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8
Teaching and Learning Functions
Mindy Kalchman and Kenneth R. Koedinger
This chapter focuses on teaching and learning mathematical functions.1
Functions are all around us, though students do not always realize this. For
example, a functional relationship between quantities is at play when we
are paying for gasoline by the gallon or fruit by the pound. We need func-
tions for financial plans so we can calculate such things as accrued income
and interest. Functions are important as well to interpretations of local and
world demographics and population growth, which are critical for economic
planning and development. Functions are even found in such familiar set-
tings as baseball statistics and metric conversions.
Algebraic tools allow us to express these functional relationships very
efficiently; find the value of one thing (such as the gas price) when we know
the value of the other (the number of gallons); and display a relationship
visually in a way that allows us to quickly grasp the direction, magnitude,
and rate of change in one variable over a range of values of the other. For
simple problems such as determining gas prices, students’ existing knowl-
edge of multiplication will usually allow them to calculate the cost for a
specific amount of gas once they know the price per gallon (say, $2) with no
problem. Students know that 2 gallons cost $4, 3 gallons cost $6, 4 gallons
cost $8, and so on. While we can list each set of values, it is very efficient to
say that for all values in gallons (which we call x by convention), the total
cost (which we call y by convention), is equal to 2x. Writing y = 2x is a
simple way of saying a great deal.
As functional relationships become more complex, as in the growth of a
population or the accumulation of interest over time, solutions are not so
easily calculated because the base changes each period. In these situations,
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352 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
algebraic tools allow highly complex problems to be solved and displayed
in a way that provides a powerful image of change over time.
Many students would be more than a little surprised at this description.
Few students view algebra as a powerful toolkit that allows them to solve
complex problems much more easily. Rather, they regard the algebra itself
as the problem, and the toolkit as hopelessly complex. This result is not
surprising given that algebra is often taught in ways that violate all three
principles of learning set forth in How People Learn and highlighted in this
volume.
The first principle suggests the importance of building new knowledge
on the foundation of students’ existing knowledge and understanding. Be-
cause students have many encounters with functional relationships in their
everyday lives, they bring a great deal of relevant knowledge to the class-
room. That knowledge can help students reason carefully through algebra
problems. Box 8-1 suggests that a problem described in its everyday mani-
festation can be solved by many more students than the same problem
presented only as a mathematical equation. Yet if the existing mathematics
understandings students bring to the classroom are not linked to formal
algebra learning, they will not be available to support new learning.
The second principle of How People Learn argues that students need a
strong conceptual understanding of function as well as procedural fluency.
The new and very central concept introduced with functions is that of a
dependent relationship: the value of one thing depends on, is determined
by, or is a function of another. The kinds of problems we are dealing with
no longer are focused on determining a specific value (the cost of 5 gallons
of gas). They are now focused on the rule or expression that tells us how
one thing (cost) is related to another (amount of gas). A “function” is for-
mally defined in mathematics as “a set of ordered pairs of numbers (x, y)
such that to each value of the first variable (x) there corresponds a unique
value of the second variable (y).”2 Such a definition, while true, does not
signal to students that they are beginning to learn about a new class of
problems in which the value of one thing is determined by the value of
another, and the rule that tells them how they are related.
Within mathematics education, function has come to have a broader
interpretation that refers not only to the formal definition, but also to the
multiple ways in which functions can be written and described.3 Common
ways of describing functions include tables, graphs, algebraic symbols, words,
and problem situations. Each of these representations describes how the
value of one variable is determined by the value of another. For instance, in
a verbal problem situation such as “you get two dollars for every kilometer
you walk in a walkathon,” the dollars earned depend on, are determined by,
or are a function of the distance walked. Conceptually, students need to
understand that these are different ways of describing the same relationship.
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TEACHING AND LEARNING FUNCTIONS
Good instruction is not just about developing students’ facility with per-
forming various procedures, such as finding the value of y given x or creat-
ing a graph given an equation. Instruction should also help students de-
velop a conceptual understanding of function, the ability to represent a
function in a variety of ways, and fluency in moving among multiple repre-
sentations of functions. The slope of the line as represented in an equation,
for example, should have a “meaning” in the verbal description of the rela-
tionship between two variables, as well as a visual representation on a graph.
