Best resources for self-studying math from K-12?

[Zephyyr441]

Hi, everyone!

I'm planning on teaching myself math all the way back from the basics, up to high school stuff.

It has been a long time since I studied math. I've never been a great student back in school. I daydreamed most of the time, and when it came to math, I always rote learned and memorized things just enough to get a passing grade, sometimes I got lucky and managed to get a perfect score, but those moments were rare.

Because of that rote learning, and memorization, I have forgot almost everything about math other than the basic arithmetic, and even there, it takes me a bit of time to calculate simple problems :(

However, now, that I'm older, I found out just how useful and great math really is. I would honestly love to go through it again, and hopefully with your help I can manage to do it!

What I'm looking for are (in your opinion) the best learning math resources that you can recommend. Anything that covers the materials from K up to Grade 12.

The type of resource is irrelevant, I'm not looking for only one type, it can be anything, whether it's a book, a website, videos, etc. Free, or paid also doesn't matter.

All that's important to me is to get a good book/video/etc. that properly explains the concepts I'm learning. I want to make sure that I actually understand the concepts I'm reading about and to be able to think logically on how to solve the problems for the topic I studied. Which means no more rote learning, and no more memorization!

If necessary, for better understanding of the concept I'm learning, it would be nice to also learn the history behind that concept I'm learning about, that is, how did we manage to come up or create something like which we use to this day, what problem did the people back in the day need to solve in order to come up with such things, etc. (eg: numbers, base 10 system, shapes, algebra, logarithms, calculus, trigonometry, probability, statistics, etc.)

While I'm studying math right now mostly for fun, and because I find math to be such an amazing subject now than I did back before while I was at school, I do plan on taking my studies seriously, and actually trying to learn it, instead of what I was doing in the past.

Aside from the resource recommendation, I would also like to ask you guys:

What do you believe is the best / proper way to study math? Instead of cramming or memorizing formulas and methods like I did back in school, what do you think is the most effective way to study to "make it stick", that is, how should one study math to better understand the content you're learning?

What are some common mistakes people do when trying to learn math?

Additionally, is there like a roadmap, or a guide for studying math? That is, from which topics should I start and how should I progress?

Also, apologies if this is posted in the wrong subforum, moderators feel free to move this thread to where you think is appropriate.

Thanks in forward, and apologies for the long post!

 

[jedishrfu]

One site that might meet your requirements is www.mathispower4u.com it covers middle school to first year college in a collection of 10 minute videos. The presenter will post a problem and then go on to solve it. Being a video you can stop it and try to do it yourself and identify the tricky parts. By watching the video, you'll know where you went wrong.

The home page starts off with video playlists for common core grades 3/4,5/6 and 7/8 followed by high school and beyond:

http://mathispower4u.com/

As far as learning strategies, you have to do and keep doing problems to retain your knowledge not unlike going to the gym and exercising continuously.

The other problem you may run into is already knowing some things, which makes the initial lessons boring and encourages you to skip over something you may need later. Don’t!

Lastly, there is no royal road to mathematics mastery, a quote atributed to Euclid.

 

[berkeman]

I'm planning on teaching myself math all the way back from the basics, up to high school stuff.

Welcome to PF.

Be sure to look through the links in this very helpful post by our Mentor @fresh_42

https://www.physicsforums.com/threads/self-teaching-gcse-and-a-level-maths.933639/#post-5896947

 

[malawi_glenn]

I have also quite recently taking up math as a self-study hobby - but I think I am on a quite higher level than you so I can not give any kind of recommendations regarding books unfortunately. But I think the same strategies applies nevertheless. Here are my personal reflection.

- Set up a concrete goal, what is that you want to learn, and why? Having a clear goal makes it easier to not fiddle around too much. Also evaluate your progress towards your goal now and then.

- Study continously, it is waay better to do say 30 min / day than 3.5h one day and nothing in 6 days. Even if you do not have time to push in an "ordinary session" - just read one page, or do 10 min review of something you did last week / two days ago.

- Watch videos, there are tons of math tutorial videos on youtube - I try to watch at least one per day, even if it is not about what I am studying at the moment, it either makes me review a certain definition or method, or it can inspire me - something to look forward to understanding.

- Take your own notes, when you write something yourself - it sticks in your mind/memory better. After each chapter in a book, I write my own summary about the important theorems and methods and perhaps fill in some details that was omitted in the book. I organize my notes in a ring binder. Sometimes I type my notes in LaTeX and print them out.

