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What Does Conceptual Understanding Mean? Florence S. Gordon fgordon@nyit.edu Sheldon P. Gordon gordonsp@farmingdale.edu


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Title: What Does Conceptual Understanding Mean? Florence S. Gordon fgordon@nyit.edu Sheldon P. Gordon gordonsp@farmingdale.edu

What Does Conceptual Understanding
Mean? Florence S. Gordon fgordon_at_nyit.edu Sheldon
P. Gordon gordonsp_at_farmingdale.edu
CUPM Curriculum Guide
  • All students, those for whom the (introductory
    mathematics) course is terminal and those for
    whom it serves as a springboard, need to learn to
    think effectively, quantitatively and logically.
  • Students must learn with understanding,
    focusing on relatively few concepts but treating
    them in depth. Treating ideas in depth includes
    presenting each concept from multiple points of
    view and in progressively more sophisticated

CUPM Curriculum Guide
  • A study of these (disciplinary) reports and the
    textbooks and curricula of courses in other
    disciplines shows that the algorithmic skills
    that are the focus of computational college
    algebra courses are much less important than
    understanding the underlying concepts.
  • Students who are preparing to study calculus
    need to develop conceptual understanding as well
    as computational skills.

AMATYC Crossroads Standards
  • In general, emphasis on the meaning and use of
    mathematical ideas must increase, and attention
    to rote manipulation must decrease.
  • Faculty should include fewer topics but cover
    them in greater depth, with greater
    understanding, and with more flexibility. Such
    an approach will enable students to adapt to new
  • Areas that should receive increased attention
    include the conceptual understanding of
    mathematical ideas.

NCTM Standards
  • These recommendations are clearly very much in
    the same spirit as the recommendations in NCTMs
    Principles and Standards for School Mathematics.
  • If implemented at the college level, they would
    establish a smooth transition between school and
    college mathematics.

Associates Degrees in Mathematics
In 2000, P There were 564,933 associate
degrees P Of these, 675 were in mathematics
This is one-tenth of one percent!
Bachelors Degrees in Mathematics
In 2000, P There were 457,056 bachelors
degrees P Of these, 3,412 were in mathematics
This is seven-tenths of one percent!
The Needs of Our Students
The reality is that virtually none of the
students we face are going to be math
majors. They take our courses because of
requirements from other disciplines. What do
those other disciplines want their students to
bring from math courses?
  • Voices of the Partner Disciplines
  • CRAFTYs Curriculum Foundations Project

Curriculum Foundations Project
A series of 11 workshops with leading educators
from 17 quantitative disciplines to inform the
mathematics community of the current mathematical
needs of each discipline. The results are
summarized in the MAA Reports volume A
Collective Vision Voices of the Partner
Disciplines, edited by Susan Ganter and Bill

What the Physicists Said
  • Conceptual understanding of basic mathematical
    principles is very important for success in
    introductory physics. It is more important than
    esoteric computational skill. However, basic
    computational skill is crucial.
  • Development of problem solving skills is a
    critical aspect of a mathematics education.

What Business Faculty Said
  • Courses should stress problem solving, with the
    incumbent recognition of ambiguities.
  • Courses should stress conceptual understanding
    (motivating the math with the whys not just
    the hows).
  • Courses should stress critical thinking.
  • An important student outcome is their ability to
    develop appropriate models to solve defined

What the Engineers Said
  • Undergrad engineering education should provide
    students with the conceptual skills to formulate,
    develop, solve, evaluate and validate physical
  • The math required to achieve these skills
    should emphasize concepts and problem solving
    skills more than emphasizing the repetitive
    mechanics of solving routine problems.

Conceptual Understanding
Everybody talks about emphasizing Conceptual
Understanding, but
  • What does conceptual understanding mean?
  • How do you recognize its presence or absence?
  • How do you encourage its development?
  • How do you assess whether students have
    developed conceptual understanding?