The third principle of How People Learn suggests the importance of
students’ engaging in metacognitive processes, monitoring their understand-
ing as they go. Because mathematical relationships are generalized in alge-
bra, students must operate at a higher level of abstraction than is typical of
the mathematics they have generally encountered previously. At all levels of
mathematics, students need to be engaged in monitoring their problem solv-
ing and reflecting on their solutions and strategies. But the metacognitive
engagement is particularly important as mathematics becomes more abstract,
because students will have few clues even when a solution has gone terribly
awry if they are not actively engaged in sense making.
When students’ conceptual understanding and metacognitive monitor-
ing are weak, their efforts to solve even fairly simple algebra problems can,
and often do, fail. Consider the problem in Figure 8-1a. How might students
approach and respond to this problem? What graph-reading and table-build-
ing skills are required? Are such skills sufficient for a correct solution? If
students lack a conceptual understanding of linear function, what errors
might they make? Figure 8-1b shows an example student solution.
What skills does this student exhibit? What does this student understand
and not understand about functions? This student has shown that he knows
how to construct a table of values and knows how to record in that table
coordinate points he has determined to be on the graph. He also clearly
recalls that an algorithm for finding the slope of the function is dividing the
change in y(∆y) by the change in x(∆x). There are, however, significant
problems with this solution that reveal this student’s weak conceptual un-
derstanding of functions.
Problem: Make a table of values that would produce the function
seen on page 356.
First, and most superficially, the student (likely carelessly) mislabeled
the coordinate for the y-intercept (0, 3) rather than (0, –3). This led him to
make an error in calculating ∆y by subtracting 0 from 3 rather than from –3.
In so doing, he arrived at a value for the slope of the function that was
negative—an impossible solution given that the graph is of an increasing
linear function. This slip, by itself, is of less concern than the fact that the
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354 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
Linking Formal Mathematical Understanding to Informal
BOX 8-1
Reasoning
Which of these problems is most difficult for a beginning algebra
student?
Story Problem
When Ted got home from his waiter job, he multiplied his hourly wage by
the 6 hours he worked that day. Then he added the $66 he made in tips
and found he had earned $81.90. How much does Ted make per hour?
Word Problem
Starting with some number, if I multiply it by 6 and then add 66, I get 81.9.
What number did I start with?
Equation
Solve for x:
x * 6 + 66 = 81.90
Most teachers and researchers predict that students will have more diffi-
culty correctly solving the story or word problem than the equation.4 They
might explain this expectation by saying that a student needs to read the
verbal problems (story and word) and then translate them into the equa-
tion. In fact, research investigating urban high school students’ perfor-
mance on such problems found that on average, they scored 66 percent
on the story problem, 62 percent on the word problem, and only 43 per-
cent on the equation.5 In other words, students were more likely to solve
the verbal problems correctly than the equation. Investigating students’
written work helps explain why.
Students often solved the verbal problems without using the equa-
tion. For instance, some students used a generate-and-test strategy: They
estimated a value for the hourly rate (e.g., $4/hour), computed the corre-
sponding pay (e.g., $90), compared it against the given value ($81.90),
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TEACHING AND LEARNING FUNCTIONS
and repeated as needed. Other students used a more efficient unwind or
working backwards strategy. They started with the final value of 81.9 and
subtracted 66 to undo the last step of adding 66. Then they took the
resulting 15.9 and divided by 6 to undo the first step of multiplying by 6.
These strategies made the verbal problems easier than expected. But
why were the equations difficult for students? Although experts in alge-
bra may believe no reading is involved in equation solving, students do in
fact need to learn how to read equations. The majority of student errors
on equations can be attributed to difficulties in correctly comprehending
the meaning of the equation.6 In the above equation, for example, many
students added 6 and 66, but no student did so on the verbal problems.
