Algebra Curric that explains WHY

[Garga]

I'm helping a homeschooling friend find an algebra curriculum that explains why algebra works the way it does.  Her son is using the McGraw Hill algebra text and feels like it doesn't give him enough background to understand the point to it.

The mom says that in general her son is the kind who likes to know the big picture first, and then learn the details of how it got that way.  He doesn't like learning the little details and having to wait for the big reveal of how it all joins together.  

An example of a why he'd like to know: when he was learning PEMDAS he asked her why it has to work that way.  Why multiplication before addition?  Yes, he can see that it gets a different answer when you do it a different way, but why?  How did the original person who discovered PEMDAS know that it had to work that way?  It sounds like the book just says, "Use PEMDAS.  It works," but didn't tell him why it works.

He's in 10th grade and is in Alg I.  He spent two years on pre-Alg because it just wasn't clicking. When they do tests, he's getting a high C average.  His mom doesn't feel like the curric is really working for him, but doesn't know how do find something else.  Her son is increasingly hating math.

Does anyone have suggestions?  I would not think that AoPS would work for him as it sounds very advanced.  I wonder if something like the Jousting Armadillo series would be good?  Except that when I look at the website, it is geared for 6th - 8th grade and this student is in 10th, so it might not be appropriate for his level.  

He does great with Geometry, if that means anything to anyone.  I've heard that some people are very strong in one but not the other, as in this boy's case.

 

If anyone could just toss out some names of curricula that might work, then his mom would have a starting point to do some research.  Thank you.

[rdj2027]

Funny, I was going to suggest Jousting Armadillo because it is like AOPS but on an easier level.

[Garga]

1 minute ago, rdj2027 said:

Funny, I was going to suggest Jousting Armadillo because it is like AOPS but on an easier level.

 

I'm wondering at this point that maybe they should just use what works and forget the "level."  He could use the JA series on the side of whatever it is he's doing now.  That's what I did when my son was in 7th grade--we had a regular math curric, and then read through JA on the side.

[Tanaqui]

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An example of a why he'd like to know: when he was learning PEMDAS he asked her why it has to work that way.  Why multiplication before addition?  Yes, he can see that it gets a different answer when you do it a different way, but why?  How did the original person who discovered PEMDAS know that it had to work that way?  It sounds like the book just says, "Use PEMDAS.  It works," but didn't tell him why it works

 

 

Well, it's not something we discovered. It doesn't have to work that way. We could've decided to do it completely differently. But it's better for us all if we use the same method when deciding which operations to do first, second, third. So one method was picked and it became the standard.

Annoyingly, that's the answer to a lot of "why" questions. "Because we had to pick a method, and this is the one we picked." Has anybody explained this general principle to him? It's as true in math as it is anywhere else. Maybe understanding that some things were really just arbitrary human decisions will allow him to stop looking for reasons when they aren't forthcoming.

Quote

I'm wondering at this point that maybe they should just use what works and forget the "level." 

 

That sounds like an excellent idea. Grade levels are also somewhat arbitrary.

[rdj2027]

That would be my recommendation.  We don't do spiral review but frequently work with different books because they forgot something or need a little more depth.  #3 son is very much a visual big picture person.  First he needs to have an answer as to why do I need it and what does it do?  Then we do a general overview of the path that leads to the solution and then go into the nitty gritty details.  He is now in 10th grade and I still sometimes have to use visuals from him to truly understand why a mathematical rule is what it is and why it is. He does well in math but needs to get down the last detail to have the feeling he understood the material.  It is getting harder in the upper grades, a couple of books we found helpful were Calculus Better Explained and Math Better Explained by Kalid Azad.  If the student is in Algebra1, he is a long way away from pre-calc and calc, just thought I throw those in for good measure.  Both books are fairly thin paperbacks.

[EKS]

22 minutes ago, Tanaqui said:

Well, it's not something we discovered. It doesn't have to work that way. We could've decided to do it completely differently. But it's better for us all if we use the same method when deciding which operations to do first, second, third. So one method was picked and it became the standard.

Annoyingly, that's the answer to a lot of "why" questions. "Because we had to pick a method, and this is the one we picked." Has anybody explained this general principle to him? It's as true in math as it is anywhere else. Maybe understanding that some things were really just arbitrary human decisions will allow him to stop looking for reasons when they aren't forthcoming.

This is why it is always helpful to have a human in the room who can answer questions.  Simply handing a kid a textbook, especially one that was written to be used in conjunction with a trained teacher, is not the best plan for many, many high school students.  IMO, the idea that this is what homeschoolers should strive for is a destructive myth.

