int. j. math. educ. sci. technol., 1999, vol. 30, no. 1, 39± 46
Di c u ltie s in kn ow le d ge in te gration : re v isitin g Ze n o’s
parad ox w ith irration al n u m be rs
IRIT PELED
School of Education, University of Haifa, Haifa, 31905, Israel
e-mail: iritp@netvision.net.il
and SARA HERSHKOVITZ
Centre for Educational Technology, 16 Klausner St., Ramat Aviv, 39513, Israel
e.mail: sara_h@cet.ac.il
( Received 15 July 1997 )
The study investigates sources of di culties exhibited by student teachers
in tasks involving the construction of an irrational length segment, and other
irrational number tasks. The results show that student teachers know the
de® nitions and characteristics of irrational numbers, yet fail in tasks that require
a ¯ exible use of their knowledge and in tasks that involve making connections
between di€ erent representations. Thus, for example, students say that irrational numbers are real numbers, yet many think they have noplace on the real
number line. Their explanations indicate that misconceptions about the limit
concept, that relate to the dilemma in one of Zeno’s paradoxes, are a main
source of di culty. These ® ndings stress the importance of creating tasks that
facilitate the integration of di€ erent knowledge pieces.
1.
In trod u c ti on
Once in a while we ® nd ourselves deliberating for asurprisingly long time over
a question relating to a concept that we thought we had understood. Such an
experience often makes us reorganize our knowledge, and integrate di€ erent items,
resulting in a better understanding of the subject following this experience. When
students are asked to cope with such a problem in a test, they might either think
they have learned something new, or complain that the test was unfair.
Our interest in investigating how student teachers integrate knowledge that
relates to irrational numbers with their knowledge about the number system arose
as a result of a problem-solving situation which caused such deliberations:
Students were given the following problem:
John gave the carpenter a5m by 1m wooden board, asking him to use all the
wood to make a square table top. The carpenter thought for a while and then
said: No problem. I can build your table by making just a few cuts. Do you
think he can do it? How?
While solving this problem, most students had arrived at the conclusion that if
such atable exists, its side has to be Ï 5. This conclusion was problematic for some
students,who believed that such a length cannot be measured.
0020± 739X/99 $12´00 Ñ
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40
I. Peled and S. Hershkovitz
Di culties related to the concept of irrational numbers were discussed by
Fischbein et al. [1], who put the blame on the failing of the school curriculum to
link the various number sets within the number system, as well as the limited
number of examples of irrational numbers shown to students. This latter issue is
discussed by Mason and Pimm [2], who claim that examples (even if intended to
be generic examples) might be perceived as the only extensions of agiven concept.
The researchers quote MacHale [3] who lists topics for which textbooks tend to
use the same small set of speci® c examples. For irrational numbers, the examples
given in most texts are Ï 2 and Ï 5, possibly evoking the misconception that these
are the only extensions of the set of irrational numbers.
Going back to our original puzzling carpenter problem, and looking at
students’ claims that `the carpenter cannot measure a length of Ï 5’, we wondered
whether their di culties stemmed from a lack of understanding of the irrational number concept or whether they had a problem with the concept of
measurement.
At this point we were curious to know what do all studentsÐ not only those
who claimed it was impossible to measure an irrational lengthÐ know about
irrational numbers. The students involved in this study had at least two years of
college mathematics, and had studied irrational numbers, but could still have a
limited understanding of this concept. Their knowledge could be restricted to a
speci® c learned representation, or toformally recited de® nitions, rather than being
a more ¯ exible conception that would constitute evidence of their understanding.
As Lesh et al. [4] put it: `Part of what we mean when we say that a student
understands an idea, like 1/3, is that: (1) he or she can recognize the ideaembedded
in a variety of qualitatively di€ erent representational systems; (2) he or she can
¯ exibly manipulate the idea within given representational systems; and (3) he or
she can accurately translate the idea from one system to another.’ (p. 36)
The main questions of the study were:
(1) Do all students have a basic knowledge of irrational numbers, i.e. do they
know how the set of irrational numbers relates to a variety of number sets
such as whole numbers, rational numbers, and real numbers?