Besides providing some insight into how students think about alge-
braic problem solving, these studies illustrate how experts in an area such
as algebra may have an “expert blind spot” for learning challenges begin-
ners may experience. An expert blind spot occurs when someone skilled
in an area overestimates the ease of learning its formalisms or jargon and
underestimates learners’ informal understanding of its key ideas. As a
result, too little attention is paid to linking formal mathematical under-
standing to informal reasoning. Looking closely at students’ work, the
strategies they employ, and the errors they make, and even comparing
their performance on similar kinds of problems, are some of the ways we
can get past such blind spots and our natural tendency to think students
think as we do.
Such studies of student thinking contributed to the creation of a tech-
nology-enhanced algebra course, originally Pump Algebra Tutor and now
Cognitive Tutor Algebra.7 That course includes an intelligent tutor that
provides students with individualized assistance as they use multiple rep-
resentations (words, tables, graphs, and equations) to analyze real-world
problem situations. Numerous classroom studies have shown that this
course significantly improves student achievement relative to alternative
algebra courses (see www.carnegielearning.com/research). The course,
which was based on basic research on learning science, is now in use in
over 1,500 schools.
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356 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
a
b
FIGURE 8-1
student did not recognize the inconsistency between the positive slope of
the line and the negative slope in the equation. Even good mathematicians
could make such a mistake, but they would likely monitor their work as they
went along or reflect on the plausibility of the answer and detect the incon-
sistency. This student could have caught and corrected his error had he
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TEACHING AND LEARNING FUNCTIONS
acquired both fluency in interpreting the slope of a function from its equa-
tion (i.e., to see that it represents a decreasing function) and a reflective
strategy for comparing features of different representations.
A second, more fundamental error in the student’s solution was that the
table of values does not represent a linear function. That is, there is not a
constant change in y for every unit change in x. The first three coordinates in
the student’s table were linear, but he then recorded (2.5, 0) as the fourth
coordinate pair rather than (3, 0), which would have made the function
linear. He appears to have estimated and recorded coordinate points by
visually reading them off the graph without regard for whether the final
table embodied linearity. Furthermore, the student did not realize that the
–3
equation he produced, y = x – 3 , translates not only into a decreasing line,
2.5 –3
but also into a table of numbers that decreases by for every positive unit
2.5
change in x.
At a surface level, this student’s solution reflects some weaknesses in
procedural knowledge, namely, getting the sign wrong on the y-intercept
and imprecisely reading x-y coordinates off the graph. More important, how-
ever, these surface errors reflect a deeper weakness in the student’s concep-
tual understanding of function. The student either did not have or did not
apply knowledge for interpreting key features (e.g., increasing or decreas-
ing) of different function representations (e.g., graph, equation, table) and
for using strategies for checking the consistency of these interpretations (e.g.,
all should be increasing). In general, the student’s work on this problem
reflects an incomplete conceptual framework for linear functions, one that
does not provide a solid foundation for fluid and flexible movement among
a function’s representations.
This student’s work is representative of the difficulties many secondary-
level students have with such a problem after completing a traditional text-
book unit on functions. In a study of learning and teaching functions, about
25 percent of students taking ninth- and eleventh-grade advanced math-
ematics courses made errors of this type—that is, providing a table of values
that does not reflect a constant slope—following instruction on functions.8
This performance contrasts with that of ninth- and eleventh-grade math-
ematics students who solved this same problem after receiving instruction
based on the curriculum described in this chapter. This group of students
had an 88 percent success rate on the problem. Because these students had
developed a deeper understanding of the concept of function, they knew
that the y-values in a table must change by the same amount for every unit
change in x for the function to be linear. The example in Figure 8-1c shows
such thinking.
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358 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
c
FIGURE 8-1
Problem: Make a table of values that would produce the function
seen above.
This student identified a possible y-intercept based on a reasonable
scale for the y-axis. She then labeled the x- and y-axes, from which she
determined coordinate pairs from the graph and recorded them in a table of
values. She determined and recorded values that show a constant increase
in y for every positive unit change in x. She also derived an equation for the
function that not only corresponds to both the graph and the table, but also
represents a linear relationship between x and y.
