What are some common fallacies students make when they learn $X$ concept?

What are some common mistakes students often make, which may seem logical at first?

I'm a student myself, but I'm curious of what some of the most frequent mistakes which happens. I'm thinking of purely mathematical incorrect beliefs like $(a+b)^n = a^n+b^n$, because $(a\times b)^n = a^n\times b^n$, or misunderstanding notation like $(f\circ g)(x)$ thinking it means $g(f(x))$, because you do calculations left to right, right? Or just abuse of notation and writing $(f(g(x)))'$ instead of $(f\circ g)'(x)$, thinking $'$ is a function on $f(g(x))$ (a value).

Also, how do you explain these mistakes?

• This is a very broad question. Could you try to narrow it down a bit, for example, by level of the students or rough subject. – quid Jan 18 '15 at 21:16
• @quid I was hoping for a broad variety of answers to compile a somewhat long list, but if that doesn't fit the rules I'll change. How about students at grade school level (Primary and secondary school), or is that still too wide? – Frank Vel Jan 18 '15 at 21:26
• It still feels pretty wide, but we can give it a try. – quid Jan 18 '15 at 21:40
• The linked papers have examples. Moreover, you may be interested in MESE 926. – Benjamin Dickman Jan 18 '15 at 22:02
• This almost seems like a duplicate of mathoverflow.net/questions/23478/…. – KCd Jan 24 '15 at 3:43

Some people omit zeros from numbers. For example, they treat all of the following as being equivalent:

• 015
• 15
• 150
• 105

In many computer programming languages, all of these are different. (Many languages default to base (5+5), but will use base (5+3) if there is a leading zero, or base (5+5+5+1) if there is a leading 0x.) Most non-programmers expect the first two examples to mean the same thing, and the last two examples to mean different things.

Conflating tens and hundreds. Some children, when learning to count to large numbers, try to do this...

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, …

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, …

Confusing partial derivatives with full derivatives.

Confusing demonstrating that something is true for a special case with proving that it is true for the general case. Of course, demonstrating that something is false for a special case does disprove the general case of a claim -- unless special exceptions are carved out. But it is not enough to prove that the general case is always false.

Thinking that units don't matter. If I travel 120 miles in 2 hours, what is my average speed? 1? 60? Um, no. Those answers should be 1 mile per minute, or 60 miles per hour.

Mis-squaring units. How large is a square mile? 1 mi² = 5280 ft² ? Um, no. 1 mi² = 1 * (5280 ft)² = 1 * 5280 ft * 5280 ft = 5280 * 5280 ft² = 27,878,400 ft².

Thinking that the calculator will tell them the correct number of significant figures. What is 1,000,000 - 100,000? 899,999.99999999? Um, no. The calculator had a rounding error. The precise answer is 900,000; rounding to the correct number of significant figures will produce either 900,000 or 1,000,000.

Rounding off intermediate values, instead of final answers. Computers are especially prone to this problem, which is why there are entire books on Numerical Methods. Intermediate values should be absurdly precise. Care needs to be exercised in interpreting them.

Throwing away the remainder. What is 10 / 9? 1? 1.1? 1.11? Um, no. It is 1 + 1/9, or 1.11… . That … is a number, and it is not zero. It might not be worth calculating right now, but if we multiply it by 9, we will get something useful. (This also makes it easier to understand why 10 = 9.99…; this … equals 9 * 0.01 / 9, which equals 0.01 .)

