48

All four of your options lead with "They are told..." Consider asking the student questions instead. At the very least, this shows interest, and they may end up catching their own mistakes as they try to explain to you what they had glossed over in their own heads. When I have the opportunity, I like to challenge my students to explain EVERY step of their ...


46

It depends on how much time I can afford to spend on the problem. I check, as you did, whether I see a mistake. If not, I try to explain the math in a different way: a different conceptual approach, which is good for everyone anyway; perhaps replace the variables with numerical values, if appropriate; or replace a general function $f$ by a specific ...


42

The problem you describe is well-known in mathematics education research. I cite the paper of De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334. and give some ...


38

I don't view these common mistakes as 'universal linearity' assumptions. The mistake that $(a+b)^2=a^2+b^2$ is just a visually appealing statement. It is mistaken to be correct because it looks nice. Our brains tend to like things that look nice. Similarly, $\sqrt{a+b}=\sqrt a+\sqrt b$ is visually appealing and it resembles the correct formula $\sqrt {ab}=\...


36

I have no evidence to back this up, nor do I know how I could obtain such a thing. But I strongly believe this is a vocabulary issue, and I would like to see the term "series" phased out of usage in this context. To many minds, "sequence" and "series" convey the same thing: a list of items. In modern parlance, we speak of a television series (a sequence of ...


29

A colleague of mine includes on all his tests a line that reads (something like) "you will be graded on what you actually write down, not what I think you may have meant by what you wrote." He spends time carefully discussing what this means (examples like the one you gave are among them) before and after testing times. Normally after the second round (of ...


29

It is actually wrong to say that parenthesis means multiplication. In $(2)(5)$ it is the lack of an operator between the parenthesis that implies multiplication, NOT the parenthesis. The parenthesis are "needed" because $25$ means the number twenty-five, the parenthesis are purely for grouping. "No operator means multiplication" is an extremely common ...


27

Maybe, your students have a belief problem. They will rarely (maybe never) have encountered problems where something was not well-defined. If you have never been in trouble since everything you were shown was well-defined, then you don't even understand the problem! (Even harder: after proving that something is well-defined, the world looks right like it ...


27

While all of your students at this point will have done (extensive) units on manipulating and simplifying expressions with exponents, this is the limits unit. When doing limits questions, most students are only searching their brains for limits techniques - they're extremely unlikely to come up with the $\frac{x^a}{x^b} = x^{a-b}$ rule that had been driven ...


27

Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just occurred. Otherwise I would let them complete their process (unless obviously totally infeasible) and then compare to a faster way afterward.


24

The root of the difficulty is that $x$ appears free in $f(z)$, but we are trying to "capture" it with $g(x)$, which is illegal. When we substitute $g(x)$ into $f(g(x))$, we have a variable clash: $$ f(g(\color{red} x)) = 3^{5\color{blue}x + 1} $$ The red (first) $x$ is a different variable from the blue (second) $x$. This is clearer if we rename the bound ...


23

Johann Wolfgang von Goethe: "By seeking and blundering we learn." Original German, 1825. Albert Einstein: "Anyone who has never made a mistake has never tried anything new." (However, attribution to Einstein is weak. See quoteinvestigator.com.) Jo Boaler: "When I have tutored people in math, I've always started by saying, 'By the way, I just want you to know ...


22

I second Andrew Sanfratello's answer—there's just no getting around the fact that technical terms have technical meanings which differ from their everyday meanings. And when I teach series, I start each class by writing "SERIES MEANS SUM" at the top of the board every day. (At one previous institution, I often taught in a room with several large ...


22

Edit 9/5/14: It has recently come to my attention that another helpful paper is: Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. Review of research in education, 3-56. Link. Though there are sure to be more technology-related errors today, you can find a late 70s article on this subject from ...


22

Personally, I don't think we attend to this sufficiently in lower-level mathematics (where it's actually needed most). Students need that vocabulary to interface with books, future teachers, tutors, other students, etc. I run questions on it in weekly quizzes; and if I had my druthers, it would be a major component of all tests (in addition to application-...


21

I have had this problem before with students who always think they're right. If a student continues to insist you made a mistake, when you know that you haven't, then tell the student to hold the thought and ask them to discuss it with you after class. Once the class is over, write the original problem on the board and ask the student to solve it. If ...


21

I like your second option the best: ...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part? Instead of just mentioning the easier way, however, you could first applaud them for using good algebra to find the right answers, then remind them of the new ...


20

They are not usually well-prepared, but factoring is not a big issue. I would like students to be able to: Make meaning from graphs. [I want to get to introduce the connection between speed graphs and distance graphs, though, so I'd be happier if you focused on other graphs.] See what is algebraically sensible, and what's not. e.g. you can't cancel from ...


20

I agree with @kathleen, that being a young (and female) instructor can lead some students to be less respectful. In this case, we can use that to our advantage... Long ago, when I was less secure as a teacher, I would be so embarrassed and bothered by making a mistake that it would decrease my ability to teach well for the rest of that class period. I knew ...


19

This became to big to be a comment. Layman's opinion. Where does it come from? It comes from the fact that universal linearity is useful to move forward in calculations even if it's wrong. Psychologically this is very attractive. The other option is being stuck. Moving forward has the added incentive that it can be right, that maybe the student can get ...


19

As a first guess, I'd give the following categories: Overgeneralization of known theorems or techniques. Application of these theorems or techniques in unsuited situations. Overconfidence in intuition Making intuition into false theorems. Applying intuition where it contradicts known theorems. Nonapplication of known theorems or techniques. (Mistakes ...


19

To answer the ultimate question ("Can anybody explain where this writing tradition comes from?"): It's explicitly taught that way by many U.S. instructors and textbooks. Examples: From the otherwise excellent Martin-Gay Prealgebra & Introductory Algebra (sec 1.5): The × is called a multiplication sign... The symbols ∙ and () can also be used to ...


18

When it comes to convincing younger students of something, I find that analogy can be quite useful, even if you have to squint a bit to make it technically rigourous. I imagine a younger student reasoning something like this: "Hey, I know 5 is different from 50, and 50 is different from 500. When you add zeros at the end of number, it changes the number. So ...


18

Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes are basic to our understanding of everything. Where mathematics comes from, by George Lakoff and Rafael E. Núñez. Basic Books, 2000. Exploring the role of ...


18

I suspect that the issue is not so much the ellipsis per se but a problem with notation in general, and in particular with the correct use of the equals sign. At the risk of repeating what I wrote in this answer, students often regard the equals sign not as a symbol meaning "these two expressions are the same" but rather as a symbol separating a question (...


17

I disagree that it is "terribly harmful". Do not prevent them from writing $(2)(5)$. Instead prevent them from writing things that are actually wrong. Thinking that $\sin x$ is $\sin$ times $x$ can happen (and does happen) even without parentheses. I agree $(2)(5)$ looks strange, but if they can write $(4-2)(4+1)$, what is wrong with $(2)(5)$?


17

What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever is $b$. This is about not being defined. Still... why is $\frac 1 0=\infty$ not so completely wrong? Because they can see that the smaller is $a$ then the ...


17

One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$. The ratio of the sum of numerators to sum of denominators is $\frac12$. However, we can also write $a=\frac03$ and $b=\frac22$. Now the ratio is $\frac25$, which is not equal to $\...


16

As a student myself, I'd say that, while I'm not representative of all students, some of it is the intimidating or dismissive way some lecturers or teachers might treat questions relevant to simple things like these, or not teach off the exam specification. This leads to a subconscious or even conscious bias against asking these sort of questions which we're ...


Only top voted, non community-wiki answers of a minimum length are eligible