How might one teach to achieve this kind of understanding? The
goal of this chapter is to illustrate approaches to teaching functions that
foster deep understanding and mathematical fluency. We emphasize the
importance of designing thoughtful instructional approaches and curricula
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TEACHING AND LEARNING FUNCTIONS
that reflect the principles of How People Learn (as outlined in Chapter 1), as
well as recent research on what it means to learn and understand functions
in particular. We first describe our approach to addressing each of the three
principles. We then provide three sample lessons that emphasize those prin-
ciples in sequence. We hope that these examples provide interesting activi-
ties to try with students. More important, these activities incorporate impor-
tant discoveries about student learning that teachers can use to design other
instructional activities to achieve the same goals.
ADDRESSING THE THREE PRINCIPLES
Principle #1: Building on Prior Knowledge
Principle 1 emphasizes the importance of students and teachers con-
tinually making links between students’ experiences outside the mathemat-
ics classroom and their school learning experiences. The understandings
students bring to the classroom can be viewed in two ways: as their every-
day, informal, experiential, out-of-school knowledge, and as their school-
based or “instructional” knowledge. In the instructional approach illustrated
here, students are introduced to function and its multiple representations by
having their prior experiences and knowledge engaged in the context of a
walkathon. This particular context was chosen because (1) students are fa-
miliar with money and distance as variable quantities, (2) they understand
the contingency relationship between the variables, and (3) they are inter-
ested in and motivated by the rate at which money is earned.
The use of a powerful instructional context, which we call a “bridging
context,” is crucial here. We use this term because the context serves to
bridge students’ numeric (equations) and spatial (graphic) understandings
and to link their everyday experiences to lessons in the mathematics class-
room. Following is an example of a classroom interaction that occurred
during students’ first lesson on functions, showing how use of the walkathon
context as an introduction to functions in multiple forms—real-world situa-
tion (walkathon), table, graph, verbal (“$1.00 for each kilometer”), situation-
specific symbols ($ = 1 * km), and generic symbolic (y = x * 1)—accom-
plishes these bridging goals. Figures 8-2a through 8-2c show changes in the
whiteboard as the lesson proceeded.
Teacher What we’re looking at is, we’re looking at what
we do to numbers, to one set of numbers, to
get other numbers. . . . So how many of you
have done something like a walkathon? A
readathon? A swimathon? A bikeathon?
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360 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
[Students raise their hands or nod.] So most of
you…So I would say “Hi Tom [talking to a
student in class], I’m going to raise money for
such and such a charity and I’m going to walk
ten kilometers.”
Tom OK.
Teacher Say you’re gonna sponsor me one dollar for
every kilometer that I walk. So that’s sort of the
first way that we can think about a function. It’s
a rule. One dollar for every kilometer walked.
So you have one dollar for each kilometer
[writing “$1.00 for each kilometer” on the
board while saying it]. So then what I do is I
need to calculate how much money I’m gonna
earn. And I have to start somewhere. So at
zero kilometers how much money do I have
Tom? How much are you gonna pay me if I
collapse at the starting line? [Fills in the
number 0 in the left-hand column of a table
labeled “km”; the right-hand column is labeled
“$”.]
Tom None.
Teacher So Tom, I managed to walk one kilometer
[putting a “1” in the “km” column of the table
of values below the “0”]. . . .
Tom One dollar.
Teacher One dollar [moving to the graph]. So I’m going
to go over one kilometer and up one dollar
[see Figure 8-2a].
FIGURE 8-2a Graphing a point from the
table: “Over by one kilometer and up by one
dollar.” The teacher uses everyday English
(“up by”) and maintains connection with the
situation by incorporating the units “kilometer”
and “dollar.”
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TEACHING AND LEARNING FUNCTIONS
FIGURE 8-2b The teacher and
students construct the table and
graph point by point, and a line
is then drawn.
