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"Life is open book."

With the advent of widely accessible, inexpensive (or even free) computational tools and Computer Algebra Systems (TI-89, Wolfram|Alpha, etc.), much of what traditionally comprises a high school math curriculum can now easily be done by almost everyone. Factoring polynomials, solving inequalities, graphing linear equations, differentiation and integration -- these are the types of skills high school math students spend most of their time learning, and yet all of it can be done for free by anyone with a web browser.

What does this mean for the high school math curriculum? On the one hand, we could leave it more-or-less the same, insisting that today's student learn what we learned decades ago, while banning or carefully regulating the use of these new tools. On the other hand, we could embrace the tools and the opportunities they create to spend more math class time on different topics and skills, perhaps focusing more on analytic and synthetic problem solving and less on mechanical symbolic manipulation -- but at the risk of students never learning some basic foundations.

So how about it? Binomial coefficients? The angle-addition formulas for trig functions? The conditions under which a function has an inverse? Basic computer programming? Keeping in mind that the vast majority of high school students do not go on to become professional mathematicians, what should the high school math curriculum consist of?

Btw, I post this question (inspired by this discussion) here because this is a community of thoughtful mathematicians. I recognize this discussion may belong in a different forum, but I don't what/where that forum is. Any suggestions are welcome.

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    $\begingroup$ Please set this to community-wiki if it does not get voted to close... (subjective yada-yada?) I liked Theo Gray's little rant when I first read it, though. :) $\endgroup$ – J. M. is a poor mathematician Aug 23 '10 at 5:12
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    $\begingroup$ To a large extent, I think the premise is false--the skills (generally meaning symbolic manipulations) that can be replaced with CAS are not the core of secondary school curricula today. Further, the notion that secondary school curricula today are the same as they have been for the past decade or two or three is just not true. There is a somewhat cyclical nature to curricula on a national level in the U.S., but even within that, there are shifts, and local variance in the U.S. is very high (and then there's international curricula). $\endgroup$ – Isaac Aug 23 '10 at 5:22
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    $\begingroup$ On the other hand, there is the persistent accusation that things are getting "dumbed down" even more, every year. Having been off academia for quite a while, it might be great if answers to this question can address that, or better yet, the ones fresh from or are still in the "affected" levels can chime in (under the presumption that since you hang around here, you have an idea on what is or isn't missing in your education). $\endgroup$ – J. M. is a poor mathematician Aug 23 '10 at 5:35
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    $\begingroup$ ...and probably the lesson I most want to be hammered home to students: don't (blindly) trust the computer/calculator. Verify, verify, verify! $\endgroup$ – J. M. is a poor mathematician Aug 23 '10 at 12:12
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    $\begingroup$ Statistics. it was an optional class in my HS, but if there is anything in math that is used more often and more maliciously to defraud people and persuade the public opinion I can't think of it. People should be aware of the limitations and the meaning of statistical results. $\endgroup$ – crasic Oct 30 '10 at 22:58
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I strongly believe that even if something can be done by calculator, a high school student should understand some nuts and bolts of how it's being done.

I spent some time doing volunteer tutoring at CRLS (Cambridge Rindge and Latin School). I still remember the girl at 8th or 9th grade who was studying inequalities, yet she couldn't compute $2-1$ without the help of calculator. The most frustrating case was the kid in 9th or 10th grade who didn't know multiplication tables, he couldn't compute, say, 3 times 7 in his head. I was supposed to help him with square roots. Well, how do you explain that $\sqrt{8} = 2\sqrt{2}$? First of, you have to realize that $8 = 2 \times 2 \times 2$, and that's exactly what he couldn't do. I ended up spending some extra time with him, afterward been told that he passed the test or exam and was very happy with my help - but I wasn't! I felt we were wasting lots of time and just because he didn't know multiplication tables, he couldn't break down numbers into factors, and calculator wasn't really help.

