Friday, February 15, 2008

Top Ten Lists of Common (Student) Math Errors!

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

If you enjoy this post, you may want to read a newer post on developing Ratio Sense for Middle Schoolers.]

In truth, we could drop the 'student' from the title and simply enumerate common math errors or, even better, fallacies. The latter term is more the flavor of this post, rather than simply careless student errors. This is because fallacies imply that there is a plausibility to some of these errors, i.e., they are somewhat natural errors to make if one doesn't fully grasp the ideas. Experienced math educators anticipate these errors and caution students about them during the lesson, preferable to commenting on these errors on their tests! In fact, it is possible that students can learn considerable mathematics by being asked to comment on these and explain each error.

IMO, helping students understand the underlying concept in each of these common mistakes is an essential part of teaching and learning mathematics. This is my approach in this post, rather than a "Can you believe I saw a student do this once!" attitude.

There are many such lists easily found by searching the web although most refer to common college errors (in reality, they include many precollegiate errors). Here is one of the best from a wonderful professor at Vanderbilt University. It is thorough, contains excellent discussion and categorizes the errors. Rather than copy from these sources, I wrote a few off the top of my head. I will also include Eric's which he posted in a recent comment to the 16/64 = 1/4 post.

Eric's Excellent List (and I know he has a few hundred more):

1. √(a²+b²) = a+b

2. (-x)ⁿ = - xⁿ

3. (fg)ʹ = fʹgʹ

These are wonderful. #1 and #3 could be classified as 'everything distributes' errors, although in verbal form, it could be interpreted as an 'everything commutes' error:
"The derivative of a product is the product of the derivatives" error.
Similarly for #1:
"The radical of a sum is the sum of the radicals" error.
#2 is an order of operations type of error and I included this in my list in a particular form.

Here's my initial offering for Grades 7-12. Feel free to bring your own list to the table for other grade levels. Common errors in calculus and beyond are fair game as well. I will try to classify some of these...

Radical Errors
√49 = ±7 type; similarly 161/2 = ±4
√(n2) = n

Operation Errors (Exponents)
-42 = 16
an = bn ↔ a = b
x2/3 = 16 → x = 163/2 or 64
[One possible correct method: x2/3 = 16 → x2 = 163 → x = ±√(163) = ±64]
Fraction Errors - Algebraic or Arithmetic (limited to the top 10000 errors please!)
(12x+7)/(4x+9) = (3+7)/9
[What name would you give this one? 'Cancelling error', 'Cancelling terms not factors error'?]

Rather than continue my list, I welcome offerings from our readers. A fairly thorough compilation of these could become a book! Perhaps, in a serious vein, a small monograph that could be helpful to both students and math educators...


Anonymous said...

In particular, the error

(f/g)' = f'/g'

seems to increase after the students get exposed to L'Hospital's rule.


Anonymous said...

I'm not a math teacher but dividing
and multiplying inequalities with
negative values is often a source of

-2 > 2x --> 1 > -x

Anonymous said...

Another one:


Cancel (x-2) from both sides, and get x=-2.


Anonymous said...

So is the error of using l'Hôpital's rule without checking that the numerator and denominator have limits 0 (or ∞).

Dave Marain said...

for one who is not a math teacher, you found a 'classic!'

tc, eric--
We could have an entire section devoted to L'Hopital's and limits in general!
A classic: (1+1/n)^n → 1
Ah, those fun indeterminate forms!

Keep 'em coming folks.
This list should be endless...

Jackie Ballarini said...

Simplifying errors with radicals:

((6)^.5)/2= 3^.5

Dave Marain said...

Luv it!
Sometimes, students simply don't simplify far enough. They will leave
an answer with √1 in their answer! And, of course, √0 is undefined.
Who is keeping track of these!

Anonymous said...

canceling part of a sum. Do it in my class, and all partial credit on that question is voided.

How about √(75) = 3√5 ?

But of course the most common student error is the magically disappearing minus sign...

Dave Marain said...

good one, jonathan!
I attempted to prevent this common transposition by requiring that students write the extra step:
√25 ⋅ √3, but no instructional technique works with all students all the time! (It did work some of the time with some students...)

We could probably devote an entire book to negatives, which is why I often told my algebra students:

Anonymous said...

∫_a^b f(g(x))g'(x) dx = ∫_a^b f(u) du.

Anonymous said...

Isn't the function f(x) = sqrt(x) defined at x=0??, i.e., sqrt(x) = 0.

Dave Marain said...

Yes, √0 = 0. I meant that students think it's undefined! Sorry for my lack of clarity. I was listing common errors but I should have made that clearer so one doesn't take it as a fact.