Questions tagged [students-mistakes]

For questions investigating students mistakes, clarifying their origin and asking for advice to fix or improve the mistakes.

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13
votes
4answers
401 views

Analyzing an answer to the following problem: Give meaning to $\frac{4}{5} + \frac{2}{3}$

Case: Exam Problem given to student at university: Give a problem/context illustrating the operation $\frac{4}{5} + \frac{2}{3}$ Answer by student: Anna and Beatrice buy flowers for grandpa for his ...
22
votes
7answers
3k views

How do you coach students who often make small errors?

Some students are prone to making small calculation errors. Not errors in understanding, but errors like adding or multiplying integers incorrectly, or dropping a negative sign. Unsystematic errors in ...
25
votes
5answers
7k views

How does a teacher come up with plausible wrong answers for multiple choice tests?

When taking a MOOC in calculus the exercises contain 5 options to select from. I then solve the question and select the option that matches my answer. Obviously only one of the options is correct. But ...
4
votes
0answers
111 views

How to explain the "less than yearly compounded interest" concept?

What difficulties can be met while teaching the "less than yearly compounded interest" concept? Based on my own (learning) experience, an objection arose when I was presented with the formula: $$A ...
4
votes
3answers
312 views

Students writing $f(x^2+1)$ when they probably mean $f(x)=x^2+1$

Over the past years teaching freshmen calculus I've repeatedly seen students make the following type of error: Suppose they have to express some quantity $y$ as function of $x$, when the relation ...
31
votes
4answers
985 views

Taxonomy of bad proofs

I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in ...
5
votes
3answers
243 views

Algebra/trig/precalculus review questions that elicit common student errors

This semester I have decided to give students a simple question or two at the beginning of every calculus class that will trap them into making the most common errors that we all know...e.g. the ...
10
votes
8answers
3k views

How to explain that the sums of numerators over sums of denominators isn't the same as the mean of ratios?

I am a teaching assistant for an intro programming course. One assignment asked for the averages of a certain ratio, but most students, rather than returning $$\frac{\text{sum of all ratios}}{\text{...
31
votes
24answers
7k views

Quote to show students don't have to fear making mistakes

I have some high school students which seem to be afraid of making mistakes. They are hesitant to make exercises in class because they want their course notes to be super clean, without any mistakes. ...
5
votes
1answer
445 views

Propositional and predicate logic, with quantifiers: Is there any research when it is ideal to explicitly teach in mathematics education?

In terms of helping students to understand propositional and predicate logic, with quantifiers, is there any research regarding when it is most advantageous for students studying mathematics, to first ...
5
votes
7answers
373 views

Category mistakes regarding symbols and their impact on math (mis) understanding. ( Object symbol/ sentence symbol confusion)

A friend of mine that teaches math has made many times the following experiment : drawing two circles on the blackboard representing two sets A and B such that A and B are disjoint writing on the ...
9
votes
3answers
580 views

What are some common ways students get confused about finding an inverse of a function?

What are some common ways students get confused about finding an inverse of a function? One I can think of is conflating multiplicative inverses of rational numbers with functional inverses. e.g. ...
12
votes
9answers
3k views

How to explain what's wrong with this application of the chain rule?

Yesterday a student in my calculus class attempted something like this: Problem statement: Find the derivative of $3^{(5x+1)}$ with respect to $x$. Proposed solution: Let the inner function be ...
14
votes
4answers
595 views

How to deal with "Why can't I just do ......" in real analysis?

I'm teaching introductory real analysis at a large public university in the US. A common question from students is of the form "Why can't I just do it like this?". i.e. Often a student has come up ...
11
votes
4answers
880 views

Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ -- why?