(2) Do they understand the meaning of the irrational number beyond its
formal de® nition? For instance, can they handle the fact that although an
irrational number has an in® nite and `unordered’ number of digits in its
decimel representation, it still has its place on the number line?
2.
Proc e d u re
Following identi® cation of the problem described here in a methods course,
this study was conducted in twogroups, both of which tookthe course. One group
comprised 55 students, and the second 15 students. The students were in their
second or third year of college mathematics, taking courses towards a mathematics
teacher certi® cation.
The study consisted of two parts. Part 1: The students were asked to solve the
carpenter’s problem. Part 2: The students were asked to complete a questionnaire
that dealt with knowledge related to irrational numbers and their place within the
number system.
41
Irrationals and knowledge integration
First group
Second group
Total
Incorrect answers
Correct
answers
irrational
measure
No
answer
38
69%
1
7%
39
56%
3
5.55%
4
27%
7
10%
3
5.5%
8
53%
11
16%
11
20%
1
7%
12
17%
Other
N
55
1
7%
1
1%
15
70
Table 1. Distribution of answers in the carpenter’s problem.
3.
Re su lts
3.1. Part 1: The carpenter’ s problem
This part was given as homework to the ® rst group and as class work to the
second group (allowing the students to work for about an hour). The distribution
of the answers in both groups is shown in table 1.
A correct answer was given by 38 out of 55 students in the ® rst group, and by
one out of 15 students in the second group. The correct solution was accompanied
by a sketch of the cut up board and the new squared table top consisting of the
rearranged pieces as presented in ® gure 1.
The students in both groups encountered two main problems: di culties
relating to irrational numbers, and di culties relating to a general measuring
misconception.
Speci® cally, students who had di culties with an irrational number (three out
of 55 in the ® rst group and four out of 15 in the second) thought that the carpenter
cannot measure a length of Ï 5 m because of its in® nite decimal representation.
Students who had a general measuring di culty (three in the ® rst group and eight
in the second) thought that the board had tobe cut intoawhole number of squares.
Figure 1.
The problem solution.
42
I. Peled and S. Hershkovitz
Since the board was 5m long they tried to divide it into ® ve squares and move
these around. Their next step was to divide each square into smaller squares or
rectangles thinking that eventually they would be the right size.
3.2. Part 2: Questionnaire
A knowledge questionnaire related to integrating irrational numbers within the
number system. The questionnaire consisted of:
(a) Questions that deal with students’ declarative knowledge on Ï 5, asking
whether this number is a whole, a rational, an irrational, or a real number,
and whether it has an in® nite number of digits in its decimal representation.
(b) Questions which ask the students to identify and mark numbers on a given
real number line, and then to refer to the unassigned numbers and explain
why these numbers have not been placed on the line. The numbers in the
number line task consisted of the following list (given in this order):
Ï 4, 1 /100, 0.12, 0.25, p , Ï 5, 1 /9, 0.3333 . . .
The number list had been carefully chosen. In fact, most of the numbers were
supposed to serve as clues to remind the students that there are other units of
measurement, for example 19, besides the decimal units. This number, 19, was also a
clue for representing 0.3333 . . . as 13 and then easily putting it on the line. The fact
that 0. 3333 . . . has aplace on the real number line could then trigger the realization
that being a number with an in® nite number of places beyond the decimal point is
not su cient to make a number `unplaceable’ on the number line.
The results of part (a), which deals with the characteristics of Ï 5 are presented
in table 2.
As can be seen in table 2, about three-quarters of the students answered all
declarative questions correctly. All of the students (except one in the second
group) could tell that Ï 5 is not a whole number and most of the students (except
one in each group) knew that Ï 5 is an irrational number. Some confusion was
observed regarding knowledge related to the decimal representation of Ï 5 (16%of
all 62 students answered incorrectly) and with identifying Ï 5 as a real number
(13%of all 62 students gave an incorrect answer).
