The A student who gets a C in Calculus I: what changed

What “good at math” meant in Secondary 5

CEGEP Science asks for Secondary 5 math in the Technical and Scientific (TS) or Science (SN) option, the two high-school math streams that lead into it. A student who earned top marks there usually has three real strengths:

• Speed. They calculate quickly and rarely slip.
• Memory. They keep a method after seeing it once or twice.
• Recognition. They see what kind of question it is and which method it calls for.

Those strengths still matter. The catch is that a student can go a long way on them without often meeting a problem they do not yet know how to start, so they never build a routine for that moment. Calculus I is full of that moment.

What Calculus I asks for instead

Among the performance criteria the ministry sets for Calculus I, competency 0M02 of the Science program, are “demonstration of rigorous mathematical reasoning through the use of concepts, properties and theorems,” “application of appropriate methods of differential calculus” and “accurate interpretation of results.” In practice that changes the shape of a question. You do not need to follow the algebra below to see how.

Same idea, two questions

A high-school question: find the minimum value of $f(x)=x^2-10x+21$. The function is a familiar object, a parabola, and a strong student knows the method at once: the vertex is $(5,-4)$, so the minimum is $-4$. It takes under a minute.

A Calculus I question: does $g(x)=x^3-6x^2+12x$ have a maximum or a minimum? Justify your answer. Nothing says which tool to use. The fast, memorized move is to set the derivative equal to zero, find $x=2$, and announce a turning point there. That answer is wrong, and the reasoning that shows why is the whole question:

1. Differentiate: $g^{\prime }(x)=3x^2-12x+12$.
2. Factor: $g^{\prime }(x)=3(x-2{)}^2$.
3. Read the sign: a square is never negative, so $g^{\prime }(x)$ is positive everywhere except at $x=2$, where it is zero.
4. Conclude: $g$ is increasing everywhere. At $x=2$ the tangent line is horizontal, but the graph rises on both sides of that point, so it is not a turning point, and $g$ has no maximum or minimum.

None of the arithmetic is harder than in high school. What is harder is that the student has to decide what to do, carry it out, and then argue from the result, and the answer is only as good as the weakest link in that chain.

Why speed and memory stop being enough

• The pace. The whole competency is taught in 75 periods of instruction. A new idea arrives before the last one is automatic, so there is no time to store a method for every type of question.
• The types multiply. Every new tool combines with every earlier one. A student who knows why a method works can handle a combination they have never seen; a student who memorized it cannot.
• Results come with conditions. A theorem applies only when its conditions hold, and checking them is part of the answer. Memory brings back the formula, not the conditions.
• The algebra has to be automatic. Factoring, algebraic fractions, the laws of exponents and the properties of logarithms are on the program's list of “previously acquired concepts” for Calculus I. Algebra that was quick but shaky now sits in every line.
• “I understood it in class” stops meaning much. Following a solution is not the same as producing one, and strong students are used to the two arriving together.

What the fix looks like, week to week

Not more hours of the old method, but a different routine:

1. Exercises within a day or two of the class, not in a block before the test.
2. Every exercise starts with the notes closed. Look at an example only when stuck, then close it and redo the exercise from the beginning.
3. Write the reasons, not only the steps. A short phrase beside each step, such as “a square is never negative, so…”, is how a student shows the reasoning that criterion asks for, and writing it is how they find the link they skipped.
4. A list of mistakes, each with its cause: an algebra slip, a misread question, a condition not checked. Patterns show within a couple of weeks.
5. Once a week, a mixed set. Three or four problems from earlier topics, without knowing in advance which tool each one needs. A textbook section, where every exercise uses that section's method, never practises the choosing.
6. One specific question to the teacher, early, rather than a general “I'm lost” later.

The free CEGEP Calculus I worksheets follow the course one topic at a time. Doing the current topic's sheet from a blank page, notes closed, is a fair test of whether a method has really been learned.

What a parent can do

• Ask about the method, not the mark: “Which exercises could you do this week without the notes?”
• Treat the first C as information about how they study, not about how able they are. A student who has always had top marks can easily read it the second way, and that is often when avoiding the course begins.
• Keep one mark in proportion. The R-score (cote R), the measure Quebec universities use to compare applicants coming from CEGEP, is calculated course by course and then averaged, each course weighted by its credits. One mark is one entry in that average, not a verdict.

When outside help makes sense

• When three or four weeks of that routine have not moved the marks.
• When your teen can follow every solution but cannot start a problem alone.
• When the trouble turns out to be high-school material, such as algebra, trigonometry or logarithms, rather than calculus. A fast student can carry a gap like that for years without noticing it.

For a strong student the first job is telling those apart, because a gap in Secondary 4 or 5 material and a gap in the new course need different work. With me, that is what the opening written diagnostic is for. After it, any practice between sessions is written for the exact topic we covered and marked before we meet again.

Free worksheets to practise from a blank page

• CEGEP Calculus I: The Derivative worksheet
• CEGEP Calculus I: Analysis of Functions and Curve Sketching worksheet
• CEGEP Calculus I: Optimization worksheet