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Thoughts on Math Theory/Instruction


stm4him
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Recently I have been trying to decide between Saxon and Strayer-Upton math for our family.  While I waited for my Strayer-Upton copy to arrive, I have been reading about math theory from sources like The Eclectic Manual of Methods and How to Tutor by Samuel Blumenfeld.  I am familiar with Right Start (having used it in the past) and a little bit familiar with Singapore.  I have owned many programs in the past but not used them for very long.

 

There are so many conflicting ideas about math that it is becoming very difficult to decide how to go about teaching it (even after having taught math for the last 10 years or more in classroom and homeschool settings).  In the past I have always liked the idea of Right Start but it was never practical for me to teach once I added in more kiddos.  I even wanted to use it as a supplement but that never happened because Saxon always took so much time.  We have mostly used Saxon Math but sometime in the fall I dropped the teaching of my third-fifth child in math and allowed my oldest two to continue unchecked in their Saxon 5/4 and 6/5 books.  When life became a little more stable I attempted to use Christian Light (which I liked) but decided that it was back to using a Teacher's Manual for several kids and I didn't want that.  Then I used Monarch but it was horrible.  So we went back to Saxon for the oldest two, having them test their way through 5/4 and my oldest began in her lessons today at Lesson 45.  I discovered a few holes where she was not understanding a few concepts so for tomorrow I have assigned for her to go back and work on those.  My second one is still testing his way through the book but I can tell that he will soon have to stop.  This is just to give you an idea of our background and where we are currently.

 

Now on to theory discussion.  One thing that is being debated is word problems.  Some books seem to claim that they are worthless and others that they are vital to making math real.  Samuel Blumenfeld seems to think they are pointless unless they are about a child's allowance or some other very relevant topic.  Others supplement with vintage texts just for the word problems in them.  

 

Another debate is whether it matters if children understand concepts or whether they learn things by rote.  I have a child (my third one) who naturally does in his head what I would have taught him with Math U See or Right Start.  He manipulates numbers in his head in ways that I have never taught him and he understands the relationship between them.  But as much as he understands them, he still can not rattle off his math facts quickly, so I find it cumbersome to work with him at times because he has to think his way through to the answer.  My oldest annoyed me because she still had to do it on her fingers or otherwise counting backwards and such (and probably still does).  They all rely on skip counting despite the fact that they have been doing Saxon Math drills with every lesson for years.  So in a way skip counting has been a crutch, but we actually use it all the time and there are times when I don't know how people could teach number patterns and other things without it.  I definitely think that understanding comes naturally eventually when a child has worked with numbers long enough.  I do think that the process Samuel Blumenfeld takes kids through with his tables really makes sense in aiding their understanding of the relationships between numbers and also in rote memorization.  I don't understand the tables he shows in Step 6, 14, and 39.  The rest have all made sense to me so far.  Hunter, maybe you can explain it to me?

 

The MoM suggested to teach children to visualize numbers instead of counting and Right Start says the same thing.  But other sources discuss the importance of learning to count and to a large degree I think that teaching my son to copy a 100 number chart when he was little helped form his understanding of the relationships between numbers.  Teaching him to recognize two-digit numbers (starting with the chart and moving to number flashcards) helped him with his skip counting.  I have followed a similar path with my fourth one but at a slower pace.  Samuel Blumenfeld talks about the practicality of teaching them to count on their fingers to begin with since it is an obvious built in part of our bodies.  Right Start uses fingers too to teach things like six is five and one.

 

I have always been accustomed to longer math lessons because of using Saxon for so long.  I could see how using something like Strayer-Upton at a normal pace could mean 2-4 pages a day done orally and with a composition book for working out problems.  This would need to be done together I think whereas with something like Saxon you train a child how to learn it by themselves.  On the one hand I see sitting together (each with our own little book and composition book) for a few pages of math a day to be a warm and cozy way to enjoy math.  On the other hand, I think having them sit independently to do their work and study and then just checking it at the end is easier even if it takes THEM longer to do.  Would they have more time to read if math took less time?  Do I want to go back to having to sit one on one for 30 minutes or more per child every day?  Robinson says two hours of math per day.  Can one accomplish just as much in 30 minutes to an hour?  Or does the amount of time equal the amount of proficiency similar to music practice?  

 

Samuel Blumenfeld talks about how arithmetic and mathematics (abstract, theoretical) are two different things and we have moved to a time of mixing the two instead of having solid arithmetic skills before tackling the more abstract mathematics.  This makes sense to me and I definitely think I would have done better in Algebra 1 (and math life skills in general) if more time had been spent on arithmetic and everyday uses of math in the grown up world (balancing a checkbook, taxes, business math, etc.) before working on higher level math.  This is why a program like Strayer-Upton that focuses so heavily on arithmetic appeals to me.  At the same time, I can't think of what one would take out of Saxon Math that doesn't count as arithmetic besides geometry related things.  What about problems with missing addends?  Isn't that beginning algebra?  And yet I have seen that in vintage math books too.

 

Samuel Blumenfeld talks about how it is not necessary to focus on the spelling of number words, but I have also seen this in vintage math texts.  It would seem strange to me if an older child didn't know how to spell numbers correctly.

 

Another debate is whether children care about why we learn something.  Samuel Blumenfeld says they don't as long as it works and is useful to them.  Strayer-Upton says it always explains why something is useful for motivation purposes.

 

Saxon Math says they should always show all their work when working out a problem.  I have never made mine do that (though we aren't really in higher math yet).  MoM says this isn't necessary once they have proven that they know the correct process.  Why can't they just show their work for the ones they've missed?  

 

All these debates are swimming in my head.  There are two things I know.  One is that I want my kids to have a very firm grasp of arithmetic.  Secondly I want them to know their facts by rote regardless of if they understand it naturally or not since I believe that will come one way or another in time.  Maybe Samuel Blumenfeld's tables are the answer since just drill and flashcards alone haven't made them fast enough yet.  But I also think that once they figure out the process of the table they may be able to fill in the answers in the workbook without internalizing it.  My oldest can copy anything without internalizing it!  Maybe she would have to copy and recite?  

 

As you can see, I don't have all the answers to these but they are all questions floating around in my head as to what is the best way.  I have used Saxon for so long because I believe it is thorough, independent in third or fourth grade for the most part, and spiral.  I like the idea that they combine Algebra and Geometry in high school.  And I am sure there are other reasons, too.  I dropped the lower levels because the scripted lessons took too long times the number of young kids I have.  I feel I am at a cross-roads for deciding what I believe about math instruction and how to proceed with my younger children (and maybe what is important to focus on in the remaining years with my oldest two).  I thought starting this discussion might help me think through some of this.  I look forward to hearing your opinions from experience on these different math topics.....

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You asked for opinions...So here is mine, for a few points you raised:

 

Word Problems: I think word problems are vital, because without the ability to apply math to a specific situation, being able to solve prepackaged equations is pointless, since life does not serve up prepackaged equations. The ability to solve a linear equation is not useful unless you can set up that equation for your scenario.

 

Conceptual: I consider conceptual understanding vital, because without, there will be no long term retention. Just to take the example of fractions: a student who does not understand why dividing fractions means multiplying with the inverse will only recall "we have to flip", but will be unable to remember in which situations to do that. A student who does not understand where the quadratic formula comes from will eventually forget, but a student  who understood conceptually will be able to re-derive the formula at any later point.

I have seen plenty of students who learned math by rote memorization and who struggle once they get to college, because they can't remember. Those who understood can reproduce all derivations and do not have to rely on memory.

 

Showing work: Absolutely essential. This trains the student for problems he can not solve mentally. The student WILL have to work on problems that can only be solved by writing out solutions, sometimes very long ones. A student who has not learned how to do this will be unable to document his work in a way he himself can understand and in a way that helps him organize his thoughts so he can solve a hard problem. (And on a practical note, he will need to show work in college because the process is what is evaluated, not merely a final result).

 

I concur that arithmetic skills need to be solidified before algebra. The students I see struggling with higher math usually have not mastered arithmetic with fractions and negative numbers. There is nothing gained by rushing to more abstract concepts without a solid basis.