Part (b) deals with the representation of numbers on the real number line. It
was meant to expose implicit de® nitions of irrational numbers. The ® ndings were
All
correct
First group
Second group
Total
36
77%
9
60%
45
73%
Incorrect (not exclusive)
not whole irrational
Ð
1
7%
1
2%
1
2%
1
7%
2
3%
in® nite
real
N
5
11%
5
33%
10
16%
7
15%
1
7%
8
13%
47
Table 2. The distributions of answers on Ï 5 characteristics.
15
62
43
Irrationals and knowledge integration
First
group
Second
group
Total
Ï 4
1/100
0.12
0.25
48
100%
15
100%
63
100%
48
100%
15
100%
63
100%
48
100%
15
100%
63
100%
48
100%
15
100%
63
100%
p
22
46%
6
40%
28
44%
Ï 5
1/9
0. 3333 . . .
N
25
52%
6
40%
31
49%
38
79%
12
80%
50
79%
26
54%
4
27%
30
48%
48
15
63
Table 3. Percentage of correct answers for each number in number line task.
Did not place these numbers on number line:
Placed
all p , Ï 5, 1 /9, 0. 3333 . . . p , Ï 5, 0. 3333 . . . p , Ï 5 other comb.
First group
Second group
Total
16
33%
1
6.6%
17
27%
6
12.5%
Ð
6
10%
6
12.5%
6
40%
12
19%
7
15%
1
6.6%
8
13%
13
27%
7
47%
20
32%
N
48
15
63
Table 4. Complete pro® les of number line answers.
analysed ® rst for each number separately, noting the proportion of students who
placed a given number on the real number line (table 3). They were then analysed
again by categorizing the students according to the characteristics of the set of
numbers not placed on the real number line (table 4).
As can be seen in table 3, no students encountered problems with placing 0.25,
0.12, 1/100, Ï 4 on the number line. However, even the number 1/9 confused 21%
of the students in the context of this task.
The numbers p , Ï 5, 0.3333 . . . were more problematic. Each of these numbers
could not be placed on the real number line by about half of the students. Table 4
may shed some light on the source of students’ problems by looking at a more
global picture.
Only 27%of the students could correctly place all given numbers on the
number line. The rest of the students experienced di€ erent kinds of di culties,
resulting in various combinations of numbers which they could not place correctly.
Each number combination comes with an explicit explanation revealing the source
of the di culty.
Three main combinations can be observed:
One group of students, 10%of the total, thought that p , Ï 5, 1 /9, 0.3333 . . .
cannot be placed on the number line. Some explained that these are irrational
numbers and therefore cannot be placed on the real number line. It is interesting to
note that those students had just claimed in part (a) that Ï 5 is a real number.
Others in this group explained that these numbers have an in® nite number of
digits in their decimal representation and therefore cannot be placed on the line.
A second group, 19%of the total, could not place p , Ï 5, 0.3333 . . . on the
number line. Basically this group used the same `in® nite number of digits’
44
I. Peled and S. Hershkovitz
argument, except that they did manage to place 19, using their fractions knowledge
yet not identifying 0.3333 . . . as the rational number 1 /3.
A third group, 13%of the total, could not place p , Ï 5 on the line. These are
students who did manage to represent 0. 3333 . . . as one-third (13).
In addition to these three groups, several other combinations were found.
Among these were students who managed to place some of the problematic
numbers by number estimation. For example, instead of putting 0.3333 . . . on
the number line, they placed 0.33 on the line. Instead of placing p , they used the
number 3.14 or 22/7. Most of these students explicitly wrote that p equals 3.14 or
that 0.3333 . . . equals 0.33, denoting that they did not see it as an estimation but as
the actual value of the number in the task.
