Questions tagged [mathematical-pedagogy]
For questions on general considerations and problems of teaching mathematics, such as issues specific to teaching mathematics that are relevant in various contexts or courses.
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How to ask a student a question to get the answer '...integer not continuous...'
Context: a very basic level statistics package computer lab. A scatter plot is produced for one integer variable versus another integer variable. The students are asked why the points form a grid ...
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Developing mathematical stories
In a comment on a recent post, Steven Gubkin pointed out that in doing mathematics he likes to develop stories. This motivation for mathematics is perhaps familiar to many practicing mathematicians. ...
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Notation Conflict between Teachers and Textbooks
In mathematics notation plays an important role in clarifying the subject. A bad notation could be confusing. Recently I use a logic textbook which has a very nice approach and content but an ...
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Why are proofs by contradiction counterintuitive?
And how to make them intuitive?
We are tasked to prove $P \implies Q$. So we assume $P$ and are trying to
prove $Q$. We assume not-$Q$ ($\neg Q$) and derive a contradiction, establishing $Q$.
There ...
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Ethics of looking at other proofs before submitting work
I am in my third year of undergraduate math, and now that classes are becoming more proof-based, many of my homework questions are proofs of relatively basic concepts that can be found with a quick ...
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Is proof-based exercise-oriented math course without solution an effective way to teach pure math?
In recent years I have seen several courses in pure math in the undergrad level (year 2, 3, 4) such as real analysis and topology where the entire course consists of:
notes written during the lecture ...
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When are partial fractions taught? [closed]
Recently I had taken the SATs, and a question came up that involved partial fractions decomposition.
$$\frac{x^2-4x+5}{x-3}$$
This is not the exact problem but a similar one. If the SAT math is ...
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How to get through the boring stuff?
It frequently happens that there's some material I have to cover which is, frankly, boring. The subject itself may be boring, or it may be the particular exercises, but in any case I have to get ...
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Common Core Question: What is included and excluded in high school mathematics?
I took pre-calculus in high school, and I did not get to learn about matrices, and conic sections, vectors law of sines and cosines, and etc. I took geometry as well and found that matrices were also ...
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Monty Hall challenge
Thinking about the counterintuitive Monty Hall Problem (stick or switch?),
revisited in this ME question,
I thought I would issue a challenge:
Give in one (perhaps long) sentence a convincing ...
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Is the current education system as bad as most critics and famous pure mathematicians try to convey? [closed]
Throughout elementary, middle and high school mathematics is quite merely about memorizing concepts and formulas, understanding the theorems (without their proofs) and applying acquired knowledge in ...
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A Lexicon of Math Mistakes
Neil Postman wrote an interesting (and freely available) article called "The Educationist as Painkiller." I highly recommend you read the article for your own enjoyment and as a background to this ...
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Calculation versus writing in mathematics
Writing mathematics is an important activity of the mathematician. In trying to write one's mathematics, one finds ways to balance intuition and rigor and to efficiently communicate concepts and ideas ...
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Group theory for high schoolers, want the opinion of other educators
So I am going to be teaching the basics of group theory to high schoolers in a few weeks, and I want to hear what the Stack Exchange network has to say on the matter.
What are the applications and ...
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What methods successfully identify and eliminate severe math anxiety?
What methods are effective in identifying and eliminating severe math anxiety, this most terrible and unfortunate part of modern mathematics education? This question is not about ordinary math anxiety ...
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How is cooperative learning being used in vector calculus, and what are the origins of this work?
I'm doing some research about cooperative learning in vector calculus.
It seems like what cooperative learning in calculus is referred to varies over time. In 1987, there was an MAA book, Calculus ...
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Immersive attention when learning mathematics
In Jennifer Roberts' article The Power of Patience: Teaching students the value of deceleration and immersive attention she talks about intentionally slowing down to contemplate deeply a work of art. ...
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In what ways can educators introduce polynomials in grades 7 to 9?
Q: Is there a way we can teach polynomials, avoiding the "watch me do it & now you do it" training method, that will allow students to anticipate and predict the existence or formulation of ...
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A good antonym for reducing/simplifying equivalent fractions
I am looking for a good antonym for reducing/simplifying equivalent fractions: 'reduce' and 'simplify' both make sense to me when dividing, but I'm struggling to name what it is we do when we multiply ...
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Is there any research on the value of extra credit in the college mathematics classroom?
After teaching mathematics for a year, where in each class I had opportunities for my students to earn extra credit, I am reflecting on whether this has any value.
The reason why I am questioning ...
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Reference request for studies on gender in math examples, homework problems, or math exams
I am looking for a study or reference on gender in math problems given in mathematics.
In math texts or even on math exams, if there is a word problem involving people, these people or "characters" ...
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Polya's "Nearby Problem" Heuristic and Inquiry Based Learning
I've often wondered about the "devise a plan" part of Polya's "How to solve it" outline. What we call "problem solving" can be thought of as what to do when you have no idea what to do. From this ...
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Are precise drawings important in geometry?
In Finnish middle school (yläkoulu) the students learn to measure distances and angles, draw geometric figures and do certain calculations (area, volume, surface measure, trigonometry). There are also ...
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Mathematical concepts and techniques that **pay off the most**? [closed]
There is a smart way of learning, and it consists in first finding out what are the most valuable pieces of knowledge to acquire. The ones that will give you the highest value for your investment in ...