 

ETA: Just in case my background is relevant: I have a degree in theoretical physics and teach physics at a university. I use math on a daily basis.

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Hmmm...ok

 

1) Word problems.  I think they are vital.  They demonstrate an ability to apply math conceptually.  If a student cannot decipher a word problem, decide which operation to use, the information that is being sought, the information that is being given, ignore impertinent information...than his or her ability to complete the MATH is meaningless.

 

What is math, anyways, without application?  As for whether or not word problems should directly relate to the student...well...as much as possible, sure.  

 

2) Conceptual understanding vs. rote understanding.  Ok...right off the bat, I want to say conceptual understanding is absolutely vital.  And...I believe that it is.  However, there should be a caveat here.  My oldest child has severe dyscalculia.  Math is extremely difficult for her.  Her number sense is very very poor.  At 9 years old, she still struggles with counting, especially backwards.  For her, and other kids that are very weak in math...plug and chug may be the only way for them to actually function mathematically in the real world.  My goals for DD are to get her a point of mathematical understanding, where she can function in a grocery store, manage money, and manage time.  I am finding that, as we move along and she gets older...her ability to actually UNDERSTAND math concepts she has memorized by rote, has actually improved.  Oh...she still struggles, yes.  But I am seeing signs of her actually being able to apply her knowledge.  

 

For most kids...if they do not have conceptual understanding...what exactly is the point of learning it in the first place?  If they have any aspirations at all, of algebra, science, etc...they need to have a conceptual understanding of math.  This is why I support the "common core" math.  Because it does require conceptual understanding.  

 

3) Spelling number words....ummmm...yeah, important.  My husband (dyslexic) cannot spell to save his life and is embarrassed to write checks because he cannot spell number words.  

 

4)  Showing work...generally, yes, I want to see my students' work.  I want to see the process that they use.  Especially if they got the problem incorrect.  My oldest son and I battle on this.  He is a math natural (lucky duck) and doesn't NEED to use bar models, or even to write equations, in order to solve some of his math problems.  I want to see him demonstrate his conceptual understanding of HOW to solve the problem.  I want to see the bar model.  And I want to see all of the intermediate steps.  For example, a problem today required a before and after bar model.  He only gave me the after bar model.  I made him redo it.  He wasn't happy...and I understand why.  But like I explained to him, the PROCESS is more important right now.  As he develops mathematically, he will encounter harder and harder math problems which will eventually require him to utilize those bar models or something similar!  

 

As for solidifying math facts...yes...it really is essential.  Its kind of the bridge that the rest of math stands on.  

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I think that it is essential to have both conceptual understanding and rote memory of math facts. It should go without saying that kids need to have good number sense and grasp the "why" behind what they are learning. Otherwise they won't be able to apply their factual knowledge creatively to any other problem. But, I think its just as important to drill, drill, drill math facts until they are absolutely mastered and can be answered with instant recall. That is the best way to prepare them for algebra and higher level math. I really think that emphasizing only one of these methods to the exclusion of the other will cripple their learning later on.

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Well as a practical thought, there is only so much time in a day. The mother of a large family cannot do multiple math programs to cover all bases unless they are rotated or something. Many moms who are able to do long one on one math lessons or multiple programs have fewer children. What program do you believe covers all of this?

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This might just add to the confusion, but I just read this article called Learning Math Without Fear. The gist of the article is that having number sense is much more important than using rote memorization. They are not advocating that you shouldn't teach math facts, only that it's as important to teach students to enjoy numbers and learn how to manipulate them intuitively.

 

Here is the link:

http://news.stanford.edu/news/2015/january/math-learning-boaler-012915.html

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People with conflicting worldviews are going to have conflicting ideas about math. There is no "right" worldview or "right" way to do math.

 

If you can find a worldview that you can ALLOW yourself to settle into without guilt and fear–one that is practical and doable for YOUR family–then you will have the option to find a math worldview that matches your general worldview. If you haven't settled into a doable general worldview and have a math worldview that conflicts with or is in excess of your already undoable general worldview, you are in for a rough ride.

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My worldview is pretty set. But you'll have to clarify how various worldviews line up with math worldviews because I bet people with very similar worldviews would disagree with each other on math.

Yes, for example, SpyCar and I agree on math but our overall worldview are different.

 

I personally like Singapore, I used a bit of RightStart with my daughter but switched to Singapore. It takes less time than RightStart and combines conceptual teaching with enough hands on practice to solidify learning. The Standards editions are easier to use because the HIGs are much better.

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My worldview is pretty set. But you'll have to clarify how various worldviews line up with math worldviews because I bet people with very similar worldviews would disagree with each other on math.

I don't think most people do have their worldviews set as much as they think they do. I don't mean that as a challenge or an insult or anything negative, but it's something I deeply believe that I am observing and have experienced myself.

 

The people that I have observed who I do believe are dead set in their worldview and have chosen one that is realistic and possible for them are able to choose their math without a lot of insecurity and hopping. For example the ultra-conservative Mennonites. Most of them know what they want to teach for math and how and it lines up tightly with their very rigid worldviews. Moderate Mennonites not so much. Some use CLE and some use all sorts of other stuff. The moderates I have know are often a bit lost in their worldviews.

 

If a daddy is a tradesman or an accountant and super content with his job, neighborhood, peer group and church, he looks through the books, picks a curriculum and wham bam, I've seen the family use it for 20 years and 5 kids. The tradesman or accountant who is not content or questioning what he was raised with and still finding his way is going to struggle more with math choices.

 

I've seen the same thing for college professors that are entirely content within THEIR world. It's a whole different curriculum than the one chosen by the daddy that is a plumber, but it's done with the same confidence. Unless they too are questioning things or not content.

 

Of cource there are exceptions. I'm just saying that those will rigid deadset worldviews that are actually POSSIBLE for THEIR families don't worry about how one author contradicts another. They know what THEY think about math and what they want to use for THEIR children. Often they think EVERYONE should use what they are using and that works so well in their little world with little understanding of the lives and abilities and realities if others. That certainty can cause the uncertain to flounder.

 

I do not think the Mennonite kids should start using what the kids of the scientists use, or vice versa. I don't believe there is a "right" way to do math anymore than there is a "right" literature list for all children.

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Yes, for example, SpyCar and I agree on math but our overall worldview are different.

 

I personally like Singapore, I used a bit of RightStart with my daughter but switched to Singapore. It takes less time than RightStart and combines conceptual teaching with enough hands on practice to solidify learning. The Standards editions are easier to use because the HIGs are much better.

I did NOT say that people using the same math curriculum have the same worldview. :lol: Whoa Nelly! I did NOT say that!

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I'm not disagreeing with you, but I think sometimes they are comfortable only because it is all they know (meaning all that is in their comfort zone based on their personal experience.)  My husband is happy in his career as a realtor and if he were choosing he may have no problems.  I don't know that he is unhappy about the math education he received.  I, on the other hand, was raised in public schools but also went to church 3 times a week and had to wrestle with the conflicting views of the two for years before finding my way.  I am still learning to filter and discern between some of these ideas and even what my church believes vs. what is found in various Christian resources.  I know that I was not happy with math instruction I received even though I went through Math Analysis/Trig in my 11th grade year.  I felt I didn't understand it at all.  The only math I ever liked was 6th grade (lots of coordinate planes graphing with a fun teacher) and Algebra 2 because Algebra finally made sense to me.  But I do enjoy teaching elementary math and I bet I will enjoy higher level math when we get there in our homeschool.  So even though I know I didn't like what I got, I don't know what we even used and I have no idea what the end result of using various curricula are so it is all a certain level of guesswork.  I am working with a struggling learner, a child who was very much like me (who can plug in answers to get good grades but doesn't necessarily understand what he is doing), and a natural math kiddo.  And my fourth is average so far and more inclined to be a natural with design and shapes vs. numbers.  I suspect number 5 will be like number 2 and I have no idea about the youngest two and the one coming up obviously.  