The bottom line: I think in school students still should learn basic arithmetic before moving to more advanced stuff. I think it's a pure idiocy to introduce the kids to set theory while having them relying on their calculators to compute $1 + 1$. Honestly, the advanced stuff can wait till college.

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  • $\begingroup$ I've seen some very advanced calculators nowadays with buttons that takes me some time to understand, which is a lot considering how good I am at math, but I get what you mean. It truly can be frustrating. $\endgroup$ – Simply Beautiful Art Feb 6 '16 at 1:00
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I would include more probability, such as how to conduct a simple probabilistic experiment using statistical tests. Once you understand that, you can do a lot by simply looking up various tests. Without understanding probability, people are easily manipulated using statistics or make poor choices.

I would include definitely include more use of technology, including Wolfram Alpha and basic mathematical programming. These tools allow people to achieve much more.

Some basic mathematical philosophy would be good as well. Game theory can be very thought provoking and is quite simple as well. We could look at other areas even though we wouldn't be able to examine them rigorously. I think students should understand that maths is based on axioms, that not all maths problems can be solved (assuming ZFC is consistent) and that many problems can't be solved efficiently (P vs NP).

I would focus more on exploring various mathematical puzzles or unexpected results: Card doubling paradox, 3 hats puzzle, Monty hall problem and other results like this.

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    $\begingroup$ In my opinion the things you are suggesting in the last two paragraphs are more of the "I wish I learned this in high school" variety. As enrichment topics for especially bright and interested students, they are attractive. But it makes me think you are not aware of the skill level of the average (or even college-bound) high school student.... $\endgroup$ – Pete L. Clark Aug 23 '10 at 13:14
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    $\begingroup$ ...For instance, suppose that a 7-inch pizza costs $5. How much should a 14-inch pizza cost? I would like a high school student to be able to quickly recognize that $11, say, is a great price, and that if the price is any more than $20, they are being cheated. I don't think most high school students have internalized the necessary math for this, so let's a bit risible to bring ZFC into curricular discussions. $\endgroup$ – Pete L. Clark Aug 23 '10 at 13:22
  • $\begingroup$ @Pete: I don't believe that everything in the syllabus needs to be examinable $\endgroup$ – Casebash Aug 24 '10 at 13:01
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I spent over a decade thinking that "mathematics" is about completing page after page of identical long division problems. Only after I left school did I discover that mathematics actually has more to it than that. (And boy, am I glad I did!)

My opinions on the matter:

  1. I think there will always be people who like mathematics, and people who don't. However, I do think the number of people who "like mathematics" could be drastically increased if it was taught better.

  2. I don't think it's super-critical exactly which particular mathematical topics you teach. I think it is super-critical that you teach people what mathematics actually is. (A surprising number of people don't understand this.)

  3. I do not think that modern technology somehow makes understanding how to do X or how to do Y by hand "obsolete". Nor do I subscribe to the idea of "banning" calculators, computers and so forth. By all means, use the technology. (But don't let us fall into the trap of just blindly pushing symbols around without comprehending their meaning.)

  4. I think it might be useful to teach a wider range of mathematical subjects. Not in exhaustive detail, obviously. In many cases, that requires a whole heap of complicated ground-work. But introduce the interesting ideas and explore the main properties.

Personally, I don't see why you couldn't do something like group theory with school-age children. (Obviously, we're talking older children here.) "You know how to add and subtract, right? OK, so let's just throw the rule book out the window and invent our own brand-new system of adding and subtracting objects. Hey, look, we invented this from scratch, but now all these interesting properties keep appearing..."

My other pet idea is to take some of the computer software out there which does stuff with user-defined functions. Let the kids play with it, and see what neat stuff they can come up with. If you just type random gibberish, nothing interesting happens. But if you understand coordinate systems and functions and have an intuition for how simple mathematical operators affect numbers, you can make the computer draw trippy stuff.

