# Tag Info

26

It may be as simple as the fact that the word quadratic derives from the Latin quadratus which means square. A square has 4 sides, but the word squared is used to mean "to the power of 2". To answer the follow on - No. Changing anything means re-writing the complete body of text to use the new designation, and un-learning the old definitions for all ...

21

The situation for quadratic equations is in one sense exactly analogous to the one for cubic equations, except that for reasons of the development of the language some discrepancy between square, squaring and related and quadratic arose. I am not well placed to discuss this in English but see on the English.SE site for the etymology of "quadratic": Why ...

20

I'm nearly sure I did this with my child when she was young. First, establish that she understands that a number, like three, is equal to $1+1+1$. Hold three fingers up and ask her "how many is this"? Then spread them out and ask the same question. Are we adding $1+1+1$? Try holding up eight fingers (keep your thumbs down, for example) and ask her to ...

15

Making a course easier to pass is not the same as making a good course. There is a certain corpus of knowledge with which practitioners are expected to have instant recall (a.k.a: "automaticity"). If one were to be looking up all of these basic facts constantly, one would not be able to accomplish anything practical. These basic topics are the ones mostly ...

13

I think that allowing students to use resources on exams can work well in certain situations, such as when the knowledge they can pull from those resources is peripheral (not of core importance to the course) or terminal (not going to be built upon in the future). However, I think it is very rare for all of the material in a math class to fall into one of ...

12

When something has a standard name, use the standard name, especially when it's something so fundamental. Calling quadratic equations anything other than quadratic equations just means you end up with a class of students who become confused when they discuss quadratic equations with anyone who wasn't taught by you (i.e., almost everybody else on the planet). ...

12

I think it is usually counterproductive. Instead of building skills, it rewards students who are good at information retrieval. And I say that as one of those advantaged. Still remember a P-chem course where prof said he would do this AND he also curved the course. I ended up with highest grade, but did not "master" the course as I had calculus, freshman ...

11

At my university, the usual multivariable calculus class is split into two semesters: the first does geometry in $\mathbb{R}^3$ and then derivatives (gradients, Lagrange multipliers, etc.) while the second focuses on integration (Green's theorem, Stokes' theorem etc.) In the second semester there are usually about three free weeks or so at the end of the ...

10

I allow students to have one index card of notes during their exams. I believe this allows me to test for understanding and skills that are more important than the ability to memorize. I also believe that it reduces student anxiety and creates a level playing field (instead of having only a few students using "cheat sheets," all of them are given this ...

10

Presumably, you are trying to reduce the chance that someone will get a correct answer by guessing. In that case, I think that 10 options are more than you need. Suppose that we have an $N$ question multiple choice exam, each of which has $M$ options. Then a randomly guessing student gets $Q$ questions correct, where $Q$ is sampled from the binomial ...

8

Echoing @AndreasBlass' remark, and having experienced somewhat similar episodes, it is already precarious enough to make such choices _for_oneself_. So, to directly answer your question: I think "no, do not encourage others to (too violently) disconnect from the math curriculum at school". I don't think it's about problem-solving versus calculus, at all. And,...

7

I am not a research-level mathematician, but this absolutely happens to me. There is so much in math that we can think more deeply about. Just last week, a student came into my office to get help on a derivative problem that used the quotient 'rule'. He was getting zero in the numerator. I showed him his mistake, and said "You won't get zero in the ...

7

Quoting Wikipedia: The adjective quadratic comes from the Latin word quadrātum ("square"). Summarizing succinctly: Consider objects of side-length $a \in \mathbb{R}^{+}$. $a^1$ gives the size ("length") of a line segment, hence linear. $a^2$ gives the size ("area") of a square, hence squared. $a^3$ gives the size ("volume") of a cube, hence cubed.

7

The following is written as if I were giving my own best answer to a student. It's probably an accident of history that Aristotle defined classical logic in a specific way, and that classical logic has been almost exclusively used as the foundation of mathematics ever since. Common sense tells us that Aristotelian logic is an oversimplification of how we ...