Additional ® ndings relate to the students’ consistency. Three relations were
investigated:
(1) The relation between a students’ answer in the carpenter problem and the
ability toplace Ï 5 on the number line. This issue is relevant only to the ® rst group
of 55 students (as only one student in the second group solved the problem
correctly and placed the numbers correctly on the number line). In this group, 38
students solved the carpenter problem correctly, showing how to cut the wooden
board in such a way that results in wooden pieces that have a Ï 5 long side. Still, 16
of them (42%) did not place Ï 5 and explicitly declared that it cannot be placed on
the number line. Two common reasons were given: (a) `. . . because it is an
irrational number’; (b) `. . . because it has an in® nite number of digits in its
decimal form’.
(2) The relation between a students’ claim that Ï 5 is a real number and a
willingness to place Ï 5 on the number line. In the ® rst group, 47 students
answered (correctly) that Ï 5 is a real number, yet 20 of them (43%) did not
place it on the number line. In the second group, 14 answered that Ï 5 is a real
number, yet 8 of them (57%) did not place it on the number line. In both groups
together, 46%of the students who had a correct declarative knowledge still could
not place Ï 5 on the number line. This amounts to 28 students (40%) of the total of
70 students in the sample that exhibited this type of inconsistency.
(3) A di€ erent type of inconsistency was involved with regard to rational
numbers. As it turned out the student’s ability to place a rational number on the
number line depended to a great extent on the number’s representation. speci® cally, 10 students out of 47 in the ® rst group and 9 out of 15 in the second group,
could place 19 on the number line, but could not place 0. 3333 . . . (the decimal
representation of 13 on the number line. This amounts to 31%(19 out of 62) in both
groups.
4.
Di sc u ssion
This research was originally planned toinvestigate sources of di culty with the
number Ï 5 that were revealed when some students solving the carpenter problem
claimed that a length of Ï 5 cannot be measured. The ® ndings show that these
students did not have a real-world measurement problem, e.g. they did not claim
that this number cannot be measured because there is no ruler than can measure it
accurately. Rather, their explanation was given in the mathematical world. In fact,
there were quite consistent as they claimed that Ï 5 cannot be placed on the real
number line.
Irrationals and knowledge integration
45
While looking into this issue, additional problems were revealed. Other
students, who did solve the carpenter problem correctly, seemed to have di culties with irrational numbers. Speci® cally, about 40%of the students of the ® rst
group, who solved the carpenter problem, could not place Ï 5 on the real number
line.
We wondered whether this problem indicates di culty with formal knowledge
relating to irrational numbers. Fischbein et al. [1]found that children and student
teachers had di culties both with the de® nition and the classi® cation of numbers
as rational (or irrational) and as real. Our ® ndings show that students’ declarative
knowledge relating to irrational numbers is not the main problem, only a few
students did not know that Ï 5 is an irrational number, and did not categorize it as
a real number.
Rather than formal knowledge of irrational numbers, students’ explanations
seemed to indicate that the main source of di culty involves the concept of limit.
Most of the explanations claimed that Ï 5 cannot be reached because of the in® nite
number of digits in its decimal representation. For the same reason, some students
also claimed that 0.3333 . . . cannot be placed on the number line.
Di culties with the concept limit have been discussed by many researchers [5±
7]. Williams [8] lists mental models of limit and beliefs about it held by college
students. One of these beliefs is the notion that a limit is something one can get
closer and closer to, but can never reach (actually this is true according to some of
the early de® nitions of limit).
This notion of getting closer and closer but never really reaching acertain point
is also the essence of one of Zeno’s paradoxes [9], where Achilles competes against
the turtle but supposedly cannot bear the turtle’s 10mhead-start although he runs
ten times faster (10m/s versus 1m/s). The claim is that by the time Achilles covers
these 10m, the turtle has advanced by 1m, and by the time Achilles runs the 1m,
the turtle has moved forward by 0.1m, and soon and on. Through this description
of the process, the turtle is always ahead of Achilles, although the distance between
them becomes smaller and smaller and approaches zero. When an algebraic
solution comparing the two distances versus time functions is introduced, it is
obvious that Achilles overtakes the turtle at some easily calculated time (if x stands
for time in seconds, then the equation 10 + x = 10x results in Achilles overtaking
the turtle after 10/9 seconds, at which time they are both at exactly the same
distance from the starting point). In fact, the di€ erence between the two viewpoints is similar tothe di€ erence between representing 119 as arational number and
using its decimal representation, 1. 1111 . . .. In both cases, although we are
convinced about the equality between the ® nite and in® nite representations,
there is still something unsettling about the `eventual convergence’ of the in® nite
series.
