What I’ve Learned from Many Years of Teaching Calculus
to
AMTNJ Conference
Keeping Math on Track:
Bridging the Gap
Between High School and College Mathematics
January 14, 2005
Brookdale Community College
Joseph G. Rosenstein (Rutgers-New
Brunswick)
Why do so many first-year students have difficulties with
calculus when they seem to be well prepared?
The last time I taught first semester calculus 41% of the
61 students in the class ended up with grades of C or worse.
Here’s some more data.
82% of the students in the class had a semester or more of calculus in
high school, and 73% had a year or more of calculus in high school. (35% even had a year of AP Calculus.) By all accounts, this is a group that is
well-prepared for college calculus.
The bad news is that if we put the data together, we can
conclude that at least 23% of my students had taken a calculus course in high
school yet had not managed to do better than C in a calculus course in
college.
Why are there so many students who have taken substantial
math courses in high school but are unsuccessful in calculus?
In this brief talk, I will discuss three kinds of issues –
content, process, and personal issues – all in about 15 minutes and then open
the floor for discussion.
Under “content”, the main issue is not that students don’t
understand the concepts of calculus,
it is that they don’t have facility with arithmetic and algebra.
One student once observed that a particular problem
involved what he called “intense algebra” – by which he meant that he had to
draw on a lot of his algebra knowledge to carry out the calculations in a
single problem. This happens, for
example, when students need to find the derivative of the function f(x) =
1/(x+3) from the definition – that is, as in the transparency, they need to
find the limit of a difference quotient, as h goes to zero. Think about the steps they need to take to
solve this problem. They need to:
write a correct expression for f(x+h) given the equation
for f(x);
combine two fractions in the numerator into a single
fraction;
combine a sum and difference of terms;
transform a fraction that has a fraction in the numerator
into one that doesn’t;
find a common factor of the numerator and denominator and
cancel it properly;
take a limit and express the result in the appropriate
form.
Not only do they have to be able to carry out each of these
steps individually, they also need a functioning high-level monitoring system
that sees the “big picture” that’s involved in finding the derivative and that
tells them what they need to do at each step.
Yet many of them are still making errors that have been
persistent since middle grades – for example, improperly canceling terms in
fractions.
A question like this is given on the midterm and on the final
exam, and although they all know that this is what they are expected to be able
to do, many of them are unable to complete the task correctly.
When we talk about the NCTM standards, we often act as if
the process standards have replaced the content standards, that understanding
has replaced facility. That is not the
case. We do want to focus on reasoning
and problem solving, but we also want our students to have the appropriate
facility with mathematical operations.
What facility is “appropriate”? That depends on the student. Those who are going to end up taking several
semesters of calculus in college definitely are at a disadvantage if they have
difficulty with arithmetic and algebra.
On the other hand, those who are unlikely to continue to calculus
will not need intense algebra. But
assuming that a particular student is in that category may end up being a
self-fulfilling prophecy.
A digression on algebra skills. Many people have criticized Rutgers’
placement test on the grounds that is not aligned with the standards, with our
reform efforts, since it focuses on skills.
But I must tell you that it is a good measure of the likelihood of
success in precalculus and calculus, and has been so over the past 20 years
since we introduced it. It measures
students’ facility with the prerequisite skills … because facility with the
prerequisites is essential for success in these courses.
I will share with you my personal experience. A few years ago, one of my daughters scored
just below the cut-off for precalculus.
Since I have a little bit of influence, I was able to enroll her in
precalculus, reasoning that if she was having difficulties, she had access to a
good tutor. That was correct … but it
was also a mistake – I ended up doing a lot
of tutoring. She wasn’t ready for
precalculus.
That’s the end of the digression.
Now a clarification is needed. When we say that facility in algebra is
essential to success in calculus, we don’t mean just learning rules for
algebraic manipulations. Facility in
algebra also means understanding the mathematics that underlies those
rules. When students make errors, they
are often a result of misunderstanding the mathematics, and we all need to
spend more time uncovering the mistaken ideas that led to those errors, and
helping the students replace them with more accurate mathematical
understandings. That means discussing
errors in class and with students individually, and not just marking them
incorrect on their homework and tests.