I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than ...
15
votes
2answers
1k views

"Always/Sometimes/Never" vs. "True/False" questions for mathematical reasoning

Has anyone performed a study on the differences between student interpretations of these words? Background: When I taught high school geometry and later undergraduate precalculus, I noticed that ...
3
votes
0answers
178 views

How to remedy the "freshman's dream"? [duplicate]

I am teaching a mid-level calculus course, and I see my students making the freshman's dream mistake of thinking that every function is a homomorphism. In particular, they think that exponents can be ...
20
votes
5answers
675 views

How students write their work, and learning outcomes

While teaching students mathematics, I have noticed that some seem sloppy in the way that they write down their work. For example, a student is given a question: What is the area of the rectangle? <...
15
votes
13answers
3k views

Mnemonics for some properties in mathematics

I am looking for various mnemonics which help students to remember some important properties or theorems. Very often students confuse signs or relations such as $\leq$ and $\geq$ in some expressions. ...
2
votes
0answers
108 views

Why does result depend on procedure in my calculation of surface area using Guldin? [closed]

At present, I teach Guldin's rules for surface and volume of rotation, and give an example task from the textbook. The textbook uses procedure 1 (below) for calculation (below), but I advocate that ...
10
votes
4answers
285 views

Algebra best practices for students

One thing I notice frequently is that students don't have 'best practices' for doing algebra. Let me given an example: If students are trying to differentiate, say, $f(x) = (x^2 + x)^2$, they will ...
6
votes
0answers
166 views

Adding one to numbers bigger than ten

If someone asks you Tell me the next number (add one) after the number one million two hundred thirty-one thousand ninety-nine, do you known if it is a common error that the first number that ...
12
votes
6answers
729 views

teach that $\frac10$ not defined properly

there're some students, who belive that $$\frac10 = \infty $$ I need to teach them that this is not true and $\frac10 $ is undefined, mathematically and give a good picture (for their minds) what is ...
28
votes
10answers
8k views

How should a student's inefficient calculation be pointed out?

One time I watched a student solve the equation $0 = (x-2)^2-9$ for $x$ like this. $$\begin{align*} 0 &= (x-2)^2-9 \\0 &= (x^2-4x+4)-9 \\0 &= x^2-4x-5 \\0 &= (x+1)(x-...
4
votes
1answer
202 views

How much credit for a single arithmetic mistake?

A question related to this one, on an equivalent problem on a community-college College Algebra exam. In the prior question most readers observed the error as a critical conceptual issue. Now consider ...
6
votes
8answers
577 views

How much credit to give a short exam question with one error?

In comments here there seemed to be significant difference of opinion on how much credit to give to a short exam question with a single critical error in it. Consider the example of student work ...
19
votes
3answers
748 views

Constructive refutation of student misconception

Although @Gareth Shepherd recently posted Addressing fundamental math errors close to the issue, I experienced my problem of misunderstanding in class, where two good K10 students were asked to ...
16
votes
1answer
376 views

Addressing fundamental math errors

I am looking for ways I can correct fundamental math mistakes. I am currently tutoring someone taking a course which is a cross between first year calculus and grade 12 functions. In high school he ...
17
votes
4answers
1k views

Common misconceptions in high school probability curriculum

I am teaching probability to high school students. The material we are covering is pretty standard and includes: Introducing how to calculate the probability of events, e.g. coin flips, card draws, ...
21
votes
5answers
3k views

Grating mathematical phrases---How to correct?

As mathematics educators, we all have come across students using mathematical notation incorrectly (looking at you, $\frac{d}{dx}$ vs $\frac{dy}{dx}$ or $\frac{\infty^2}{\infty}$). My question focuses ...
10
votes
2answers
756 views

"Good" and "Bad" student intuitions when teaching and learning mathematics

I'm a college math/science tutor and I'm really interested in STEM education. I'm currently starting work on a project I hope to present in a couple of months at a tutoring conference and I was ...
12
votes
2answers
3k views

How are students messing up in this Khan Academy surface area problem? (right triangular prism: 3-4-5 triangular base, height of 11)

The following question appeared in this Khan Academy exercise in January 2017, and the actual data on the ten most frequent incorrect responses is shown below the question. What mistakes did students ...
3
votes
0answers
218 views

How do i deal with students who make these mistakes? [closed]

I came across some interesting mistakes in many area of mathematics with my students and do not let me also to tell you for university students level, I would like to know How do i deal with ...
16
votes
5answers
348 views

