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Do the math: too much calculus? (2012)

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Re: Do the math: too much calculus? (2012)

#211

Post by baldfat »

hprotagonist wrote: post #65
baldfat wrote: post #8

Earlier quoted context omitted.

I have had the pleasure of pushing many female students into taking calculus but my success rate was about 50%. Each one of my children's friends LOVED STEM and were thinking about going into engineering. All of them said that they were the only girl in the class and were intimidated and the other said it was REALLY boring. Calculus should be easily made fun just exploring the actual concepts in real life similar to…

As it happens, my teacher was female. She was a firm believer in "teach to the test", "do these exercises and follow the magic rules of differentiation, shut up and just do it don't ask why it works". That's not my jam.

I think many "Math" people get the mechanics and like the functions and miss out on the application.

I remember trying to figure out how the Cha Cha Slide could be a decent Calculus problem. How does the song change the pattern?

Re: Do the math: too much calculus? (2012)

#212

Post by euroclydon »

ivansavz wrote: post #171
euroclydon wrote: post #26

I have a 5th and a 6th grader. I'm going to buy them an educational copy of Maple soon, so they can have a beautiful platform to plot 2-D and 3-D graphs, solve equations, do derivatives and integrals, and factor algebraic expressions. I want them to do all this in high school, whether or not they take Calculus, so they can get an intuition for it. Then they can memorize the trig rules in college.

You should also consider SymPy, available as a webapp http://live.sympy.org/ or in jupyter notebook form. Plotting function graphs is a bit harder (matplotlib) but overall I find SymPy to be much more logical and easy to grok than Maple's weird syntax. Here is a short tutorial on how to do basic math using SymPy: https://minireference.com/static/tutorials/sympy_tutorial.pd...

I love Jupyter. I set up pydot in a notebook, and made high level functions to create and print trees and graphs. My 10 yr old picked it up quickly while we killed time at his brother's swim meet.

But, I'd love for them to learn to use a CAS engine. I appreciated, in college, being able to quickly factor a polynomial, solve an equation, and find derivatives.

Re: Do the math: too much calculus? (2012)

#213

Post by pdm55 »

jpfed wrote: post #46
ThatGeoGuy wrote: post #31

I am not sure how much of this is specific to the American education system, as I am Canadian, and have not experienced it, but my own take after going through high school to my current graduate studies is that 1) We teach algebra, geometry, and their relationship wrong in many ways 2) We don't teach enough calculus at lower levels of mathematics, instead opting to prefer a "poor man's" algebra. I'm sure there's too…

>Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason I recently-ish helped my stepdaughter with an algebra assignment where she was asked to find the maximum of some polynomial. I totally blanked on how to do this with only the tools that she was supposed to have. I didn't know how to do it or explain it without calculus.

Use a tool like Symbolab. It shows you the steps for different methods. Very useful.

Further, we should all remember to draw a graph as our first step. Symbolab, Desmos and Geogebra (my favourite) are all fantastic graphing software.

Also, for a quadratic if you know the zeroes (roots), it's half-way between them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the mid-point (maximum) is at x=3. Your graphing will confirm this.

Your stepdaughter might also have learnt that the mid-point of a quadratic ax^2 + bx + c = 0 is x=-b/2a. (This can be derived by setting the derivative equal to zero, but is generally just given to the students.) Her teacher may be expecting her to use that formula.

Alternatively the students might be required to determine and plot various data points using, say, Excel, and find the maximum this way.

The other point is that solutions often do not spring to mind immediately. That is why I ask students to send me their problems before our tutoring sessions. I often need time to think about them.

The fun of math is this exploration to find an answer. So try to get problems and give yourselves time (days) to explore them together.

Re: Do the math: too much calculus? (2012)

#214

Post by ivansavz »

euroclydon wrote: post #212
ivansavz wrote: post #171

Earlier quoted context omitted.

You should also consider SymPy, available as a webapp http://live.sympy.org/ or in jupyter notebook form. Plotting function graphs is a bit harder (matplotlib) but overall I find SymPy to be much more logical and easy to grok than Maple's weird syntax. Here is a short tutorial on how to do basic math using SymPy: https://minireference.com/static/tutorials/sympy_tutorial.pd...

I love Jupyter. I set up pydot in a notebook, and made high level functions to create and print trees and graphs. My 10 yr old picked it up quickly while we killed time at his brother's swim meet. But, I'd love for them to learn to use a CAS engine. I appreciated, in college, being able to quickly factor a polynomial, solve an equation, and find derivatives.

> But, I'd love for them to learn to use a CAS engine

SymPy is the best CAS I know of. You can use the methods like expand, factor, collect, and solve on any expression.

Re: Do the math: too much calculus? (2012)

#215

Post by angry_octet »

So let me get this straight: kids who didn't study enough math in middle school have difficulty doing harder math later on, so they rote learn it to get into college, then drop out of math. Okay?

And the kids that studied math, e.g. because they enjoy it and/or have good teachers early on, take AP math and go on to do well in math in college.

Is this a Captain Obvious moment? When will people learn that there is no shortcut and they actually have to do the work, and pay a living wage to teachers?

It reminds me of the 'coding in schools' thing. Who exactly is going to take a massive pay cut to teach computing? And do we just give them a walled garden to code in?

