Calculus for autistic student

I have signed up to teach (at my home) calculus to an autistic high school sophomore. He has been home schooled all of his life but his Mom felt completely lost teaching him Trig and absolutely can't handle calculus. I have agreed to waive any pay.

Reportedly, the young man loves mathematics and believes that Math is the basis for everything in the world. I have been told that he has been impatient and has started reading the first chapters of the calculus text on his own. He has passed some standard competency tests for certifying completion of HS geometry and trigonometry and (I'm unsure of this one) for some pre-calc material. He is planning to graduate to a local community college either at the end of his sophomore or junior year of high school.

Over the years, I have tutored science and math (college statistics) to family and friends- usually with very good success. So, I'm confident of my basic teaching skills. However, I have never tutored an autistic child, much less an autistic gifted child -and it has been decades since I took calculus. I will have lesson plans and we will use a standard text book, though.

I will post updates on this thread. I may ask for help with calculus problems or specific calc topics, if I need it.

Any advice or suggestions about this? Teaching calc? Teaching a gifted autistic child?

No advice but thanks for being one of the good guys.
I hope your paid back X10.

Quote: gordon888

Any advice or suggestions about this? Teaching calc? Teaching a gifted autistic child?

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First, talk with his Mom and find out all you can about the child's personality traits, i.e., personal likes/dislikes, what makes him happy/sad/frustrated/angry, how best to communicate, attention span, and personal needs. Be patient and supportive and find out what makes him laugh. You will be great.

tuttigym

I have an autistic granddaughter age 9 . lives in Utah. Wondered about diagnosis. My daughter heavy into conspiracies. Spent March last year with her and the stockpile of N95 masks. Sanitizers, etc. Now she has an AK47, several handguns, a sniper rifle.bullet maker and 4000 rounds. The entire month we wete there all I saw was a very happy outgoing little girl.
Back to Utah March this year. Bought gifts for all the kids. DeeDee got off school bus and came crying into house. Some boy on bus teased her. She ran into a corner crying . My wife got her to come into front room to open package. That is when DeeDee went autistic. Sat next to my wife turned her head to the left and just stared into space.Took almost an hour before she came back. One day in school she peed herself on school yatd. Zoned out and teacher had to het her sister Sunshine from another class to retrieve her.
So good luck on your mission of mercy. As for Deedee's mom Mary Jo she is looming to move to Lander Wyoming. Needs at least 80 acres. Her husband is a mechanic. Hoping to get a job with a guy who collect military vehicles. The ones that a magnetic pulse will not disable in case of attack by Russia or the aliens I guess. I just hope she has enough bullets fot an Zombie attack. But afraid?to ask her HaHa

Is he autistic or Asperger's? I can speak personally on the later not the former. And yes, I know they renamed Asperger's, so whatever they call it now.

Quote: AZDuffman

Is he autistic or Asperger's? I can speak personally on the later not the former. And yes, I know they renamed Asperger's, so whatever they call it now.

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Duff’s question is paramount. It all depends on the level of dysfunction. ‘Autism’ is like ‘cancer’ in that the breadth of possibilities is huge. Without specifics it’s hard to give advice. But it is a great thing you are doing!

Are you doing t for free because the student is autistic or because you have a relationship with the family?

The student is autistic, its not Asperger syndrome.

Quote: DRich

Are you doing t for free because the student is autistic or because you have a relationship with the family?

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We do have a relationship with the family, but the Mom is someone we haven't seen in almost 10 years and I have never met her son. When I have tutored in the past, I have always tutored for free. This will be a much bigger time commitment than tutoring, and the Mom (whom is single) has several times offered to pay me. But honestly, my wife and I are financially comfortable and there is great ease, satisfaction and peace of mind in just helping people without being a paid contractor. I am hoping it will be a good experience for both the youngster and I.

Hmmmm, I wonder if this kid has any special talent at picking NFL games against the spread . . . . . j/j

Quote: gordonm888

When I have tutored in the past, I have always tutored for free. This will be a much bigger time commitment than tutoring, and the Mom (whom is single) has several times offered to pay me. But honestly, my wife and I are financially comfortable and there is great ease, satisfaction and peace of mind in just helping people without being a paid contractor.

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I have some family who did broadly this sort of thing.
When the people being helped insisted on paying, it was graciously accepted, and then at the end, returned in the form of a savings bond (or similar) for the child's future benefit.

Quote: gordonm888

The student is autistic, its not Asperger syndrome.

Quote: DRich

Are you doing t for free because the student is autistic or because you have a relationship with the family?

link to original post

We do have a relationship with the family, but the Mom is someone we haven't seen in almost 10 years and I have never met her son. When I have tutored in the past, I have always tutored for free. This will be a much bigger time commitment than tutoring, and the Mom (whom is single) has several times offered to pay me. But honestly, my wife and I are financially comfortable and there is great ease, satisfaction and peace of mind in just helping people without being a paid contractor. I am hoping it will be a good experience for both the youngster and I.

Hmmmm, I wonder if this kid has any special talent at picking NFL games against the spread . . . . . j/j

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That is fantastic. You are an admirable person.