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Do iPhones help students in their math class?
While the question is stated with reference to the iPhone, my actual question is about phones in general. Just as there was much talk about the use of Computers in the classroom over the past fifty ...
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What is a variable?
There are two kinds of answers I'm looking for:
What do students think a variable is?
What do YOU, the teacher, think a variable is?
I'm also interested in why you think a variable is what you think ...
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Should one justify formulae in middle school?
Consider two possible lesson outlines:
Check homework.
Show a visual demonstration for the area of a circle, e.g. https://tube.geogebra.org/student/m279
Calculate the area of a circle as an example.
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Pedagogical Purpose in Making Students Do Problems in A Less Efficient Way First
Let's assume that a group of students need to learn to solve a certain type of mathematical problem for which there is two general methods of solving it, $X$ and $Y$. We also assume that $Y$ is more ...
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Rigor in secondary mathematics
If rigor doesn't mean more challenging problems, then what does it mean?
There is a big push for rigor in common core mathematics, but I'm not sure exactly what rigor means (I'm pretty sure it has to ...
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In Text Exercises
For an undergraduate mathematics textbook, what are the pitfalls of inserting all of the exercises in the text? (As opposed to grouping every exercise at the end of the section).
IMO I feel it is ...
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Research on the use of outlined / structured proofs in instruction
Has there been any research into comparing the effectiveness of using "structured proofs" or "outlined proofs" in higher level mathematics education, compared to traditional "prose" proofs?
For the ...
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Logic in symbols or words
Making precise logical statements is an important part of teaching and learning mathematics.
There are many ways to write such statements, and let me divide them into two main types1: writing in ...
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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 ...
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Helping a reluctant 12 year old
How can I help my 12 year old daughter strengthen her math skills?
My strategy up until a year or so ago had been relaxed. I subscribe to the idea that the best motivation for learning is the ...
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Specific examples (like elementary proofs,or simple problems) which appear rich in abstractions when observed through the lens of abstraction
I am looking for pedagogically motivated examples (like elementary proofs,or simple problems) of "abstraction in action" ? I am looking for good specific examples (pre-university level or first year ...
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Cost and benefits of compartmentalization in k-12 curriculum
This is a soft question perhaps not well suited for the format of the site but I'm interested to hear opinions from this community on this topic.
K-12 mathematics textbooks (understandably) divide ...
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How to use false theorems or proofs?
I would like students to be critical and not believe that every proof they see is correct.
Lecturers make mistakes and students should not think: "That must be a valid argument/proof/syntax because it ...
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When did the term and taught technique 'cross multiplication' enter into common use?
The title says it all, I suppose. I'm interested to know when/where the term/technique cross multiply came into use. Sources would be nice.
In case it's unfamiliar to anyone, or in case the usage of ...
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What are your opinions of a flipped classroom at the secondary level?
Warning: a lot of this post borrows heavily from education theory. I'm in my student teaching semester right now, so a lot of what I explain is taken from research papers and things like that. So how ...
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Placement tests for middle school math
Our district has changed its approach to placing students in grades 6 and 7 math classes. Students considered for placement above grade level must now take a test composed of problems drawn from the ...
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Method for teaching factorization
A while back I stumbled on teacher's website that advocated a different way to teach factorization. Rather than jumping straight to factorization practice, the teacher first had their student's ...
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How to present $\Bbb Z/n\Bbb Z$ to highschool level audience
I have the oportunity to talk to a highschool class about mathematics, the topic I got to present are the integers modulo $n$, ie, $\Bbb Z/n\Bbb Z$ , however I don't want to be very heavy and formal, ...
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What things should one know in order to enjoy their undergraduate degree?
From looking at undergraduate mathematics programmes it's quite apparent that mathematics degrees are demanding, one could even say the work load is gruelling. However I'm certain that there are ...
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Making co-ordinate geometry interesting for XI grade students
I am presently teaching eleventh grade (XI standard) students an introductory course in co-ordinate geometry with a focus on preparations for competitive exams. I have seen books like S.L.Loney's co-...
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Traditional "long" method of multiplication versus grid and partial products -- evidence of better outcomes?
I'm not a math teacher but am actively involved in teaching my children mathematics (elementary age). I learned the traditional "long" approach to multiplication, but the school systems now emphasize ...
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Using original texts while introducing new concepts in class
I'm still a undergrad math student, and my experience in education in math is very limited, however I've been lucky enough to meet teachers that encourage students who are interested in teaching, like ...
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Textbooks for mathematical/computing/physics teaching that are based on empirical research [closed]
I am looking for any (and all) books (on math, physics and Computer Science) that discuss how to teach and selects methods based on empirical research and solid evidence. My biggest interest right now ...
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Can a constructivist learning model be applied to online lower division math courses?
I'm using the word constructivist as it is used in this paper, not in the sense used in logic. The abstract should be sufficient to understand at least roughly what the model is. The important part, ...
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Impact of philosophy of mathematics upon effectiveness of instructor
Is there any research out there on how an instructor's philosophical beliefs about mathematics might affect some aspect of his or her impact as a teacher? My intended meaning of 'impact' is broad, ...
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The word "and" rather than "or"
I asked my students the following question.
Q: Express $\cos(\pi+x)$ in terms of $\sin$ and $\cos$.
A: $-\cos(x)$.
Students: Yeah, but where is the $\sin$ part? If I got this in an exam then I'd ...
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