This
section is divided into two parts discussing the results of the research in
terms of the two research questions directly addressed by the current study.
As was noted in Section I, this research reported here provides
background for future efforts to address Question 1 of the PRISM project (In
what ways can the mathematical reasoning of primary school students and the
needs associated with their reasoning differ from those of secondary school
students?) but this question is not directly addressed here.
To what
extent do the models of needs to reason developed in previous research
describe students’ reasoning at various school levels, and how might they be
improved?
Recall
that the model described in Reid (1995) includes four categories of
descriptors for reasoning: The need motivating the reasoning, the type
of reasoning, the degree of formulation of the reasoning, and the
degree of formality in the communication of the reasoning.
The students' reasoning observed in the context of games at the grade 2
level led to modifications in the way the model describes two types of
reasoning: deductive reasoning and reasoning by analogy.
Attempting to make fine distinctions in the reasoning of young children is very difficult. They usually leave much of their thinking unarticulated and
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make connections that are not easily
understood by adults who are constrained by a history of developing standards
of reasoning appropriate to their communities of discourse.
Even so it is possible to observe episodes of deductive reasoning in
the mathematical activity of young children, and to detect differences between
sub-types of deductive reasoning. Describing
precisely what those differences are, however, is more difficult.
The following sub-types of deductive reasoning were observed: Simple
one step deductive reasoning, Simple multi-step deductive reasoning,
Specialisation, Specialisation with multiple attributes, Hypothetical one step
deductive reasoning, and Hypothetical multi-step deductive reasoning. They are
tentative and will have to be confirmed by additional analysis and comparison
with the reasoning of students of other ages.
February
3, Mastermind game, MAT 4.3
|
1 |
bl |
o |
y |
g |
3w |
|
2 |
g |
br |
y |
o |
2w |
After she gave Maurice the two white pegs for his second guess the teacher asked him if he knew anything new. He said "It's blue. Cause if there's three [white pegs] there [in guess one]. I changed the blue and I only got two." His reasoning is simple in the sense of not involving any assumption of a hypothesis. Counting the number of steps in a deduction is impossible in practice as the number of steps depends on how much information is combined in each step and how many assumptions are allowed to remain
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implicit.
Maurice articulates one step here, which is sufficient for his argument
to be understood. That is the
criterion used to classify it here.
The
subjectivity of this classification is clear when an effort is made to analyse
the structure of Maurice's deduction in more detail. There is more than one possible chain that links what he
knows to his conclusion. One
involves a consideration of four cases:
1.
The three correct colours in guess one are:
o, y, g
2.
The three correct colours in guess one are:
bl, y, g
3.
The three correct colours in guess one are:
bl, o, g
4.
The three correct colours in guess one are:
bl, o, y
Three
of these cases are compatible with the result of guess two: 2, 3, and 4.
All of these include blue, so blue must be correct.
This chain fits the description of "simple" given above; it
does not involve making a hypothesis. It
does stretch what counts as a single step deduction however, seeming to
involve the testing of four cases and then drawing a conclusion from a common
feature of three of them. Another
chain begins by assuming that blue is not correct, and then determining that
there should be three white pegs awarded to guess two (because the other three
colours must be correct). Because
this is not the case, the hypothesis (that blue is not correct) must be
rejected, leading to the conclusion that blue is correct.
Although this seems to be a more compact argument (fitting the
description "one step" better than the consideration of cases), it
isn't simple, because it involves a reducio
ad absurdum from a hypothesis.
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February
10, Tic Tac Drop, MAT 7.3
Needed
to win: 3 in a row
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4 |
1 |
3 |
5 |
2 |
Immediately
after placing his first piece, Maurice said "If he don't put it next to
me, I won." While Maurice was an experienced Connect Four player, and
seems to have had a set of standard strategies for Connect Four, this
situation was unusual in that the winning condition was now three in a row
instead of four in a row. We
cannot know how Maurice came to his conclusion that he would win if the
computer didn't place a piece next to Maurice's but it seems plausible that he
deduced this new general rule either from the strategies he already knew from
playing Connect Four (one of which involves making a line of three counters
with free ends) or from analysing the new situation in terms of general
features (e.g., two in a row with free ends).
January
22, Connect Four, MAT 12.3
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10 |
14 |
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9
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13 |
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16 |
3 |
5 |
8 |
12 |
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11 |
1 |
2 |
7 |
6 |
After
Ira played move 16, Maurice put his arms behind his head and said, "He
got me." The teacher asked
why and Maurice whispered to her the two possible ways that Ira could win
(playing to the left of move 11 or above move 14).