Perhaps the biggest problem with mathematics is that most people don't understand it. Many people think that mathematics "is about numbers". (That's like saying that science "is about test-tubes"!) Many think that mathematics "is one subject". (Again, that's like claiming that science is only "one subject".)

The way that I do mathematics is that it's this fascinating journey of exploration, experimentation and discovery. But the way the textbooks do it is "Here is a type of problem. Here is the exact procedure for solving this type of problem. Here are 300 identical problems of this type. Go solve them. Your teacher will then compare your answers to the published answer book, and give you some ticks." Wow, how thrilling. :-P

Having said all that, we do seem to currently live in an age where being stupid is something to be proud of. This doesn't just affect mathematics. How to solve that larger meta-problem, I have no idea....

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In high school in the 1970s I got tremendous benefit out of a one-term elective course called Mathematical Logic, which was nothing fancy, just basic prepositional stuff. Understanding the basic structure of logical arguments empowers one to learn anything more efficiently. If college-bound students could ditch half of their AP Calculus for a course in Logical Reasoning that includes basic material from Daniel Velleman's book How to Prove It and some basic facts about statistics, they would be in a far better position to pursue many subjects, not just math.

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    $\begingroup$ Yes! I agree completely. A course in logic or problem solving skills will be much more useful. (Personally I am also of the opinion that calculus [in highschool] should not be taught as a class in itself, but as part of physics.) $\endgroup$ – Willie Wong Oct 30 '10 at 20:14
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Whatever you think the high school mathematics curriculum "should" be, in the United States a curriculum has been put in place, known as the Common Core Standards (CCS), which will significantly - I believe - change American mathematics education. (The CCS extend to all K-12 mathematics.)

http://www.corestandards.org/the-standards/mathematics

I don't think these changes are for the better. A short summary of my personal views are available here:

http://www.education.umd.edu/MathEd/conference/vbook/public-perceptions_Malkevitch.pdf

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I recall the Asimov short story about humanity that has completely lost the knowledge of how to do simple arithmetic without use of a computer, called "The Feeling of Power". The point is, despite computers being able to do these things for us, it is still important and valuable to know how to do simple operations. At times, we are amazingly close to that point in this very day, certainly in the current crop of kids growing up now.

Basic mathematical skills are tremendously valuable in a variety of places. Just because you CAN factor a polynomial by computer does not mean that at least understanding the tools to do so are no longer necessary for 99% of the people in the world.

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  • $\begingroup$ +1.We need people to state the obvious sometimes to keep us from confusing the forest for the trees. $\endgroup$ – Mathemagician1234 May 3 '12 at 16:18
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Students should learn math. You know, the stuff that most of us do. When we use the word "math" we certainly don't have in our minds adding, multiplying, factoring polynomials, or even calculus. So why do we teach children that this is what math is?

This view has been in several published places like Devlin's The Math Gene. He makes a good argument that the concept of a group isn't any harder than other stuff done in grade school. The problem is that it is introduced later.

The best thing I've read about math education is A Mathematician's Lament by Paul Lockhart.