 

I can see the beauty and theory behind all of the different ones and in an ideal world I could use multiple curricula to hit all the points (because I am one to really think EVERYTHING is important).  But I know that is not my reality so I have to use something that works and covers most of what I want in a math curriculum.  To some degree I have to trust those who have come before me and are "math" people such as Leigh Bortins and Art Robinson (and many others I could list).  I like Saxon a lot, but I also know that either I am expecting things to come together that are just a matter or time in their brains, or maybe we are focusing on the wrong things (in other words not narrowly focusing enough on arithmetic).  I definitely don't think that a more conceptual math program is the way to go for us because they take too long to teach and don't sit well with me for different reasons.  But I do still have enormous respect for the ALAbacus and will always own one.  I just wish I had the time to learn to use it well enough to pick it up and teach with it for more than addition and subtraction.  I wish a self-taught Right Start book existed as a supplement.  

 

Gotta go feed my little one.....maybe more later....

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I can see the beauty and theory behind all of the different ones and in an ideal world I could use multiple curricula to hit all the points (because I am one to really think EVERYTHING is important).  But I know that is not my reality so I have to use something that works and covers most of what I want in a math curriculum.  To some degree I have to trust those who have come before me and are "math" people such as Leigh Bortins and Art Robinson (and many others I could list).  

 

Generally, when someone settles completely into a worldview, they no longer think EVERYTHING is important. And they no longer trust that everything an "expert" says will be applicable to them.

 

Some worldviews imply that the whole world should adopt that worldview. Others are just for the ones choosing/able to live by that worldview.

 

Settling into a worldview that includes some boundaries can be peaceful, sometimes not. But it often can make choosing math easier.

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There are so many conflicting ideas about math that it is becoming very difficult to decide how to go about teaching it (even after having taught math for the last 10 years or more in classroom and homeschool settings).  

 

[...]

 

All these debates are swimming in my head.  There are two things I know.  One is that I want my kids to have a very firm grasp of arithmetic.  Secondly I want them to know their facts by rote regardless of if they understand it naturally or not since I believe that will come one way or another in time.  Maybe Samuel Blumenfeld's tables are the answer since just drill and flashcards alone haven't made them fast enough yet.  But I also think that once they figure out the process of the table they may be able to fill in the answers in the workbook without internalizing it.  My oldest can copy anything without internalizing it!  Maybe she would have to copy and recite?  

 

As you can see, I don't have all the answers to these but they are all questions floating around in my head as to what is the best way.  I have used Saxon for so long because I believe it is thorough, independent in third or fourth grade for the most part, and spiral.  I like the idea that they combine Algebra and Geometry in high school.  And I am sure there are other reasons, too.  I dropped the lower levels because the scripted lessons took too long times the number of young kids I have.  I feel I am at a cross-roads for deciding what I believe about math instruction and how to proceed with my younger children (and maybe what is important to focus on in the remaining years with my oldest two).  I thought starting this discussion might help me think through some of this.  I look forward to hearing your opinions from experience on these different math topics.....

 

Quickly, as I am supposed to be doing something else. It has been a while since I have read Lockhart's Lament so I am not going to be able to quote from it verbatim but it was so validating to read it all those years ago just after I had started homeschooling. There should be a free pdf online if you google Lockhart's Lament.

 

Second, I haven't read other replies in detail so apologies if anyone mentioned this. We taught number sense without teaching the numerals or spelling out words first. We taught what "one" means by showing one apple, one orange, one tree etc. We did this very young, not intending to hothouse but because it is very ingrained in my DH and me to teach and share what we love. Similarly, we showed DS what something means before expecting him to understand or memorize it. Then we taught him to look for patterns.

 

I have an only child and it's easier, I agree. Of all the curriculum we used when he was younger, my favorite is MEP because it really develops math understanding from young. It makes you think.

 

Sorry, I have to run but I hope something here helps you.

 

ETA...why I mention Lockhart...he is a mathematician AND a math educator.

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OP, I understand the difficulties of having to school multiple children.  I have four...each of them very different.  Oldest...DD9...has dyscalculia, as I mentioned above.  DS8, is math-gifted.  DS7...well, I *thought* he was going to be average in math, but he has really struggled this year.  With all content areas, actually...and I'm thinking it may be a maturity thing.  It may also be an anxiety thing.  DS6(almost) is only in K and hasn't shown a proclivity towards math-giftedness or math struggles.  I'm hedging my bets that he will be an average math kiddo.  But...who knows!  He's only now starting to get into the basic meat and potatoes of number sense and conceptual understanding of how numbers work.

 

 

Anyways, my point is...my goals for each kiddo are going to be different.  For DD, I need to get her to a point of functionality.  I will never say that a certain math topic is barred for her but....I would be shocked if she ever opens a calculus book, unless its her brothers.  But primary goal #1...function with math in the real world.  Primary goal #2...UNDERSTAND, conceptually, what the numbers mean and how they work together.  We'll go from there.  I use MUS with her because MUS presents a basic math education.  There is very little extra.  Yet, they do have an avenue to discuss conceptual understanding and mental math.  

 

Eldest DS...well...my goals are so very different.  I use Singapore with him.  I use Beast Academy with him.  I use Competitive Math with him.  He will likely end up with AoPS.  At some point, I anticipate I will need to outsource his math instruction because I'm not sure I'll be able to teach him Calculus (despite taking it in college).  

 

As a result of their differences, we have a LOT of math curricula hanging around.  To help with affordability, I reuse as much as I can.  The workbooks...I three hold punch them and the kids use wet erase markers from Vis-A-Vis and dry erase pockets.  This way, I can pass down workbooks.  

 

It DOES consume a lot of time.  To help...we school year round.  Summer school is mostly review, but for DD, because she is "behind" grade level, she just continues all summer with her MUS.  But schooling year round gives us the ability to space out lessons a bit...which helps a lot with the time constraints.  

 

As far as the worldview aspect of this discussion...you know, I hadn't ever really thought about it.  Hunter does have a point with regards to the mennonites.  And I know that was just an example but...we live in an old order mennonite community and I've seen her example first hand.  The families that *were* old order but are struggling with their specific beliefs in regards to that world view...they tend to be more flighty with their curricula choices.  But not always!  In fact...we have an Amish family that attends our church...they recently left an old-order Amish church.  They are still very strict in regards to curricula choices for their homeschool kids.  Maybe that will change, as they are only recently out of their church.  

 

Overall...I'm not entirely sure the relationship between one's allegiance to a worldview and one's allegiance to a math theoretical approach.  Unless we want to consider the personality aspect of holding allegiance to a specific worldview/theoretical approach.  Folks that DO hold tightly to their worldviews...are probably more likely to hold tightly to their theoretical opinions in other areas, too.  Likewise...folks that are a bit more wishy washy on their worldviews...may also be more wishy washy on other theoretical views.  

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The people that I have observed who I do believe are dead set in their worldview and have chosen one that is realistic and possible for them are able to choose their math without a lot of insecurity and hopping. For example the ultra-conservative Mennonites. Most of them know what they want to teach for math and how and it lines up tightly with their very rigid worldviews. Moderate Mennonites not so much. Some use CLE and some use all sorts of other stuff. The moderates I have know are often a bit lost in their worldviews.

 

If a daddy is a tradesman or an accountant and super content with his job, neighborhood, peer group and church, he looks through the books, picks a curriculum and wham bam, I've seen the family use it for 20 years and 5 kids. The tradesman or accountant who is not content or questioning what he was raised with and still finding his way is going to struggle more with math choices.

 

I've seen the same thing for college professors that are entirely content within THEIR world. It's a whole different curriculum than the one chosen by the daddy that is a plumber, but it's done with the same confidence. Unless they too are questioning things or not content.

 

Of cource there are exceptions. I'm just saying that those will rigid deadset worldviews that are actually POSSIBLE for THEIR families don't worry about how one author contradicts another. They know what THEY think about math and what they want to use for THEIR children. Often they think EVERYONE should use what they are using and that works so well in their little world with little understanding of the lives and abilities and realities if others. That certainty can cause the uncertain to flounder.

 

I do not think the Mennonite kids should start using what the kids of the scientists use, or vice versa. I don't believe there is a "right" way to do math anymore than there is a "right" literature list for all children.