[Students continue to provide the dollar
amounts for each of the successive kilometer
values. Simple as it is, students are encour-
aged to describe the computation—”I multiply
two kilometers by one to get two dollars.” The
teacher fills in the table and graphs each
coordinate pair. [The board is now as shown in
Figure 8-2b.]
Teacher Now, what I want you to try and do, first I want
you to look at this [pointing to the table that
goes from x = 0 to x = 10 for y = x] and tell me
what’s happening here.
Melissa You, like, earn one dollar every time you go up.
Like it gets bigger by one every time.
Teacher So every time you walk one kilometer you get
one more dollar, right? [Makes “> 1” marks
between successive “$” values in the table—
see Figure 8-2c.] And if you look on the graph,
every time I walk one kilometer I get one more
dollar. [Makes “step” marks on the graph.] So
now I want to come up with an equation, I
want to come up with some way of using this
symbol [pointing to the “km” header in the
left-hand column of the table] and this symbol
[pointing to the “$” header in the right-hand
column of the table] to say the same thing, that
for every kilometer I walk, let’s put it this way,
the money I earn is gonna be equal to one
times the number of kilometers I walk. Some-
one want to try that?
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384 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
pairs of students then show their functions to the whole group. Samples of
student work are shown in Box 8-4. We also challenge students to work
“backwards,” that is, to find what the starter offer would have to be if
the slope were 10, or what the slope would have to be if the starter offer
were 20.
Summary of Principle #2 in the Context of Learning y-Intercept. We
have used a lesson on y-intercept to illustrate how students connect their
factual/procedural and conceptual knowledge within the instructional bridging
context of a walkathon. The walkathon context is intended to help students
relate their new and existing knowledge within an organized conceptual
framework in ways that facilitate efficient retrieval of that knowledge. The
idea of a “starter offer” gives students a reasonably familiar situation that
provides a context for learning y-intercept—ordinarily a relatively abstract
and difficult mathematical topic that is often confused with slope in stu-
dents’ understanding of linear function. In our approach, students still learn
the notations, symbols, words, and methods necessary for identifying the y-
intercept of a function (linear or nonlinear). However, they acquire that
knowledge in context and initially without algorithms, and with a depth of
understanding and attribution of meaning that minimize the procedural and
conceptual difficulties many students experience with the topic.
Example Lesson 3: Operating on y = x2
After the first four lessons, which take place in the classroom, students
move to a computer environment where they work with spreadsheet tech-
nology to consolidate and apply the concepts introduced in the classroom
instruction and to extend their understandings to new situations. The par-
ticular lesson we use for illustrating principle 3, developing metacognitive
skills, is the fourth in the series of computer activities.
The Lesson. Pairs of students use prepared spreadsheet files to work with a
computer screen such as that seen in Figure 8-3. Students are asked to
change specific parameters in the function y = ax2 + b to move the graph
through preplotted colored points. The file is designed so the students can
change the value of just the exponent, the coefficient of x2, the y-intercept,
or any combination of these. With each change, the graph and table of
values change instantly and automatically to reflect the numeric (tabular)
and graphic (spatial) implications of that change. For example, students are
asked to describe and record what happens to the graph and the “Y” column
of the table of values when the exponent in y = x2 is changed to 3, to 4, and
then more generally to any number greater than 2. Students are then asked
to describe and record what happens to the curve when x2 is multiplied by
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TEACHING AND LEARNING FUNCTIONS
Two Different Student Solutions to an Open-Ended Problem
BOX 8-4
a value larger than 1, smaller than 1 but greater than 0, and less than 0. They
are then asked to compare the tables and graphs for y = x2, y = 2 * x2, y = 3
* x2, y = 4 * x2, etc. and to describe in words what patterns they find. Finally,
students are asked to compare the table of values for y = 2 * x2 and y = -2 *
x2 and describe what they notice.
