# s/o Saxon Math- Early Impressions--Now Problem Solving

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Wapiti wrote the following on the Saxon Math- Early Impressions thread:

I am thinking about ways to work on problem solving. It is not easy to tap into the motivation to do the hard thinking that is necessary. Brain-tickling, for lack of a better term, would be one such angle, if anyone knows what I mean by that. I would guess that there might be a fair amount of problem solving involved in programming,say, though perhaps it is not coincidental that some discrete math is involved there too. Maybe a spin off thread is in order - I would enjoy hearing more ideas.

Thank-you, ~h

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I can't link right now but there are a few old threads that might be a good start.

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Games :D

For problem solving and to hopefully keep dementia at bay try Chess and Mahjong. There is math embedded in both games and the game will teach the player :)

ETA:

Math and music has always been interlinked since ancient times.

"It is generally not mentioned that Pythagoras (569-480 BC ) was also a musician and that he used his mathematical knowledge to build and play his own instruments, tuned in 3:4:5 ratios. "

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Have you looked at Hoagies' Gifted Education Page? Lots of math and programming links.

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As my sons are advancing in classical guitar, I'm really seeing how math, puzzles and patterns are everywhere in music and especially certain instruments. For example, on the guitar, you can find the same "E" in many different places on different strings and frets. Also there in violin, but no frets to make a grid - you have to visualize the grid getting smaller and smaller the higher up you go. It's "puzzles" and patterns for the eyes, ears, fingers and mind! Love it.

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I can't say that an intentional look at what Saxon lacked motivated us, but we added in strategy puzzles (Perplexors by mindware), coding, chess, music, and number patterns to our regular course of study last year.  This year, we're starting off with Beast Academy, and will dip into Saxon 5/4 eventually, which is a total shift from our Saxon faithfulness.  We plan on continuing with Saxon, but in small chunks interspersed with these other activities that he enjoys.

There should be something a kid can enjoy in their math studies, and I can't say that happened with Saxon alone.  It made for a very unhappy school dynamic, unfortunately.

Stella

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Ok, finally got some time to post.  Here are my observations based on experiences teaching at the university:

My answer depends upon the goal you seek.  Are you looking to enhance competition-style skills, creativity, or skills needed for advanced mathematics (post-calculus)?

For competitions, there are quite a few good resources discussed on this board.  These skills do not directly help with advanced mathematics, though.  Tricks and shortcuts don't go very far after the SAT.

For creativity, puzzle books, music, and the like as mentioned above actually help.  A mathematician needs to have a high degree of creativity to be successful.  It isn't enough to just do practice, though -- you have to study the methods behind the arts.

For preparation for advanced math, there really is no alternative to learning the roots of the field.  That means set theory, symbolic logic, number theory, and the like.  Programming is helpful because it encourages logical thinking, but that advantage diminishes over time.

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For preparation for advanced math, there really is no alternative to learning the roots of the field.  That means set theory, symbolic logic, number theory, and the like.

Very interesting. DS the older loves number theory, but has never done any set theory or symbolic logic.  Are there any introductory books to these topics that would take just a few months to complete, rather than a year long course?

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For my younger, I am actually working on frustration tolerance in other subjects hoping it will transfer into the less-liked subject, math.  So Latin and violin right now, and moving into programming soon.

I am also working on a bit of self teaching in mathematics, because problem solving / deep thinking often requires referencing other ideas.  My older who is quite good at this type of thinking is often digging out lots of different types of books and reading up on an idea that he was only minimally confident with.  But to do this you need to be able to *read* the explanations in math books.  My younger cannot and is completely reliant on me to explain everything, which I think will restrict his movement forward in mathematics.

I also do some mathematical investigations with my younger.  You can find them in MEP secondary (free online), singapore's NEM, and Jacobs a human endeavour. They are about exploring a concept, like prime numbers, and typically guide you through the process.  They don't require the insight that difficult problem solving does, but they do help a student to think differently about math.

I'm using these ideas as stepping stones to help my younger cross the bridge to harder problem solving.  He can do word problems just fine, but the moment you ask for any insight or out-of-the-box thinking, he's sunk and gives up very quickly.

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What about puzzles, knitting, sewing, and building things (with wood, Legos, Zome tools)?

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Very interesting. DS the older loves number theory, but has never done any set theory or symbolic logic.  Are there any introductory books to these topics that would take just a few months to complete, rather than a year long course?

