MODELING THE DEVELOPMENT OF THE CONCEPT OF FUNCTION[1]
|
Mindy
Kalchman OISE/University
of Toronto |
Robbie
Case[2] OISE/University
of Toronto |
As children move from a
process-based to an object-based understanding of mathematical functions, we
suggest they pass through three phases of processing: (1) procedural; (2)
interval; and (3) object-based. Analysis of a microgenetic instructional study
showed that each of these processes is developed through a particular stage in
an experimental curriculum. We present here the (i) psychological processes
that characterize each phase of processing, (ii) the component of the
curriculum that supports each phase, and (iii) examples of students’ reasoning
within each phase of processing. We suggest that standard curricular approaches
to the teaching of functions do not adequately address the many psychological
connections children must make in understanding functions.
Much
has been written about the importance of the process to object conceptual shift for developing a deep
understanding of mathematical functions (e.g. Sfard, 1992). However, analyses
that characterize students’ psychological processes when going from a process-
to object-based understanding are lacking. In this paper, we present the
results of such an analysis.
We
consider a process understanding to be one where students use a procedural or
computational (Sfard, 1992) approach for deriving pairs of values, which are
then used to represent a function in a tabular and graphical way; and we regard
an object-based understanding as one where a function is conceptualized as
something to which processes may be applied (Harel & Dubinsky, 1992). Three
phases of psychological processing have emerged from our analyses of children
going from a process to object understanding of function: (1) procedural; (2)
interval; and (3) object-based. Because we see a circular relationship between
structural modeling and curricular design (Kalchman, Moss & Case, in
press), we derive our cognitive models from the analysis of instructional
studies (Kalchman & Case, 1998, 1999).
Three sixth-grade
female students attending a university’s laboratory school participated in the
study. They spent nine 45-minute sessions engaged in an experimental
instructional unit on functions. The first five days were spent in a classroom
setting; the next four were spent using spreadsheet technology; and the final
day was spent meeting on a one-to-one basis. On days 1, 2, 5, and 9, students
were asked the same set of items either at the beginning or the end of the
session. This repeated questioning was to identify at which points in the
curriculum students made conceptual gains. Students worked at their own pace on
the computer for days 6, 7, and 8.
Phases of
Processing: Curriculum, Phase Attributes, and Students’ Thinking
Table 1 summarizes
the central curricular features that support each phase of processing, the
cognitive attributes that characterize children’s thinking in each phase, and
examples of how children conceptualized a particular item while working within
each level of understanding. The item analyzed was one where students were
shown a graph of the function y = x + 7, seen only in the upper right
quadrant of the Cartesian grid. Both axes were labeled from 0 to 10, with the
points (0, 7), (1, 8), and (2, 9) plotted within the line. Students were asked
to give an equation for a function that would pass through the linear function
seen, and then to explain their reasoning.
The Procedural
Phase
In the first phase of
processing we identified, children go from no understanding of what a function
is to a procedural understanding. They become able to generate a table of
numbers and a graph for any function of the form y = f(x) from x = 0 to x = n (where n is a positive integer). We
support the development of this understanding by beginning instruction with the
context of a walkathon, where a specified amount of money earned per kilometer
walked is symbolized, calculated, and then graphed. For example, if the
sponsorship arrangement is earning $4.00 per kilometer walked, students first
construct an algebraic symbolic representation for the rule, e.g., $ = km
* 4. They then calculate the amount of money earned at each kilometer walked by
multiplying each kilometer by 4, e.g., 0 km = $0.00, 1 km = $4.00, 2 km =
$8.00, etc. They then represent the results of these calculations in a tabular
fashion. When the table has been completed for 10 km walked, students place
markers at each coordinate point on a Cartesian
Table 1. Phases of processing, their attributes,
and students’ thinking.
