Conjecture activities for comprehending statistics terms through speculations on the functions of imaginary spectrometers.
Liu, Shiang-tung ; Ho, Feng-chu
Introduction
The purpose of this study was to describe students' problem
solving performance when they make conjectures to comprehend three
statistics terms. Teachers are key figures in changing the ways in which
mathematics is taught and learned in schools. Mathematics teachers are
supposed to design meaningful tasks to motivate students' interest
and to enhance students' communication and reasoning. Within
various contexts, if meaningful tasks are designed for students to work
on, then students should benefit more from those contexts of problem
solving. For example, in this unit for learning three statistics terms,
i.e., median, mode and range, the authors provided opportunities for
students to conjecture, verify, and modify their rules rather than
directly telling them the rules to find those three statistics terms.
Such a learning process might result in better student performance.
The National Council of Teachers Mathematics (NCTM, 2000) stressed
that middle grade students should have the opportunities to make
conjectures and design experiments or surveys to collect relevant data.
In the middle grades, students should learn to use the median, the mode
and the range, to describe distribution of a data set. Students finally
need to understand that the median indicates the 'middle' of a
data set, the mode highlights the most prevalent sample of a data set,
and the 'range' is one measure of the spread of a data set.
Students often fail to comprehend the meaning of those statistics terms.
Thus, mathematics teachers have an important role in providing
experiences for students to comprehend statistics terms.
Inductive inference is expected to appear during the process of
conjecturing. An induction is an inference of a generalised conclusion
from particular instances. Inductive processes will produce a net
increase in knowledge, yet must be severely constrained if they are to
produce plausible conclusions. What is more, an induction is risky in
the sense that it may not be true, even if the premises are true
(Bisanz, Bisanz & Koepan 1994).
In addition, the form of activity, the kind of thinking required,
and the way in which students are led to explore the particular content
all contribute to the kind of learning opportunity afforded by the task.
To capitalize on this opportunity, teachers should deliberately select
tasks that provide them with windows into the students' thinking.
Teachers should consider what they know about their particular
students not only from a mathematics learning perspective, but from
psychological, cultural, and sociological ones as well. Sensitivity to
the diversity of students' backgrounds and experience is crucial in
selecting worthwhile tasks.
The purpose of this article is to introduce the design of
conjecture activities, to describe students' dialogues in
activities and to discuss the implications from these activities.
The design of conjecture activities
A spectrometer is an instrument used to disperse radiant energy or
particles into a spectrum and measure certain properties such as
wavelength, mass, energy, or index of refraction. To conjecture, verify,
and modify the relationships between data and answers, three statistics
terms were first temporarily replaced by three imaginary spectrometers
in the conjecture activities. We label these as
'spectrometers' because the spectrometers have the functions
of transmission, scattering, concentration and relocating. The
spectrometers were not real instruments, but the name is being used
metaphorically. This design has three stages. First, the source data and
the target answers were shown (Figures 2, 3, and 4) to students. The
functions of those imaginary spectrometers were conjectured by students.
Second, based on the analyses of the relationships between data and
results, students conjectured, discussed and negotiated the functions of
the three spectrometers. Finally, students were asked to do a matching
activity to connect three statistics terms and three spectrometers. In
addition, students were asked to find the statistical definitions of the
three based on their functions.
[FIGURE 1 OMITTED]
The conjecture activities
Ms Ho is an expert teacher. Her fifth grader's classroom had
finished the unit of arithmetic average. She found some students could
find the average but failed to understand the meaning of average. She
then tried to utilise conjecture strategies to teach the other
statistics terms. There are three stages during the conjecture
activities.
Stage 1: Commencement
Ms Ho said, "In our class, some student assistants have
special responsibilities to help the teachers. One student is in charge
of the discipline in the classroom when the teachers are absent, another
student is in charge of cleaning the classroom, and another student
looks after sport equipment. Each assistant has his/her own
function."
Ms Ho introduced three spectrometers by drawing an analogy with the
duty of assistants in the classroom. As every student assistant has
his/her own function, each of three spectrometers performs its own
functions. Let us find their functions from the following conjecture
activities.
Stage 2: The dialogues in conjecture activities
The dialogues of one group which had a lot of debates and
refutation has been recorded and analysed. The symbols S1, S2, represent
different students in this group. The functions and names of three
imaginary spectrometers which represent median, mode, and range will be
conjectured by students. As the author expected, students could
comprehend statistics rules from conjecturing activities and inductive
speculation.
Episode 1: No. 1 Spectrometer for Learning Median
S1: Finding the common function from those data seems difficult. It
is similar to guessing the security password of a software package.
S2: Let start from this set (1, 3, 5, 7, 9)! (1, 3, 5, 7, 9). The
answer is 5. Is 5 obtained by deleting from both sides 1 and 9, then 3
and 7?
S3: It seems right. Let us check another set of data.
