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Discrete Mathematics: An Open Introduction [pdf]

discrete.openmathbooks.org

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Re: Discrete Mathematics: An Open Introduction [pdf]

#31

Post by chongli »

rak1507 wrote: post #29
chongli wrote: post #20

Earlier quoted context omitted.

Here’s a sample problem from discrete math when I took it in university: For any integer n ≥ 0, let Cn be the set of all integer compositions of n with odd number of parts, and each part is congruent to 1 modulo 3. Prove that: |Cn| = [x^n] (x - x^4)/(1 - x^2 - 2x^3 + x^6) Where [x^n] indicates the coefficient of the x^n term in the formal power series generated by the rational function (rational representation of the…

Why not? All that is really required is knowing 1/(1-x) = 1+x+x^2+... and a bit of algebraic manipulation.

And the idea of a formal power series. And integer compositions. And combinatorial enumeration (counting sets in different ways for a proof). And a bit of set theory (cardinality of sets).

There is a whole lot of background stuff here that elementary school students do not have. Way more than what you’ve stated.

Re: Discrete Mathematics: An Open Introduction [pdf]

#32

Post by rak1507 »

chongli wrote: post #31
rak1507 wrote: post #29

Earlier quoted context omitted.

Why not? All that is really required is knowing 1/(1-x) = 1+x+x^2+... and a bit of algebraic manipulation.

And the idea of a formal power series. And integer compositions. And combinatorial enumeration (counting sets in different ways for a proof). And a bit of set theory (cardinality of sets). There is a whole lot of background stuff here that elementary school students do not have. Way more than what you’ve stated.

You definitely don't need to know any of that background to be able to arrive at the answer. To fully understand everything maybe, but all it takes is:

a = x^1 + x^4 + x^7 + ... = x(1 + x^3 + x^6 + ...) = x/(1-x^3)

a + a^3 + a^5 + ... = a(1 + a^2 + a^4 + ...) = a/(1-a^2)

Substitute + simplify. I don't think this is beyond a (fairly smart) elementary school student.

Re: Discrete Mathematics: An Open Introduction [pdf]

#33

Post by defrost »

redczar wrote: post #30
defrost wrote: post #19

Earlier quoted context omitted.

Just for interest. We covered part of this material to a similar level in high school in Australia in the early 1980s along with Calculus, probability and statistics. In the math II/III streams ("advanced" for those wanting to do Enginering, Medicine, STEM, hard trades courses that need a grasp of math, etc). This was a public high school in a remote area BTW, which I attended pretty much straight off an outback catt…

I’m speaking from the perspective of the U.S. I still contend though that the vast majority of people ages 12 - 18 are incapable of learning this book. Note that I’m talking about this book as it is in it’s totality. This book includes counting principles in it so one can always claim that aspects of this book can be taught to people between ages of 12 - 18. We do teach people how to count at a young age. The subject…

> the vast majority of people ages 12 - 18 are incapable of learning this book.

Sure.

That doesn't prevent it being a high school text taught in some high school streams though.

Math I was taught to bulk of upper school students in Australia (in years 11 and 12), Math II/III was taught to those students interested in STEM at a university, Math IV was remedial math to bolster the poorer math students that didn't exactly pick up primary school math.

Again, it's similar in depth to how I was taught math in high school and how my peers were taught in high school, but I guess we were more math orientated than others. *

What makes it not a high school text (for advanced high school students here) is length rather than depth, while a breadth of math subjects were introduced along with concepts such as proof, reasoning, various notations, etc it was uncommon for a single subject such as discrete mathematics to be dealt with at great length, perhaps a third of the material presented here at a similar depth would be more common (again, for some high school students, not all high school students).

* eg: https://profiles.imperial.ac.uk/j.gauntlett , https://mysite.science.uottawa.ca/tschmah/ , https://hpi.uq.edu.au/profile/388/dominic-hyde are three of the people in my very small (12 in total) first year university math class who all had texts not disimilar to this in their senior math streams at their various high schools at the same time I attended high school. The entire group of twelve are not dissimilar.

Re: Discrete Mathematics: An Open Introduction [pdf]

#34

Post by redczar »

defrost wrote: post #33
redczar wrote: post #30

Earlier quoted context omitted.