Di culties with the concept of limit are only a part of the story, which relates
to the decimal (and in® nitely long) representation of irrational numbers. We are
still left wondering why students solved the carpenter problemyet could not create
a similar length on the number line: how could they tell that an irrational number
is areal number and still claim that it does not have aplace on the real number line,
and why did they not realize the implications of the fact that one-third has a place
on the number line even though it has in® nitely long decimal representation?
These inconsistencies are evidence of some incomplete understanding. Hiebert
and Carpenter [10] argue that `Understanding occurs as representations get
46
Irrationals and knowledge integration
connected into increasingly structured and cohesive networks’ (p. 69). The
students in our study failed in making connections between di€ erent aspects of
the irrational number concept, its de® nitions and representations, its place within
the number system, and its relation to other concepts, such as limit.
The educational implications of these ® ndings are mainly directed towards the
need to go beyond de® nitions and help students build ¯ exible knowledge. As
mentioned earlier (and cited from [4]), the ¯ exibility of perceiving a concept in
di€ erent representations is considered to be proof of understanding this concept.
Not having this ¯ exibility had resulted, for example, in students’ inability to
perceive one-third in its di€ erent representations, and to make the right connections between them.
Vinner [11] suggests using non-routine problems in order to facilitate the
extension of the learner’s concept image. The carpenter problemcan be considered
an example of anon-routine problem, as it managed to bring up some problematic
issues. Giving it to students and then using it as a basis for discussion might cause
the creation of new insights. Our educational goal should be to create many such
opportunities in order to encourage the construction of ¯ exible and connected
knowledge.
Re fe re n c e s
[1] Fischbein, E., Jehiam, R., and Cohen, D., 1995. Educ. Stud. Math. , 29, 29± 44.
[2] Mason, J., and Pimm, D., 1984, Educ. S tud. Math. , 15, 277± 289.
[3] MacHale, D., 1980, Amer. Math. Monthly , 87, 9, 752.
[4] Lesh, R., Post, T., and Behr, M., 1986, Representations and translations among
representations in mathematics learning and problem solving. In C. Janvier (ed.)
Problems of Representation in the T eaching and L earning of Mathematics (Hillsdale,
NJ: Lawrence Erlbaum Associates).
[5] Fischbein, E., Tirosh, D., and Hess, P., 1979, Educ. S tud. Math. , 10, 30± 40.
[6] Davis, R. B., and Vinner, S., 1986, J. Math. Behavior, 5, 281± 303.
[7] Tall, D., 1992, The transition to advanced mathematical thinking; functions, limits,
in® nity, and proof. In D. A. Grouws (ed.) Handbook of Research on Mathematics
Teaching and L earning (New York: Macmillan), pp. 495± 511.
[8] Williams, S. R., 1991, J. Res. Math. Educ. , 22, 219± 236.
[9] Salmon, W. C., 1970, Zeno’ s Paradoxes (Indianapolis: Bobb-Merrill).
[10] Hiebert, J., and Carpenter, T. P., 1992, Learning and teaching with understanding.
In D. A. Grouws (ed.) Handbook of Research on Mathematics Teaching and L earning
(New York: Macmillan), pp. 65± 97.
[11] Vinner, S., 1991, The role of de® nitions in the teaching and learning of mathematics.
In D. Tall (ed.) Advanced Mathematical Thinking (Dordrecht: Kluwer Academic),
pp. 65± 81.