My Journey With Mathematics

Sounds perfectly reasonable. In other words "teach thinking"! We get entirely too many questions on this board asking "what is the formula for this problem".
 
harpazo said:
… What do you say?
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I think Pólya was a smart guy who had a lot of insight, but he grew up in a different era. (Although, he likely did shock some academics, in his time.)

I would say that when the solving of word problems is the focus, then, yes, gaining the ability to take statements written in words and convert that given information into mathematical expressions and to show relationships (write equations) is a high priority. Yet, math education is more than solving word problems -- especially now, as education shifts to the digital age. I'm not going to argue against Pólya, but I sense that presenting math at the secondary level as an interconnected group of concepts and allowing students the freedom to realize those patterns in their own ways are also important. At some point, artificial intelligence will allow machines to teach themselves how to solve math problems. Getting there will first require a lot of humans to view mathematics as much more than a list of solution strategies.

Here is a video you may find interesting, harpazo.

YouTube video
 
I was only surprised that he would specify equations; I would expect him to talk about problem solving in general - that is, thinking.

I did some searching and found that the quote is from his book Mathematical Discovery, which I'd like to read; and another place that quotes it also looks worth reading: http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf
 
Otis said:
I think Pólya was a smart guy who had a lot of insight, but he grew up in a different era. (Although, he likely did shock some academics, in his time.)

I would say that when the solving of word problems is the focus, then, yes, gaining the ability to take statements written in words and convert that given information into mathematical expressions and to show relationships (write equations) is a high priority. Yet, math education is more than solving word problems -- especially now, as education shifts to the digital age. I'm not going to argue against Pólya, but I sense that presenting math at the secondary level as an interconnected group of concepts and allowing students the freedom to realize those patterns in their own ways are also important. At some point, artificial intelligence will allow machines to teach themselves how to solve math problems. Getting there will first require a lot of humans to view mathematics as much more than a list of solution strategies.

Here is a video you may find interesting, harpazo.

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Most exams leading to city jobs in NYC involve applications. Employers are not interested in the ability of applicants to solve linear equations like 2x + 5 = 20. They seek people who can reason their way to the answer.

Word problems, for example, prevented me from landing good-paying jobs back in my youth, including full-time teaching. I did work as a substitute teacher for 8 years in NYC but state certification is not required for subbing. Even today, after many years of studying math and answering textbook questions, I greatly struggle with word problems.

I believe that learning how to set up equations from word problems is what truly separates a math person from someone pretending to be a math person. This skill leads to better employment. It is a great art, a joy to be form an equation or in some cases a system of equations from applications.

What do you think has kept me from landing a part-time math tutor job with Kaplan, Princeston, Kumon, Big Apple Tutoring, Sylvan Learning Center and the rest of the more popular tutoring companies in NYC? The companies listed here pay very well for tutoring. A part-time job as a math tutor (most companies pay over 20 dollars an hour) would greatly increase my finances and put my feet on higher ground in one of the most expensive cities in the USA.
 
harpazo said:
What do you think has kept me from landing a part-time math tutor job with Kaplan, Princeston, Kumon, Big Apple Tutoring, Sylvan Learning Center and the rest of the more popular tutoring companies in NYC?
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Insufficient alphabet soup on your resume?
 
harpazo said:
… after many years of studying math and answering textbook questions, I greatly struggle with word problems …
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Have you had any thoughts about why you experience difficulties setting up word problems? That is, have you identified some possible reasons?

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I think that for Americans advanced calculus is the hardest course they will take. Americans seem to do better in algebra classes (Linear Algebra, Abstract Algebra).
In other countries students do better in analysis (ie advanced calculus) than algebra.

I do agree that if you can pass a rigorous advanced calculus class you can earn a degree in math.

Now the real question, if you believe what I wrote above, is why do students in one country tend to like/do better in analysis and worse in algebra and in another country it is reversed?
 
Otis said:
Have you had any thoughts about why you experience difficulties setting up word problems? That is, have you identified some possible reasons?

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Some possible reasons for difficulty solving word problems:

1. Perhaps over thinking a problem.

2. The language is often obscure and fuzzy.

3. I do not know what to let x or y be.

4. I get lost if the word problem is lengthy and involves too many numbers.

5. Back in my school days, I found myself reading each word problem twice or more, which is not good when pressed for time during standardized and classroom exams.

6. My academic history is pathetic. For example, I did not graduate from middle school. I was bullied out of middle school.

7. Read 6 again. As a result, I went to high school with a poor reading, writing and math background. I was placed in modified courses, which is more like repeating middle school. I DID NOT take algebra 1, algebra 2, geometry and trigonometry in high school. Modified courses are mainly arithmetic reasoning and basic math operations and little to no applications.
 
Jomo said:
I think that for Americans advanced calculus is the hardest course they will take. Americans seem to do better in algebra classes (Linear Algebra, Abstract Algebra).
In other countries students do better in analysis (ie advanced calculus) than algebra.

I do agree that if you can pass a rigorous advanced calculus class you can earn a degree in math.

Now the real question, if you believe what I wrote above, is why do students in one country tend to like/do better in analysis and worse in algebra and in another country it is reversed?
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There are many reasons why certain students do better in one class than another.

1. Weak math background.

2. Assigned A students to modified courses in public schools. In other words, placing smart students in remedial courses and vice-versa.