Ruth, have you looked at the eIMACS Elements of Mathematics and advanced math logic courses? The former might be too easy for him (sorry, no personal experience) and the latter is billed as a 40-week course but a couple of people have mentioned to me that their teens were able to finish them over a summer. The latter is pricey as an online course, but if you write to them, they will email you a booklist and order form and you could see if it's something you are willing to purchase/ pay shipping for. The courses are based on the books.

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What about puzzles, knitting, sewing, and building things (with wood, Legos, Zome tools)?

Do you mean in the context of problem solving?

My boys can knit and they knitted their own winter wear. We crochet many hyperbolic corals.

http://crochetcoralreef.org/Content/makeyourown/IFF-CrochetReef-HowToHandout.pdf

Their Legos is use for physics (catapults) and robotics rather than math.

For zome tools below linked book was recommended by someone but we didn't use it. My older made a Darth Vader mask with his zome tools :lol:

http://www.amazon.com/Zome-Geometry-Hands-Learning-Models/dp/1559533854

Not strictly wood building but we enjoyed building the pine derby cars and my kids optimize their cars by trail and error.

If your child like building things, toothpicks and glue goes a long way to creating pieces of art.

http://www.toothpickworld.com

Not math but we are looking at building a simple earthquake table since we are in earthquake territory (ETA: just realized we are also staying in earthquakes (soccer) territory).

http://jclahr.com/science/earth_science/shake/index.html

ETA:

https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf

Fun hands on problem solving activities suggestions

http://web.stanford.edu/group/ree/archives/archive07/usa/notes/2004-897_Final.pdf

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Very interesting. DS the older loves number theory, but has never done any set theory or symbolic logic.  Are there any introductory books to these topics that would take just a few months to complete, rather than a year long course?

You know, I don't necessarily shy away from the real deal, even with DS7.  We started learning some tenets of abstract algebra together, and I just filtered out the problems requiring linear algebra (matrices, etc).  Basically, I read the book, and taught him independently.  He had no difficulties understanding ANY of it, largely because he had no preconceptions.  He loved the idea of making up his own mathematical operations and testing to see if his sets and operations were open, closed, formed a group, a ring, a lattice, a module, or an algebra (it gets tougher the farther down the chain you go).  He had only just finished learning his multiplication tables at the time.

Symbolic logic is a lot of fun for visual kids.  They like the "secret language" aspect of it, and enjoy the puzzle-style problems.  You don't need any prerequisites for the course, other than the ability to read and write and perform basic addition and subtraction.  It is the foundation for modern programming algorithms, and can go a long ways toward developing a budding computer scientist.

The nice thing about axiomatic mathematics is that almost any book on the topic will work.  They're consistent, because axioms can't change from text to text.  So, look for one with a writing style that suits your tastes, and your comfort for reading ahead.  Symbolic logic is relatively easy to keep up with yourself (relative being a loaded term), but abstract algebra texts can get pretty advanced.  It's better to back off to elementary set theory, or step back to an old introductory text in the Dover series.  Definitely preview before you buy!

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I have a friend who teaches high school math, and he teaches his remedial Algebra students to play cribbage.  He says this helps them with algebra more than anything else he has tried.

He also fantasizes about writing a book for parents of littles encouraging them to play games with their kids - any game requiring logical thinking, planning, multi-step decision making - yahtzee, chess, rummicube, whatever.  He finds that what his students lack isn't the ability to perform operations, it's the ability to think logically and to plan multiple steps to get to a goal.

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Mike, I need more direction please.  When I think of logic, I think of truth tables with nand/nor/or gates.  I also think of set theory with VENN diagrams and statistics.  Should I be looking at basic beginner books in these areas?

Also, I am not clear about your definition of axiomatic math.  I keep thinking of geometric and algebraic proofs.

When I work with my DS and we encounter math difficulty, I like to fall back on what we know and ask questions about the real direction of the problem and what is necessary to figure out the answer.  I can't think of an example right now.

My end goal is to have a student that can handle a basic college level stats class.

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My end goal is to have a student that can handle a basic college level stats class.

Have a good grounding in calculus too to understand statistical distributions.

ETA:

http://forums.welltrainedmind.com/topic/453395-calculus-based-statistics/

ETA

For axioms, Euclid's The Elements

More axioms to look at

http://mathworld.wolfram.com/topics/Axioms.html

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