What Does the Slope Mean?
Comparison of student response to a problem on
the final exams in Traditional vs. Reform College
Algebra/Trig Brookville College enrolled 2546
students in 1996 and 2702 students in 1998.
Assume that enrollment follows a linear growth
pattern. a. Write a linear equation giving the
enrollment in terms of the year t. b. If the
trend continues, what will the enrollment be in
the year 2016? c. What is the slope of the line
you found in part (a)? d. Explain, using an
English sentence, the meaning of the slope. e.
If the trend continues, when will there be 3500
Responses in Traditional Class
  • 1. The meaning of the slope is the amount that
    is gained in years and students in a given
    amount of time.
  • 2. The ratio of students to the number of years.
  • 3. Difference of the ys over the xs.
  • 4. Since it is positive it increases.
  • 5. On a graph, for every point you move to the
    right on the x- axis. You move up 78 points on
    the y-axis.
  • 6. The slope in this equation means the students
    enrolled in 1996. Y MX B .
  • 7. The amount of students that enroll within a
    period of time.
  • Every year the enrollment increases by 78
  • The slope here is 78 which means for each unit of
    time, (1 year) there are 78 more students

Responses in Traditional Class
10. No response 11. No response 12. No
response 13. No response 14. The change in
the x-coordinates over the change in the
y- coordinates. 15. This is the rise in the
number of students. 16. The slope is the average
amount of years it takes to get 156 more
students enrolled in the school. 17. Its how
many times a year it increases. 18. The slope is
the increase of students per year.
Responses in Reform Class
  • 1. This means that for every year the number of
    students increases by 78.
  • 2. The slope means that for every additional
    year the number of students increase by 78.
  • 3. For every year that passes, the student
    number enrolled increases 78 on the previous
  • As each year goes by, the of enrolled students
    goes up by 78.
  • This means that every year the number of enrolled
    students goes up by 78 students.
  • The slope means that the number of students
    enrolled in Brookville college increases by 78.
  • Every year after 1996, 78 more students will
    enroll at Brookville college.
  • Number of students enrolled increases by 78 each

Responses in Reform Class
  • 9. This means that for every year, the amount of
    enrolled students increase by 78.
  • 10. Student enrollment increases by an average
    of 78 per year.
  • 11. For every year that goes by, enrollment
    raises by 78 students.
  • 12. That means every year the of students
    enrolled increases by 2,780 students.
  • 13. For every year that passes there will be 78
    more students enrolled at Brookville college.
  • The slope means that every year, the enrollment
    of students increases by 78 people.
  • Brookville college enrolled students increasing
    by 0.06127.
  • Every two years that passes the number of
    students which is increasing the enrollment into
    Brookville College is 156.

Responses in Reform Class
17. This means that the college will enroll
.0128 more students each year. 18. By every
two year increase the amount of students goes up
by 78 students. 19. The number of students
enrolled increases by 78 every 2 years.
Understanding Slope
Both groups had comparable ability to calculate
the slope of a line. (In both groups, several
students used ?x/?y.)
It is far more important that our students
understand what the slope means in context,
whether that context arises in a math course, or
in courses in other disciplines, or eventually on
the job.
Unless explicit attention is devoted to
emphasizing the conceptual understanding of what
the slope means, the majority of students are not
able to create viable interpretations on their
own. And, without that understanding, they are
likely not able to apply the mathematics to
realistic situations.
Further Implications
  • If students cant make their own connections with
    a concept as simple as the slope of a line, they
    wont be able to create meaningful
    interpretations and connections on their own for
    more sophisticated mathematical concepts. For
  • What is the significance of the base (growth or
    decay factor) in an exponential function?
  • What is the meaning of the power in a power
  • What do the parameters in a realistic sinusoidal
    model tell about the phenomenon being modeled?
  • What is the significance of the factors of a
  • What is the significance of the derivative of a
  • What is the significance of a definite integral?