Maurice claimed earlier (Jan. 22 MAT 11.2) that he was good at Connect
Four because he played at home, and here he seems to be using a general rule
to guide his play. He stated his
rule later: "get three either way" (Jan. 22 MAT 21.1) and his
application of this rule to Ira's situation amounts to a very simple type of
deduction, specialisation, which is the use of a general rule to determine
something about a specific instance of the situation the general rule
describes. Here the general rule
could be written "If you have a row of three counters with both ends
free, then you can win" and the specialisation is "Ira has a row of
three counters (moves 16, 4 and 10) with free ends, so Ira can win.
The game of Set involves determining if three cards satisfy the conditions that define a "Set." In the standard game those conditions are that the three cards be identical, or all different, in four attributes: colour, number, shading, and shape. (See Appendix A for details). On January 11 the children played a variation in which the striped shading was removed from
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the deck, and the open shading was treated as a fourth colour: white.
The teacher defined a Set as three cards that were different colours,
shapes, and numbers.
After
playing for a while, the teacher asked the children to explain why 1 red oval,
3 green diamonds and 2 white squiggles are a Set (MAT 12).
Cynthia replied, "They're all...different. They're all different
shapes. They're all different colours."
Alison added, "They're all different numbers." And Jared
summed up, "Different colour, number and shape."
Their reasoning here is a specialisation from the general rule
expressed in Jared's summary, but this rule is more complex than Maurice's
rule in the previous example of specialising, as it involves evaluating three
separate conditions.
On
January 27 the teacher led a discussion in a small group of the story The
Doorbell Rang, by Pat Hutchins. As
the story progresses increasing numbers of children arrive to share twelve
cookies. When the number had
increased to twelve children, Preston explained that each would get one cookie
by saying, "If you took two then you'd have to take one away from one
person and then it wouldn't be fair because six people wouldn't have cookies.
And six people would have two cookies. So it has to be one."
His reasoning involves hypothesising an alternative (that each would
get two cookies) and then showing that would result in a contradiction (not
everyone could have a cookie). He
implicitly assumes that the only possible answers are one and two, which means
that by showing the answer can not be two he has shown it must be one.
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February
3, Mastermind, MAT 2.7-2.10
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1 |
y |
o |
bl |
g |
1w , 1bk |
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2 |
br |
g |
bl |
o |
2w |
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3 |
bl |
o |
y |
g |
2w |
After
giving Kyla two white pegs for her third guess, the teacher asked her which
one she thinks might have been in the right place. Kyla pointed to the blue peg in the first row and then changed her mind.
"I never got a black one right there. (Pointing to the blue in the second
turn.)" She then indicated that the green can not be correct either in
the first try. "Cause on this one (turn three) I didn't get a
black." After that
Kyla stated that the orange one on turn one must be in the correct spot
but then realised it can not be. "Cause I got a black one right here --
no! Oh my! It's yellow." Kyla's
reasoning includes three hypotheses: That the Blue peg is in position three,
that the Green peg is in position four, and that the Orange peg is in position
two. Having arrived at
contractions from each of these hypotheses in turn she concludes that the one
remaining case (the Yellow peg in position one) must be correct.
On January 25th when discussing the story The Doorbell Rang the teacher asked the students to figure out how many scoops of ice cream each group member would get if she brought in ten scoops. (5 people, 10 scoops) (January 25 MAT 11). After the teacher restated the problem once Alicia stated that each person would get 2 each because "6 plus 6 is the same as 5
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plus 5." In other words, she made a connection to the underlying
structure of a similar situation in the story, in which 12 cookies had to be
divided among 6 children. In this
analogy the surface features are difference (10 vs. 12; 5 vs. 6) but the
relationships are the same.
In
contrast, most of the other analogies the children made were based on surface
features. Alicia, shortly before
the episode quoted above, made the following analogy (January
25 MAT 9.3): "It's like they're counting by twos."
The teacher had asked if they had noticed anything about the numbers
she was recording on a chart as the story progressed, specifically the number
of people:
|
Number of people |
Number of cookies each person got |
|
2 |
6 |
|
4 |
3 |
|
6 |
2 |
Alicia's
analogy depends on the surface features of two situations being identical (the
2,4,6 on the chart and the 2,4,6 of counting by twos.)
The underlying structures however are quite different.
In the story the number of people could have increased by any amount
(in fact the next time the doorbell rings, it jumps to 12 people).
In counting by twos however the numbers always increase by two.
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How can
teaching interventions and other features of learning mathematics in schools
foster emotional orientations toward reasoning and create occasions for
deductive reasoning and the formulating of that reasoning?