 

This perspective seems incredibly limiting to me. Why should it be OK to limit a child to what the parents were able to achieve, instead of enabling the child to go beyond this? Why should a plumber's child receive a less rigorous math education than a scientist's child?

 

While evaluating a literature curriculum may have a lot more room for opinion, the quality of math education can be evaluated much more objectively. This still does not mean that one curriculum fits every student, but mastery of math skills and concepts are specific and can be evaluated objectively. Why settle for less than the student is capable of - irrespective of the family's socioeconomic situation?

 

I find it important that parents being content with their place in life (which is a wonderful thing) does not translate into narrowing the child's life to just this same possibility. One can be content and secure in one's worldview and still search hard for the best educational opportunities for one's child - especially since children are different. The math curriculum that may work fine for one kid may not be appropriate for another sibling. I fail to see how making a quick choice based on "feeling secure" gives the kid the best possible education. How can a gifted sibling and a sibling with dyscalculia be taught with the same curriculum without compromising one or both kids' math education? Where does worldview even come into play when choosing how to best teach mathematics?

(The only way I can see worldview play a role is by deliberately limiting educational goals to less than the student's abilities, for whatever ideological reasons.)

 

But maybe I do not understand what you are trying to say here, Hunter.

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Ok, a couple of points.  

 

Regarding worldview:  I think I do have a pretty set worldview and believe in absolute truth.  I just am self-aware enough to know that I haven't studied out many of my questions so I haven't come to conclusions.  I believe the Bible is the final authority on everything, but I know that even though I attend the church that I most identify with, no denomination could possibly have it all correct in terms of their interpretation and application of the Scriptures.  So, maybe that means in math I do believe there is a RIGHT way, but I don't have enough experience yet to discern what that right way is.  With denominations I can see how they came to the conclusions they come to on different issues and I respect that.  In terms of what I actually follow myself I often look to my husband and my pastor for guidance.  I want to someday study the Bible in its original at which point I may have a little bit more confidence if I believe my husband or pastor has something wrong.  For now I choose to trust (to a degree) in the TEACHINGS of those in authority over me but that doesn't mean that I think their character is infallible.  That is a whole other issue.  But it shows you that where I am doubtful of my own knowledge in an area I look to someone else to follow that seems right to me until I have time to put in the effort to study something out for myself.  I don't trust or follow everything ANYONE says, but I can agree with their general guidance.  And so often in math that general guidance has been Saxon.  I don't follow that blindly but do see the merits and practicality myself after dabbling in many other things.  But that doesn't make it perfect and when confronted with another program that possibly meets my fundamental goals for math I am open to considering it, which is why there are some programs I have zero interest in and know that instantly at this point, and others that have the capability to strike my fancy.  Honestly, I am surprised that S-U did strike my fancy.  I TOLERATED Monarch very briefly because in my freaking out temporarily I was willing to give up my convictions about solid curriculum for something that might actually get done consistently.  But I came to my senses quickly.  CLE I still believe is wonderful but because of my own personal quirkiness I can't do the math and no other part of CLE.  I think the only reason S-U did strike my fancy is that when I fell in love with RLTL and ELTL (because they were the embodiment of what I would have liked to have created myself but knew I never would), it made me think that the coziness of sitting with my child to do math as well might be really enjoyable.  Whether it is practical for us in the long haul remains to be seen.  With Saxon, I was/am buying into the methods over the long-haul as compared with other modern programs.  Bringing in vintage books in comparison opens a whole other realm of comparing and contrasting and different yardsticks even.  That is what sparked all of this.

 

Different programs for different kids:  I just can't do it.  I am comfortable figuring out how to adjust my teaching to fit the child, and I am comfortable adjusting the pace, but for my sanity I need there to be one (or at most one and a supplement) program or progression of programs that I can trust to carry us through and that I can get to know intimately as the teacher as well as confidently recommend to others.

 

Goals:  I am learning to adjust my goals now but it is really hard.  I wanted all my kids to reach the top somehow someway but then I have moments where I would be happy if we reach bare minimum for a regular diploma with my oldest.  I don't know if I can truly move my standard, but I can begin to be realistic about how far we can get towards my standard before they are 18 (or graduate) and then I honestly think I will let go and trust that they will get whatever it is they need from what is available to them at the time.  I will always be here to help them to that end after 18 if they wish, but I will NOT push my standard on them after they graduate.  My parents did that and I hated it.

 

Year Round: We have always schooled year round.  I couldn't do it any other way.

 

 

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  But I do still have enormous respect for the ALAbacus and will always own one.  I just wish I had the time to learn to use it well enough to pick it up and teach with it for more than addition and subtraction.  I wish a self-taught Right Start book existed as a supplement.  

 

Gotta go feed my little one.....maybe more later....

 

The 'Activities for the AlAbacus' book shows how to use it for all the arithmetic functions.....  have you seen it?

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Yes, I actually own it.  I just have never ended up getting through it all before deciding we're spending way too much time on math.  But now that I have dropped Saxon in the younger years there is a slight possibility I would consider picking it up again :-)

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Overall...I'm not entirely sure the relationship between one's allegiance to a worldview and one's allegiance to a math theoretical approach. 

Here's how I think of it:  my approach to math is likely to reflect my overall educational approach, and my educational approach reflects my educational goals; in turn, my educational goals reflect my overall goals in life.  What does it mean to live a good life?  What is my purpose in life?  How do I live a meaningful life?  Those questions are fairly core to a person's worldview; sometimes they are answered by a person's religious beliefs, but there are non-religious answers to those questions, too.

 

Anyway, here's a personal example of how the two intersect.  I used to be in the "math for math's sake" camp - education is about seeking the true, the good, and the beautiful, and so math should be presented and taught so that the inherent truth/goodness/beauty of math is brought out, and so I only looked to educators/mathematicians/curricula that were in line with this view, which vastly simplified things.  Concepts before procedures; understanding before memorizing, with memorization primarily in context; proofs, not calculations, as the focus of math; focus on flexible thinking and seeing multiple ways to solve a given problem - all answered because of the choice of approach.  And I would be equipped to modify programs to fit kids, because I knew what my ultimate goal was.

 

However, I recently learned that my religious beliefs and my educational beliefs (re: education being about seeking the true, good, and beautiful) may actually be at odds.  Because the idea that education should be about seeking the true/good/beautiful has as its basis that belief that the highest aspiration in *life* is the contemplation of the true/good/beautiful.  And my religious tradition, as it turns out, does *not* consider contemplation of the divine as the highest human aspiration, or as the life God means us to live.  (In fact, I recently realized that I actually have *no idea* what my religious tradition teaches about the highest goal of life, or how our beliefs impact daily life, or anything of that nature.  Which is why I was borrowing from other, apparently contradictory, traditions.  The funny thing is that *all* the traditions in question are rooted in the classical tradition :lol:.)

 

Anyway, a re-think in my approach to the "Big Questions" of life had direct impact on my theoretical approach to math.  "Math for math's sake" is no longer an option (and actually, after many years of equating them, it turns out that "math for math's sake" is at odds with "seeking truth/goodness/beauty", something else I didn't know till recently), and I haven't replaced it with anything yet, because the answers to the big questions are still in flux.  And it is totally unsettling.  Because right now I'm just doing math because it's probably necessary no matter what ;), but I'm not aiming at anything in particular (other than mastering the content in the books we have), because I no longer know what to aim for - I'm just keeping the status quo until I know where to go next. 

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Believe me....it was extremely difficult to get to a place with DD where I acknowledged that my math goals for her were going to be very different.  Especially with her being my oldest...and my attitude towards schooling being more "gung-ho" if you will...I expected ALL of my children to finish high school, attend college, get a degree, etc.  Algebra, Trig, Geometry, Calculus...yeah...that was pretty much all a part of my expectation.  

 

DD really helped to open my eyes in a lot of different ways.  Again, I'll never say never but...it would be an amazing miracle if she were able to progress through math past Algebra.  Honesty?  I don't even feel positive about Algebra for her.  Right now, I just want her to be able to add and subtract and count backwards.  And count money.  And tell time.  