I’m speaking from the perspective of the U.S. I still contend though that the vast majority of people ages 12 - 18 are incapable of learning this book. Note that I’m talking about this book as it is in it’s totality. This book includes counting principles in it so one can always claim that aspects of this book can be taught to people between ages of 12 - 18. We do teach people how to count at a young age. The subject…

> the vast majority of people ages 12 - 18 are incapable of learning this book. Sure. That doesn't prevent it being a high school text taught in some high school streams though. Math I was taught to bulk of upper school students in Australia (in years 11 and 12), Math II/III was taught to those students interested in STEM at a university, Math IV was remedial math to bolster the poorer math students that didn't exact…

Length + depth determines the appropriateness of a book at a given level. They go hand in hand and my comment was about this book. Not 1 or 2 chapters of it and not 1 or 2 chapters of it dumbed down a bit.

Re: Discrete Mathematics: An Open Introduction [pdf]

#35

Post by redczar »

rak1507 wrote: post #29
chongli wrote: post #20

Earlier quoted context omitted.

Here’s a sample problem from discrete math when I took it in university: For any integer n ≥ 0, let Cn be the set of all integer compositions of n with odd number of parts, and each part is congruent to 1 modulo 3. Prove that: |Cn| = [x^n] (x - x^4)/(1 - x^2 - 2x^3 + x^6) Where [x^n] indicates the coefficient of the x^n term in the formal power series generated by the rational function (rational representation of the…

Why not? All that is really required is knowing 1/(1-x) = 1+x+x^2+... and a bit of algebraic manipulation.

You obviously have not taught mathematics to high school students.

Re: Discrete Mathematics: An Open Introduction [pdf]

#36

Post by defrost »

redczar wrote: post #34
defrost wrote: post #33

Earlier quoted context omitted.

> the vast majority of people ages 12 - 18 are incapable of learning this book. Sure. That doesn't prevent it being a high school text taught in some high school streams though. Math I was taught to bulk of upper school students in Australia (in years 11 and 12), Math II/III was taught to those students interested in STEM at a university, Math IV was remedial math to bolster the poorer math students that didn't exact…

Length + depth determines the appropriateness of a book at a given level. They go hand in hand and my comment was about this book. Not 1 or 2 chapters of it and not 1 or 2 chapters of it dumbed down a bit.

Again, while part of the book might be presented in a high school courses I took, a number of students that took such courses would read the entire book out of interest.

I'm unsure how you were taught but we in Australia frequently had textbooks that were only partially taught in high school, leaving the rest as untested and for interest.

This, as I'm sure you follow, means it would be fine as a text book despite only part of the text being taught.

Many of my peers were recommended a list of texts to read in high school that were never fomally taught in high school. A couple of people that were in our circle at that time read such things before high school.

Re: Discrete Mathematics: An Open Introduction [pdf]

#37

Post by redczar »

defrost wrote: post #36
redczar wrote: post #34

Earlier quoted context omitted.

Length + depth determines the appropriateness of a book at a given level. They go hand in hand and my comment was about this book. Not 1 or 2 chapters of it and not 1 or 2 chapters of it dumbed down a bit.

Again, while part of the book might be presented in a high school courses I took, a number of students that took such courses would read the entire book out of interest. I'm unsure how you were taught but we in Australia frequently had textbooks that were only partially taught in high school, leaving the rest as untested and for interest. This, as I'm sure you follow, means it would be fine as a text book despite onl…

It is quite obvious that you have not taught mathematics at a non specialized high school. You think the experience you and 3 other people each of whom got a Ph.D. at a good university in science and/or philosophy is somehow normative. You think this book could be taught to 15 year olds to anything more than a very small number of people is insane. Please go spend time in a classroom as a mathematics teacher before asserting such things. You might be right but your evidence is lacking. You clearly have no idea about the state of things in the average high school.

EDIT: My wife is a physician who dropped out of high school. There are people who think that because she did it this proves that more than an extremely small number of such people could become physicians. They are wrong. She got lucky and the vast majority of high school dropouts can’t become physicians.

People think that their experience going through high is indicative of what it is like on average and they advocate for positions based on their experience. It’s the “I did it therefore you can too” fallacy.

You found 12 people who could learn this book in high school. You found them in university. They didn’t all go to the same high school. Each high school has very, very few students who could understand this book. High schools don’t have the resources to teach a class for just 2 or 3 students.

Re: Discrete Mathematics: An Open Introduction [pdf]

#38

Post by defrost »

redczar wrote: post #37
defrost wrote: post #36

Earlier quoted context omitted.

