# Tag Info

2

As Benjamin Dickman, I am also not aware of any texts or other resources aimed directly at high-school (or earlier) teachers, but there are a number of APOS oriented studies at these levels that should be accessible and may contain some concrete ideas on structuring e.g. lesson plans. Here is a small selection of texts that you and your colleagues may find ...

1

I think it's too early to be showing this sort of topic and discussing the connection to equations. So resist that mathy temptation to only think of the topic analytically. These kids haven't even done the equation of an ellipse yet, let alone analytical geometry. Given you have to do this topic, the way to approach it is descriptively, not analytically. ...

2

This may be a bit of a stretch, but once you introduce the sphere, you could at least mention how to shade a sphere for 3D graphics: if $\theta$ is the angle between the light source ray and the normal to the sphere at point $p$, then shade $p$ with a light intensity proportional to $\cos \theta$. This is known as Lambertian shading. So when $\theta =0$, ...

1

My own experience in math classes (as a student in Germany) and tutoring my peers would lead me to the following conclusion regarding your question: Many students have problems with the generalized formulas, because they find it rather unintuitive to calculate with "letters" rather than numbers and they would frequently ask for real world examples and ...

5

Expanding upon my comments to the question, now that I have time to: All of the courses you listed as looking at are ones that I think every high school math teacher should take and be exposed to. Calculus and Statistics are also courses that you may very well be expected to teach at the high school level, so these make the most sense of your list to take ...

3

I got my NYS certification in Math 7-12 and Generalist Special Education 7-12 (with the grades 5-6 extension in both) about two and a half years ago. The test you are describing is what Sue's website calls the Content Specialty Test in Mathematics. (I also know it by that name, and didn't realize that the name had changed.) Your basic coursework will ...

4

If you want to teach high school math, then of course you'll need to know high school math. And if you want to get a teaching credential, you'll need to take whatever courses your state says are necessary. Beyond that, it's just a matter of getting a deeper understanding of the material, rather than just knowing enough go through the steps. Real Analysis ...

3

You want to be a high school math teacher in the state of New York. I searched on "high school math teacher new york state requirements", and found this site. I'm a bit leery of this site, because I can't tell who is responsible for it. But the information seems reasonable. You will want to talk to someone at a teacher preparation program. (This ...

1

I think the difference between the two approaches is the goals. Your solution is exactly what I would do, but I'm an engineer. In high school, the goal of learning is usually more about understanding the general concept and gaining some practice with the mechanics. I would think that plugging in the values early on is "easier" to grasp the ideas than ...

5

I don't think this is a nationalistic difference (I'm in the US), but I also don't think your example is optimal. As an example where the correct technique is more well defined, let's say we have a physics problem like this: A bug starts from rest and accelerates with constant acceleration for 0.53 s, traveling 1.37 m. Find the bug's acceleration. I would ...

2

You can't generalize a single, isolated question, which is what your high school question about collinear points is. Of course you can invent a set of similar problems, create a general solution for the whole set, and then solve the particular problem given - but why do all that unnecessary work? If your child's homework set contains several similar ...

9

Calculate when you want an answer. Solve algebraically when you seek patterns. If the problem was "show that for any two points, you can find a third point along the y-axis that is collinear with them," then symbolic logic is the right way to go. But if you have the points, just plugging them in simplifies the problem dramatically and makes it easier to ...

15

Speaking from an American perspective, your son's approach strikes me as much more natural. For instance, to solve your problem We have three points $M(7;-2)$, $N(0;t)$, $P(3;1)$. Find $t$ so that they are aligned. my students would start by observing that the slope (gradient) of $\overline{MN}$ would have to be the same as that of $\overline{MP}$ and ...

1

A compromise approach could be to give a problem (with parts), as you might on a guided worksheet. Such as: A function $f$ has domain $(2,4)$. We define $g$ by $g(x) = f(x-2)$. a) Is $g(3)$ defined? b) Is $g(5)$ defined? c) What is the domain of $g$? You might even give a follow up problem (perhaps as extra credit) along the lines of: A function $f$ ...

9

I think the given example is highly appropriate. You cannot cover every possible combination of ideas in class. Students display understanding of a concept (rather than "recipe following") by showing the ability to adapt at least a little bit to novel conditions. I think the problem you gave is a great homework problem. I personally like homework to be a ...

9

I would frame this issue a little differently than you have. I think it's unreasonable, at least in the context of courses which aren't well into a math major, to ask students to do something they have not been taught to do. That is, the problems on an assessment should be the same as problems they've seen already. The catch is that "the same" is actually ...

7

My advice is to minimize the amount of such synthesis required. Don't make it a large fraction of your tests, if at all. Teach the students the methods you expect them to display on the exam. Not something requiring some spark of creativity. Program for success. Creativity is tough in general and even tougher under test conditions. If you push too ...

3

Both of my sons are learning division right now (or rather, just finished the section), one in a public school using Eureka Math (in 4th grade level math), one in a Montessori (in Primary, at 1st grade age). The public school focused on long division certainly, but (either because of the curriculum, or more likely because of the teacher, as I don't ...

1

Doing a quick google search for "why do students struggle with division?" made me realize myself why long division is difficult for a lot of students. Common reasons claimed include: The long division algorithm is long. Remembering the steps can be difficult. The long division algorithm may not be intuitive. We can tell them that it's just "how many groups ...

5

Not a teacher here, but I noticed when my kids went to school there was far less emphasis than I remember on techniques that require above average insight or intuition. I think there's more pressure these days toward making sure most students achieve a predetermined minimum performance, and less on helping high-performing students stretch their capabilities. ...

6

If you are tutoring, it's important to value whatever algorithms work. Your frustration with the (new to me) lovely algorithm you show concerns me. It shows why each step makes sense, which is much better than the form of long division I learned. "In fact a lot of kids find it [short division] amazing to learn." If you show an individual a new process at ...

26

It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a slower or ...

8

I think it is very possible to have real, meaningful research projects at all levels. As an example, I had 2 of my students in Calc 2 work on a project which started with the idea "Can we define a tangent circle instead of a tangent line?" They started by finding a formula for the center and radius of a circle given three non-colinear points: https://www....

5

Perhaps a research project could be replaced by an independent study project. The idea is to learn more math, in a more independent way. The student would still have to produce something to show what they've learned (although if the teacher found it easier, they could instead require an oral explanation of the content learned). I learned a lot on the ...

3

Opinion: I think they get enough exposure to writing reports in other subjects. Think the time is better used for drill and exams, than for writing a report. The one benefit might be learning to search the literature, but even that is probably much better done in history or science where, although rather difficult, the papers are still comprehensible. (...

3

Anecdotally, my father's generation (1950s-60s, UK) all learned logarithms in primary school and used them extensively for calculations. All children had their own slide rule which they carried with them and used in classes other than maths classes. There were no electronic calculators. My father became a radar engineer. My generation (1980s-90s, UK) only ...

1

If you teach the chain rule with Leibniz notation I recommend this suggestion of Steven Gubkin. It makes computations more explicit and straightforward, and students pick it up fairly well in my experience. For the remainder I'll address some of the subtleties involved with derivative notation, the function concept and how that relates to the chain rule. ...

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