In the context of this study, the
main aspect of this question that can be addressed is how
teaching interventions and other features of learning mathematics can create
occasions for deductive reasoning. The
main difference in the contexts for learning observed are related to the
activities.
Table
1
summarises the number of cases of reasoning observed for each
activity, and what percentage of those cases was classified into each kind of
reasoning. It should be noted that the intent of this study is not detailed
statistical analysis, and so all the figures in the table should be treated as
approximate. Minimal attempts
were made to validate the precise figures.
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|
|
Cases |
Analogy |
Induction |
Multi-step deduction |
Single step deduction |
Aesthetic |
Empirical |
|
Set |
122 |
3 (2%) |
2 (2%) |
6 (5%) |
111 (91%) |
0 (0%) |
0 (0%) |
|
Connect 4 |
45 |
0 (0%) |
5 (11%) |
4 (9%) |
36 (80%) |
0 (0%) |
0 (0%) |
|
Doorbell |
74 |
3 (4%) |
6 (8%) |
23 (31%) |
42 (57%) |
0 (0%) |
0 (0%) |
|
Mastermind (Feb.) |
19 |
0 (0%) |
0 (0%) |
2 (11%) |
17 (89%) |
0 (0%) |
0 (0%) |
|
Connect 4 on computer |
51 |
0 (0%) |
2 (4%) |
5 (10%) |
44 (86%) |
0 (0%) |
0 (0%) |
|
Literature |
14 |
0 (0%) |
9 (64%) |
5 (36%) |
0 (0%) |
0 (0%) |
0 (0%) |
|
Base 10 |
14 |
0 (0%) |
5 (36%) |
0 (0%) |
9 (64%) |
0 (0%) |
0 (0%) |
|
Mastermind (March) |
52 |
0 (0%) |
1 (1%) |
17 (33%) |
31 (60%) |
3 (6%) |
0 (0%) |
|
Patterns |
21 |
1 (5%) |
11 (52%) |
0 (0%) |
9 (43%) |
0 (0%) |
1 (5%) |
|
Paper Folding |
31 |
1 (3%) |
15 (48%) |
0 (0%) |
15 (48%) |
0 (0%) |
0 (0%) |
|
Geoboards |
14 |
1 (7%) |
1 (7%) |
0 (0%) |
12 (86%) |
0 (0%) |
0 (0%) |
|
All |
457 |
9 (2%) |
57 (12%) |
62 (14%) |
326 (71%) |
3 (1%) |
1 (0%) |
Table 1: Reasoning by Activity
Of all the kinds of reasoning observed in all the activities combined (457 cases in total), 71% were single step
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deductions and 14% were multi-step deductions.
This is not surprising given that the activities were chosen for their
potential to elicit deductive reasoning.
The teacher and the researchers involved in coding and analysing the
reasoning were also focussed on deductive reasoning, which would explain the
high proportion of deductions observed.
Four
activities are unusual in that the number of cases of deductive reasoning is
lower than the average. In the
Literature activity, the Base 10 activity, the Patterns activity and the Paper
folding activity, reasoning by induction was used more often than in the other
activities.
In the Literature activity the teacher read a book called The 512 Ants on Sullivan Street by Carol A. Losi. The students made predictions about the number pattern that was present in the book and later had the chance to create their own number patterns. The pattern presented in the book was a doubling pattern. The deductive reasoning they engaged in had to do with ways of figuring out sums of two digit numbers (e.g., 16 plus 16).
In the base 10 activity, students were given a chart consisting of three rows and 25 columns. The rows were used for (1) the number to be made using base ten blocks, (2) a diagram of the base ten blocks used and (3) the number of blocks used. Once students had modeled numbers from 100 to 124, they were asked if they could see any patterns. It was hoped that students would eventually discover that the number of blocks they had to use (if they always traded ten of a smaller unit for 1 of a larger unit) would be the sum of the digits. After they had examined the numbers for patterns, they
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were asked to make predictions about the number of blocks that would be needed to make particular numbers and then told to test their guesses by modeling them with base ten blocks. In this activity the student must identify the pattern in the numbers before they can make any deductions from the pattern. Most of the students did not come to a stable understanding of the way the number of blocks increased, or why the pattern of increasing by one changed at the end of each decade. None understood that the number of blocks is equal to the sum of the digits. This left them with few opportunities for deductions. The single step deductions they did make were to establish the total number of blocks by adding the number of blocks of each type.