Facility in algebra also means being able to draw on one’s
entire mathematical experience to figure out an appropriate next step in a
problem – that’s what I referred to above as monitoring one’s progress …
knowing what to do next.
That brings us to “process issues”.
We all have a tendency to compartmentalize what we learn –
in part because we come across new information linearly and we have to store it
somewhere. But it is very important that
the learning be connected. Anything we
can do as teachers to make connections between topics, to focus the students on
the big picture, is very important.
Giving examples and homework problems that link different
concepts is important, as is giving regular cumulative examinations. Otherwise, students learn what they need to
know for this week’s quiz and then forget it.
In some schools, students’ success is rewarded by exempting
them from midterm and final exams. I
believe that this practice is a serious mistake – the students don’t get a chance
to pull together the different pieces of knowledge they have acquired. Moreover, it doesn’t prepare them for the
cumulative examinations that are routine in college. Along those lines, a report released three
weeks ago noted that taking AP calculus in high school was not a predictor of
success in college, although scoring well on the AP exam was.
We need to help our students get the big picture. One part of that involves
decompartmentalizing and integrating knowledge, as we have discussed. But there are a few other aspects as well.
One is encouraging students to have multiple
perspectives. For example, they should
be familiar with different aspects of the idea of a function – as an equation,
as a rule, as a graph, as a table, as an input-output machine – and be able to
move back and forth easily among these representations.
Similarly, they should be able to move back and forth
between algebra and geometry. When
discussing the solution of simultaneous linear equations, they should recognize
that that’s the same as asking where two lines cross. When you give a quadratic function they
should be able to visualize the parabola that it defines – maybe not all of the
details, but they should certainly be aware that it is does define a parabola,
and know whether it opens up or down.
Not only should they be able to visualize a parabola, they should
actually do it. The equation and the
graph should be two views of the same object.
And when you find the solutions of a quadratic equation,
they should be able to translate that with facility to the graph of the
quadratic function – so that if the roots of a quadratic function are, for
example, 3 +/- sqrt2, they should be able to picture about where the graph of
the function crosses the x-axis.
On the first day of class, I give students a small scrap of
paper – 1/8 of an 8.5x11 sheet and ask them to find the tangent of the angle
whose sine is 3/5. Some of the students
draw a triangle; almost all of them
then get the right answer. Some of the
students do not draw a triangle; none of them get the right answer.
Since I don’t ask them to put their names on the papers, I
can’t relate solutions to this problem to their grades in the course, but my
guess is that there would be a high degree of correlation. Students who can visualize algebra, who can
move easily from algebra to geometry and back, are likely to be successful in
calculus.
At the second class, I report to the students the results
of this experiment and reinforce the importance of visualization. I encourage them to turn on their
visualization switch so that they draw a picture in their mind of each
algebraic expression that’s in their book or on the board.
I point out that a picture can contain a lot of
information. For example, if they can visualize
and interpret the graphs of the sine, cosine and tangent functions then they
only need to remember three facts – that sin 30 = ½ , that tan 45 =1, and that
sin2x + cos2x =1 .
Just about everything else they need to know about trigonometry can be derived
from these. In particular, they don’t
need to memorize lots and lots of facts.
That is what they will have to do if they don’t understand the pictures. Some find this hard to believe, and persist
in trying to remember lots of facts about trigonometric functions. It’s no wonder that they sometimes feel that
their heads are full.
There are about a dozen pictures that encapsulate much of
first semester calculus – if you understand and can explain what’s in those
pictures, then you will do very well in calculus. They find this hard to believe as well.
Another issue that I will mention briefly is that students
need to have a better sense of whether an answer that they generate is
reasonable. A prerequisite of that, of
course, is that they actually ask themselves whether their answers are
reasonable. Actually, if they ask
themselves the question, they are likely to respond appropriately. So the goal is to get them to ask that
question – is that answer reasonable?
