PROSPECTIVE SECONDARY MATHEMATICS TEACHERS’
KNOWLEDGE OF MATHEMATICS AND
THEIR ABILITY TO UNDERSTAND NONSTANDARD REASONING
Daniel Siebert
Brigham Young University
dsiebert@math.byu.edu
This study examines eight PSTs’ ability to make sense of
nonstandard explanations of division of fractions problems after receiving six
weeks of instruction on this topic. The
PSTs have extreme difficulty understanding the nonstandard explanations. Three
categories are proposed to describe PSTs’ newly acquired knowledge. This study
suggests that PSTs’ knowledge of the mathematics they are going to teach is
problematic, and not easily remedied.
Past studies of prospective secondary
mathematics teachers (PSTs) have shown that PSTs have critical gaps in their
understanding of secondary school mathematics topics (Ball, 1990b; Cooney, Wilson, Albright, & Chauvot, 1998;
Even, 1993). In particular, Ball (1990a; 1990b) showed that PSTs have difficulty creating
representations for division of fractions or articulating the connection
between division of fractions and whole number division. A question currently
facing teacher educators is the following: How difficult is it to repair the
gaps in PSTs’ knowledge of a mathematics topic such as division of fractions so
that PSTs will have the mathematical understanding necessary to engage in
instruction that focuses on student reasoning and sense-making? One might
assume that since many PSTs are mathematics majors in college, helping them fix
the gaps in their knowledge of division of fractions—a mathematics topic that
is typically taught for the first time in elementary school, but is nonetheless
reviewed repeatedly in junior high and high school—would be a relatively quick
and easy task. However, the research reported in this paper suggests otherwise.
Surprisingly, even after the eight PSTs in this study learned multiple ways of
representing and solving division of fractions problems, most still lacked the
mathematical understanding to make sense of nonstandard explanations of
division of fractions. This study examines the mathematical understandings of
the eight PSTs for clues as to why understanding nonstandard reasoning was so
difficult for them, even after receiving substantial instruction to develop rich
conceptions of division of fractions. In particular, this paper identifies
characteristics of the PSTs’ mathematical knowledge that seemed to influence
their ability to make sense of nonstandard reasoning about division of
fractions problems.
Method and
Data Sources
This study was part of a larger
qualitative study designed to investigate the changes that occurred in PSTs’
knowledge of and beliefs about mathematics as the PSTs participated in a
semester-long capstone mathematics course specifically designed to challenge
and invoke change in their mathematical knowledge and beliefs (Siebert, 2000). Eight subjects were recruited to participate in the
study, representing approximately a fourth of the class. All eight subjects
were either juniors or seniors majoring in mathematics, and typically had two
or fewer math courses remaining to take before graduation.
The first unit of the capstone course focused on division of
fractions, and lasted approximately six weeks. During this unit, the instructor
engaged the PSTs in developing meaning for division of fractions and in
identifying and articulating the important mathematical concepts underlying
this topic. PSTs were required to draw pictures to both solve division of
fractions problems and to justify the invert and multiply (IM) rule, write
story problems for division of fractions number sentences, analyze and discuss
videotaped and written accounts of student thinking about division of
fractions, and create conceptual analyses of division of fractions.
During the division of fractions unit, I
interviewed the subjects for 45 minutes each week to determine what changes
they were making in their knowledge of division of fractions and their beliefs
about mathematics. I also videotaped the class sessions and collected copies of
their written assignments. The qualitative analyses of the data was guided by
my desire to create coherent, theoretical models of the PSTs’ knowledge of and
beliefs about mathematics. Like Simon and Tzur (1999), I assumed that the subjects in my study were acting
in ways that were sensible to them. Thus, in my accounts of their knowledge and
beliefs, I tried to capture the sensibility and coherence the PSTs perceived in
what they did and said.