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FIGURE 8-3 Sample computer screen. In this configuration, students can change the value of
a, n, or b to effect immediate and automatic changes in the graph and the table. For example,
if students change the value of b, just the y-intercept of the curve will change. If students change
a or n to a positive value other than 1, the degree of steepness of the curve will change. If
students change the value of a to a negative value, the curve will come down. All graphic
patterns will be reflected in the table of values.
Students must employ effective metacognitive strategies to negotiate and
complete these computer activities. Opportunities for exploring, persever-
ing, and knowing when and how to obtain help are abundant. Metacognitive
activity is illustrated in the following situation, which has occurred among
students from middle school through high school who have worked through
these activities.
When students are asked to change the parameters of y = x2 to make it
curve down and go through a colored point that is in the lower right quad-
rant, their first intuition is often to make the exponent rather than the coef-
ficient negative. When they make that change, they are surprised to find that
the graph changes shape entirely and that a negative exponent will not
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TEACHING AND LEARNING FUNCTIONS
satisfy their needs. By trying a number of other possible alterations (perse-
vering), some students discover that they need to change the coefficient of
x2 rather than the exponent to a negative number to make the function curve
down. It is then a matter of further exploration and discovery to find the
correct value that will make the graph pass through the point in question.
Some students, however, require support to discover this solution. Some try
to subtract a value from x2 but are then reminded by the result they see on
the computer screen that subtracting an amount from x2 causes a downward
vertical shift of the graph. Drawing students’ attention to earlier exercises in
which they multiplied the x in y = x by a negative number to make the
numeric pattern and the graph go down encourages them to apply that
same notion to y = x2. To follow up, we suggest emphasizing for students
the numeric pattern in the tables of values for decreasing curves to show
how the number pattern decreases with a negative coefficient but not with a
negative exponent.
Following is a typical exchange between the circulating teacher and a
pair of students struggling with flipping the function y = x2 (i.e., reflecting it
in the x-axis). This exchange illustrates the use of metacognitive prompting
to help students supervise their own learning by suggesting the coordina-
tion of conclusions drawn from one representation (e.g., slope in linear
functions) with those drawn from another (e.g., slope in power functions).
Teacher How did you make a straight line come down
or change direction?
John We used minus.
Teacher How did you use “minus”?
Pete Oh yeah, we times it by minus something.
So . . . how about here [pointing at the x2]?
Teacher
We could times it by minus 2 [typing in x2 • -2].
John
There! It worked.
Without metacognitive awareness and skills, students are at risk of miss-
ing important inconsistencies in their work and will not be in a position to
self-correct or to move on to more advanced problem solving. The example
shown earlier in Figure 8-1a involves a student not reflecting on the incon-
sistency between a negative slope in his equation and a positive slope in his
graph. Another sort of difficulty may arise when students attempt to apply
“rules” or algorithms they have been taught for simplifying a solution to a
situation that in fact does not warrant such simplification or efficiency.
For example, many high school mathematics students are taught that
“you only really need two points to graph a straight line” or “if you know
it’s a straight line, you only need two points.” The key phrase here is “if you
know it’s a straight line.” In our research, we have found students applying
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388 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
FIGURE 8-4
that two-point rule for graphing straight lines to the graphing of curved-line
functions. In the example shown in Figure 8-4, an eleventh-grade advanced
mathematics student who had been learning functions primarily from a
textbook unit decided to calculate and plot only two points of the function
y = x2 +1 and then to join them incorrectly with a straight line. This student
had just finished a unit that included transformations of quadratic functions
and thus presumably knew that y = x2 makes a parabola rather than a
straight line. What this student did not know to perform, or at least exer-
cise, was a metacognitive analysis of the problem that would have ruled out
the application of the two-points rule for graphing this particular function.