Further Implications
If we focus only on developing manipulative
skills without developing conceptual
understanding, we produce nothing more than
students who are only Imperfect Organic Clones of
a TI-89
Developing Conceptual Understanding
Conceptual understanding cannot be just an
add-on. It must permeate every course and be a
major focus of the course. Conceptual problems
must appear in all sets of examples, on all
homework assignments, on all project assignments,
and most importantly, on all tests. Otherwise,
students will not see them as important.
Should x Mark the Spot?
All other disciplines focus globally on the
entire universe of a through z, with the
occasional contribution of ? through ?. Only
mathematics focuses on a single spot, called
x. Newtons Second Law of Motion y mx,
Einsteins formula relating energy and mass y
c2x, The Ideal Gas Law yz nRx.
Students who see only xs and ys do not make
the connections and cannot apply the techniques
when other letters arise in other disciplines.
Should x Mark the Spot?
Keplers third law expresses the relationship
between the average distance of a planet from the
sun and the length of its year. If it is
written as y2 0.1664x3, there is no
suggestion of which variable represents which
quantity. If it is written as t2 0.1664D3 ,
a huge conceptual hurdle for the students is
Should x Mark the Spot?
When students see 50 exercises where the first
40 involve solving for x, and a handful at the
end involve other letters, the overriding
impression they gain is that x is the only
legitimate variable and the few remaining cases
are just there to torment them.
  • Some Illustrative Examples
  • of Problems
  • to Develop or Test for
  • Conceptual Understanding

Identify each of the following functions (a) -
(n) as linear, exponential, logarithmic, or
power. In each case, explain your
reasoning. (g) y 1.05x (h) y x1.05
(i) y (0.7)x (j) y x0.7
(k) y x(-½) (l) 3x - 5y 14
(m) x y (n) x y
0 3   0 5
1 5.1   1 7
2 7.2   2 9.8
3 9.3   3 13.7
For the polynomial shown, (a) What is the
minimum degree? Give two different reasons for
your answer. (b) What is the sign of the leading
term? Explain. (c) What are the real roots? (d)
What are the linear factors? (e) How many
complex roots does the polynomial have?
Two functions f and g are defined in the
following table. Use the given values in the
table to complete the table. If any entries are
not defined, write undefined.
x f(x) g(x) f(x) - g(x) f(x)/g(x) f(g(x)) g(f(x))
0 1 3        
1 0 1        
2 3 0        
3 2 2        
Two functions f and g are given in the
accompanying figure. The following five graphs
(a)-(e) are the graphs of f g, g - f, fg,
f/g, and g/f. Decide which is which.
The following table shows world-wide wind power
generating capacity, in megawatts, in various
Year 1980 1985 1988 1990 1992 1995 1997 1999
Wind power 10 1020 1580 1930 2510 4820 7640 13840

(a) Which variable is the independent variable
and which is the dependent variable? (b) Explain
why an exponential function is the best model to
use for this data. (c) Find the exponential
function that models the relationship between
power P generated by wind and the year t. (d)
What are some reasonable values that you can use
for the domain and range of this function? (e)
What is the practical significance of the base in
the exponential function you created in part
(c)? (f) What is the doubling time for this
exponential function? Explain what does it
means. (g) According to your model, what do you
predict for the total wind power generating
capacity in 2010?
Biologists have long observed that the larger the
area of a region, the more species live there.
The relationship is best modeled by a power
function. Puerto Rico has 40 species of
amphibians and reptiles on 3459 square miles and
Hispaniola (Haiti and the Dominican Republic) has
84 species on 29,418 square miles. (a)
Determine a power function that relates the
number of species of reptiles and amphibians on a
Caribbean island to its area. (b) Use the
relationship to predict the number of species of
reptiles and amphibians on Cuba, which measures
44218 square miles.
The accompanying table and associated scatterplot
give some data on the area (in square miles) of
various Caribbean islands and estimates on the
number species of amphibians and reptiles living
on each.
Island Area N
Redonda 1 3
Saba 4 5
Montserrat 40 9
Puerto Rico 3459 40
Jamaica 4411 39
Hispaniola 29418 84
Cuba 44218 76

(a) Which variable is the independent variable
and which is the dependent variable? (b) The
overall pattern in the data suggests either a
power function with a positive power p lt 1 or a
logarithmic function, both of which are
increasing and concave down. Explain why a power
function is the better model to use for this
data. (c) Find the power function that models
the relationship between the number of species,
N, living on one of these islands and the area,
A, of the island and find the correlation
coefficient. (d) What are some reasonable
values that you can use for the domain and range
of this function? (e) The area of Barbados is 166
square miles. Estimate the number of species of
amphibians and reptiles living there.