Results
The unit on division of fractions was
very successful in helping the PSTs develop new understandings of division of
fractions. When the PSTs began the unit, none of them could provide an
explanation for why the IM rule worked or draw a picture of x ÷ 1/y,
where x and y are nonzero whole numbers. The PSTs also had difficulty writing
story problems for division of fractions number sentences. By the end of the
unit, however, all eight PSTs could use pictures to solve division of fractions
problems, illustrate the IM rule using pictures, create story problems for
division of fractions number sentences, and identify and explain several of the
important concepts underlying division of fractions.
To illustrate the type of understanding
the PSTs had developed for the topic of division of fractions, consider, for
example, Josie’s explanation for why the IM rule works for the division of
fractions problem 5 ÷ 2/3. Josie recognized that one way to think
about this problem was to ask the question, “How many 2/3s are in 5?” To answer
this question, Josie first found how many 2/3s were in 1 by drawing and
reasoning about the following diagram:
Josie discovered that there are 1 1/2, or 3/2,
two-thirds in 1. She then made the
following argument for why 5 ÷ 2/3 = 5 ´ 3/2:
There’s three-halves of two-thirds in one…. If you have five wholes, um, you multiply by the three-halves, because that’s how many you have in one, so it’s five times that amount…. I’m thinking of it visually, like, OK, if I had three-halves of something, one and a half of something, and I have five of those [three-halves of] something, then how many will I have? It’s a multiplication problem.
Josie’s argument for why the IM rule works consisted
of first noting that 3/2, the reciprocal of 2/3 , is the number of 2/3s in 1.
Since there are 5 ones in the dividend, and there are 3/2 two-thirds in each 1,
it follows that there are 5 ´ 3/2 two-thirds in
5. Thus, 5 ÷ 2/3 = 5 ´ 3/2. This line of
reasoning about the IM rule was common in the class, and all eight PSTs
demonstrated this reasoning during the interviews.
Despite the progress the PSTs made during
the unit, when I asked them in the last week of the unit to grade hypothetical
students’ explanations for the division of fractions problem 1/4 ÷ 5/8,
only one of the PSTs recognized that the following two explanations were
correct:
Carlos: There are eight 1/8s in 1. There are only one-fifth as many 5/8s in 1 as there are 1/8s in 1. So there are eight-fifths 5/8s in 1. There are only one-fourth as many 5/8s in 1/4 as there are in 1. So the answer is 1/4 ´ 8/5 = 2/5.
Kristin:
There are two one-eighths in 1/4. I
need five one-eighths to make 5/8. So
1/4 has two of the five one-eighths that I need to make 5/8. So 1/4 ÷ 5/8 = 2/5.
This result was surprising to myself and the course
instructor; both of us had felt this task was well within the grasp of the
PSTs, particularly since the ideas in these explanations had arisen in class
during the six weeks of instruction. The PSTs’ poor performance on this task
revealed important characteristics of their newly acquired knowledge of
division of fractions, namely that this knowledge was often rigid, fragile, and
missing pieces of conceptual understanding.
Rigidity
in PSTs’ Solution Methods for Division of Fractions. Six of the eight PSTs had difficulty interpreting
the first three sentences in Carlos’s explanation. This difficulty seemed to be
caused at least in part by Carlos’s use of a different multiplicative
comparison than what the PSTs were accustomed to using to solve division of
fractions problems. To arrive at a fact like “there are eight-fifths 5/8s in
1,” the PSTs had learned to draw a picture to determine how many copies of 5/8s
fit in 1 whole, as Josie did above. This involved making a comparison between
the relative sizes of 5/8 and 1. Carlos’s reasoning, in contrast, compared the
relative sizes of 1/8 to 5/8, and used this relationship to derive the number
of 5/8s in 1. Three of the six PSTs who struggled to make sense of the first
three sentences of Carlos’s explanation explicitly noted that Carlos was making
a multiplicative comparison between the number of eighths and the number of
five-eighths in one, but they could see no way for Carlos to know that there
were 1/5 as many 5/8s in 1 as there were 1/8s in 1 without first actually finding the number of 5/8s in 1:
Roberto: But there are only one-fifth as many five-eighths, to me it’s like, OK, how did he know that? How do you know that without, without doing a proportion? So he said, well, how many five-eighths, how many five-eighths are in one? Well, eight-fifths. Then you could see that the proportion of eight-fifths to eight equals one-fifth. So then you say, yeah, there’s one-fifth as many.
