The quadratic formula is a "messy, silly thing?" :( :( :( I think we'll have to agree to disagree on this one.
I have a question for you. My understanding is that the quadratic formula is a big challenge for many students, and that a substantial portion of high-school students have real difficulty plugging numbers into the expression and then correctly evaluating it. Is that your experience? I'm not making a claim about the exact percentage -- is it 80%? 50%? 20%? I don't know. But it's not 1%, it's some non-trivial proportion of students cannot handle this level of complexity.
Personally, I would think that this formula is exactly the sort of thing that should be memorized. But maybe that just means that I'm not suited to being a middle-school math teacher.
I think a perfectly reasonable problem is "Here is a quadratic and here is the quadratic formula, find the roots." Students should be able to evaluate the expression. (Note that there's some stuff they need memorized here, like what a, b, and c mean.) Asking students to use the quadratic formula while **giving students the quadratic formula** is very different from saying "Here is a quadratic, find the roots" and expecting students to have the formula memorized.
Students should know how to use the formula, no argument there. The debate is whether the teacher should give students the formula on a reference sheet on assessments, or expect students to know it off the top of their head.
But I'm still wondering -- do you find that some non-trivial proportion of students struggles to use this formula? That is, even if you don't require them to memorize it, and instead provide it to them on a formula sheet, do you find that some students struggle just to perform the computation?
I haven't taught it in a few years but when I did, yes absolutely. I don't know how well I could generalize a percentage but it's significant. There are lots of things a student has to understand before using the quadratic formula makes sense. (That's one reason I'm opposed to making kids memorize it, if you lack all that background knowledge it's way harder to memorize.) It's definitely a non-trivial proportion of students.
What's remarkable about this is that in one sense using the quadratic formula seems to be super-simple: you just read a, b, and c off, and then perform a sequence of arithmetic operations. Nine operations are required to obtain one root, and all of them are available on any standard calculator. Yet even this is really challenging for many people.
It's easy for people who are good at math to take something like this for granted, but it's important to realize that for many students this is a big deal. I appreciate your real-world insights and experiences!
The quadratic formula is a "messy, silly thing?" :( :( :( I think we'll have to agree to disagree on this one.
I have a question for you. My understanding is that the quadratic formula is a big challenge for many students, and that a substantial portion of high-school students have real difficulty plugging numbers into the expression and then correctly evaluating it. Is that your experience? I'm not making a claim about the exact percentage -- is it 80%? 50%? 20%? I don't know. But it's not 1%, it's some non-trivial proportion of students cannot handle this level of complexity.
Personally, I would think that this formula is exactly the sort of thing that should be memorized. But maybe that just means that I'm not suited to being a middle-school math teacher.
I think a perfectly reasonable problem is "Here is a quadratic and here is the quadratic formula, find the roots." Students should be able to evaluate the expression. (Note that there's some stuff they need memorized here, like what a, b, and c mean.) Asking students to use the quadratic formula while **giving students the quadratic formula** is very different from saying "Here is a quadratic, find the roots" and expecting students to have the formula memorized.
Students should know how to use the formula, no argument there. The debate is whether the teacher should give students the formula on a reference sheet on assessments, or expect students to know it off the top of their head.
Thank you for you very prompt reply!
But I'm still wondering -- do you find that some non-trivial proportion of students struggles to use this formula? That is, even if you don't require them to memorize it, and instead provide it to them on a formula sheet, do you find that some students struggle just to perform the computation?
I haven't taught it in a few years but when I did, yes absolutely. I don't know how well I could generalize a percentage but it's significant. There are lots of things a student has to understand before using the quadratic formula makes sense. (That's one reason I'm opposed to making kids memorize it, if you lack all that background knowledge it's way harder to memorize.) It's definitely a non-trivial proportion of students.
Thanks. That's very helpful.
What's remarkable about this is that in one sense using the quadratic formula seems to be super-simple: you just read a, b, and c off, and then perform a sequence of arithmetic operations. Nine operations are required to obtain one root, and all of them are available on any standard calculator. Yet even this is really challenging for many people.
It's easy for people who are good at math to take something like this for granted, but it's important to realize that for many students this is a big deal. I appreciate your real-world insights and experiences!