[Farrar]

I agree with the Arbor school books, but only if he likes to write.

I also think Jacobs sort of does this.

But mostly a person who can walk him through it is essential.

[RootAnn]

I think Jacobs helps the student understand why what they are doing works they way it does, but I think it works in little pieces & builds - so I'm not sure if that is what the kid is looking for. It doesn't start with the big picture, IMO. It starts with what the kid might already know (like what fractions really mean and how to work with fractions) before showing how to it works with algebra (equivalent algebraic fractions, dividing algebraic fractions, or simplifying algebraic fractions).

[daijobu]

Wikipedia is actually a good starting place for research of this sort.  For example, if you want to know the origin of the order of operations, Wikipedia links to this interesting Dr. Math explanation:

"The basic rule (that multiplication has precedence over addition) 
appears to have arisen naturally and without much disagreement as 
algebraic notation was being developed in the 1600s and the need for 
such conventions arose. Even though there were numerous competing 
systems of symbols, forcing each author to state his conventions at 
the start of a book, they seem not to have had to say much in this 
area. This is probably because the distributive property implies a 
natural hierarchy in which multiplication is more powerful than 
addition, and makes it desirable to be able to write polynomials with 
as few parentheses as possible...."

We take parentheses for granted, but apparently there were other methods to indicate grouping.  (This all reminds me of conventions used in knitting instructions!)

Another arbitrary mathematics rule is that when a decimal ends in 5, you round up.  Who came up with this rule and why?  A trip to Wikipedia reveals that while this is the most commonly taught algorithm, it isn't the only one in common use and one could argue that it isn't even the best, since there is asymmetry because 0.5 always rounds up, which may have an impact if your data set is very large.   (I did exactly this when I was tutoring a student in Beast Academy and we were discussing rounding.)

I suggest that whenever something seems arbitrary, start with Wikipedia because you can quickly uncover some interesting history.  

[CAJinBE]

Maybe Shorman math would work. At least he would have the videos to watch. Dr. Shormann will also answer questions via email. We like his DIVE Science programs and his Saxon math videos. 

[MamaSprout]

Holt series with Burger. https://www.amazon.com/Holt-Algebra-1-Student-2007/dp/0030358272/ref=sr_1_1?ie=UTF8&qid=1524487731&sr=8-1&keywords=holt+algebra+1+burger

The videos are mostly the same as Thinkwell and do often give a short proof of why something works. The videos are keyed to the books, so it's easiest if you pick up a cheap student copy to be able to look up topics. The problem sets are just okay, but in text examples and videos make a nice supplement to another course.

[Momto6inIN]

We like Video Text because it really explains the "why" very well. Not sure I'd have a student who was done (or almost done) with Algebra I start over with it though.

[MarkT]

On 4/22/2018 at 12:07 PM, Garga said:

I'm helping a homeschooling friend find an algebra curriculum that explains why algebra works the way it does.  Her son is using the McGraw Hill algebra text and feels like it doesn't give him enough background to understand the point to it.

The mom says that in general her son is the kind who likes to know the big picture first, and then learn the details of how it got that way.  He doesn't like learning the little details and having to wait for the big reveal of how it all joins together.  

An example of a why he'd like to know: when he was learning PEMDAS he asked her why it has to work that way.  Why multiplication before addition?  Yes, he can see that it gets a different answer when you do it a different way, but why?  How did the original person who discovered PEMDAS know that it had to work that way?  It sounds like the book just says, "Use PEMDAS.  It works," but didn't tell him why it works.

PEMDAS is just a way to remember the order of operations which is the accepted precedence from long ago.

https://en.wikipedia.org/wiki/Order_of_operations

[kbutton]

On 4/23/2018 at 8:52 AM, MamaSprout said:

Holt series with Burger. https://www.amazon.com/Holt-Algebra-1-Student-2007/dp/0030358272/ref=sr_1_1?ie=UTF8&qid=1524487731&sr=8-1&keywords=holt+algebra+1+burger

The videos are mostly the same as Thinkwell and do often give a short proof of why something works. The videos are keyed to the books, so it's easiest if you pick up a cheap student copy to be able to look up topics. The problem sets are just okay, but in text examples and videos make a nice supplement to another course.

What videos? Does the book have a link to videos online?

[Zoo Keeper]

2 hours ago, kbutton said:

What videos? Does the book have a link to videos online?

The videos are linked to the book on the publisher's site….

[Lori D.]

Math U See? Jacobs Algebra?