Pitfalls in teaching and learning calculus

Oliver Knill

5/20/2005, working draft

Abstract:

Students studying calculus often make the same mistakes. Similarly, teachers teaching calculus have patterns of mistakes. It is much easier to tell how things should not be done then how things should be done. I collect in this document some "no-go's". This is a living document and is expected to be updated and adapted in the future.

$\parbox{10cm}{\bf{ \Large{ General teacher mistakes }}}$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Lack o... ...tion leads to mess, the characteristic polynomial has no apparent root. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Improv... ...not work like for a Lissajou curve $\vec{r}(t) = (\cos(5t), \sin(7t))$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Diffic... ...with fixed surface area and maximal volume using Lagrange extremization. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Politi... ...here are students of almost any political orientation in the classroom. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Lectur... ...e students of the next class have to wait. The next class gets delayed. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Too la... ...m}{ {\bf Example:} At Harvard, classes start 7 minutes after the hour. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf No int... ...Harvard are 90 minutes. This is a long time to keep the attention span. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Wrong ... ...ic definitions which are necessary for the homework are not introduced. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Outfit... ...ut clothing issues should not become a distraction or an embarressment. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Untest... ...ctor or trying to fix a line of Mathematica code in front of the class. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Misuse... ...er point presentation 3) use the overhead projector to sketch a figure. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Except... ...uch counter examples may be amusing but have also a pathological touch. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Uninsp... ...: a teacher copies problems from the book directly onto the blackboard. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Ignore... ...hat.'' or ''You should have payed more attention to what I just sayd''. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Mumble... ... back of the teacher. Worse, the teachers body covers the written part. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Giving... ...and leave the other $\sin^2$. Hint 3: use then a trigonometric identity. }} }$

$\parbox{10cm}{\bf{ \Large{ Mistakes by Course-head }}}$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Lack o... ...he homework useful, the section leaders can not prepared early enough. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Too mu... ... point out possible pitfalls, student questions, common misconceptions. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Distri... ...e, holidays, conflicts, lecture hall reservation), this is a disaster. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Use of... ...nse because some CAs will have to grade the exams of their own friends. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Change... ...erial, reduce the complexity of the examples or ask students to preread. }} }$

$\parbox{10cm}{\bf{ \Large{ Mathematical teacher mistakes }}}$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Ambigu... ... taking bold face letter, such features can get lost on the blackboard. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Uncomm... ...in the plane and let $P,Q,R$\ denote the line segments connecting them. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Unusua... ...rbox{7cm}{ {\bf Example:} use a left handed orientation $e_1,e_3,e_2$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Incorr... ... {\bf Example:} like a vector is an object with length and orientation. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Overki... ...on $x$\ close to $y$, then $f(x)$\ close to $f(y)$\ is more intuitive. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Too di... ...t of all students would get very frustrated when assigned this problem. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Incorr... ...nabla g$. (This is incorrectly stated as a theorem in many text books.) }} }$

$\parbox{10cm}{\bf{ \Large{ Course assistant mistakes }}}$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Not co... ... he knows the material enough, fails to come to lectures. He got fired. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Not gr... ... to do it. The teaching fellows spend an afternoon grading the papers. }} }$

$\parbox{10cm}{\bf{ \Large{ General student mistakes }}}$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Do not... ...empting, it needs stamina to make up the learning time away from class. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Do not... ...hing is not clear to one student, it is not clear to the entire class. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Copy h... ...ng. The process of struggling with a homework problem is important too. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Come u... ... room time is to have already partly absorbed the material before hand. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Cheat ... ...ciously. A student uses a calculator in an exam, where none are allowed. }} }$

$\parbox{10cm}{\bf{ \Large{ Mathematical student mistakes }}}$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ Improper us... ... = \int_0^{1/\sqrt{2}} u^2 \; du$, where $u=\sin(x), du=\cos(x) \; dx$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ Improper ha... ...$\int_{-\pi}^{\pi} 1/x \; dx =0$\ so that $\int_0^{\pi} 1/x \; dx = 0$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ Matrix alge... ... \fcolorbox{yellow1}{yellow1}{\parbox{7cm}{ $(A+B)^2 = A^2+2AB + B^2$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ Wrong linea... ...ox{yellow1}{yellow1}{\parbox{7cm}{ ${\rm det}(2 A ) = 2 {\rm det}(A)$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Too li... ... \sqrt{y}, e^{x+y} = e^x+e^y, (x+y)^2=x^2+y^2$. $1/(x+y) = 1/x + 1/y$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Too co... ...{\parbox{7cm}{ $\log\sqrt{x} = \sqrt{ \log(x)}, \sin(5x) = 5 \sin(x)$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf A sin}... ... \fcolorbox{yellow1}{yellow1}{\parbox{7cm}{ $\frac{\sin x}{x} = \sin$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Inappr... ...$((x+1) + 5x^2)/(x+1) = 1+5x^2/(x+1)$. $\frac{3+x}{x} = 3+x/x = 3+1=4$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Unclea... ...{yellow1}{\parbox{7cm}{ {\bf Example:} $3x/5 = 3/5 x = 3/5x = 3/(5x)$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Unit c... ...(10 c)^2 = (\$0.1 \$)^2 = 0.01 \$ = 1c$. (source: Eric Schechter, 2004) }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Divisi... ... Example:} $x (x^2-1) = x$\ implies $x^2=2$\ and so $x = \pm \sqrt{2}$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Square... ...b) = 1 (a^2-a b)$ and canceling $(a^2-ab)$\ from both sides gives $1=2$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Signs ... ...yellow1}{yellow1}{\parbox{7cm}{ {\bf Example:} $-a<-3$\ implies $a<3$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Miscon... ...}{yellow1}{\parbox{7cm}{ {\bf Example:} $\frac{d}{dx} x^x = x x^{x-1}$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Integr... ...rbox{7cm}{ {\bf Example:} $\int x \cos(x) \; dx = x \sin(x) - \sin(x)$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Polar ... ...e have $r=\sqrt{2}$\ and $\theta = \arctan(-1/-1)= \arctan(1) = \pi/4$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf An SAT... ...eq 0^2, 1 \leq 1^2, 2 \leq 2^2, 3 \leq 3^2$ one concludes $x \leq x^2$. }} }$

$\parbox{16cm}{ \fcolorbox{green1}{green1}{\parbox{7cm}{ {\bf Sloppi... ...\parbox{7cm}{ {\bf Example:} $\int dx/(1-x) = \log\vert 1-x\vert + C$. }} }$

Literature:

• Eric Schechter, "The most common errors in undergraduate mathematics education"
  http://atlas.math.vanderbilt.edu/ schectex/commerrs
• http://tutorial.math.lamar.edu/AllBrowsers/CommonErrors/CalculusErrors.asp
• Andy Engelward, "do's and don'ts for Classroom style", talk at Bok Center Fall Teaching Orientation Sept. 14th, 2004