Write a possible formula for each of the
following trigonometric functions

The average daytime high temperature in New York
as a function of the day of the year varies
between 32?F and 94?F. Assume the coldest day
occurs on the 30th day and the hottest day on the
214th. (a) Sketch the graph of the temperature
as a function of time over a three year time
span. (b) Write a formula for a sinusoidal
function that models the temperature over the
course of a year. (c) What are the domain and
range for this function? (d) What are the
amplitude, vertical shift, period, frequency, and
phase shift of this function? (e) What is the
most likely high temperature on March 15? (f)
What are all the dates on which the high
temperature is most likely 80??
Building Conceptual Understanding
We cannot simply concentrate on teaching the
mathematical techniques that the students need.
It is as least as important to stress conceptual
understanding and the meaning of the mathematics.
To accomplish this, we need to stress a
combination of realistic and conceptual examples
that link the mathematical ideas to concrete
applications that make sense to todays students.
This will also allow them to make the
connections to the use of mathematics in other
Building Conceptual Understanding
This emphasis on developing conceptual
understanding needs to be done in classroom
examples, in all homework problem assignments,
and in test problems that force students to think
and explain, not just manipulate symbols. If we
fail to do this, we are not adequately preparing
our students for successive mathematics courses,
for courses in other disciplines, and for using
mathematics on the job and throughout their
Recognizing Conceptual Understanding
In a college algebra class, one student asked
"Is it true that every cubic is centered at its
point of inflection?" "Well, if you start at
the point of inflection and move in both
directions, don't you trace out the identical
Recognizing Conceptual Understanding
In precalculus we assign a project based on a set
of temperature measurements for Dallas taken
every two weeks over the course of a year. The
students have to construct a sinusoidal function
that models this data. They usually come up with
a variety of schemes for doing this. A typical
formula looks like In one written report where
student was explaining his reasoning in creating
each of the parameter values was "The frequency
was the next value to determine. This was
deceptively simple."
Recognizing Conceptual Understanding
Given the graph of the derivative f , where does
the function f achieve its maximum and minimum?
I expected f is mostly positive, so f is
mostly increasing, and its minimum is at the
left and its maximum is at the right. Of the 28
students, 9 gave this line of reasoning for a
problem they had never seen before. 14 came up
with the idea of using the graph of the
derivative to sketch a graph of the actual
function (reversing the process of graphical
differentiation they had seen). More
significantly, under the pressure of an exam,
these 14 students created the concept of the
antiderivative, a notion which had not previously
been mentioned in class.
Recognizing Conceptual Understanding
Early in calculus, I introduced the notion of
Taylor approximations as an extension of local
linearity, long before introducing any derivative
formulas. But the class focused on this
concept. A couple of weeks later, I discussed
how NASA could use local linearity to calculate
the path of a spaceship and the weakest student
in the class asked Couldnt they improve on
that path process by using a Taylor polynomial
instead of the tangent line? Yes, you can!
Its called the Improved Euler Method for the
numerical solution of differential equations.
Recognizing Conceptual Understanding
A month later, when I first introduced Newtons
Method, another student asked Couldnt you
improve on the accuracy by using a Taylor
polynomial instead of the tangent line? Yes, you
can! This result is known as the Euler
correction formula.
What we value most about great mathematicians is
their deep levels of conceptual understanding
which led to the development of new ideas and
methods. We should similarly value the
development of deep levels of conceptual
understanding in our students. Its not just the
first person who comes upon a great idea who is
brilliant anyone who creates the same idea
independently is equally talented!
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