The PSTs’
skill at solving this problem in a particular way seemed to get in their way of
seeing a different approach to the problem.
Fragility
of Their Knowledge of Division of Fractions. As the PSTs engaged in making sense of nonstandard reasoning, their
knowledge of division of fractions seemed to be challenged in a unique way.
Occasionally, their attempts to make sense of nonstandard reasoning seemed to
cause them to temporarily loose sight of correct reasoning about division of
fractions. For example, while trying to make sense of Kristin’s reasoning, two
of the PSTs spontaneously began to investigate how many 2/8s were in 5/8, which
would yield the answer to 5/8 ÷ 1/4, not 1/4 ÷ 5/8.
Consider, for example, Julie’s reasoning about Kristin’s explanation:
Julie: I don’t understand where Kristin is getting fifths [in the answer]. There’s no explanation. I was trying to reason through it, and think there are five parts here [in the 5/8], but if you’re grouping it in [groups of] two [one-eighths], then there would be one of two left over, instead of one of five.
Interviewer: Why are you grouping it in two?
Julie: Because it takes two eighths to get into a fourth. So how many eighths do you need out of 5/8s to get in, that will go into a fourth?
One might argue that perhaps Julie didn’t
know how to reason about a problem where the dividend is smaller than the
divisor, but Julie had demonstrated earlier in the interview that she could
interpret remainders in division of fractions problems, thus indicating she
understood that when the divisor is greater than the dividend, the meaning for
division switches from “how many of
the divisor fit in the dividend” to “how much
of the divisor fits in the dividend.” A more plausible explanation is that she
got caught up in making sense of Kristin’s comparison of 2/8 and 5/8 and lost
track of how the division problem should be interpreted. The task of
coordinating the acts of (a) making sense of someone else’s reasoning and (b)
maintaining a correct understanding of the problem seemed to put too great of a
strain on her understanding of division of fractions.
Missing
Pieces in Conceptual Understanding.
Six of the PSTs had difficulty making sense of the last two sentences in
Carlos’s explanation. This difficulty may have been caused by their tendency to
view multiplication as repeated addition. The PSTs developed their explanations
for the IM rule by examining division of fractions problems involving whole
number dividends. Like Josie, they tended to justify the step of multiplication
in the IM rule by reasoning that for each 1 in the dividend, they had a certain
number of the divisor, and instead of repeatedly adding this number to itself,
they could multiply that number by the dividend. For 5 ÷ 2/3, this
meant that for each 1 in 5, there were 3/2 two-thirds. To get the answer, one
could multiply 5 times 3/2 instead of adding 3/2 five times. When the PSTs were
asked to justify the IM rule for non-whole number dividends, they had
difficulty justifying the step of multiplication; they often tried
(unsuccessfully) to use the repeated addition meaning for multiplication in
their explanations, despite that this interpretation for multiplication breaks
down when neither factor is a whole number. It is likely that Carlos’s
explanation for why he multiplied did not match with their understanding of
what multiplication means, and thus did not make sense to them.
Conclusion
By studying how PSTs’ respond to
nonstandard reasoning, this research offers insight into important
characteristics of PSTs’ newly acquired mathematical knowledge, namely that
this knowledge can be rigid, fragile, and missing pieces of conceptual
understanding. The difficulty PSTs experienced in making sense of nonstandard
explanations, despite receiving significant instruction and developing new
understandings, suggests that helping PSTs fix the gaps in their understanding
of the mathematics they will teach can be a formidable task.
References
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