49
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., $x^3 + 3x^2 y - 41x + 2z$ rather than $(x^3) + (3(x^2)y) - (41x) + (2z)$.
However, what really matters is that the notation is clear and unambiguous, so expressions like $...
47
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 such examples often requires mathematics that is far harder than the actual content of the course.) However, I also show them the following slide:
I also ...
14
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 is wrong when some crooked formula/value shows up" you mention is more a feature in my opinion. It is not a bad thing to do heuristic checks that suggest one ...
12
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, as you are writing at the board, you do not just carry on with the next step yourself, but rather, you say, "and then...?" until the students say "add three to ...
11
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 avoided in order for the language parser to create the
correct compiled (or interpreted) machine code to implement the expression.
For example, the expression $4+...
9
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 rarely think about why they're doing a calculation---they're doing it because they were told to.
If you actually want your students to make their own choices ...
9
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 solving problems from a variety of representations (symbolic, graphical, numerical, verbal), and on translating between these representations.
Example: Here is a ...
8
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), all of which do not have differing precedence of operators at all, and write expressions in different ways.
Given 4 parallel translations of the same ...
8
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, this is not actually a rule -- this is a mnemonic device that's supposed to help students remember the actual rules of the order of operations (in the traditional ...
8
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 things is important, not actual calculations. Even if the students feel that something is wrong with the numbers: As long as his/her way was correct, it is only ...
7
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 with lecturers skipping the "simple" calculations
Students often talk to me about how frustrating it is when a lecturer skips steps, or at least when the ...
6
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 computers themselves are getting more and more sophisticated in doing things that were hard for them previously. But the existence of a soul (whether or not true) ...
6
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 problem is to divide the six-bit number 110000 into the thirteen-bit number 1000100010000. Scanning the bits of the second number(the dividend) from left to ...
5
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 follow complicated proofs or discussions, and may pay more attention to examples. If you perform an example with concrete numbers, then a lot of students will try ...
5
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 with. Students are often hesitant about volunteering that information, so I created an anonymous email account for them, so that they could provide feedback ...
5
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 eigenvalue problems, let me propose the following trick I picked up from a senior colleague.
Use a problem, where the precise eigenvalues are not needed, only their ...
4
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 examples should omit the easiest steps. I think a rule of thumb should be to ask yourself "Can the students show the missing steps in example N by looking at ...
4
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 agreement the above resolves to 20. There's no more complicated origin than this.
3
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 and engineering problems.) Look at all the questions here about the frustration of dealing with kids that can't perform algebra.
It can be good to look at ...
3
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 multiply and check, either one. Take $8$ for an example. $48\times8=384$ and $436-384=52$.
By looking at what we have left, we know that we need only one more $48$ ...
3
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 to provide the missing transformations or steps.
Give them a solution with mistakes in several places. Ask them to find and correct the solution. You should ...
3
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 following anymore. Or start the other way, adding steps if you see lost faces.
Ask them to do (a part of) an example. The level of detail shown is a clue.
2
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 detail than you'd want to see from them.
2
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 factoring and require use of the quadratic formula arise in chemistry.
2
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 calculation, I generally display a set of slides for it, which I then make available on the class web page. (This is similar to, but I think also meaningfully ...
2
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 gives one step, I write that down then ask another random student for the next step.
In this way I see whether or not this kind of problem is truly easy for the ...
2
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 significant digits you can through the process. But then, it's important to know when to get back to a reasonal number of significant digits. Say you have an ...
2
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 is built-in.
a = Mod(3,7)
print -a
print a^-1
This gives 4 and 5, respectively.
1
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 said they lack mastery. Please correct me if I'm wrong but I interpret this as they cannot reliably perform said computations and use said pre-calculus tools ...
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