[FIGURE 2 OMITTED]
S2: Look at this set, (4, 8, 12, 16, 20, 24). If we delete data
from both sides of this set, 4 and 24 then 8 and 20, it still has two
numbers 12 and 16. Which one is correct? The answer is 14, neither 12
nor 16.
S4: Is 14 the 'most' middle number in the set?
S3: Yes, it might be possible. Let us use this strategy to check
the third set. 25,10, 5, 20, 15). By deleting from both sides twice, the
answer is 15, rather than 5. Why?
S4: 15 is also one of the numbers in that set. Is it not?
S3: Let us rearrange the order of this set by the sequence from
small to large. We get the new set, (5, 10, 15, 20, 25).
S1: Oh, I got it! The function of No. 1 is to find the middle
number of the data sequence.
S4: It's possible. Let us check this conjecture using the
other data sets.
T: Do you think that the function of No. 1 spectrometer is to
cancel both numbers from the largest and smallest ones and what remains
(the most middle number) is the answer?
S3: Yes!
T: After deleting by pairs, did you finally find a single number
was left in one set of data?
S3: It depends! When there are even numbers in a set, the final
results deleted by pairs are two numbers rather than one.
T: Then, how do you get the exact one?
S3: We can use the 'average' strategy to find the most
middle one for the two numbers!
T: Is it true for all cases?
S4: Yes, give the 'average' for the two numbers from the
even sequence data! Then, it works for every set.
From the process of conjecturing, students found the pattern once
the set had been rearranged by order from smallest to largest. Students
are pleased to say so during the process of conjecturing. They then
immediately test another set having even numbers of data to confirm
their conjecturing.
Ms Ho then asked students to name the spectrometer. Students gave
different names to No. 1 spectrometer, such as the middle number, the
centre number, the balance number, Libra, and Impartial God. The first
three names are similar to mathematics terms. The last two names seem to
reflect the role of helpers. The reason they chose Libra is that it
stands on the middle position on the pivot of a balance. The explanation
for why they chose the 'Impartial God' is that it stands on
the centre of the data sequence and is impartial to any side.
Episode 2: No. 2 Spectrometer for Learning Mode
S1: Here is another spectrometer! Its function may be different
from No 1. This number sequence, (2, 2, 4, 5, 9, 2, 7), passed through
No. 2 and got the result 2. What is this pattern?
Students tried the smallest number, the common factors, and the
least common factor. Then Ms Ho suggested students consider several sets
of data and answers together.
S2: Let's see! (2, 2, 4, 5, 9, 7, 2) turned to 2, (3, 8, 8, 5,
8, 6, 13, 8, 3) turned to 8, (10, 15, 5, 10, 10, 20, 10, 5) turned to
10. It seems they have repeated numbers in each set.
S3: Is possible to find the 'repeated number' as the
result?
[FIGURE 3 OMITTED]
S1: Let's check the other sets.
S4: Yes, No. 2 spectrometer is to find the most prevalent numbers!
After Ms Ho encouraged students to take several sets together, they
thus found the pattern and used their own words to stand for the
repeated numbers. They name No. 2 spectrometer "the copy
devil," "the identical number," "multiple
fetus" and "most frequent number." Finally, they agreed
the function of No. 2 is to find the most prevalent number.
Episode 3: No. 3 Spectrometer for Learning Range
Students utilised the previous strategies to guess No. 3. In
addition, most of the ranges, being the distances from the smallest one
to the largest one; do not appear in the data set. Students tried using
the middle number, 5, added 2, then got 7; or the repeated number
subtracted by the smallest number. Ms Ho thus suggests that they
rearrange the data sequence from the smallest to the largest and look
for the pattern.
[FIGURE 4 OMITTED]
S4: I got it! It's that the largest number minus the small
one. Let's check the other sets.
After students rearranged the data sequence, they got the answers.
In fact, the titles of No. 3 given by students are far from the term
'range'. However, a suitable term, the difference calculator,
has been represented during the process of identifying function. In the
long run, students discovered that the No. 3 spectrometer finds the span
between the largest and the smallest data entries.
If students have experiences in locating numbers on number lines
and identifying that the difference is the distance between any two
points on the number line, they might more quickly identify the
functions by their own words and might use the closer term of
'range' during the conjecture activity. Students finally find
the functions of three imaginary spectrometers from the process of
conjecturing, and induction.
Stage 3: A matching-up activity and giving meaning for terms
Students can understand basic concepts of three terms from the
process of giving names. After they experienced the activities of
conjecturing and naming, students realised the role of three statistic
terms. During the third stage, the authors asked students to carry out a
matching up activity between three statistics terms and three imaginary
spectrometers. Students quickly connect the term 'median' to
No. 1 spectrometer because median in Chinese possesses the meaning of
'centre position'. They also easily assigned the term of
'mode' to No. 2 spectrometer due to the term's meaning in
Chinese indicating plenty or abundance. English speaking children would
have difficulty in catching the meaning of these terms. Although they
were less confident in naming No. 3 spectrometer than in naming the
other two, the corresponding matching up activity was still easy for
them to solve because after they chose the two terms of No. 1 and No. 2
spectrometers, the third term, range, was naturally matched to No. 3.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
Discussion and reflections
As Lobato, Clarke and Ellis (2005) point out, just telling students
how to deal with algorithms is likely to foreclose the discussion
without contributing much to their understanding of the concepts.
