They Already Know the Formula — They Just Don't Know Its Name Yet

Hand a student L(x) = f(a) + f′(a)(x−a) cold, and most of them will do exactly what you'd expect: stare at the letters. Not because they can't do the arithmetic sitting underneath it — they can, easily. Because nobody told them the arithmetic came first.

This is an account of a small bet I made in a supported section of Math 1530, differential calculus, at LSU: delay the formula. Build its content out of arithmetic students already trust, one familiar piece at a time, and only attach the notation once there's something for the notation to name.

01 The Class

Who this was for, and why it existed

Regular Math 1530 sections run 3 credits, 3 hours a week. This section was also 3 credits, but met 5 hours a week — a pilot for students whose ALEKS placement scores fell just short of the standard cutoff. Twenty-five students, mostly headed into pre-medicine, pre-nursing, biology, and chemistry.

The extra two hours weren't for covering more material. They were for covering the same material differently — with room to build ideas up instead of handing them down. The goal was never just to get students through one exam. It was short-term support for a long-term outcome: graduating on time, on the same timeline as students who placed directly into the standard sequence.

02 Two Ways to Run Fifty Minutes

What changes when you stop lecturing at students

Traditional lecture has a simple shape: I write on the board, students copy it down. It's efficient, and it's honest about who's doing the thinking — which is part of the problem for students who are already unsure whether they belong in the room.

Every class meeting ran on a handout designed around one idea: concreteness before abstraction. We solved it together. Early in the semester, I carried most of the weight — explaining, prompting, filling gaps. By the back half, students were doing the heavy lifting themselves, solo or in groups, and I was mostly getting out of the way.

I do → we do → you do — solo or in groups.

03 What Linearization Actually Is

The topic this case study is built around

Linearization is one of the first times students use the derivative for something beyond slope: estimating a quantity you can't easily recompute — a drug concentration, a population, a distance traveled — using its value and rate of change at one known instant. It lands right after derivatives are introduced, which makes it an early test of whether the abstraction actually landed.

Shown cold, the formula is just symbols. So we didn't show it cold.

04 The Same Equation, Three Times

The case study, step by step

Step 1 — an equation they already trust. A nurse records a drug concentration of 2.5 mg/L, increasing at 0.8 mg/L per hour. What should we expect 30 minutes later? No function, no derivative — just arithmetic. By the end, the class has written, in their own words:

Step 2 — the same question, harder. Now the concentration is a function: f(t) = 2.5 + 0.8t − 0.1t². Same question, same equation from Step 1 — the only new question is what Initial and Rate even mean now. Plug in t = 0 for the initial value. For the rate, we've said it all semester: rate is derivative. Plug t = 0 into f′(t).

Step 3 — renaming what they already know. Nothing new happens here. Every piece already exists — it just needs a name that works for any function, at any point a:

Final quantity → L(x)  ·  Initial value → f(a)  ·  Rate of change → f′(a)  ·  Time elapsed → (x−a)

05 Why Delay the Notation

The actual dilemma this design solves

When students see L(x)=f(a)+f′(a)(x−a) first, most of them focus on decoding the symbols instead of understanding the idea underneath. The fix isn't a clearer explanation of the formula. It's sequencing:

1. Build the idea.
2. Connect familiar pieces.
3. Introduce the notation.

Notation goes on last, not first.

06 What We're Seeing

Early results from the pilot

07 A Reusable Pattern

The same build — easy version first, formula assembled from pieces the class already trusts, notation attached last — recurs across the section's daily handouts, from related rates to root approximation. Rigor doesn't drop; students still reach the general formula and apply it cold. What changes is which door they walk through first.

None of this required new mathematics. It required deciding, deliberately, that the formula is a destination — not a starting point.