3. Learning math from an unqualified teachers. For example, passing teacher certification exams does not lead to great teaching ability. Kids are not taught effectively.

4. Learning math from an OUT OF SUBJECT teacher. NYC public schools are notorious for hiring gym teachers to teach math, music teachers to teach history, science teachers to teach computers, etc. It is IMPOSSIBLE to teach math effectively and correctly if the teacher has not taken sufficient math courses.

5. Spending more time "teaching the test" to students who are yearning, in some cases, to learn HOW TO LEARN.
 
firemath said:
Welcome, 2020.
Many tech deadlines are supposed to materialize in you. Do not disappoint.
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What are your thoughts on Advanced Calculus? I am NOT talking about Calculus 3. Watch this clip. Tell me what you think.

 
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Jomo said:
I think that for Americans advanced calculus is the hardest course they will take. Americans seem to do better in algebra classes (Linear Algebra, Abstract Algebra).
In other countries students do better in analysis (ie advanced calculus) than algebra.

I do agree that if you can pass a rigorous advanced calculus class you can earn a degree in math.

Now the real question, if you believe what I wrote above, is why do students in one country tend to like/do better in analysis and worse in algebra and in another country it is reversed?
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Watch Lesson 1 in its entirety and tell me what you think. Did you ever take this course? I have never taken calculus at its most basic level aka Calculus 1. Please, watch this video lesson and share your input. This video lesson is beyond Calculus 3. I find it interesting but insanely hard to comprehend.

 
harpazo said:
Some possible reasons for difficulty solving word problems:

1. Perhaps over thinking a problem.

2. The language is often obscure and fuzzy.

3. I do not know what to let x or y be.

4. I get lost if the word problem is lengthy and involves too many numbers.

5. Back in my school days, I found myself reading each word problem twice or more, which is not good when pressed for time during standardized and classroom exams.
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One thing that may help is to focus on a different kind of "reason". Forget the past and blaming others, no matter how valid that might be; you want to move forward, so you need to think about things you can change now. How can you improve your approach to a problem?

I'll take the other points in reverse order:

5. Forget about time issues; you should now be taking all the time you need to solve problems, and leave speed as a later goal. Reading through a problem twice is strongly recommended! You should skim once to get a sense of the overall problem (so you won't start of in the wrong direction), then again collecting information and writing it in an organized list; then again as you work through it, and a final time when you check your answer against the words of the problem.

4. Organizing the information reduces the complexity and makes it manageable. Your goal, in a sense, is to rewrite the problem so it is easier to comprehend. Pictures, tables, or mere lists of facts pull the data out of the paragraph so you can see them spread out.

3. If necessary, you can just assign a name to every unknown quantity in the problem! Don't worry at first about which variables are really needed; as you work, extra ones will be eliminated. But if you are uncomfortable with too many variables, you can do a little thinking first, so that some quantities will be named not with their own variable, but with an expression in terms of others. (For example, if you are told that Ann is 5 years younger than Beth, rather than use variables A and B, you can use B and B - 5.)

2. I often reword a problem, paraphrasing complicated parts, sometimes in several phases to make sure I'm not changing the meaning. I do this mostly in explaining a problem to others, but sometimes when working on a problem on my own.

1. Think about what specific kinds of "overthinking" you tend to do; that is a catch-all category that doesn't really mean much. If you can identify particular errors you tend to make, you will be better able to correct them than just trying not to think too much (which is not the way to solve a problem!).
 
harpazo said:
What are your thoughts on Advanced Calculus? I am NOT talking about Calculus 3. Watch this clip. Tell me what you think.
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Advanced Calculus is deep stuff. I think that such things are good and useful (and even required, depending on what you are going to teach) to have mastered if you are going to tutor or teach math, but are rather beyond most other disciplines only remotely related to math.

This fellow is well spoken. Might I ask where he teaches?
 
harpazo said:
… The language is often obscure and fuzzy …
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That's a big issue, when it comes to translation. I'm not sure what to suggest, other than asking for comprehension help with those specific sentences -- before starting on the mathematics. If help isn't readily available, perhaps skip such problems until later and work with exercise statements that you do understand.

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I didn't do a Math degree but I'm teaching myself some basic Topology (amongst other topics) and it's kicking my butt. When I went through I would have said that Linear Algebra was the worst, but now I think it's one of the easiest! (In the Physics curriculum the hardest was Statistical Thermodynamics but it's less of a problem now than Electromagnetism. For some reason I've developed a bit of a block when it comes to EM.)

-Dan
 
Otis said:
That's a big issue, when it comes to translation. I'm not sure what to suggest, other than asking for comprehension help with those specific sentences -- before starting on the mathematics. If help isn't readily available, perhaps skip such problems until later and work with exercise statements that you do understand.

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I plan to solve word problems throughout 2020 more than anything else in math. I mean, is there anything more important than applications in terms of mathematics? Of course, I plan to show my effort to each problem. I will break down most problems sentence by sentence to simply state what I think each sentence means. I will this task next week.
 
firemath said:
Advanced Calculus is deep stuff. I think that such things are good and useful (and even required, depending on what you are going to teach) to have mastered if you are going to tutor or teach math, but are rather beyond most other disciplines only remotely related to math.

This fellow is well spoken. Might I ask where he teaches?
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I'm not sure where James teaches. He does begin the course with a prayer. It might be a Christian university.
 
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