• Maybe the child you mention (who counts by 'tens' like '$\dots , 70, 80, 90, 100, 200, \dots$') isn't at fault but rather is running on a floating-point processor with one significant figure! – Vandermonde Nov 4 '15 at 9:39

A common error is canceling something "completely" instead of leaving the "trivial" term $1$. What I mean is the following type of incorrect simplifications:

$$\frac{4x + 2}{2} =2x$$ or

$$4x^3 + 3x^2 + x = x(4x^2 + 3x).$$

A way to counter this can be to insist that the factor is first written out at each term, like: $$\frac{4x + 2}{2} = \frac{2 \cdot 2x + 2 \cdot 1}{2} = 2x + 1$$

• The standard conversation I have with a student on the first simplification: "What is left when you divide 4x by 2?" "2x is left." "So 4x divided by 2 is 2x." "Yes." Okay, good so far -- now, "What is left when you divide 2 by 2?" "Nothing is left." Ah. "So 2 divided by 2 is nothing." "Yes." "2 divided by 2?" "Oh..." – Chris Cunningham Jan 20 '15 at 15:07
• Well, the result is a multiplicative nothing/identity... – Vandermonde Jan 24 '15 at 21:51
• 'Nothing' is a dangerous word in beginning mathematics anyway. I have a terrible time convincing my calculus students that there is a difference between "the curve has no slope" (like the absolute-value function at 0) and "the curve has 0 slope" (like the squaring function at 0). – LSpice Sep 30 '15 at 17:41
• @LSpice: Unsure whether the comment is directed here, but I'm not saying you have to call any particular concept 'nothing', and even if you did, anyone reasonable would realise what's going on after a bit of explanation if necessary. The 'nothing' in 'a ham sandwich is better than nothing' is of a completely different character than that in 'nothing is better than eternal happiness', and everyone I've met understands that the implied conclusion is a joke; the two 'nothings' merely happen to be spelled alike. – Vandermonde Nov 4 '15 at 9:31
• Namely, the former instance genuinely means nothing as an object, along the lines of 'a ham sandwich is better than the contents of an empty box', and (as a litmus test of sorts) would also make sense if perturbed to 'something negligible', whereas the second hides a quantifier and would not make sense under such a replacement. This parallels the matter of 'zero slope' vs. 'no (real) number satisfies the definition of slope'. – Vandermonde Nov 4 '15 at 9:35

the freshman mistake is what a colleague of mine used to call the error that every function is linear. Like $$\sin(3x) = 3 \sin x$$ another popular version $$\frac{1}{x+y} = \frac{1}{x} + \frac{1}{y}$$

• (As mentioned in the comments to the OP.) This sort of linearity problem is discussed specifically at MESE 926. – Benjamin Dickman Jan 19 '15 at 17:23
• Worse than that is $1/\sqrt{x^2+y^2} = 1/x + 1/y$. Today a colleague told me that he saw that on a student's multivariable calculus exam. – KCd Jan 24 '15 at 3:39

Indeed, it is a very very broad question with a very very long literature behind it since for a long time Math Ed research was just about recognizing such mistakes. Thus, I narrow down my answer to a general phenomenon and a favorite example of mine. The general phenomenon is to project (mistakenly) the properties of an old domain to a new domain. An often quoted example is the idea of "multiplication makes bigger". Working with vectors as scalars is another example. There are too many. A favorite of mine, happened once in one of my classes, is the one about complex numbers.

We have just seen the geometric interpretation of complex number, and the very fact that $i$ is one unit above the origin on the y-axis. Then one student used the Pythagorean Theorem for the right triangle with the "legs" $i$ on the y-axis, and $1$ on the x-axis, and concluded that the length of the hypotenuse is zero: $i^2+1^2=0$!!

I think in part you answered your own question. Students make these mistakes because they seem logical. What 'seems logical' to a student will be the product of a host of factors such as age, maturity (social and mathematical) experience and many more besides. Whether you make these mistakes or not is not necessarily a sign of mathematical ability. The student that writes $a^2=a2=a\times2$ has a right to be confused if they have taken on board the idea operations don't need to be written down after all small handwriting was not part of the deal and my English teacher is forever telling me to write on the line! Sir obviously has the same problem.

• I might have been unclear, but I want specific 'why's for specific fallacies. – Frank Vel Jan 24 '15 at 18:17