Summary of Principle #3 in the Context of Operating on y = x2. The
general metacognitive opportunities for the computer activities in our cur-
riculum are extensive. Students must develop and engage their skills involv-
ing prediction, error detection, and correction, as well as strategies for scien-
tific inquiry such as hypothesis generating and testing. For instance, because
there are innumerable combinations of y-intercept, coefficient, and expo-
nent that will move y = x2 through each of the colored points, students must
recognize and acknowledge alternative solution paths. Some students may
fixate on the steepness of the curve and get as close to the colored points as
possible by adjusting just the steepness of the curve (by changing either the
exponent or the coefficient of x2) and then changing the y-intercept. Others
may begin by selecting a manageable y-intercept and then adjust the steep-
ness of the curve by changing the exponent or the coefficient. Others may
use both strategies equally. Furthermore, students must constantly be pre-
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dicting the shapes and behaviors of the functions with which they are work-
ing and adjusting and readjusting their expectations with respect to the math-
ematical properties and characteristics of linear and nonlinear functions.
SUMMARY
Sometimes mathematics instruction can lead to what we refer to as “un-
grounded competence.” A student with ungrounded competence will dis-
play elements of sophisticated procedural or quantitative skills in some con-
texts, but in other contexts will make errors indicating a lack of conceptual
or qualitative understanding underpinning these skills. The student solution
shown earlier in Figure 8-1a illustrates such ungrounded competence. On
the one hand, the student displays elements of sophisticated skills, including
the slope formula and negative and fractional coefficients. On the other
hand, the student displays a lack of coordinated conceptual understanding
of linear functions and how they appear in graphical, tabular, and symbolic
representations. In particular, he does not appear to be able to extract quali-
tative features such as linearity and the sign of the slope and to check that all
three representations share these qualitative features.
The curricular approach described in this chapter is based on cognitive
principles and a detailed developmental model of student learning. It was
designed to produce grounded competence whereby students can reason
with and about multiple representations of mathematical functions flexibly
and fluently. Experimental studies have shown that this curriculum is effec-
tive in improving student learning beyond that achieved by the same teach-
ers using a more traditional curriculum. We hope that teachers will find the
principles, developmental model, and instructional examples provided here
useful in guiding their curriculum and teaching choices.
We have presented three example lessons that were designed within
one possible unifying context. Other lessons and contexts are possible and
desirable, but these three examples illustrate some key points. For instance,
students may learn more effectively when given a gradual introduction to
ideas. Our curriculum employs three strategies for creating such a gradual
introduction to ideas:
• Starting with a familiar context: Contexts that are familiar to students,
such as the walkathon, allow them to draw on prior knowledge to think
through a mathematical process or idea using a concrete example.
• Starting with simple content: To get at the essence of the idea while
avoiding other, distracting difficulties, our curriculum starts with mathemati-
cal content that is as simple as possible—the function “you get one dollar for
every kilometer you walk” (y = x).
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• Focusing on having students express concepts in their own language
before learning and using conventional terminology: To the extent that a
curriculum initially illustrates an idea in an unfamiliar context or with more-
complex content, students may be less likely to be able to construct or
invent their own language for the idea. Students may better understand and
explain new ideas when they progress from thinking about those ideas us-
ing their own invented or natural language to thinking about them using
formal conventional terms.
A risk of simplicity and familiarity is that students may not acquire the
full generality of relevant ideas and concepts. Our curriculum helps students
acquire correct generalizations by constructing multiple representations for
the same idea for the same problem at the same time. Students make com-
parisons and contrasts across representations. For example, they may com-
pare the functions y = .5x, y = 2x, and y = 10x in different representations
and consider how the change in slope looks in the graph and how the table
and symbolic formula change from function to function. We also emphasize
the use of multiple representations because it facilitates the necessary bridg-
ing between the spatial and numerical aspects of functions. Each representa-
tion has both spatial and numerical components, and students need experi-
ence with identifying and constructing how they are linked.
As illustrated earlier in Figure 8-1a, a curriculum that does not take this
multiple-representation approach can lead students to acquire shallow ideas
about functions, slope, and linearity. The student whose response is shown
in that figure had a superficial understanding of how tables and graphs are
linked: he could read off points from the graph, but he lacked a deep under-
standing of the relationship between tables and graphs and the underlying
idea of linearity. He did not see or “encode” the fact that because the graph
is linear, equal changes in x must yield equal changes in y, and the values in
the table must represent this critical characteristic of linearity.