Again, while part of the book might be presented in a high school courses I took, a number of students that took such courses would read the entire book out of interest. I'm unsure how you were taught but we in Australia frequently had textbooks that were only partially taught in high school, leaving the rest as untested and for interest. This, as I'm sure you follow, means it would be fine as a text book despite onl…

It is quite obvious that you have not taught mathematics at a non specialized high school. You think the experience you and 3 other people each of whom got a Ph.D. at a good university in science and/or philosophy is somehow normative. You think this book could be taught to 15 year olds to anything more than a very small number of people is insane. Please go spend time in a classroom as a mathematics teacher before a…

Twelve, not three, in my first year university math 100 class, I gave three typical examples of the twelve.

It was almost the only university in the state at that time, two others opened in the year or so prior and these were (at the time) free universities for any that got a sufficient TAE score (high school exam for tertiary entrance).

All in all many more people were educated in mathematics or in their particular interests in this small state, and that began in primary and high school.

I attended a non specialized high school in a remote corner of a large state (3x size of Texas) with a small population (less than 2 million at that time).

I don't think that the book can be taught to any 15 year olds, I think it can easily be taught to 15 year old students interested in math, as other similar books have been.

In a not disimilar manner I know others who started olympic level swimming training in high school as part of their high school curriculum. Not for everyone, just for those high school students with aptitude.

I have spent time in a classroom as a mathematics assistant tutor, it was good pick up money during the five years I attended university.

Naturally many people that were awarded Ph.D. attended high school, I'm unsure why you would choose to exclude them from the population of high school students.

The state of things in average state run Western Australia high schools in the 1980s was that most student got mainstream education and students that showed promise in any number of different ares would get moved to specialised stream or invited to subsidised speciality camps; these existed for math, literature, theatre, music, sports, machining, etc.

eg. Heath Ledger got more theatre exposure in his High School years in this state than the majority of other high school students .. the fact that they didn't go on to play the Joker in a major Hollywood production doesn't negate what he did in high school.

You clearly have no idea about the average high school in the place and time to which I refer.

My own son attended a state high school (free public education) that had an aviation course, he and his classmates built an aircraft over two years and then took turns flying it. ( https://www.kentstreetshs.wa.edu.au/aviation )

It's sad that you seem to want to homogenize the high school experience to the least common denominator.

EDIT: “I did it therefore you can too” - not a claim I made, please stop strawmanning.

Again, there is nothing preventing a university level text being taught to high school students with aptitude and this actually happens in some education systems outside your ken.

In parallel other advanced subjects and skills can also be taught to high school students with aptitude .. and some education systems do this.

I'm sorry you apparently have not experienced such a system.

Re: Discrete Mathematics: An Open Introduction [pdf]

#39

Post by chongli »

rak1507 wrote: post #32
chongli wrote: post #31

Earlier quoted context omitted.

And the idea of a formal power series. And integer compositions. And combinatorial enumeration (counting sets in different ways for a proof). And a bit of set theory (cardinality of sets). There is a whole lot of background stuff here that elementary school students do not have. Way more than what you’ve stated.

You definitely don't need to know any of that background to be able to arrive at the answer. To fully understand everything maybe, but all it takes is: a = x^1 + x^4 + x^7 + ... = x(1 + x^3 + x^6 + ...) = x/(1-x^3) a + a^3 + a^5 + ... = a(1 + a^2 + a^4 + ...) = a/(1-a^2) Substitute + simplify. I don't think this is beyond a (fairly smart) elementary school student.

The question doesn’t ask for an answer, it asks for a proof. You can’t just write a bunch of algebra and call it a day. You have to justify all of your arguments.

Re: Discrete Mathematics: An Open Introduction [pdf]

#40

Post by WhitneyLand »

redczar wrote: post #4
WhitneyLand wrote: post #2

Think this would be a great course for high school or even middle school. No plug and chug that makes it a grind, plus a great intro to proofs and deeper mathematical thinking.

I taught mathematics for 30 years at the college level. This is a college level textbook and it is not appropriate for either high school or middle school. Very few students at that level would be able to understand this material.

I meant more the subject itself rather than this particular textbook, but I’m curious about your opinion in general.

I came to this opinion after taking it in college and not recalling very much in the way of needed prerequisites, but maybe this is a selective memory…

What are some of the biggest things needed beyond algebra?

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