maurice makes a 3 x 3 square and quickly counts that there are nine squares in it.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-167
To Beginning of Section | To End of Section
Activity: Working with geoboards.
Notes:
The video tape shuts off with a minute or two left.
There does not appear to be too much of interest on this tape...I could be wrong of course!
1.0 Ira is asked to make a square on the geoboard. He quickly makes a 1x1 square. (3:25)
1.1 Ira is asked to make a square bigger than the first on the geoboard. He creates a 4 x 4 square.
1.2 The teacher asks him how he knew how far to stretch the elastic band to make a square.
1.3 Ira: "Because there's four (corners) and two on each side." EWT Simple Deduction
1.4 The teacher makes a 3 x 4 rectangle on the geoboard and asks if it is a square. Ira nods yes.
1.5 The rectangle is altered to create a trapezoid. Ira is asked if it is a square. He quickly nods no. When asked why he explains, "Cause that got a curvy line going up. Like a hill. " EWT Simple Deduction
1.6 The teacher asks Ira how he would describe a square.
1.7 Ira: "It's like a wrestling ring."
1.8 The teacher prompts for further description. Ira says, "A house can (be like a square)." he continues to describe a square by comparing it to other objects.
EWT analogy
2.0 A 3 x 3 square is created on the geoboard. Ira is asked to figure out how many small squares could be fit into the large square. He guesses two, then is instructed to find out by using the elastic bands. (3:28)
2.1 He figures out that the square has nine squares.
2.2 The teacher asks him why he said two at first. Ira points to the geoboard and explains, "Cause there's two here, two here, two here, two here and one left."
EWT Simple Deduction
2.3 The teacher asks is Ira can make any other squares inside his large square.
2.4 Ira does not make another square.
3.0 The teacher moves the elastic on the 3 x 3 square to create a triangle half the size. (3:31)
3.1 The teacher asks how many little squares could fit into the triangle. He is told that he can cut up some squares.
3.2 Ira: "But I can only fit three squares." (3:31) EWT Simple Deduction
3.3 The teacher tells Ira that he can cut up squares to fill up the remaining spaces that are left after he has placed three squares.
3.4 Ira indicates that he could use three more squares. It is actually three half squares he needs.
4.0 The teacher makes a diamond that is two squares big. The teacher asks Ira how many squares could be placed in the diamond.
4.1 Ira indicates that there is only one “hole” on the geoboard for the elastic bands.
4.2 The teacher asks Ira to think about how many squares he could use if he could use parts of squares.
4.3 Ira makes a square, one half that is in the diamond. When asked how much of the square is inside the diamond and how much is outside the diamond, he correctly identifies that there is one half of a square both in and out of the diamond.
4.4 Ira is asked if he can make some more squares “around” the diamond like the one he has already made. (3:34)
4.5 Ira makes a second square that intersects the diamond. When questioned about how many squares are in the diamond, he states, “two halves.”
4.6 Ira quickly makes the third and fourth halves in the diamond. “Four halves and I can’t make anymore,” Ira says without prompting. EWN Simple Deduction
4.7 The teacher asks Ira how many squares he could make with the four halves.
4.8 Ira shrugs his shoulders.
4.9 The teacher points out the 1x1 square that is divided into two halves and asks how many halves are in it.
4.10 Ira says that there is only one half in the square, “cause it’s only one triangle (he says this as he pulls the elastic which divides the square).” EWT Simple Deduction
4.11 The teacher gives Ira one elastic and asks him to make half a square. Ira makes a 1x1 square and then gets another elastic to divide it.
4.12 The teacher asks how many halves are in the square he just created. Ira says “two.” The teacher then asks how many halves are the first 1x1 square. Ira says “one.”
5.0 Kyla is asked to make a square. (3:38)
5.1 Kyla makes a 4x4 square. The teacher asks Kyla what makes the figure she has created a square.
5.2 Kyla: “Squares have four (she points to the corners).” EWT Simple Deduction
5.3 The teacher moves one of the four points so the figure is no longer a square and questions Kyla about the shape.
5.4 Kyla says that the figure is not a square, “cause it goes up like that (she points to the slanted side of the figure).” EWT Simple Deduction
5.5 The teacher moves the elastic to create a 3x4 rectangle and asks if the new figure is a square. Kyla says “yes” it is.