Instead of telling the rules, then asking students to copy and practice
the rules of statistics terms, the teacher conducted the conjecturing,
discussion, and induction activities to introduce the statistics terms.
In addition, when students encountered difficulty, the teacher posed
questions such as: "Why don't you rearrange the data set from
the smallest to the largest one?" or "Why don't you
consider several data sets together." These questions really help
students to scaffold their learning. Under this atmosphere of scaffolded
teaching, students were aware of what they were doing, and frequently
adjusted their strategies as they solved problems
The process of conjecturing hinges on being able to recognise a
pattern or analogy, in other words, on being able to make a
generalisation. As Bisanz, Bisanz, and Korpan (1994) stressed, some
conclusions induced from conjecturing might be wrong and those temporary
conclusions have to be checked by the other data sets. During the
process of the conjecturing activities in stage 2, students met with
many difficulties, such as, "Why did the hypothesis fit the former
data set and not the latter data set?" Students are encouraged thus
to make new hypotheses to fit the whole data. Giving up the original
hypotheses and setting up a new hypothesis for survival is a process of
adaptation. In addition, getting another data set to fit the new
hypothesis is a process of accommodation. During the process of
conjecture, students were engaged in the process of adaptation and
accommodation; hence they develop basic concepts of those statistics
terms.
In Episode 2, students used the "smallest number,"
"common factors," and "least common factor." In
Episode 3, a student found the pattern may be from the smallest number
subtract the repeated number. As Bisanz, Bisanz, and Koepan (1994)
emphasised, inductive processes must be severely constrained if they are
to produce plausible conclusions. From the above episodes, the
temporarily wrong conclusions are generated when students make
inferences from very few data sets.
As Mason, Burton, and Stacey (1985) claimed, generalisation
reasoning involves focusing on certain aspects common to many examples,
and ignoring other aspects; the process of generalizing is that of
moving from a few instances to making informed guesses about a wide
class of cases. To generalise their findings, students engage in the
cyclic process of making a hypothesis, checking the answer and
generating another hypothesis during the process of generalisation. This
kind of mathematics activity resembles that of scientific investigation
(Bisanz, Bisanz & Korpan, 1994).
As NCTM (1991) stressed, mathematics instruction needs to be
orientated away from an emphasis on mechanistic answer-finding, and
towards conjecturing, and problem solving. This study showed that
students could infer statistics rules from inductive speculation and the
functions of three spectrometers from the process of conjecturing,
verifying and modifying. In addition, students could provide intuitive
terms for the functions of the three spectrometers and they then perhaps
realise the basic concepts of the three formal statistic terms from the
process of giving names.
If students found the process of conjecturing problematic, how did
they proceed? What information did they use? What misconceptions did
they have? Students' problematic experiences can serve as a
springboard to pose new problems for students to improve their
mathematical ability. From this article we found that conjecture
activities are beneficial for initiating statistics lessons. The
conjecture activities can stimulate the mechanism of adaptation and
accommodation. Given these findings, the question we must ask ourselves
is: what kinds of lessons are suitable for utilising conjecture
activities?
References
Bisanz, J., Bisanz, G. & Korpan, C. A. (1994). Inductive
reasoning. In R. Sternberg (Ed.), Thinking and Problem Solving. San
Diego: Academic Press.
Chazan, D. & Ball, D. (1999). Beyond being told not to tell.
For the Learning of Mathematics, 19(2), 2-10.
Labato, J., Clarke, D. & Ellis, A. (2005). Initiating and
eliciting in teaching: A reformulation of telling. Journal for Research
in Mathematics Education, 36(2), 101-36.
Mason, J., Burton, L. & Stacey, K. (1985). Thinking
Mathematically. Menlo Park, CA: Addison-Wesley Publishers.
National Council of Teachers of Mathematics (1991). Professional
Teaching Standards for Teaching Mathematics. Reston, VA: NCTM.
Polya, G. (1954). Induction and Analogy in Mathematics. Princeton
University Press.
Stein, M. K., Schwan-Smith, M., Henningsen, M. A. & Silver, E.
A. (2000). Implementing Standards-based Mathematics Instruction: A
Casebook for Professional Development. New York, NY: Teachers College
Press.
Shiang-tung Liu
National Chiayi University, Taiwan
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[email protected]>
Feng-chu Ho
Yan-Shuei Elementary School of Tainan County, Taiwan