I met with the mother of the student I will be teaching (but have not yet met the student.) I learned that the student is 14, and is a freshman in high school. He communicates well, and is usually highly focused on whatever he is doing, and especially on math.

I also got the textbook I will be teaching from. The first 356 pages is Advanced Algebra with these topics:

- Coordinate System in Three Dimensions (18 pages)
- Matrices (48 pages)
- Problem Solving and Variation (38)
- Sequences and Series (36)
- Complex Numbers (36)
- Conic Sections (42)
- Radicals, Exponents, Logarithms and Rational Expressions (44)
- Exponential and Logarithmic Function (24)
- Rational and Radical Functions (32)
- Writing Models for Functions (20)
- Analyzing Functions (41)

Then comes Calculus (301 pages)

- Limits and Continuity (48 pages)
- Derivatives (32)
- Applications of Derivatives (44)
- Integration (28)
- Applications of Integrals (32)
- Differential Equations (18)
- Logarithmic, Exponential and Transcendental Functions (18)

Interesting that when i was a student some of these topics were first introduced in college math courses (e.g., Matrices, Complex Variables, Series) after Calculus, but are now high school content before calculus. I guess I'm showing my age.

The textbook is excellent - much much better than anything I had as a student.

I NEED SOME HELP.

After a lesson on solving logarithmic equations, my autistic-gifted student went home and tried applying the techniques I had shown him to a very famous equation, and came up with an impossible/incorrect result. He recognized it as being incorrect and expressed that he was “troubled.”
He showed this (below) to me today, and I sat there puzzled until I had to admit that I had no clear answer as to what he had done wrong.

The very famous equation is:


First he squares both sides:

Then he writes:

Then he essentially takes the natural log of both sides:

So, this result is obviously incorrect. He then squared both sides to get:

Which troubled him because he recognized this could not be right.

I considered whether he went outside the domain of log(x), or had introduced an ambiguity in squaring functions, but I can’t seem to find the invalid operation.
Any help on this would be greatly appreciated.

I NEED SOME HELP.

After a lesson on solving exponential and logarithmic equations, my autistic-gifted student went home and tried applying the techniques I had shown him to a very famous equation, and came up with an impossible/incorrect result. He recognized it as being incorrect and expressed to me that he was “troubled.”
He showed this (below) to me today, and I sat there puzzled until I had to admit that I had no clear answer as to what he had done wrong.

The very famous equation is:


First he squares both sides:

Then he writes:

Then he essentially takes the natural log of both sides:

So, this result is obviously incorrect. He then squared both sides to get:

Which troubled him because he recognized this could not be right.

I considered whether he went outside the domain of ln(x), or had introduced an ambiguity by squaring functions, but I can’t seem to find the invalid operation.
Any help on this would be greatly appreciated.

Yeah, I don't think you can SBS here, but can't really tell you why.

ETA: if you go to the original (famous) equation.. I don't think there is a way to write -1 in the form of e, to equate both sides of exponents (other than what is given).

Per Wolfram Alpha messing around the issue is the step where you take the natural log.

ln(e^2ipi) = 0 =/= 2ipi

I don’t think algebra always applies to i the same way it does to “normal” numbers

This Wikipedia page has an explanation: https://en.m.wikipedia.org/wiki/Complex_logarithm

Scroll down to the section “ Problems with inverting the complex exponential function”. It even has a cool 3D graph to illustrate.

Heart of the problem is: “e^(w+2kπi) = e^w for any complex number w and integer k, since adding iθ to w has the effect of rotating ew counterclockwise θ radians.”

ETA: Gordon, in your problem it’s easy to see how above is your formula where w=0 and k=1.

Quote: unJon

This Wikipedia page has an explanation: https://en.m.wikipedia.org/wiki/Complex_logarithm

Scroll down to the section “ Problems with inverting the complex exponential function”. It even has a cool 3D graph to illustrate.

Heart of the problem is: “e^(w+2kπi) = e^w for any complex number w and integer k, since adding iθ to w has the effect of rotating ew counterclockwise θ radians.”

ETA: Gordon, in your problem it’s easy to see how above is your formula where w=0 and k=1.
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I think this is the answer I was looking for. Let me re-state it. A natural logarithm (ln) function is not the inverse of a complex exponential function of form e^(w+2kπi) because:
- in order for a function to have an inverse it must map distinct values to distinct values. Given e^(w+2kπi), every integer value of k maps e^(w+2kπi) to the same value, Therefore, one cannot take ln(e^(w+2kπi)) and assert that it is equal to (w+2kπi).

As unJon (and his reference) says, e^(w+2kπi) is a periodic or circular function. If you try to take the natural logarithm you destroy the periodical (or circular) nature of the function.

Many thanks, unJon!

I think a related concept is that x^1/2 is not single-valued, therefore x² and x^1/2 are only inverse functions if you restrict the domain of x^1/2 to "all non-negative numbers" or "all negative numbers."

Clearly you cannot take the square of x = -1 and calculate the result as 1 and then take the square root of 1, and calculate that result as +/-1 and somehow claim that -1 = +/-1.