 

But with DDs struggles came another realization to me.  My kids don't HAVE to go to college.  Why do they have to?  (I'm not saying you implied that they did...this was just a realization to me over the last couple of years).  

 

We live in a rural farming community.  There are folks here that are "uneducated".  And by uneducated I mean...they never got more than a 9th grade education.  But they are VERY educated in how to make a living.  They have trades.  They farm.  They have a savvy business sense and can build a dairy farm from the ground up.  There are electricians.  There are pipefitters.  There are highway mechanics.  None of them are wealthy.  But they are all comfortable.  

 

I started to think about how I viewed education when I was a teen.  That I HAD to go to college so I could get an education, a degree, and a well-paying job.  And here I am..."educated", with a degree, and making absolutely no income at all.  I don't rue the education I earned.  But my point is that...my motivations and expectations of education were skewed...and remained so until I began to think about my daughter's future. 

 

Then...add in that I'm here now in my mid 30's and...many many of my high school friends are returning back to college because...the education they received right out of high school turned out to not be what they wanted to do for the rest of their lives.  And I think that's pretty common...most folks will have more than one career.  

 

I think, I'm going to strongly encourage all four of mine to seek a trade while in their mid to late teens.  Something that they can get licensed for, certified, etc. within two years and then work this career for awhile.  Experience the real world.  And then, once they have some real world experience, if they wish to go on to college...then by all means. 

 

But at least, by then, they'll have an established career/licensing/credentials they can always fall back on.  

 

That's my theory anyways...we'll see how it all pans out, lol.  

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I have known for quite some time that I don't expect them to go to college.  In fact, I don't even WANT them to unless they KNOW why they are there.  In my case I did happen to know exactly why I was there and had since I was probably in preschool.  But when I wanted to change my major to drop my certification and just get a minor in psychology so I could stay home with my baby my parents flipped out because I had a full scholarship and in their minds HAD to finish as planned.  I almost felt I wouldn't be worthy as their daughter unless I did.  My sister went to college with no idea why she was going and ended up flunking out and taking FOREVER to finally finish her associates.  She still has this deep need to finish her four year degree and get a diploma even though she probably still doesn't know what exactly she wants to study or do with it.  Somehow it validates her worth.  My husband went to college and got a psychology degree while selling books door to door in the summers to pay for it.  He ended up going back to school later to get his teacher's license.  He taught for a few years and then ended up writing for a think tank and eventually landed in real estate which he loves and has stuck with for over ten years.  Ironically, there were signs from his childhood that that is what he would end up doing, but he just didn't see them until he was a mature adult.  He has no intention of changing careers.  And he didn't need his degree or his certification to do what he does now.  And we have all this stupid debt from my living expenses during college and his student loans for the teacher certificate.  And I don't make an income either just like you.  It is all kind of funny.  But I have always been someone who thinks that if my children do go to college they need to have a way to support themselves (a trade or real estate license under their daddy's guidance, or something of that nature) and that even if they are a plumber they should learn calculus :-)  Now I'm realizing that they don't have to know calculus to be well educated or see the beauty of math :-)  But I would still love it if they did :-)

 

On a completely different note, one thing that I just thought about is why I burnt out.  I freaked out because I wasn't able to do what I had planned to do with them (due to health issues) so I had these self-inflicted obligations (in all subjects, including math) in my mind and I was failing.  But I couldn't let go of my ideal.  When I did feel well the last thing I wanted to do was pick up scripted lessons and worksheets for four children regardless of the fact that I believe it is a great program and was producing good results.  Now I feel that math doesn't have to be obligatory and something I do as a duty, but can be ENJOYED together.  Maybe I can look forward to our math lessons together.  Maybe a quaint little vintage math text is the tool for that.  Maybe that is more important than whether it perfectly fits my goals for math (I'm not saying it doesn't, I just don't know yet).  Also, it could be that after enjoying our quaint math book for a month or so I decide that whether or not I enjoy it it either isn't getting done consistently or isn't meeting other goals that were being met with Saxon and may want to go back.  

 

I guess I found it neat that the vintage texts were debating the same exact things that I find debated between modern programs.  I thought I had already made my decisions about these things, if not in theory, in practice by solidly choosing Saxon math.  But it turns out that I can't really get away from the theory part altogether because they have existed since at least the 1800s but probably before.  And just because I choose a program doesn't mean I can't add in other things (not necessarily with another program but just as we go about naturally from my own store of knowledge and math theory the way I do with language arts no matter what I use) or approach some parts of a curriculum in a different or multi-faceted way.  But I can't decide whether or not to do that until I solidly figure out what I believe in these different areas.  And sometimes I won't know what I believe until I have tried it out on a few kids and seen for myself whether it made any different in their understanding or progress or if it was really just a waste of time.  I often figure out my opinion by listening to other's opinions and seeing how it resonates with me (or not).  So I appreciate hearing all the ideas on here.  

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When I switched to Singapore, it made a huge difference in my daughter's understanding of math, and also my ability to teach elementary math. Reading Liping Ma's Knowing and Teaching Elementary math was also helpful.

 

I worked as a statistician and took math through Differential Equations, including 3 semesters of college calculus, upper level stats, and Linear Algebra. I tutored Algebra and Trig but did not understand how to teach elementary math well until reading Liping Ma's book and using Singapore.

 

I can teach Algebra and Pre-Algebra well, and I know much of the background for why the math works. But, owning and reading through and teaching from the early Dolciani books has made me better understand the unity and foundation behind Algebra. (I have not updated my signature yet, my daughter has finished Dolciani Pre-Algebra and we are now using Dolciani Algebra.)

 

Another good book is "Mathematics is God Silent."

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This perspective seems incredibly limiting to me. Why should it be OK to limit a child to what the parents were able to achieve, instead of enabling the child to go beyond this? Why should a plumber's child receive a less rigorous math education than a scientist's child?

 

While evaluating a literature curriculum may have a lot more room for opinion, the quality of math education can be evaluated much more objectively. This still does not mean that one curriculum fits every student, but mastery of math skills and concepts are specific and can be evaluated objectively. Why settle for less than the student is capable of - irrespective of the family's socioeconomic situation?

 

I find it important that parents being content with their place in life (which is a wonderful thing) does not translate into narrowing the child's life to just this same possibility. One can be content and secure in one's worldview and still search hard for the best educational opportunities for one's child - especially since children are different. The math curriculum that may work fine for one kid may not be appropriate for another sibling. I fail to see how making a quick choice based on "feeling secure" gives the kid the best possible education. How can a gifted sibling and a sibling with dyscalculia be taught with the same curriculum without compromising one or both kids' math education? Where does worldview even come into play when choosing how to best teach mathematics?

(The only way I can see worldview play a role is by deliberately limiting educational goals to less than the student's abilities, for whatever ideological reasons.)

 

But maybe I do not understand what you are trying to say here, Hunter.

 

I am trying not to judge, but rather to state what I have observed. I have observed that people who are deeply rooted in a world view and are content with their lives seem to choose math quickly and permanently.

 

I have observed that some people's worldviews include judging the worldviews of others as inferior. And often therefore thinking that the math choices based on those inferior worldviews are inferior math choices.

 

These statements I am making are an attempt to state observations with as little judgement as possible, even when attempting to answer questions about judgement. :lol:

 

Why settle for less than a student is capable of? The answer to that would often be deeply rooted in a person's worldview and general lifestyle choices, wouldn't it? 

 

When Early Americans chose to settle the frontier, parents made a choice that usually severely limited their children's access to many things, including academics. Were all frontier parent negligent? Were math programs designed to deal with the realities of the frontier lifestyle inferior to the ones designed to be used in private academies in the Eastern cities? Even if the frontier friendly programs were inferior, was their use justified and sensible?

 

Should what the "student is capable of" be the primary consideration in choosing math? Always? Is that a worldview? Is that worldview superior to other worldviews?

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Ok, a couple of points.  