In
the Patterns activity students were given two sheets
to do containing examples of "growing patterns." The
two patterns that they were asked to make were increasing sizes of squares and
increasing sizes of equilateral triangles, using pattern blocks of the same
shape. 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. No
deductive reasoning was called for in this activity, and the cases of single
step deductions that were observed were mostly around deciding whether a shape
that had been created continued the pattern or not.
[case of Leonard interesting]
In the paper folding activity 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
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and a specified number of holes and then asked to predict what the results would be when the paper was unfolded. The single step deductions that were observed included explaining the arithmetic use to calculate an answer, and specific predictions based on an observed pattern.
All four of these activities are essentially about noticing a pattern, and so the comparative absence of deductive reasoning in them is not surprising. Many elementary school curricula emphasise pattern noticing activities. The contrast between the results of these activities and results of other activities that provoked much more deductive reasoning suggests that such a curricular emphasis may undermine efforts to encourage deductive reasoning in the early grades.
Three activities involved an unusually large proportion of multi-step deductions: Doorbell, Literature, and Mastermind. Teacher interaction was an important part of these activities, unlike, for example, Connect 4 in which the students mostly interacted with each other. Doorbell and Mastermind are similar in that the percentage of deductions is about the same as the average for all activities, but the proportion of multi-step deductions is higher than in other activities. For the Literature activity however the percentage of deductions is must lower than the average, but all of them are multi-step deductions.
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The
Literature activity and the Doorbell activity both involve the teacher reading
a story to the students and asking them to make predictions about what might
happen next.
In the Doorbell activity the teacher read the book The Doorbell Rang by Pat Hutchins and had the students solve simple problems based on the story. The story is about a mother who has just baked a dozen cookies for her two children. The children are told they can share the cookies between them. Each time they figure out how many cookies each person will get, the doorbell rings and more children come in. The children then have to share the cookies out again.
The multi-step deductions the students made were arithmetical calculations that involved several steps. For older students these would be automatic, mechanical deductions (Reid, 1994?) but for these children they needed to think through each stage, and keep track of the overall goal of their calculation. The nature of the situation requires that the calculation be of more than one step (unless guided by someone who can keep the goal in mind). As each group of children arrives in the story, it is necessary to add them to the total already present and divide the total number of cookies among them, or to reallocate the cookies that have already been allocated according to some scheme that takes into consideration the relationship between the number of people present and the number arriving.
For example, Preston (January 27 MAT#4.10) when working out the answer for twelve people dividing the cookies, offered this explanation: "If you took
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two then you'd have to take one away from one person and then it wouldn't be fair because six people wouldn't have cookies. And six people would have two cookies. So it has to be one." He begins with a hypothesis: “If you took two [If each person was given two].” He then deduces from that hypothesis the need to take a cookie away from someone who already had one in order to give two to another person. This in turn leads to the statement that six people would not have cookies and six would have two. The contradiction between this and the requirement that the division be equal “fair” leads to the rejection of the hypothesis in favour of the conclusion that each would get one cookie.
The Literature activity was described in the previous section. The deductive reasoning the students used had to do with ways of figuring out sums of two digit numbers. These deductions were multi-step in that the children often used partial sums to get to a final answer. For example Alison (February 18 MAT 2.8) added 16 and 16 by splitting the problem into three additions: 10+10, 5+5 and 1+1. "32. Well, I knew there was two tens in -- 16. So I added ten plus ten which equals twenty. And I knew there was 5 in 6 so then I added on two fives, then I added on two ones and that made thirty-two."
With these two activities the teacher’s questions were important, along with the complexity of the calculations the children needed to perform. In the Doorbell activity they were supported in their calculations by having counters with which to represent the numbers, but they seemed to use these more to explain there calculations to others than to come up with the answers in the first place.
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When the teacher first played Mastermind with the students (Feb. 1 & 3), the reasoning used was not impressive. It consisted almost exclusively (89%) of single step deductions in response to teacher questions. Given the anticipated potential of Mastermind for provoking multi-step deductions, the activity was redesigned, and the modified activity produced different results.
In Mastermind a sequence of four pegs is chosen from pegs of six different colours. This sequence is hidden, and the challenge is to guess the sequence. Each guess is made by placing a sequence of pegs on the board, and white and block pegs are placed next to each guess to indicate how accurate it is. A white peg indicates that there is a peg that is the right colour but not in the correct spot. The black peg indicates that the peg is the right colour and in the correct spot. This is how the game was played in February. In March the students were presented with Mastermind games that had already been played. Students were asked to make their guesses about the code after seeing some or all of the "clues" available for a particular game. A higher percentage of multi-step deductions were observed in March than in February (33% versus 11% of all reasoning observed).