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  • $\begingroup$ Google "The New Math" movement of the 1960's and 1970's,Matt. They TRIED that,it was a resounding disaster. The correct way of doing math is not always the best way to learn it-especially when dealing with grade schoolers. $\endgroup$ – Mathemagician1234 Dec 2 '13 at 3:52
  • $\begingroup$ @Mathemagician1234 I think the key word in this question is "High School." I'm not proposing we get rid of the basics in elementary school for some discovery based method of teaching. Just like a musician still has to practice scales and basics, math students need to drill the basics as well. In order to stay motivated, music students still get to listen to great music and see what will come from their efforts some day. Math students don't ever get a taste of what will come from working hard, so why do should they do it? $\endgroup$ – Matt Dec 2 '13 at 7:07
  • $\begingroup$ I think they don't lie to kids that that's what math is. All they do is cause students to notice that that's the material covered in math class without telling them what "math class" really means. One student told me he was really struggling with modern algebra. That doesn't mean the concept should be have been introduced at a younger age. According to inc.com/bill-murphy-jr/…, kids learn better if they start grade 1 older. I belive that concept would have been just as hard to learn at a much $\endgroup$ – Timothy Dec 19 '18 at 21:59
  • $\begingroup$ younger age and maybe even harder. Grade 1's sometimes take so long to learn how to add. Not only that but they would have been teaching so much material before they age they claim kids become too old to learn new stuff that it's too fast paced and they move on before the students are ready and they end up learning none of what school is trying to teach them. I have aspergers and unlike a normal person, I learned the material in Modern Algebra and Modern Algebra II very easily because the book explained the material clearly and I could remember it well even though I wasn't introduced to that $\endgroup$ – Timothy Dec 19 '18 at 22:04
  • $\begingroup$ concept at all when I was younger and probably learned at a lot faster than I could have when I was in elementry school. I know Mary and Janelle from Sister wives were raised very differently and after they met each other, they were too old to learn each other's way of thinking until they got long training by a therapist but that doesn't mean everything can be learned more easily by younger people. $\endgroup$ – Timothy Dec 19 '18 at 22:06
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I am surprised people advocate the white box approach to learning math. In computing, it is imperative from programmers of lower layers(operating system, compilers) to hide their code internals to higher layers programmers. I don't see anything wrong with hiding low-level math from kids through the use of CAS(computer algebra system) and mathematical programming languages. In fact it need be such low level "algorithms" can be taught using such CAS eg "Write an algorithm that performs long division". Math assimilation has been regressive because of old methods of teaching which were clearly paper-based before the advent of computers. Einstein had to hire an "assistant" who acted basically as a computer to grind the numbers. My argument is we have computers now and let us incorporate them in learning "high" math which matters. In fact, low-level math has no applications in the real world which makes sense to relegate it to the computer while kids are taught how to use math as professionals do.

My ideal syllabus for high school, assuming they have been introduced to computer earlier on, especially programming lessons.

  1. discrete mathematics (binary, Boolean algebra, logic)
  2. recursion theory (theory of functions, classifications of functions, graphing using computer)
  3. history of mathematics
  4. calculus (allowed to use computer)
  5. mathematical modeling.
  6. computer programming.
  7. linear algebra (matrices)
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I think the problem of not knowing how to do math is mainly caused by something other than what the curriculum is. Here's my idea of how to solve the problem but I'm not sure if it's worth doing. If it can be done, it's better to make people under 18 unable to use a website to make calculations for them than to tell them not to use one because if you tell them not to, they might have to choose between concentrating hard and using their will power to avoid using one or being unable to resist using it and ruining their self esteem as a result. That way, when they think of their own math problem and get interested in solving it, they will be like "I want to know the answer so I will figure out the answer by myself." without the need for will power to resist using such a website. I think the evolutionary learning laboratories are working towards gaining the advanced research results to solve complex problems. Once they have those research results, I think it's better if everything gets reviewed by the evolutionary learning laboratories before it gets posted on the public internet to make sure that unwanted stuff doesn't get uploaded there such as a guide on how to make a gun or a web page that makes calculations for you.

Maybe in addition to that, the evolutionary learning laboratories could work towards having jobs provide their own specific training so that students won't have to learn so much material in school before university and some of the time can be used to instead train people's ability to do mental calculations and then they can do tests without using a calculator. Maybe their tests could be done on a computer so that to get full marks, they will have to do the calculation entirely mentally without even using pencil and paper to help. They should not however be made to perform lightning fast mental calculations of two numbers at least 8 digits long. That would probably cause them to put their full attention into the task of making rapid calculations when they're given that task and not spend any of their time doing creative thinking as a result. If they don't have the pressure to perform the calculation really fast, then they might discover so much in math as a result of their creative thinking and as a result naturally gain the ability to do even faster mental multiplication by using short cuts way later.

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