Re: Do the math: too much calculus? (2012)

#216

Post by jpfed »

pdm55 wrote: post #213
jpfed wrote: post #46

Earlier quoted context omitted.

>Finding the maximum of a relation using Calculus is much easier than strictly using algebra for this reason I recently-ish helped my stepdaughter with an algebra assignment where she was asked to find the maximum of some polynomial. I totally blanked on how to do this with only the tools that she was supposed to have. I didn't know how to do it or explain it without calculus.

Use a tool like Symbolab. It shows you the steps for different methods. Very useful. Further, we should all remember to draw a graph as our first step. Symbolab, Desmos and Geogebra (my favourite) are all fantastic graphing software. Also, for a quadratic if you know the zeroes (roots), it's half-way between them. So for the equation -(x-1)(x-5) = 0, the roots are x=1 and x=5, and the mid-point (maximum) is at x=3. Y…

It was a cubic, but the point is that calculus so obviates the need for any degree-specific tricks that I don't have any particular memory for them.

Re: Do the math: too much calculus? (2012)

#217

Post by greendragon »

jameshart wrote: post #100
greendragon wrote: post #45

Earlier quoted context omitted.

They're not always required, but they try to bridge what you already know (or are supposed to know) with what is to come and move at a slower pace. It's also sometimes like an extra option for the less mathematically inclined who are still required to take such a number of math courses. On the other hand the only 'pre-' course I ever took (I tested out of pre-algebra somehow) was 'pre-calculus with honors'. I don't r…

There's a difference between AP calculus in Alberta and British Columbia? :)

Hadn't thought of it like that before. :) A bit more usefully than the "one is harder one is easier" distinction, you can kind of think of the letters corresponding to semesters in a college course. A would be an introductory level (and possibly the only exposure a student might get if they're not interested in math but have to take a course), B would be a semester of Calculus 1, and C would be a semester of Calculus 2. Since high school the class goes year round you're still at a slightly slower pace than college, but if you do well on the AP BC test at the end many colleges will waive Calc 1 and Calc 2 from your requirements, whereas the AB test would only let you waive Calc 1. (A few other AP tests have different letter levels -- for physics only the C level seems to count and that's just basic Newtonian physics with very little calculus, though there is an optional E&M portion I never got to that may approach the Maxwell equations by the end... For computer science they used to have an A and an "AB", but they've dropped the "AB" since I took it. The "AB" basically got into actual algorithms and data structures, the A portion was basic programming and OOP heavy. All in Java.)

Re: Do the math: too much calculus? (2012)

#218

Post by Shorel »

We can teach intuitive integral calculus in kindergarden:

Have lots and lots of ping pong balls handy, and try to measure the volume of everyday things in ping pong balls.

Exact results are not a requirement and in fact they don't matter at all for this early age, only the intuitive visualization.

We can teach the concepts of squares, cubes and prime numbers using the ping pong balls as well: Get for example nine balls and form a square shape. Squares are numbers that can form that shape. Similar for cubes. Factorization in two factors is arranging the balls in a rectangle. A prime number is any number that can't be arranged in a rectangle.

All this is intuitive, children can understand it easily, and it will help them in the future.

And all this is before they learn about equations and algebra, it's even before fractional numbers.

Re: Do the math: too much calculus? (2012)

#219

Post by rhizome »

gohrt wrote: post #185
jameshart wrote: post #23

Earlier quoted context omitted.

As an alien to the US education system, I am always completely confused by these 'pre-' course designations. What on earth is 'pre-trig' or 'pre-calc'? I mean, I got all the way up to undergraduate level in mathematics and I never needed to learn 'pre-' anything.

"pre-trig" doesn't exist most anywhere. "pre-calculus" is whatever isn't covered in intro to algebra and geometry, but needed for differentiation and integration: https://www.khanacademy.org/math/precalculus * Trigonometric equations and identities * Conic sections * Vectors * Matrices * Imaginary and complex numbers * Sequences, series and induction * Probability and combinatorics <-- more of a side topic, usually.

I tried to post this yesterday but was getting rate-limited.

After Geometry we had Algebra 2 (quadratics, etc.), then Advanced Math (trig, pre-calc). To my detriment I made up "pre-trig" to illustrate being thrust back into a cold curriculum after (what I felt were) warmer concepts in Geometry.

Re: Do the math: too much calculus? (2012)

#220

Post by wutbrodo »

sotojuan wrote: post #43
wutbrodo wrote: post #27

Earlier quoted context omitted.

> There should be some vetting process for the students who want to take AP classes. Is this not usually the case? My high school used to make you fill out a short form and make the case for why you should be allowed in the AP class, and my senior year I had to go in and speak with a counselor who was concerned that I was taking too many. Sure, it was a pointless hurdle for me personally, but I can see how it would b…

Depends on the school district and size of the school. For example, Texas high schools are for the most part huge and can accommodate anyone who wants to take AP classes without question. We did check with our counselor but they were more than happy students took AP classes.

It wasn't a size thing as much as for "protecting" students from classes they couldn't really handle. I can't speak to whether that was necessary or not; it's always worked out very well for me to try things that I didn't think I could achieve, but maybe that doesn't generalize to everyone.
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