7

From John Stembridge's web site at http://www.math.lsa.umich.edu/~jrs/plans.html : "Are we going to have to think today, or is it going to be all math?" --a student in Phil Hanlon's Math 115 class

7

There are many reasons, but the classic explanation is that professors (especially at research universities) are picked (and compensated) for research ability versus teaching efficiency. (In general...caveat hawks.) Probably a secondary reason is that mathematics tends to be a field with a high emphasis on logic and precision. However EDUCATION is more ...

7

The article at artofproblemsolving seems silly to me. The author's idiosyncratic opinion seems to be that students who are ready to take calculus should refrain from taking calculus and instead do math contests. People are all different, and there is not just one appropriate path for a mathematically precocious student. Some people might want to take ...

6

Blocks work well for thinking about addition. Have her count out 8 blocks, and then ask her about all the addition problems that have 8 blocks as the answer. A lovely children's book which looks at all the sum pairs for 7 is Quack and Count, by Keith Baker. (You can buy it used here.) It has luscious pictures, a driving rhythm, and a lovely storyline. (“...

6

There's a fair amount of research about this, mainly from psychology and medical education as well es test development research for high stakes tests. In practical development, I usually resort to the "gold standard" and either collect sample answers analyzing the frequency and types of errors of students. Or I consult research articles on common ...

5

I do think that 10 options for a multiple-choice question is excessive. A few things to consider: This will be outside the range that standard automation tools can handle (TestGen application, Scantron sheets, premade testbanks, etc.) When I started teaching, I thought that using multiple-choice questions would be a time saver for my classes. But what I ...

5

Imagine a scenario where you don't have a hypothesis about your students' unwillingness to ask questions in class or during office hours. Your question may just boil down to: "How do you get students to interact with you in class, without expecting them to initiate?"Absent your hypothesis, you might: Ask them to (anonymously?) submit their answers to a ...

5

A good answer to this question is one that (is correct, and) she finds convincing. As kids are growing up and making sense of the world around them, experimentation is often one of their key sources of truth. Perhaps, experimentation can also serve as a reliable and effective gateway to the abstract world of mathematical truths! Here's one possible roadmap ...

4

I don't have any scholarly justification for this suggestion, but it seems very plausible that quadratic equations are so called because they split the rectangle up into quadrants as so Obviously not all quadratics can be expressed in the form $(x+a)(x+b),$ but the ones that were considered solvable when the word quadratic was first being used (i.e. before ...

4

I would say it completely depends on what is being asked. For example, something simple like "What is the derivative of $\cos(x)$ with respect to $x$?" could have as many as 10 options, because no real work or computations are needed -- either you know this one or you don't. [Assuming, of course, that you don't include both $-\sin(x)$ and \$-\frac{\pi}{180}\...

4

[VERY LONG ANSWER, needs patience to read through] I feel this is a problem many students who are good at maths face. They understand the simple tricks and patterns which are present in the school syllabus and so it is simple for them and after some practise and memorisation they are done. Then they seek out more maths and find out about topics like ...

3

Like others, I feel this is too much. It turns the question into almost a mini research project (assuming there is some reasonable amount of checking needed). It's one thing to just scan through ten choices, but then if not immediately clear (in a false, false, true, false...series), than having to look at so many options and compare them is difficult--...

3

An exam is a totally artificial scenario. In real life you will never be in a scenario with all these fact at same time: a) you can not consult external resources or colleagues; b) you have limited time; c) you do not known the activity before to start it. As conclusion, exams should be deprecated in all forms, with or without consultation of external ...

3

I once had a colleague when I was teaching in Asia who was new to teaching high school, but was a very accomplished Mathematician. In one of his first lessons he was teaching simultaneous equations, and said that he would teach them two methods to do this. He thought that offering an option would endear him to the students. A student then put his hand up, ...

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