- Work on problems, prove things on your own. Solving problems and proving things is a key thing to master the material covered. I try to use books / resources that provides some kind of solutions manual/guide, which can be a negative thing too - it is so tempting to sneak peak too early at the solution given. Also there might be more than just one way to solve the problem, which is often not provided in these guides.

- Ask questions, there are several great online communities, such as this one, where you can ask questions. Make sure to search first and see if your question or a similar one has already been asked and answered.

- Use several resources, but focus on one. Have a main-book and use other books/resources online as a "second opinon". For instance now I am reading Axlers book on real analysis, integration and measure theory. But I do watch a YT video series and have another online resource to get some more "flavor" or see things explained in another way. But I follow the structure in Axlers book.

 

[The Bill]

I hope this isn't hijacking the thread, but I have a related question the answers to which might/should be relevant to the OP.

What hardcover textbooks would you all recommend for teaching K-12 math? The closest I saw in the linked resources above were the OpenStax textbooks, but those are only available in softcover. I'm particularly interested in durably-bound hardcover texts for middle-school general math, pre-algebra, high-school algebra, proof-based geometry, and trigonometry. The key being that you know enough about the textbook and like it enough to recommend using it in a class.

 

[vdicarlo]

Hi, everyone!

A few years ago I did exactly what you say you plan to do. The best resource I found then was

https://www.edx.org/course/precalculus

In spite of the name, it covers a lot more than is in most precalculus classes. It mostly consists of using software called Aleks, which diagnoses all of your weaknesses in math from basic arithmetic to precalculus, and then leads you through the acquisition of each skill needed with presentations, problems, and further tests. One of the best things I've found on the internet. I think it could help a lot of struggling high school students.

After completing this course I was able to go on to complete MIT's online advanced placement course in differential calculus, and courses in linear algebra, machine learning, and computer science. If you don't like the Edx platform, you may be able to find Alex elsewhere.

 

[MidgetDwarf]

I hope this isn't hijacking the thread, but I have a related question the answers to which might/should be relevant to the OP.

What hardcover textbooks would you all recommend for teaching K-12 math? The closest I saw in the linked resources above were the OpenStax textbooks, but those are only available in softcover. I'm particularly interested in durably-bound hardcover texts for middle-school general math, pre-algebra, high-school algebra, proof-based geometry, and trigonometry. The key being that you know enough about the textbook and like it enough to recommend using it in a class.

Non really. They are mostly terrible. I write my own while pulling information from these run of the mill books.
For Algebra 2/Trig/Pre-Calculus. Cohen Pre-Calculus is one of the better books, excluding Lang's Basic Mathematics, which would be nearly impossible to use for todays students. Moreover, Cohen can be used for 3 classes. Since Pre-Calculus is just a rehash of Algebra 2 and Trigonometry (most schools). Problems range from easy to hard, and separated into 3 section (A,B,andC).

For proof based Geometry, there is an older book written by Moise/Downs titled Geometry. Good luck finding a classroom set, since I believe, it is not in print anymore. One of my favorite books.

 

[Muu9]

Because you want to learn math for math's sake (and not to get better at mental arithmetic), I would strongly recommend the Art of Problem Solving's Prealgebra book. It's meant to be a prealgebra book for gifted children, which you can easily exceed thanks to your superior metacognitive skills. There is a placement test here. If you can't do it, no worries, come back as soon as you're ready. You could also take a look at Gelfand's algebra book, but that's more difficulyand less"curricula"

P.S. if cost is an issue, DM me

 

[Muu9]

Non really. They are mostly terrible. I write my own while pulling information from these run of the mill books.
For Algebra 2/Trig/Pre-Calculus. Cohen Pre-Calculus is one of the better books, excluding Lang's Basic Mathematics, which would be nearly impossible to use for todays students. Moreover, Cohen can be used for 3 classes. Since Pre-Calculus is just a rehash of Algebra 2 and Trigonometry (most schools). Problems range from easy to hard, and separated into 3 section (A,B,andC).

For proof based Geometry, there is an older book written by Moise/Downs titled Geometry. Good luck finding a classroom set, since I believe, it is not in print anymore. One of my favorite books.

Just curious, why don't you think Basic Mathematics would be impossible for today's student? What do you recommend for algebra 1?

 

[MidgetDwarf]

Just curious, why don't you think Basic Mathematics would be impossible for today's student? What do you recommend for algebra 1?

