Questions tagged [students-mistakes]

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

Filter by
Sorted by
Tagged with
6 votes
3 answers
304 views

What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?

So in an advanced mathematics course for engineers, there are often problems of the type: Prove claim A. Given equation A, show that you can obtain equation Z. I am frequently faced with a problem ...
Fraïssé's user avatar
  • 737
11 votes
1 answer
1k views

Does there exist a (statistical) topology induced by students on the space of algebraic formulas? :)

It's kind of a serious question even if the title seems silly. As math educators, we all know that students link together different algebraic expressions thinking that they mean the same thing, e.g. \...
marco trevi's user avatar
2 votes
2 answers
268 views

Scepticism as the cornerstone for not making mistakes in arithmetic/algebra etc, especially for students who relentlessly make every possible error

As a maths tutor, some students I have tutored don't just make the odd mistake in arithmetic (including fractions) and algebra: they make every possible mistake and regularly. My go-to approach for ...
Adam Rubinson's user avatar
3 votes
2 answers
355 views

Common mistakes in probability

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
Botnakov N.'s user avatar
12 votes
6 answers
5k views

Is patching up a student's poor solution better than providing a good solution?

A student benefits from their attempt at a solution or proof being checked by the teacher. My own view is that, if the student's work is poor, it is best just to provide a model solution or proof in ...
John Bentin's user avatar
27 votes
6 answers
5k views

Students confusing "object types" in introductory proofs class

In my intro to proofs (and discrete mathematics) class, I see a common mistake where students make nonsensical statements because, for lack of a better term, they confuse the types of the mathematical ...
Daniel's user avatar
  • 381
5 votes
2 answers
325 views

How to teach the concept of probability distribution?

I observed that my students do not understand what a probability distribution is. We do not treat probability axiomatically on the course, so the required level of understanding is knowing all the ...
Tommi's user avatar
  • 7,144
2 votes
1 answer
899 views

Why can't students master math simply by passive reading?

Yearly, at least one student emails me this question, after wholly relying on passive reading then failing the exam. They successfully remember and can regurgitate everything from the textbook and ...
user131533's user avatar
11 votes
4 answers
6k views

How can a teacher help a student who has internalized mistakes?

John is trying to learn some field of mathematics. John's teacher Sarah gives constructive criticism to John after seeing John's attempts on exercises related to the field he studies. Later, John ...
tryst with freedom's user avatar
6 votes
3 answers
471 views

How can you elicit the $\log x = {\log} \cdot x$ error?

You know the error, when you're watching a student work through an algebraic calculation to solve for a variable trapped in the argument of a function, usually $\log$ or a trig function, and you watch ...
Mike Pierce's user avatar
  • 4,845
13 votes
4 answers
569 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 ...
Improve's user avatar
  • 1,881
25 votes
7 answers
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 ...
Mike Pierce's user avatar
  • 4,845
24 votes
5 answers
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 ...
Marcus's user avatar
  • 351
4 votes
0 answers
124 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 ...
user avatar
4 votes
3 answers
439 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 ...
Michael Bächtold's user avatar
37 votes
4 answers
3k 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 ...
Patrick Lutz's user avatar
8 votes
3 answers
409 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 ...
Jon Bannon's user avatar
  • 6,173
11 votes
8 answers
6k 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{...
JohnnyApplesauce's user avatar
32 votes
23 answers
8k 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. ...
dietervdf's user avatar
  • 431
6 votes
1 answer
713 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 ...
amWhy's user avatar
  • 2,095
5 votes
7 answers
463 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 ...
user avatar
8 votes
3 answers
1k 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. ...
Eleanor Hankins's user avatar
13 votes
11 answers
4k 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 ...
Michael Bächtold's user avatar
14 votes
4 answers
853 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 ...
SBK's user avatar
  • 301
11 votes
4 answers
1k 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 ...
user avatar
17 votes
2 answers
2k 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 ...
Opal E's user avatar
  • 4,044
3 votes
0 answers
255 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 ...
NoName's user avatar
  • 141
21 votes
5 answers
785 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? <...
ctrl-alt-delor's user avatar
15 votes
13 answers
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. ...
YukiJ's user avatar
  • 712
2 votes
0 answers
116 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 ...
Morten Engelsmann's user avatar
10 votes
4 answers
336 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 ...
Elle Najt's user avatar
  • 341
6 votes
0 answers
182 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 ...
pasaba por aqui's user avatar
10 votes
6 answers
819 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 ...
Bad English's user avatar
30 votes
11 answers
9k 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-...
Mike Pierce's user avatar
  • 4,845
4 votes
1 answer
307 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 ...
Daniel R. Collins's user avatar
6 votes
8 answers
820 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 ...
Daniel R. Collins's user avatar
19 votes
3 answers
851 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 ...
Morten Engelsmann's user avatar
17 votes
1 answer
447 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 ...
Gareth Shepherd's user avatar
19 votes
5 answers
2k 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, ...
Improve's user avatar
  • 1,881
21 votes
5 answers
4k 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 ...
erfink's user avatar
  • 1,129
10 votes
2 answers
1k 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 ...
Jorge Medina's user avatar
12 votes
2 answers
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 ...
Cameron Christensen's user avatar
3 votes
0 answers
223 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 ...
zeraoulia rafik's user avatar
17 votes
5 answers
413 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 ...
amarius8312's user avatar
29 votes
6 answers
8k 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 ...
zipirovich's user avatar
6 votes
0 answers
182 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 ...
Stephan Kulla's user avatar
12 votes
6 answers
2k 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 ...
Michael Hardy's user avatar
20 votes
2 answers
468 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 ...
Christian's user avatar
  • 544
25 votes
5 answers
1k 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 ...
Amir Asghari's user avatar
  • 4,428
17 votes
4 answers
818 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 ...
Joonas Ilmavirta's user avatar