Finally, students need to have the sense of mathematics as
a language. Mathematics has words and
symbols and rules about their use. We
often ignore the grammar of mathematics, and allow our students to speak and
write mathematics incorrectly – a practice that would not be permitted in a Spanish
class. So they end up not using
parentheses when they should and making all sorts of mistakes as a result. They don’t use the equals sign to separate
equal expressions in their mathematical sentences, and, as a result, quantities
wander out of one expression and into another.
And they are often unable to translate their answers to problems from
mathematical language into the English language. This issue requires more attention from all
of us.
And now we come to what I called personal issues. I will make four points. One is that many students come to first
semester calculus thinking that they know calculus already. That may be true – but it’s only true for
some of them. However, that is a dangerous assumption, for those who believe
this will not do anything for the first four weeks of the semester … and then
find that it’s too late to catch up.
Please warn your students that even though they may be
successful in your course, they will not automatically be successful in a
course with the same title in college.
Although both courses cover the same material, the college course goes
into more depth.
A second point is that students need to know that they will
have to work in college. Some of them
will be able to get by without too much work – in which case they should have
been taking a more difficult course – but most of them will have their hands
full with the course that they take – whether it’s calculus or precalculus or
even algebra – whether or not they got a good grade in that course in high
school.
I have learned that the best predictor of a good grade in
Calc 1 is getting a good grade on the very first exam. Look at the data in the chart. It shows that 86% of the students who scored
70% on the first exam got a grade of C+ or better for the course. On the other hand, only 17% of those who
scored less than 70% on the first exam got a grade of C+ or better for the
course. Consistent work pays off. Those who start off well and work
consistently do well.
Students in my Calculus 1 classes
Fall 1999, Fall 2000, Fall 2001, Fall 2002
|
# of students |
70% or more on first exam |
69 or less on first exam |
Total |
|
Final Grade: C+ or higher |
74 |
21 |
105 |
|
Final Grade: C or lower |
12 |
102 |
114 |
|
|
86 |
123 |
219 |
86% of
those who got 70% or better on the first test got a C+ or better in the course;
17% of those who got 69% or worse on the first test got a C+ or better in the
course
Another thing I do on the first day of class is ask each
student to make a realistic assessment of what grade he or she expects to get
in the course – taking all sorts of things into consideration – and hand that
in on another little slip of paper.
Every student, without exception, expects to get a B or better!
I report this to the students at the second class and then
show them this chart. I tell them that
they cannot start off the semester thinking that because they know the formulas
for a few derivatives they know calculus.
I tell them that they need to start off the semester working on
calculus. Perhaps it makes a
difference. I tell them that I will do
everything that I can to help each one get the grade that he or she hopes to
get – but in the end it is up to them.
That’s the third point I want to make – students need to learn
to take responsibility for their own education.
In high school you see them
every day and can cajole them into taking their studies seriously. That’s great.
But when they get to college, they are on their own, and if they haven’t
yet learned to take responsibility for their education, they will have a tough
time.
I’m not sure how to get them to take responsibility, but
here’s a modest experiment that you might try.
Tell them that you will not be collecting assignments for the next two
weeks. Then give them an exam on the
material. Some of them will not do the assignments and will do poorly on the
exam. Perhaps their performance on that
exam will convey to them that your not collecting the homework should not have
been interpreted as their not needing to do it.
Another aspect of taking responsibility for one’s education
is asking for help, and taking advantage of the opportunities that are
available to them. Fewer than 20% of my
students ever come to see me, even though I regularly encourage them to do
so. Fewer than 20% of my students ever
email me with their questions, although I tell them that they will most likely
get a response within a few hours.
Although one-third of my students will end up with a D or F, few of them
will seek out the various types of help that are available to them.
Most students have not yet learned that it’s ok for them to
seek assistance – they have not learned that if they’re having difficulties in
a course, they should seek help as soon as possible. They need to know that waiting is not a good
strategy. Perhaps your telling that to
them will make a difference.
That brings me to the end of my remarks. I have talked a bit about the content issues,
the process issues, and the personal issues that interfere with students’
success in precalculus and calculus courses, and I have given you a few
suggestions for how you might help prepare the students to overcome the
obstacles to their success.
Thank you very much for your attention, and we’ll now have
a discussion of these issues.
To see other articles and
presentations by Joseph Rosenstein, please click here.