What I’ve Learned from Many Years of Teaching Calculus
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
C+ or higher
C or lower
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.