Its a good book, and my favorite pre-calculus book. Besides S.L Loney book on trigonometry (would not recommend for someone first trying to learn trig).

Lang has lack of problems. Not enough of the computational variety. Many students first encounter with Calculus will be of the computational variety. Lang is geared towards a mathematics major, and for that, it serves its purpose.

Not enough word problems...

 

[mathwonk]

Since you specifically asked for math instruction materials for K-12, I suggest you look at the set of material developed by the School Mathematics Study Group in the 1960's. These were prepared by professional mathematicians working with professional math educators, intended to be an improved coherent collection of texts for math instruction from elementary school through high school, covering sets, arithmetic, algebra,geometry, statistics and calculus.
The earliest grade level mentioned seems to be grade 4, but that is probably far enough back for you to begin.
Here is a link where many of them are available free:
http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=School Mathematics Study Group

Having perused many of these free volumes online, my opinion is they are of somewhat variable quality. The best ones seem to be written by a group including the famous mathematicians Charles Rickart of Yale and R.J. Walker of Cornell. The volume on high school mathematics: elementary functions, seems to me perhaps the best. It is really excellent as a clear and elementary introduction to the ideas of calculus, emphasizing the case of polynomials.

in my opinion however, you might be better off just reading Euclid and Euler, for a very thorough classical grounding in geometry and algebra, that far surpasses anything available in essentially any high school in america today. Then maybe follow up with the volume above on elementary functions.

https://www.amazon.com/dp/1888009187/?tag=pfamazon01-20

https://www.amazon.com/dp/110800296X/?tag=pfamazon01-20

 

[gmax137]

It mostly consists of using software called Aleks, which diagnoses all of your weaknesses in math from basic arithmetic to precalculus, and then leads you through the acquisition of each skill needed with presentations, problems, and further tests.

I seem to recall Our @Dr. Courtney recommending ALEKS in several similar threads.

HIJACK
Hmm I haven't seen him posting lately, wonder what he's up to?
/HIJACK

 

[Muu9]

I seem to recall Our @Dr. Courtney recommending ALEKS in several similar threads.

HIJACK
Hmm I haven't seen him posting lately, wonder what he's up to?
/HIJACK

The edx course is extra nice considering ALEKS usually costs money. I'm not sure if it's available in the audit track

 

[CrysPhys]

Since you specifically asked for math instruction materials for K-12, I suggest you look at the set of material developed by the School Mathematics Study Group in the 1960's. These were prepared by professional mathematicians working with professional math educators, intended to be an improved coherent collection of texts for math instruction from elementary school through high school, covering sets, arithmetic, algebra,geometry, statistics and calculus.
The earliest grade level mentioned seems to be grade 4, but that is probably far enough back for you to begin.
Here is a link where many of them are available free:
http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=School Mathematics Study Group

Having perused many of these free volumes online, my opinion is they are of somewhat variable quality. The best ones seem to be written by a group including the famous mathematicians Charles Rickart of Yale and R.J. Walker of Cornell. The volume on high school mathematics: elementary functions, seems to me perhaps the best. It is really excellent as a clear and elementary introduction to the ideas of calculus, emphasizing the case of polynomials.

Wait! Wasn't the School Mathematics Study Group (the acronym SMSG is triggering unpleasant flashbacks) responsible for the infamous "New Math" foisted on US students in the 1960's? I was one of the guinea pigs back then. The "New Math" was a disaster. I don't know whether a historical perspective provides a kinder assessment (haven't followed it). But I would check it out carefully before following this curriculum.

 

[haushofer]

One book I really like is Calculus Lite.

 

[Muu9]

Wait! Wasn't the School Mathematics Study Group (the acronym SMSG is triggering unpleasant flashbacks) responsible for the infamous "New Math" foisted on US students in the 1960's? I was one of the guinea pigs back then. The "New Math" was a disaster. I don't know whether a historical perspective provides a kinder assessment (haven't followed it). But I would check it out carefully before following this curriculum.

One of the reasons why new math failed was because it was developmentally inappropriate for elementary aged children. It could still be useful for OP.

 

[CrysPhys]

One of the reasons why new math failed was because it was developmentally inappropriate for elementary aged children. It could still be useful for OP.

Feynman didn't think some key aspects of New Math were appropriate or useful for anyone:

https://calteches.library.caltech.edu/2362/1/feynman.pdf

I was subjected to New Math in 7th grade in what was supposed to be introduction to algebra. The New Math course material was obsessed with the underlying unifying theme of set theory. Our school dropped it after a wasted year, and we underwent intensive remedial work in 8th grade to undo the damage done. And this was in an ultra-competitive school in which students needed to perform well in an entrance exam for admission.