The curriculum presented in this chapter attempts to focus limited in-
structional time on core conceptual understanding by using multiple repre-
sentations and generalizing from variations on just a few familiar contexts.
The goal is to develop robust, generalizable knowledge, and there may be
multiple pathways to this end. Because instructional time is limited, we de-
cided to experiment with a primary emphasis on a single simple, real-world
context for introducing function concepts instead of using multiple contexts
or a single complex context. This is not to say that students would not
benefit from a greater variety of contexts and some experience with rich,
complex, real-world contexts. Other contexts that are relevant to students’
current real-world experience could help them build further on prior knowl-
edge. Moreover, contexts that are relevant to students’ future real-world
experiences, such as fixed and variable costs of production, could help them
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in their later work life. Since our lessons can be accomplished in anywhere
from 3 to 6 weeks (650 minutes), there is sufficient time for other activities to
supplement and extend students’ experience.
In addition to providing a gradual introduction to complex ideas, a
key point illustrated by our lessons is that curriculum should be mathemati-
cally sound and targeted toward high standards. Although the lessons de-
scribed here start gradually, they quickly progress to the point at which
students work with and learn about sophisticated mathematical functions at
or beyond what is typical for their grade level. For instance, students progress
from functions such as y = x to y = 10 – .4x in their study of linear functions
across lessons 1 to 3, and from y = x2 to y = (x – 2)2 + 4 in their study of
nonlinear functions across lessons 4 to 8.
We do not mean to suggest that this is the only curriculum that promotes
a deep conceptual understanding of functions or that illustrates the prin-
ciples of How People Learn. Indeed, it has important similarities, as well as
differences, with other successful innovations in algebra instruction, such as
the Jasper Woodbury series and Cognitive Tutor Algebra (previously called
PUMP), both described in How People Learn. All of these programs build on
students’ prior knowledge by using problem situations and making connec-
tions among multiple representations of function. However, whereas the
Jasper Woodbury series emphasizes rich, complex, real-world contexts, the
approach described in this chapter keeps the context simple to help students
perceive and understand the richness and complexity of the underlying math-
ematical functions. And whereas Cognitive Tutor Algebra uses a wide variety
of real-world contexts and provides intelligent computer tutor support, the
approach described here uses spreadsheet technology and focuses on a
single context within which a wide variety of content is illustrated.
All of these curricula, however, stand in contrast to more traditional
textbook-based curricula, which have focused on developing the numeric/
symbolic and spatial aspects of functions in isolation and without particular
attention to the out-of-school knowledge that students bring to the class-
room. Furthermore, these traditional approaches do not endeavor to con-
nect the two sorts of understandings, which we have tried to show is an
essential part of building a conceptual framework that underpins students’
learning of functions and ultimately their learning in related areas.
ACKNOWLEDGMENTS
Thanks to Ryan Baker, Brad Stephens, and Eric Knuth for helpful com-
ments. Thanks to the McDonnell Foundation for funding.
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392 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM
NOTES
1. The study of functions, as we define it here, overlaps substantially with the
topic of “algebra” traditionally taught in the United States in ninth grade, though
national and many state standards now recommend that aspects of algebra be
addressed in earlier grades (as is done in most other countries). Although
functions are a critical piece of algebra, other aspects of algebra, such as equa-
tion solving, are not addressed in this chapter.
2. Thomas, 1972, p. 17.
3. Goldenberg, 1995; Leinhardt et al., 1990; Romberg et al., 1993.
4. Nathan and Koedinger, 2000.
5. Koedinger and Nathan, 2004.
6. Koedinger and Nathan, 2004.
7. Koedinger et al., 1997.
8. Kalchman, 2001.
9. Schoenfeld et al., 1993.
10. Schoenfeld et al., 1987.
11. Schoenfeld et al., 1998, p. 81.
12. Chi et al., 1981.
13. Chi et al., 1981; Schoenfeld et al., 1993.
14. Kalchman, 2001.
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Representative terms from entire chapter:
conceptual understanding