5.6 Teacher: “Is it as good as a square as the one you made?”
5.7 Kyla shakes her head to indicate “no.” She is unable to explain why.
6.0 The teacher asks Kyla if she can construct a smaller square within the 4x4 square. She quickly places a 2x2 square in the centre of the 4x4 square. (3:40)
6.1 The teacher asks Kyla if she can make a smaller square within the 2x2 square. She quickly shakes her head “no.”
6.2 The teacher asks Kyla why she thinks that no more squares can be made within the 2x2 square.
6.3 Kyla: “There’s only one peg.”
6.4 The teacher asks could she do it if she reused some of the pegs she has already used.
6.5 After looking at the square for a while, Kyla states, ‘you can’t make a smaller square.” EWT Simple Deduction
6.6 Kyla shows that she is unable to make a square by placing an elastic band within the 2x2 square to create a 1x2 rectangle. “I don’t think that’s a square.”
6.7 The teacher asks Kyla if she can shrink the shape she has made. Kyla moves the elastic so that a line is created.
6.8 The teacher asks her if she can shrink the square so that it is “more square.”
6.9 She is unable to do it.
7.0 The teacher makes a unit square in the corner of the geoboard. Kyla agrees that it is a square. (3:43)
7.1 The teacher asks Kyla how many of the little squares could fit into the 2x2 square. She quickly responds, “four.” The teacher then lets Kyla “try it out.”
7.2 The teacher then asks Kyla to figure out how many little squares would fit into the 4x4 square. Kyla quickly states “four” once again.
7.3 Kyla makes a little square in the corner of the big square then says that perhaps 6 little squares could be placed into the bigger square.
7.4 Kyla creates 1x1 squares to find out how many squares fit into the 4x4 square.
7.5 When Kyla has made four 1x1 squares across the top of the 4x4 square, the teacher asks her how many “of those fours could you make?” (3:46)
7.6 Kyla: “Six.”
7.7 Kyla continues to make more squares.
7.8 Kyla covers in the square with 1x1 squares. The teacher asks her how many squares the large square has and Kyla says nine without looking at the geoboard.
7.9 The teacher instructs Kyla to count the squares she has created. She counts to discover that she has made 16 squares.
8.0 The teacher makes a diamond that Kyla quickly identifies. (3:49)
8.1 The teacher asks Kyla how many little squares could be placed inside the diamond.
8.2 Kyla guesses four. The teacher asks her if she would like to try. She begins and asks the teacher if she should place the squares outside the diamond.
8.3 Kyla quickly places four unit squares within the diamond. There are four corners that are not “covered”.
8.4 The teacher makes a square that lies both inside and outside of the diamond. Kyla correctly identifies that one half lies within and one half lies outside.
8.5 Kyla agrees that she could make other half squares in the diamond. The teacher asks her how many could be made and tells her that she can try it out. (3:52)
8.6 Kyla correctly states that eight half squares were made.
8.7 The teacher asks Kyla how many whole squares could be made if she put together the half squares.
8.8 Kyla shrugs her shoulders.
9.0 The teacher gets another geoboard. The teacher makes a half of a square that Kyla agrees is half. The teacher then makes a second half to create a unit square. The teacher asks Kyla how many of the small squares could be made with the eight halves she has. (3:54)
9.1 Kyla constructs unit squares by using two elastic bands for each. Each elastic band makes one half of the square. Kyla has created 6 squares.
9.2 The teacher asks Kyla how many half squares she has. Kyla does not look at her geoboard. She stares up in the air and then responds, “six.” She makes another half square and then says, “and one half one.” (3:57)
9.3 After Kyla has nine squares made. The teacher asks how many halves does she have made.
9.4 Kyla counts 18.
9.5 The teacher shows Kyla the original diamond that she was working on and asks how many half squares she used to completely cover in the figure.
9.6 Kyla does not respond so the teacher reminds her that she needed eight half squares. The teacher then asks her to figure out how many squares eight half squares would make. Kyla is unsure of how to find out the answer.
9.7 The teacher tells Kyla that she could use her geoboard which has the squares created using two elastic bands (two halves) to figure it out. (3:59)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-168
To Beginning of Section | To End of Section