 

[MidgetDwarf]

Wait! Wasn't the School Mathematics Study Group (the acronym SMSG is triggering unpleasant flashbacks) responsible for the infamous "New Math" foisted on US students in the 1960's? I was one of the guinea pigs back then. The "New Math" was a disaster. I don't know whether a historical perspective provides a kinder assessment (haven't followed it). But I would check it out carefully before following this curriculum.

I remember reading an article which I cannot find. Hopefully a member here is familiar with it, and can link it.

The article states that one of the major failures of the SMSG group, was that teachers would go to seminars showing them the new math. But, when they returned to the classroom, they would resort to old ways. A disconnect between what the book contains and what the teacher is saying was created.

Richard Feynman's view on new math http://calteches.library.caltech.edu/2362/1/feynman.pdf

It appears we both referenced the Feynman article. lol

https://www.maa.org/archives-spotlight-the-school-mathematics-study-group-records

The MAA link suggest places more information regarding the SMSG can be found.

 

[vanhees71]

The "new math" was also introduced in German elementary schools. It was a great disappointment for me, not to learn how to calculate (which I was expecting to learn at school), but we were given some little plastic gadgets we had to sort into Venn diagrams corresponding to certain properties. Of course, I had no clue, what I should learn from it, and it drove my parents nuts, who of course also had no idea, what this should be good for ;-)).

 

[CrysPhys]

The "new math" was also introduced in German elementary schools. It was a great disappointment for me, not to learn how to calculate (which I was expecting to learn at school), but we were given some little plastic gadgets we had to sort into Venn diagrams corresponding to certain properties. Of course, I had no clue, what I should learn from it, and it drove my parents nuts, who of course also had no idea, what this should be good for ;-)).

It was all a conspiracy. LEGO was a key sponsor of the New Math movement.

 

[vanhees71]

Well, LEGO I'd have liked ;-)).

 

[Office_Shredder]

Well, LEGO I'd have liked ;-))

Subtle correction. Lego you now like, because of your indoctrination!

 

[malawi_glenn]

The "new math" was also introduced in German elementary schools.

Same in Sweden but that was before my time

 

[mathwonk]

Let me just suggest that the many sins of the new math movement should probably not be blamed on the original SMSG books, which I believe range from good to excellent. Perhaps one should skip the set theory segments in the early grades.

Another reliable source for good high school instruction are the two books by Harold Jacobs, Elementary Algebra, and Geometry (preferably the first edition).

 

[gmax137]

How can I tell if I received grade school instruction in "new math?" I have no recollection of any specific books. I do remember the books having sets (ha!) of problems to be worked. Lots of two-digit by three-digit multiplications and "long divisions" where maybe a four- or five-digit number was divided by a two- or three-digit number. I hated those, but we did have some "word problems" that I liked.

I was in fifth grade in 1966/7
I remember "...six times six is thirty-six, six times seven is forty-two..." and on and on and on.

The next year (sixth grade)
I remember unions and intersections of sets, "ordinal" "cardinal" "number" "numeral" I remember my father being kind of baffled by "sets." Him asking, "what do you mean by a set?"
But I also remember "least common denominator" and simplifying fractions. Lots of 7/8 minus 2/3 stuff.

I haven't had any kids (yet) so I don't know what they're teaching them now.

 

[mathwonk]

forgive me, i am remided of huck finn's reminiscence of school:
"
I had been to school most all the time ... and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics, anyway.
Huckleberry Finn."

 

[terrytosh]

I agree with the other user who suggested AOPS, it focuses less on rote memorization and more so on understanding concepts, which appears to be what you're looking for.

Also Herb Gross has a great course on basic arithmetic:

There is a study guide and other resources in the description boxes of the videos.

 

[MidgetDwarf]

I agree with the other user who suggested AOPS, it focuses less on rote memorization and more so on understanding concepts, which appears to be what you're looking for.

Also Herb Gross has a great course on basic arithmetic:

There is a study guide and other resources in the description boxes of the videos.

I see AOPS as a bag of tricks...

 

[Muu9]

I see AOPS as a bag of tricks...

It has the the most challenging exercises I've seen in any basic algebra book. I'm not sure which of their books you've read, but the intro to algebra book focuses on the core concepts of algebra 1 and 2.