 

Regarding worldview:  I think I do have a pretty set worldview and believe in absolute truth.  I just am self-aware enough to know that I haven't studied out many of my questions so I haven't come to conclusions.  I believe the Bible is the final authority on everything, but I know that even though I attend the church that I most identify with, no denomination could possibly have it all correct in terms of their interpretation and application of the Scriptures.  So, maybe that means in math I do believe there is a RIGHT way, but I don't have enough experience yet to discern what that right way is.  With denominations I can see how they came to the conclusions they come to on different issues and I respect that.  In terms of what I actually follow myself I often look to my husband and my pastor for guidance.  I want to someday study the Bible in its original at which point I may have a little bit more confidence if I believe my husband or pastor has something wrong.  For now I choose to trust (to a degree) in the TEACHINGS of those in authority over me but that doesn't mean that I think their character is infallible.  That is a whole other issue.  But it shows you that where I am doubtful of my own knowledge in an area I look to someone else to follow that seems right to me until I have time to put in the effort to study something out for myself.  I don't trust or follow everything ANYONE says, but I can agree with their general guidance.  And so often in math that general guidance has been Saxon.  I don't follow that blindly but do see the merits and practicality myself after dabbling in many other things.  But that doesn't make it perfect and when confronted with another program that possibly meets my fundamental goals for math I am open to considering it, which is why there are some programs I have zero interest in and know that instantly at this point, and others that have the capability to strike my fancy.  Honestly, I am surprised that S-U did strike my fancy.  I TOLERATED Monarch very briefly because in my freaking out temporarily I was willing to give up my convictions about solid curriculum for something that might actually get done consistently.  But I came to my senses quickly.  CLE I still believe is wonderful but because of my own personal quirkiness I can't do the math and no other part of CLE.  I think the only reason S-U did strike my fancy is that when I fell in love with RLTL and ELTL (because they were the embodiment of what I would have liked to have created myself but knew I never would), it made me think that the coziness of sitting with my child to do math as well might be really enjoyable.  Whether it is practical for us in the long haul remains to be seen.  With Saxon, I was/am buying into the methods over the long-haul as compared with other modern programs.  Bringing in vintage books in comparison opens a whole other realm of comparing and contrasting and different yardsticks even.  That is what sparked all of this.

 

Different programs for different kids:  I just can't do it.  I am comfortable figuring out how to adjust my teaching to fit the child, and I am comfortable adjusting the pace, but for my sanity I need there to be one (or at most one and a supplement) program or progression of programs that I can trust to carry us through and that I can get to know intimately as the teacher as well as confidently recommend to others.

 

Goals:  I am learning to adjust my goals now but it is really hard.  I wanted all my kids to reach the top somehow someway but then I have moments where I would be happy if we reach bare minimum for a regular diploma with my oldest.  I don't know if I can truly move my standard, but I can begin to be realistic about how far we can get towards my standard before they are 18 (or graduate) and then I honestly think I will let go and trust that they will get whatever it is they need from what is available to them at the time.  I will always be here to help them to that end after 18 if they wish, but I will NOT push my standard on them after they graduate.  My parents did that and I hated it.

 

Year Round: We have always schooled year round.  I couldn't do it any other way.

 

Religion is often the beginning of a worldview, but some religions leave much still left to decide. The religious aspect can be distracting from what is still left undecided. Some religions are all about the neon colored frosting, but there is no cake never mind meat under the DayGlo. Most of the time a firm worldview does quickly answer the little daily questions quickly and consistently.

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It fascinates me that of the scientists and mathematicians it seems like half fall in the Saxon camp and half in the more Asian versions of math instruction.  It is almost like the Traditional Phonics vs. O-G/Spalding Phonics.  I see the difficulties of O-G/Spalding, but no matter what when I try to teach Traditional programs I always end up inserting Spalding because it is what makes the most sense to me even though it isn't how I was taught (actually, I wasn't taught any phonics so neither relates to my background.)  If I didn't have a curriculum, I would likely teach basic arithmetic the way I learned it and wouldn't know how to teach in an Asian style without a program in front of me and I guess I didn't spend enough time with any of those kinds of programs to internalize them to the extent I did with Spalding/O-G.  Some people may see merits in O-G/Spalding but feel more comfortable teaching Traditional phonics so they go that route.  I believe in acquiring knowledge through memorization and then layering conceptual or deeper understanding on top of that through reading and experiences.  I believe that across the board; therefore, a program with more drill and a spiral approach makes the most sense to me, though I see benefits of the other way, too.  

 

I don't think anyone is against conceptual understanding, but the question is WHEN to teach it.  Is it necessary to teach it from the beginning, or can it be explained in the dialectic years when they maybe care more or are ready to comprehend the WHY of things.  Of course, if a child WANTS to know why earlier than the dialectic years no one would say NOT to tell them (which of course implies that the teacher knows the answer to this).  But is it worth spending more time on than the memorization and practice?  I don't think so, personally.  That being said, I don't know that the WHY of the basics was EVER explained to me in school.  I would have appreciated a teacher going back to the basics and laying out the why's before trying to teach me abstract concepts.  So maybe I think that the WHY of the basics is worth learning, but may be more appropriately taught when children are a little older.  I think I am admiring How to Tutor arithmetic at the moment because it teaches the WHY in a way that simultaneously emphasizes drill.  THAT I could probably buy into even in the younger years.

 

That's all for now....

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However, I recently learned that my religious beliefs and my educational beliefs (re: education being about seeking the true, good, and beautiful) may actually be at odds.  

 

I recently realized that my educational beliefs were at odds with my general lifestyle beliefs and choices. I'm finally resolving this and I'm finding greater peace and clarity.

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I don't think anyone is against conceptual understanding, but the question is WHEN to teach it.  Is it necessary to teach it from the beginning, or can it be explained in the dialectic years when they maybe care more or are ready to comprehend the WHY of things.  Of course, if a child WANTS to know why earlier than the dialectic years no one would say NOT to tell them (which of course implies that the teacher knows the answer to this).  But is it worth spending more time on than the memorization and practice?  I don't think so, personally.  That being said, I don't know that the WHY of the basics was EVER explained to me in school.  I would have appreciated a teacher going back to the basics and laying out the why's before trying to teach me abstract concepts.  So maybe I think that the WHY of the basics is worth learning, but may be more appropriately taught when children are a little older.  I think I am admiring How to Tutor arithmetic at the moment because it teaches the WHY in a way that simultaneously emphasizes drill.  THAT I could probably buy into even in the younger years.

 

That's all for now....

 

I can't really imagine teaching kids a mathematical tool without also teaching them the concept.  And in any of the good math programs, unless your child has learning difficulties, the concepts are meant to be at a level the child can understand.  If that isn't the case, I think it is a bad program, or at least not a good fit for the child.

 

If somthing is really too abstract, I would tend to wait to teach it. 

 

That being said - I think that it is common for all students in all subjects that sometimes there will be ideas they don't grasp as well as they might, and they may get filled in later.  But it really shouldn't be too many.

 

There also seem to be some children who really struggle with memorizing math facts - I always did, and still do have a very hard time memorizing anything that involves what seem like radom sequences of numbers, or numbers and letters .  I think it is best to find solutions to help that if possible - as an adult I learned to create narratives for them - but sometimes you may have to find other ways to help the student work quickly without memorizing eveything.  I've read some things that suggested to me that this may relate to the way some brains process information, and is just normal in a percentage of teh population. 

 

I think word problems are important.  In m experience, students who cannot figure out how to deal with them don't really understand the math.

 

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Another thought I had last night was that with spelling and reading I learned the Spalding method and now I feel comfortable teaching reading and spelling without a curriculum if I had to. You study the method and then you know how to teach the subject. I think I could do the same with writing. But math seems so different. I want something that teaches me a method I can apply even if I don't have a curriculum, but I haven't found anything like that for math. I want to be able to teach with a stick in the sand and I am nowhere near that with math after ten years of teaching...

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My opinion:

The best math program is the one you teach the best. 

 

Whether that be because you can teach it without much prep work and respond to questions on the fly, or because you are very enthusiastic about the method, or because it's the way you learned it... it is the TEACHER, not the book that will matter most in student's understanding. 

 

This doesn't mean you must teach the way you were taught.  It may mean you are very enthusiastic to learn a new method, and so teach it very well and with lots of energy.  Or perhaps you have a large family, and so the program you teach best is the one that is written directly to the student, so that you can serve in tutor role and make sure each child is making small, consistant steps, every single day. 