Frequent calculus error: replacing interior part of an expression with its limit

For example $$\lim\limits_{n\to\infty}\left(1+\frac{1}{2n+1}\right)^{n} =\lim\limits_{n\to\infty}{1}^{n}=1\,.$$ Here the student has replaced the sub-part $\frac{1}{2n+1}$ with its limit $0$, but he ...
27
votes
6answers
6k views

Misuse of parentheses for multiplication

I'd like to raise the issue of constant misuse of parentheses in the U.S., and I'm wondering if anybody else shares the same feelings, has had the same issues, knows any history behind it, and can ...
5
votes
0answers
172 views

Catalog of undergraduate's misconceptions / problems while proving

Selden & Selden (2011) listed 41 difficulties their students had in an experimental proving course into 9 categories. Unfortunately I haven't found similar work. Thus, my question is: Is there ...
11
votes
6answers
1k views

Why does the widespread erroneous definition of "irrational number" persist without being taught?

Among the mathematically ignorant one often finds a mistaken proposed definition of "irrational number", which says that it is a number whose decimal expansion does not terminate or repeat. The ...
19
votes
2answers
422 views

Literature on learning from errors in mathematics

In teaching undergraduate mathematics, I implemented some strategies to encourage the students to look at errors they made or at "typical errors" in the current topic. One attempt was to compile a ...
24
votes
5answers
993 views

A Series of Unfortunate Examples!

All of us know, when teaching, the "right" choice of examples is important. Though, making the "right" choice is one of those things that are easier said than done. Here is the ...
16
votes
4answers
697 views

Do students confuse $\log_ab$ and $\log a^b$?

I recently observed a group of students being introduced to logarithms for the first time. Some of them had trouble writing $\log_ab$ properly, and it looked more like $\log a^b$. All logarithms have ...
4
votes
0answers
224 views

Is there a meeting place for fraction education stakeholders?

Is there a meeting place where all fraction education stakeholders (students, teachers, parents, researchers, fraction software developers) meet and collaborate? If not, do you have an advice on ...
4
votes
0answers
238 views

And the joke was NOT a joke [closed]

This is not a question; so I perfectly understand that it could be deleted almost immediately. Yesterday, I was attending a New Year eve party and I was introduced to a young and brilliant lawyer. ...
17
votes
4answers
2k views

Why do students only see the last term of a sum abbreviated with an ellipsis?

It's very common in learning mathematical induction to prove statements like $$ 0^2+1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.$$ I've found that very frequently, on this sort of problem, when ...
1
vote
1answer
100 views

Moving lines in schematic diagrams

I am self-teaching myself precalculus.My biggest sticking point is to move lines within a diagram, especially, functions in the coordinated plane. For instance, doing horizontal and vertical shifts of ...
30
votes
3answers
2k views

Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$

On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other ...
22
votes
2answers
2k views

Is this just a mistake or a more serious misconception?

One of my main research areas is algebraic thinking at different levels. Yet, from time to time, I observe something that I cannot relate to anything else that I know. This is the story of one of ...
14
votes
2answers
495 views

Students problems with reasoning, not exactly math

Consider the following problem: Maria always buys ice-cream when she goes to the beach. She bought ice-cream today. So, she must have gone to the beach. Obviously this statement is wrong. Maria ...
5
votes
2answers
142 views

Approaches to Teaching for Questions Around Partitioning Space

My students are preparing for their SATs and have problems with a certain type of questions, i.e., questions involving a geometrical figure and $n$ (usually $n = 2,3$) straight lines passing through ...
8
votes
1answer
898 views

Should $\varphi$ be monotone in the integration by substitution?

I'm trying to calculate $$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\sin t \cos^3 t\,dt$$ using integration by substitution $$\int_{\varphi([a;b])} f(x)dx=\int_{[a;b]} f\left(\varphi(t)\right)|\varphi'(t)|...
19
votes
3answers
659 views

How can I discourage proof by patchwork?

I have a student who is working in their spare time on proving or disproving a conjecture of the form $$\exists x.\forall y.\phi(x,y).$$ Right now their strategy is to construct an $x$ and then show ...