|
Phase
of Processing |
Central
Curricular Features |
Psychological
Attributes |
Example
of Students’ Thinking |
|
Procedural
|
Walkathon
-- calculation of dollars earned per kilometer walked; creating a table of
values; and plotting discrete points on a grid |
Mastery
of calculating numerical values for the dependent variable given a set of
values for the independent variable and arranging them in a tabular form Mastery
of plotting pairs of x and y coordinates that correspond to the
pairs of values found in the above table Recognition
that each pair of numerical values is associated with a unique point on the
graph and vice-versa Mastery
of heuristics for moving from a set of points found in a tabular form or
graph to the rule that generated them |
“Let's say someone paid you $9.00
for every kilometer you walked. So, I was thinking the equation could be x
times 9 equals money. At 0 kilometers you have $0 and then [when] x is 1 … 1
times 9 is 9 so y = $9, which has already crossed." |
|
Interval
|
Calculation
of numeric intervals between values of the dependent variable in both tabular
and graphic representations |
Recognizing
particular properties of intervals between successive y values as a constant increasing value for functions such as y = 4x (with 4 being that constant value) Recognizing
that constant intervals between y
values on a graph always result in straight lines Connecting
a concrete mathematical rule, the pattern in the sequence of numbers it
generates (e.g.., a constant interval), and the pattern seen in the graph
(e.g., a straight-line) Repeating
above for properties such as y-intercept,
non-linearity, and the negative or positive slope or “direction” of any
function |
“I started off with 3, … and each
time I’m going to move up 6. So, it’s
going to be a straight line because it’s going up by the same amount and it’s
going to be steep because it has a slope of 6 and it does pass through (the existing
line) The equation is y = x times 6 + 3.” |
|
Object-Based
|
Computer
spreadsheet activities where students operate on y = x or y = x2 |
use of
the graphic representation of y = x and the y = x2 as
mental referents, or concrete objects,
for abstracting properties and features of a function and judging the
“steepness” of a line, degree of a curve, “directionality”, and y-intercept |
“y = x times 5. It goes up by the same amount and it’s a straight line
and it’s steep enough because it goes up by 5 each time. You could also do
other steep ones but I just did this one.” |
grid. The markers are then joined to form a line. Several different
rules of sponsorship are explored in this way, including those that produce
curved lines (e.g., the amount of money earned is equal to the number of
kilometers walked times itself [$ = km2]), and those that involve
initial "starting amounts," which correspond to the y - intercept (e.g., one is given a
starting bonus of $5.00 and still earns $4.00 per kilometer [$ = 5 + 4 * km]).
The psychological attributes of this phase
are summarized in the middle column of the first row of Table 1. In this phase,
students work from a procedural standpoint. That is, they perform calculations
on the independent variable in order to obtain values for the dependent
variable and then organize their results in first a tabular and then graphic
way. General properties of these representations are not abstracted, but the
link between the coordinate pairs found on the graph and the pairs of values
recorded in the table is recognized and understood to have been generated from
a specific rule such as multiplying each x
by 4.
When asked to
provide an equation for a function that would pass through the one described
above, prior to instruction students made a series of dots at the coordinates
(0,9), (1,8), (2,7), and (3,6) but did not express the meaning of these dots as
any sort of functional relationship. Then, on day 2, CP explained: "Let's say someone paid you $9.00 for every
kilometer you walked. So, … the equation could be x times 9 equals money. At 0
kilometers you have $0 and then [when] x is 1…1 times 9 is 9 so y equals $9,
which has already crossed." With each calculation for the dependent
variable, CP plotted the coordinates on the grid.
The Interval
Phase
The second phase of
processing is one where students develop an interval
understanding of a function by noticing the second-order features found in the
tables and graphs. These second order features include the value of the numeric
and coordinate jumps found between
successive y values. The second row
of Table 1 describes the particulars of this phase.
We promote an
interval understanding of function by having students investigate what
generally determines the linearity, degree of “steepness”, and y – intercept of a function. For
example, in the tabular expression of the function $ = km
* 4, students add a column to the right of the $ column and write in the difference between successive values,
which in this case is consistently 4. Students also graph functions by
sketching the graph in intervals by moving their pencils “up by” for example 4
as they move over one unit, rather than as discrete points. Intervals are also
calculated for non-linear functions, and students realize that the degree of
the “steepness” of a curve is related to the span of the intervals. Students
eventually generalize that straight-line functions always have constant
intervals between y values, and
curved lines do not.
By noting second-order
properties found in the tables and graphs of particular functions (i.e.,
numeric or graphic jumps or
intervals), students abstract general features of functions such as (a)
linearity, (b) specific slopes, (c) the degree of an exponent, and (c) the y-intercept.