 

If you choose a program, and do it consistantly and well day in and day out, if you are competant to answer questions asked or find the answers elsewhere... then it should "work".  Great mathematicians were not made by a particular textbook or method.  They have sprung up from many different backgrounds because they had some gift for the field.  Competant math users (the minimum goal) is about using the tools consistantly enough to master them and retain them. 

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Another thought I had last night was that with spelling and reading I learned the Spalding method and now I feel comfortable teaching reading and spelling without a curriculum if I had to. You study the method and then you know how to teach the subject. I think I could do the same with writing. But math seems so different. I want something that teaches me a method I can apply even if I don't have a curriculum, but I haven't found anything like that for math. I want to be able to teach with a stick in the sand and I am nowhere near that with math after ten years of teaching...

 

The problem with math is that not only do you need a method, but you need large numbers of problems for all concepts and difficulties... and to make THOSE is a PITA.

When I started homeschooling, it was initially only for a few months to bridge a certain situation, and I thought that, since I have plenty of math expertise, I should be able to teach without a curriculum. I found out quickly, however, that it takes an insane amount of time to design a sufficient number of good practice problems that illustrate precisely the concept I intend to teach. Having subject expertise and knowing how to teach math did not suffice - I had to make problems. I tried to resort to online worksheets, but again, you have to wade through and pick exactly what you need...

I finally came to the conclusion that attempting to teach math without a curriculum is reinventing the wheel and an utter waste of my time.

 

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I don't actually want to do it.  I just want the confidence that I COULD.  I'm sure that really only comes from studying higher level math for many years, but I bet there is something I could learn about methods in general (like the tables that HTT has or how to work an abacus all the way) that would be helpful.  

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I don't actually want to do it.  I just want the confidence that I COULD.  I'm sure that really only comes from studying higher level math for many years, but I bet there is something I could learn about methods in general (like the tables that HTT has or how to work an abacus all the way) that would be helpful.  

 

Well, I have studied higher math and and I have complete confidence that I could not teach without a curriculum because it would literally be a full time job to create a sufficient number of good problems.

 

For my job, I make physics problems for practice sheets and exams and know how immensely time consuming it is to make a single good worksheet.

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I finally came to the conclusion that attempting to teach math without a curriculum is reinventing the wheel and an utter waste of my time.

 

The more I realize about the theory of designing good problems, the more I realize this is true.

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The more I realize about the theory of designing good problems, the more I realize this is true.

Also true for other subjects. It has taken years to get a good set of remedial phonics lessons for adults and older children, I adapt them with each student as things crop up. I theoretically could teach beginning phonics on my own, but why reinvent the wheel, I used only time tested methods with my children, programs that had been revised over the years and were designed for a young beginning student.

 

The more students you use a program with, the more its flaws become apparent. For example, Alphaphonics is good for most young students but for some beginners and a large percentage of older remedial students, the word family teaching leads to guessing.

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The more students you use a program with, the more its flaws become apparent. 

 

:iagree:   :sad:

 

Starting over with something else doesn't produce better results, though. I try and just think that Adam had his weeds and we have flawed curricula. I am starting to believe that there is a bigger force preventing anything less flawed from being published. Or is man really THIS stupid?

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:iagree:   :sad:

 

Starting over with something else doesn't produce better results, though. I try and just think that Adam had his weeds and we have flawed curricula. I am starting to believe that there is a bigger force preventing anything less flawed from being published. Or is man really THIS stupid?

 

 

I don't think it's necessarily that the curriculum is flawed.  Perhaps, it is that the curriculum is just wrong for that particular student?  Singapore was a perfect fit for eldest DS, and I was pretty convinced that it would be an appropriate curriculum choice for just about any other average to above average math student.  But it (so far) is not a good match for middle DS.  Its "flaws" are more apparent with him, than they were with DS. 

 

The thing is...the aspects of Singapore that are "flaws" for middle DS...were positives for eldest DS.  

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stm4him, do you own "A Guide to American Christian Education for the Home School"? It has a great outline of arithmetic.

http://www.amazon.com/Guide-American-Christian-education-school/dp/0961620110

 

Arithmetic Made Simple was the book I first used afterschooling my boys. I'm really glad this was before TWTM forum, because there was no one to tell me that I couldn't learn and teach arithmetic, or that grammar stage math was so much harder than what was in this book.

http://www.amazon.com/Arithmetic-Made-Simple-Robert-Belge/dp/0385239386

I used little more than this book and Saxon Algebra 1 and after just a few months of homeschooling my little guy went from barely above average test scores to earning the highest test scores for his grade in a town of 50, 000 people. I'm really glad I was as fully naive as I was. 

 

When you tackled phonics, you didn't try and tackle all of language arts. Try taking on just arithmetic, not all of mathematics, for now.

 

With my youngest, I got it into my head that I needed to give him a "complete" math education, because you know, he was proving to be some sort of gifted. I did my research and collected a list of all the math topics and started tracking down resources for them all. I made sure to cover topology and since we were already studying Greek of course we needed to do some of our math in Greek. This website is still up after all these years. It really was a waste of time and money. Sure it had SOME merit, BUT it took away from other things.

http://mysite.du.edu/~etuttle/classics/nugreek/contents.htm#conts

 

The 3R's are reading, writing and arithmetic; not language arts and mathematics. There is a lot of language arts and mathematics that is NOT the 3R's.

 

I do believe that you can get a bit "stick in the dirt" with arithmetic, to the level that you have with phonics. GACE (link above) and HTT are a place to start.

 

Math on the Level is expensive and bulky. You need to juggle several books with MOTL like you do with Ray's, but like Ray's it's not really that much bulk in total. I really wish MOTL came as an eBook, but I doubt it ever will.

 

No, I don't want to TOTALLY design my own 3R's curricula, but I don't buy it that it's THAT complicated to learn and do that. Design my own "complete" language arts and mathematics curricula all the way through PhD? That is a whole other story that really doesn't fully apply to basic 3R's.

 

The 1980's and 1990's was full of negligent and naive homeschooling moms whose kids just left the PS kids in the dust, despite or maybe BECAUSE we didn't know what the experts knew. And yes, we designed our own problem sets and pulled them from cheap workbooks. I didn't own a single math textbook before Saxon Algebra 1, until I found a junior college remedial basic math text years later at a yard sale that my older son used a bit.

 

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"Math for math's sake" is no longer an option (and actually, after many years of equating them, it turns out that "math for math's sake" is at odds with "seeking truth/goodness/beauty", something else I didn't know till recently)

 

I would love it if you could explain this some more... pretty please?

I'll try :). Can't find the original source :(, but a series of googlings and memory jogging has (hopefully) enabled me to say something intelligible ;).

 

The original context was about "art for art's sake", but I believe it applies to "math for math's sake", or any sort of "learning for learning's sake" or "creating for creating's sake". "Art for art's sake" says that the highest, purest motivation for making art is for the sake of art itself. Art itself becomes its own end; it no longer has a "self-transcending purpose". Pure art only has meaning as art; it does not have inherent meaning in the world outside itself.

 

But the point of seeking truth, beauty and goodness is because of the belief that those things embody transcendent truth. Art is not good in and of itself, but by whether it conforms to, reflects, transcendent truth. But art for art's sake rejects the idea that artists should conform to anything outside the demands of art itself - embodying transcendent verities is no longer seen as the purpose of art.

 

In math, my understanding is that in Ancient Greece, the first principles, the axioms, of math were meant to be fundamental truths, and finding the best and purest axiom system meant finding the one that best reflected transcendent truth. But from what I understand in modern math, the best axiom system is the smallest, most conceptually elegant one, and it makes no difference whether those "conceptually elegant" axioms reflect "real world" truth or not - it's rather beside the point. "Math for math's sake" has math as its own end, judging mathematical truth and beauty apart from any universal sense of transcendent truth (or the idea that there's any inherent moral qualities to math itself, whereas "seeking truth/goodness/beauty" believes all those qualities are unified in everything (goodness meaning moral goodness), and educators should preserve that unity in how they teach).

