49
votes
Proof of why BODMAS (or BIDMAS) works?
It's purely a matter of how we choose to define the notation. The main reason for it is that it lets us write polynomial expressions (which are extremely common) without parentheses, e.g., $x^3 + 3x^2 ...
- 4,755
48
votes
Accepted
Questions with "round" answers only?
When teaching linear algebra, I rely heavily on carefully constructed examples where everything can be done with rational numbers of small denominator. (It is faintly amusing that the construction of ...
- 2,684
14
votes
Questions with "round" answers only?
I agree that such exercises have a built in risk of giving the students the wrong idea what is the generic situation and what is quite a special case.
However, "the risk that the student knows she ...
quid♦
- 7,572
12
votes
How much detail should you show in algebra steps while teaching?
The most effective manner is to ask the students to provide the steps as you carry them out at the board and use the resulting feedback to determine how much detail to provide going forward. That is, ...
- 3,986
11
votes
Proof of why BODMAS (or BIDMAS) works?
Perhaps it is worth pointing out that every programming language defines
an operator precedence structure to avoid ambiguities.
An example table for C and C++ can be found here.
Ambiguities must be ...
- 28k
10
votes
How to teach if calculations and algebraic manipulations are off limits
I highly recommend looking at the Calculus exams from the University of Michigan.
https://dhsp.math.lsa.umich.edu/examshops.html
These problems tend to focus on extracting relevant information for ...
- 19.4k
9
votes
Precision in student work
Students have no basis for figuring out what "appropriately precise" means. How much precision is appropriate is a property of the purpose of a calculation, not the calculation itself, and students ...
- 11.3k
8
votes
Proof of why BODMAS (or BIDMAS) works?
To build on other answers, you might show how other conventions exist. Use an H.P. calculator for example (postfix), the LISP family of languages (prefix), and the APL language (all right-associative),...
- 181
8
votes
Proof of why BODMAS (or BIDMAS) works?
I understand this is not a realistic suggestion, but can you avoid "teaching" "PEMDAS" or "BOMDAS" altogether, and teach your students just the math instead? As pretty much everybody already said, ...
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8
votes
Questions with "round" answers only?
I would encourage to have a significant amount of non-"round" numbers in your homework and also in exams. Some reasons:
Except when you would teach calculations with numbers, normally the way you do ...
- 9,096
7
votes
Amount of concrete calculations on board?
I will answer you from my own experience: I am a lecturer in a Maths Learning Centre at a University. We provide a drop-in centre, and also revision lectures and other resources.
Students' problems ...
- 8,707
6
votes
If a computer could be programmed to do a math test, then should those tests be changed?
No.
There is a sort of magical thinking implicit here that something needs to stay, be different between computers and people. At the end of the day, we are "meat" computers. And the ...
- 85
6
votes
Accepted
Determining the first digit of the Quotient using hand long division efficiently?
The awkwardness of "guessing" in the division algorithm is an artifact of the base-ten representation of numbers. If you represent in binary, then your only possible "guess" is 1.
In binary, your ...
- 7,575
5
votes
Accepted
Amount of concrete calculations on board?
I find that too much detail distracts poor students, in particular when calculating numbers.
Almost every student in a lecture wants to understand something in the lecture. Some students may not ...
- 9,096
5
votes
How much detail should you show in algebra steps while teaching?
Answer 1: err on the side of too much detail. In my case at least, the students always want more detail.
Answer 2: I think the only way to know for sure is to ask your class what they are comfortable ...
- 3,613
5
votes
Questions with "round" answers only?
Like others I more often than not craft the problems to have nice solutions. Just to keep the students on their toes I occasionally insert something not so nice. As we are largely discussing ...
- 1,946
4
votes
How much detail should you show in algebra steps while teaching?
The amount you write in the calculation of things should be inversely proportional to how many times the students have seen these calculations. The first example or two should have everything. Later ...
- 451
4
votes
Proof of why BODMAS (or BIDMAS) works?
It works by avoiding the ambiguity that
2 + 3 x 6
would otherwise have. If we simply said we calculate left to right, we'd have a result of 30. With the priority, multiplication higher, we have ...
3
votes
Student-friendly / efficient approach to computing Taylor coefficients of infinite binomial series expansions?
I think the bug can actually be a feature. Kids need work on the "muscles" of computation. It's not like this is the only setting where doing long calculations is needed (can be the norm in physics ...
- 119
3
votes
Determining the first digit of the Quotient using hand long division efficiently?
Round. Check. Correct.
What we need is $436\div48$. Rounding to remove the last digit reduces the problem to $44\div5$. This gives us an educated guess of either $8$ or $9$.
From here we then ...
- 1,512
3
votes
How much detail should you show in algebra steps while teaching?
You can test, how detailed the students' understanding is, by using alternative exercises. Examples:
Give them a rather long solution without the transformations or with some steps missing. Ask them ...
- 3,352
3
votes
How much detail should you show in algebra steps while teaching?
You just have to learn to "read" your audience. Getting them to speak up is usually hard, and most of the time too late. Do one step by little step, and start omitting steps until you see they aren't ...
- 12k
2
votes
How much detail should you show in algebra steps while teaching?
By default, show the level of detail that you want to see from the average student in the class. Be willing to show more detail if students ask (at least, up to a point), but don't ever show any less ...
- 161
2
votes
Questions with "round" answers only?
It may be useful to present at least one application that is solvable but does not have a simple answer. As an example, quadratic equations with rational coefficients that cannot be easily solved by ...
- 1,735
2
votes
Amount of concrete calculations on board?
When I want to present a problem ("if a man can row so fast and walk so fast, then what is the quickest path from A to B?") in a lower-level math class that is important, but that requires detailed ...
- 758
2
votes
Amount of concrete calculations on board?
I teach high-school Calculus. When I teach a new concept, I almost always give a specific example, then I choose a student at random and ask him or her how to begin solving the problem. After he/she ...
- 2,496
2
votes
Precision in student work
There are multiple issues here. When dealing with a series of equations, taking a power, multiplying and then entering into the next calculation, it can be pretty important to hang on to all the ...
2
votes
Accepted
How to explain inverse modulo?
For the first, there are a number of apps out there, though I will recommend that you go somewhat non-simple and use Sage. Are you looking for additive or multiplicative inverse? In either case, it ...
- 5,752
1
vote
Resources for improving computational skills at the high school/university transition
The Art of Problem Solving has plenty of problems involving pre-calculus tools, some of them quite challenging.
However, the best way forward might not be computationally challenging exercises.
You ...
- 2,859
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