As an example of the use of multi-step deductions in Mastermind, consider the case of Jerome. Jerome is not typical, as he was far more likely to use multi-step deductions than the other children (50%, 12 out of 24, of cases of his reasoning were multi-step deductions, compared to the average of 14%). This example comes from his solution of the clues in Game 2 in March.
In Game B the clues given were these:
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|
Guess |
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1 |
yellow |
pink |
green |
black |
2 white |
|
2 |
pink |
yellow |
Blue |
red |
2 white |
|
3 |
black |
blue |
Red |
green |
1 black, 3 white |
|
4 |
black |
green |
Blue |
red |
4 white |
|
5 |
red |
black |
Blue |
green |
1 black, 3 white |
|
6 |
black |
red |
Blue |
green |
2 black, 2 white |
|
Code |
blue |
red |
black |
green |
|
After the teacher showed him the first two clues Jerome said: "Blue isn't the right colour." (March 2 MAT #2.2). The teacher asked him to explain, and he said, “Cause it wasn't there [in clue 1] and it was still two [correct colours]." (March 2 MAT #2.4). The teacher asked him which colours he thought might be correct, and after some thought he replied confidently, “Pink and yellow…. Cause that [green] wasn't there [in clue two], and there was still two [correct] and black wasn't there [in clue two] and there was still two [correct]." (March 2 MAT #2.6).
This is multi-step deduction, even
thought Jerome’s conclusion is wrong. He
has implicitly assumed that the simplest explanation is the correct one.
The two white pegs signalling that the two colours present in both
clues (pink and yellow) are correct is a much more simple assumption that
assuming that one or both of them stands for a different colour each time.
Errors arising from unstated (and hence unquestioned) assumptions are
much more likely to occur if reasoning is unformulated, as the reasoning of
grade two children usually is.
Jerome’s reasoning continued, as the teacher showed him the third clue (in response to which he made no audible comment) and then the fourth clue.
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He then said, “So blue can't be there [in the third position]. - - So, red can't be there [in the last position]. So green can be there, there or there. [First, third or fourth positions]." (March 2 MAT #2.10). The teacher asked him why the green could be in the position indicated and he replied, “. . .they changed green around and all of them all wrong [in clue 4]." (March 2 MAT #2.11).
It seems that Jerome’s reasoning began again after he saw the third clue, which contradicted his previous conclusions. He recognized that the result of the third clue shows that the four colours must be black, blue, red and green, and that the fourth clue showed him positions the colours could not be in. His comment about possible positions of the green pegs, however, show that he is not considering all the clues, and the first clue shows that the green pegs cannot be in the third position.
After the teacher laid down the fifth clue, Jerome said, “I think I know where green is supposed to be. They got one right there [clue three] and green was right there [position 4]. They got another one right [clue 5] and green was right there [position 4]. Blue wasn't there and red wasn't there and black wasn't there." (March 2 MAT #2.13). When he talked about the blue and red, he indicated the column he thought each should be in. Once again he is assuming that the pegs that are the same in two clues (the green one in clues three and five) accounts for the common features of the scores those clues received. In this case his conclusion is correct, but his reasoning is faulty. The black scoring peg in clue three could indicate that the red peg is in the right place, and the black scoring peg in clue five could be for the black peg in the pattern.
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When he was shown the sixth clue he called out, “This one [red] and this one [green] are perfect. I found it out!" ." (March 2 MAT #2.15). He went on to explain that assigning one black scoring peg to the red peg was consistent with the previous clues, as red had never been in that position before and that the other black scoring peg could be assigned to the green peg. While the pattern in clue six was not correct, he seemed confident that he know the correct pattern based on the clues. His exact words were: "None there. [He pointed to the column that red is in, and indicates with his finger that red has never been there before.] Two there. [He points to the green in clue 5 and 6.] And I got one perfect there [He points to clue 5.] - - I won another game!" (March 2 MAT #2.17).
The case of Jerome reveals several features of the way Mastermind was played in March that contributed to the high rate of multi-step deductive reasoning associated with it. The first and most obvious is the teacher’s questions. She both paced the game by controlling the rate at which clues were revealed (when playing in February the students could make another guess before reflecting seriously on the clues they already had) and encouraged verbalisation of the students’ reasoning. The structure of Mastermind also encourages multi-step deductive reasoning as it both requires it for winning the game, and also supports it by providing a representation of the information available. In other contexts using multi-step deductive reasoning requires keeping several pieces of information in mind at once, which is challenging unless some form of representation is used.
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