Questions tagged [students-mistakes]
For questions investigating students mistakes, clarifying their origin and asking for advice to fix or improve the mistakes.
84
questions
6
votes
3
answers
251
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 ...
- 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.
\...
- 805
2
votes
2
answers
245
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 ...
- 935
3
votes
2
answers
333
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, ...
- 179
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 ...
- 249
27
votes
6
answers
4k
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 ...
- 381
5
votes
2
answers
287
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 ...
- 6,751
2
votes
1
answer
878
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 ...
- 357
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 ...
6
votes
3
answers
461
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 ...
- 4,606
13
votes
4
answers
557
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 ...
- 1,881
23
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 ...
- 4,606
25
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 ...
- 361
4
votes
0
answers
122
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:
$$A ...
user12116
4
votes
3
answers
434
views
Students writing $f(x^2+1)$ when they probably mean $f(x)=x^2+1$
Over the past years teaching freshmen calculus I've repeatedly seen students make the following type of error:
Suppose they have to express some quantity $y$ as function of $x$, when the relation ...
- 1,961
37
votes
4
answers
3k
views
Taxonomy of bad proofs
I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in ...
- 521
8
votes
3
answers
397
views
Algebra/trig/precalculus review questions that elicit common student errors
This semester I have decided to give students a simple question or two at the beginning of every calculus class that will trap them into making the most common errors that we all know...e.g. the ...
- 6,133
11
votes
8
answers
5k
views
How to explain that the sums of numerators over sums of denominators isn't the same as the mean of ratios?
I am a teaching assistant for an intro programming course. One assignment asked for the averages of a certain ratio, but most students, rather than returning
$$\frac{\text{sum of all ratios}}{\text{...
- 229
33
votes
23
answers
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. ...
- 441
6
votes
1
answer
683
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 ...
- 2,075
5
votes
7
answers
453
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 ...
user12295
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. ...
- 175
10
votes
9
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 ...
- 1,961
14
votes
4
answers
818
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 ...
- 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 ...
user507
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 ...
- 3,996
3
votes
0
answers
251
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 ...
- 141
21
votes
5
answers
777
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?
<...
- 331
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. ...
- 712
2
votes
0
answers
115
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
4
answers
331
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 ...
- 341
6
votes
0
answers
178
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 ...
- 425
10
votes
6
answers
804
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 ...
- 201
28
votes
10
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-...
- 4,606
4
votes
1
answer
289
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 ...
- 24.4k
6
votes
8
answers
790
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 ...
- 24.4k
19
votes
3
answers
840
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 ...
17
votes
1
answer
438
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 ...
- 271
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, ...
- 1,881
21
votes
5
answers
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 ...
- 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 ...
- 533
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 ...
3
votes
0
answers
222
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 ...
- 287
17
votes
5
answers
406
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 ...
- 757
29
votes
6
answers
7k
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 ...
- 884
5
votes
0
answers
179
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 ...
- 2,021
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 ...
- 1,690
20
votes
2
answers
461
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 ...
- 546
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 ...
- 4,428
17
votes
4
answers
807
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 ...
- 3,720