In response to the item,
on day 2, HJ explained that “if your
starter offer was 1 and your next point was 6, so you go up by 5…then your next
one is 10 so each time you go up by 5.” Her interval understanding appears
incomplete at this point, however, because she did not include the $1.00
starting value in the equation or mental computation. This omission may have
been due to a still emerging understanding of how to connect the mathematical
rule, the pattern in the sequence of numbers it generates, and the pattern seen
in the graph. On day 5, however, HJ seemed to
make these connections: “I started off
with 3…and each time I’m going to move up 6.
So, it’s going to be a straight line because it’s going up by the same
amount and it’s going to be steep because it has a slope of 6 and it does pass
through. The equation is y = x times 6 + 3.”
Object-Based Phase
In this phase, students construct and abstract third-order properties such as slope and y-intercept from the second-order properties already noted. The curricular feature that supports this phase is students’ use of spreadsheet technology. Students work with a pre-configured computer screen, which displays the graph of y = x (or y = x2) on the left-hand side, and a spreadsheet with the corresponding table of values (found in columns X and Y) on the right. A console displays the equation of the function. On the spreadsheet, students are asked to change individual parameters of the function (slope or y-intercept) in order to move y = x through randomly placed pre-plotted points. All actions carried out are reflected instantly and automatically in the graph and in the numeric pattern found in the Y column. Then, using columns to the right of Y, students program their own functions with given specifications such as a slope greater than that of the original line (y = x) and a y-intercept < 0. Students also carry out these sorts of activities for curved lines.
Using y = x
and y = x2 as concrete mental objects, students are able to operate on the functions as entities
in order to produce new functions that have, for example, constant intervals
greater than 1, and thus, are steep relative to y = x. Others with
constant fractional intervals are relatively flat. Students may operate on y = x2
to produce functions whose curves are steeper than y = x2 by
making the exponent larger than 2, the coefficient greater than 1, or some
combination of the two. Likewise, for functions that curve down, the degree of
“steepness” is still described by the exponent and the coefficient, and the
“down by” is defined by the xn
being multiplied by a negative coefficient. With respect to the y – intercept, students see the addition
or subtraction of a constant value as a means to qualitatively and
quantitatively translate the base function.
On the final day, students
seemed to be using the given line as a mental referent for determining a
suitable equation for a function that would pass through the one given. HJ drew
a straight line starting from the origin and passing through (1,5) and (2, 10).
She explained: “y = x times 5. It goes up
by the same amount and it’s a straight line and it’s steep enough because it
goes up by 5 each time. You could also do other steep ones but I just did this
one.”
We attribute students’
success with this difficult topic to their engagement with the innovative
curriculum, which allowed them to work within the walkathon context to master
the necessary sorts of procedures. First, they calculated values for the
dependent variable from values for the independent variable. Then they
calculated the intervals between successive values for the dependent variable –
a calculation that essentially defines the general shape (i.e., linear or
curved) of a function. As students moved among tables, graphs, and symbolic and
verbal rules, they recognized and generalized certain important features of a
function such as y-intercept, degree
of “steepness” and linearity. We think that experts in mathematics
underestimate the time it takes to make the many psychological connections
proposed here and neglect to provide an adequate body of examples necessary for
abstracting the entailments of particular functions.
References
Harel, G. & Dubinsky, E.
(Eds.) (1992). The concept of function:
Aspects of epistemology and
pedagogy. West laFayette, IN: Mathematical Association of America.
Kalchman, M. & Case, R.
(1998).Teaching mathematical functions in primary and middle school: An
approach based on Neo-Piagetian theory. Scientia
Pedagogica Experimentalis, 35(1),
7-53.
Kalchman, M. & Case, R.
(1999). Diversifying the curriculum in a mathematics classroom streamed for
high-ability learners: A necessity unassumed. School Science and Mathematics, 99(6), 320-329.
Kalchman, M., Moss, J. & Case, R. (in press). Psychological Models for the Development of Mathematical Understanding: Rational Numbers and Functions. To appear in S. Carver & D. Klahr (Eds.), Cognition and instruction: 25 years of progress. Mahweh, NJ: LEA.
Sfard, A. (1992).
Operational origins of mathematical origins and the quandary of reification:
The case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of
epistemology and pedagogy. West laFayette, IN: Mathematical Association of
America, 59-84.