 

Does that make any sense? (Hopefully I didn't make too many gross math, history, or philosophy errors ;).)

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I think I do own it but it is in a box in the garage. I will check out the other book. They also sell Mathematics Made Simple and other titles for philosophy, biology, chemistry, physics, and spelling. Now I want to get them all... Lol.

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In math, my understanding is that in Ancient Greece, the first principles, the axioms, of math were meant to be fundamental truths, and finding the best and purest axiom system meant finding the one that best reflected transcendent truth. But from what I understand in modern math, the best axiom system is the smallest, most conceptually elegant one, and it makes no difference whether those "conceptually elegant" axioms reflect "real world" truth or not - it's rather beside the point. "Math for math's sake" has math as its own end, judging mathematical truth and beauty apart from any universal sense of transcendent truth (or the idea that there's any inherent moral qualities to math itself, whereas "seeking truth/goodness/beauty" believes all those qualities are unified in everything (goodness meaning moral goodness), and educators should preserve that unity in how they teach).

 

Does that make any sense? (Hopefully I didn't make too many gross math, history, or philosophy errors ;).)

 

No, the above makes no sense to me.

What is "real world truth" supposed to mean?

Any "conceptually elegant" axiom does, of course, represent a true statement.

 

Whether a mathematical entity even exists in "real world" is a question for philosophical debate.. one can argue than only positive integers "really exist", and that everything else is a creation of the human intellect. Functions, complex numbers, etc can be used to model reality - but one could argue they do not, in fact, exist in the "real world". But they are immensely useful to figure out things about the "real world".

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No, the above makes no sense to me.

What is "real world truth" supposed to mean?

Any "conceptually elegant" axiom does, of course, represent a true statement.

Whether a mathematical entity even exists in "real world" is a question for philosophical debate.. one can argue than only positive integers "really exist", and that everything else is a creation of the human intellect. Functions, complex numbers, etc can be used to model reality - but one could argue they do not, in fact "exist in reality".

Yes, that was a sloppy phrase. I think my point is that modern math separates, or allows for the separation of, the doing of math from the "philosophical debate of whether a mathematical entity even exists in the 'real world'". I mean, currently, what a mathematician believes about the metaphysical connection of mathematical truth to the rest of the universe is seen as irrelevant to the mathematics he does, yes? Two totally separate issues, and work done on one does not materially impact work done on the other; or at least the metaphysical question does not impinge on the math itself, though math itself might impinge on the metaphysical question. Is that more accurate?
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Yes, that was a sloppy phrase. I think my point is that "math for math's sake" separates the doing of math from the "philosophical debate of whether a mathematical entity even exists in the 'real world'". I mean, currently, what a mathematician believes about the metaphysical connection of mathematical truth to the rest of the universe is seen as irrelevant to the mathematics he does, yes? Two totally separate issues, and work done on one does not materially impact work done on the other. Is that more accurate?

I am not a mathematician - but I do not think it is the mathematician's job to ponder any metaphysical connection, but to research, and prove, the mathematical relationships and patterns within the logical system. That's what mathematicians do.

 

Other disciplines ( and the applied math people) concern themselves with the application of mathematics in the modeling of real systems. Physicists invented a lot of mathematical constructions (which the mathematicians then had to go and clean up and prove rigorously). This still does not mean that the mathematical entities themselves exist "in real world". That would be a question for math philosophers, and the answer is purely academic and not very relevant for anything but philosophical debate

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I am not a mathematician - but I do not think it is the mathematician's job to ponder any metaphysical connection, but to research, and prove, the mathematical relationships and patterns within the logical system. That's what mathematicians do.

 

Other disciplines ( and the applied math people) concern themselves with the application of mathematics in the modeling of real systems. Physicists invented a lot of mathematical constructions (which the mathematicians then had to go and clean up and prove rigorously). This still does not mean that the mathematical entities themselves exist "in real world". That would be a question for math philosophers, and the answer is purely academic and not very relevant for anything but philosophical debate

Thank you for clarifying :).

 

My overall point stands, I think - there is indeed a difference between that view of math and a "seeking truth/goodness/beauty" view of math, and it involves whether it is good or not to separate the philosophical questions of math from the discipline or practice of math.

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My overall point stands, I think - there is indeed a difference between that view of math and a "seeking truth/goodness/beauty" view of math, and it involves whether it is good or not to separate the philosophical questions of math from the discipline or practice of math.

 

Yes, that may be - but I still do not see how that would affect teaching math in any way. Any debate about the philosophical implications of math can only come after deep study of math... before the math is understood, pondering metaphysics has no basis.

 

The ability of math to model and describe a "real" situation and the intrinsic beauty of the mathematical theory are enough reason to teach mathematics, and I fail to see how math would be taught differently based on different philosophical attitudes. The derivative of a function is defined, calculated, and interpreted, in a very specific way - irrespective of whether one views a function as something "existing in the real world" or an invention of the human mind. I simply don't think it makes one iota of a difference for doing math.

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Yes, that may be - but I still do not see how that would affect teaching math in any way. Any debate about the philosophical implications of math can only come after deep study of math... before the math is understood, pondering metaphysics has no basis.

 

The ability of math to model and describe a "real" situation and the intrinsic beauty of the mathematical theory are enough reason to teach mathematics, and I fail to see how math would be taught differently based on different philosophical attitudes. The derivative of a function is defined, calculated, and interpreted, in a very specific way - irrespective of whether one views a function as something "existing in the real world" or an invention of the human mind. I simply don't think it makes one iota of a difference for doing math.

I am so woefully underequipped, but I'll take a stab at explaining (just explaining, not defending or trying to persuade; iow I'm just trying to explain the what without much in the way of the why):

 

If I am understanding you correctly, the natural world works as it works, whatever the answers to the Big Questions of life. No matter what the answer to the Big Questions, nothing about the natural world would be remotely different. So how in the world do answers to the Big Questions have anything to do with math, or teaching math? Because the Big Questions have nothing to do with the natural world, they can have nothing to do with learning about the natural world, or teaching about the natural world, so they can have nothing to say about math, or the teaching of math.

 

The thing is, the idea that the natural world is separable from the philosophical/metaphysical/supernatural realm - that's a relatively recent idea, and is far from universal. It's actually quintessentially modern; the physical and metaphysical were united in pre-modern thought, and post-modern thought is trying to figure out ways to re-unify them (albeit in very different ways from pre-modern thought). I can't explain the whys or wherefores of why people reject the strict separation of the physical and metaphysical here, not so it would help, but for the purposes of the discussion, accept that such people exist (and generally includes those who are seeking truth/goodness/beauty), people who think the answers to the Big Questions of life are inseparable from the nuts and bolts of *living* that life. (And historically there's nothing weird at all about that - modernism's separation is the outlier.)

 

In which case, for them, *everything* about life is impacted and informed by those Big Questions. And part of the everything is education, and part of that is math. It's not so much needing to have detailed answers to abstract math philosophy issues to teach math, but that one's overall philosophy of life narrows the scope of what schools of math philosophy are compatible and which ones aren't, *and* that philosophical questions and their answers are seen as explaining the nature of the natural world, too. In this view *all* ways of teaching math have some underlying assumption about the philosophical nature of math (and wrt teaching, additional assumptions about the nature of man, and the purposes of education), because it's unavoidable (physical and metaphysical truth are intertwined, not separate) - and you want to find the ways that are compatible with your views.

 

ETA: I think it's a common belief today that for a given set of answers to the Big Questions to be valid, nothing in them can go against scientific fact.  Historically, that went the other way, too - physical truth and philosophical truth needed to be in harmony, and just as physical truths could point out philosophical falsities, philosophical truths in turn could point out physical falsities.  For them, just as the natural world limits what can be philosophically true, the philosophical realm limits what can be physically true.  Sounds weird to modern ears, but it's a basis for thinking the Big Questions have direct impact on something as apparently mundane as teaching math.

 

ETA2:  And really, modernism's position that philosophical truths don't impact the natural world is a *philosophical* position ;), one that does indeed limit what can be physically true. 

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