I’m a teacher who is amazed by all your contributions to this forum. Thanks for your efforts!
I would like to teach my students to become great at math using the techniques you’re using.
Which methods of memorizing and solving problems would you advise me to give them as homework?
My own thoughts:
A lot of students make mistakes solving problems due to not having learned to prioritize (),x^z,*,a/b,+ and - correctly.
This will easily be solved through memorizing a crazy story surrounding the PEMDAS acronym which gives to following order: Parenthesis, Exponents, Multiplication, Division, Addition and Subtraction.
Learning the PEMDAS will improve their skills easily.
Teach them a simple equation solving checklist such as the following:
Simplify both sides of the equation.
Move all parts of the equation that contain the variable you’re solving for to the same side.
Isolate the variable using multiplication, division, exponentiation or by taking roots.
Check the solution.
What are your suggestions? Any good methods for them to use when gathering data to solve the problems?
On the behalf of the students and myself I thank you for helping out.
What do you consider great at math? Sorry, but that sounds rather vague… are you only interested in mnemonics like PEMDAS and the alike? I’m asking because the points you mention only cover basic arithmetic.
Isn’t this asking the wrong question? Order of operations is misleading.
Given something like “x^2 + 3(3x - 2x) - 4 = 6”, It is not necessary to evaluate parentheses first. All parentheses indicate is a grouping–that that within must be treated as a unit. So while one could reduce “3(3x - 2x)” to “3x” by evaluating in the parentheses, one could also reduce it to “9x-6x” into “3x”, which is just as valid, but never even bothers evaluating within the parentheses. Given something like “(x^y)(x^z)”, we can reduce it to “x^(y+z)”, even though we have not yet evaluated either of the exponents.
Also, order of operations is relied upon too much in the case of division, especially. For readability, noone should EVER have to apply the rule of order of operations that operations of the same priority level are evaluated from left to right. NEVER. Grouping symbols are ALWAYS preferable. With fraction bars being very much preferred if the medium doesn’t make it too hard. “3 / 6 / 12” should not be considered “1/24” (nor “6” for that matter), but should rather be considered a badly written expression which should be re-written. And yet questions like this show up on quizzes testing on understanding of order of operations despite it being a complete waste of time to even write it that way in the first place. It would be like going to an English Literature class and being tested on how well you can understand 1-semester ESL students. That might be a fine test for a class on ESL pedagogy or something, but completely irrelevant in a Literature class.
The only thing which determines how things can be evaluated in math is grouping. Some operations can be performed on groups, some cannot, sometimes groups are indicated purely as groups (parentheses/brackets/braces), other times in conjunction with another operation, such as with the fraction bar, but the only thing you need to know, is what you can do to a given group. Nothing else. Order of Operations is misleading and counterproductive.