We Can Do by Moshe Kai with guest Robert Levy discussing Saxon Math.

[Mandabplus3]

Alex might not be ready for an hour of sit down study but I don't see that as too much of a barrier to her starting Saxon. She is ready ( or very close to) so I was thinking how it could be done with our younger el kids...whose parents are happy to do calculus with 9 year olds  I have a 9 year old and I am struggling to get my brain around that kind of thinking from that age bracket. 
Anyway, if I was doing ti with a 4 or 5 year old I would break up the lessons into chunks. First thing in the morning teach the new concept and work together on the questions for the new concept.
Later in the morning do 10-15 of the 30 practice questions. If 10 is too many then do less. Have a break and finish the rest later in the day. Even doing 5 questions at a time will get you to the end of the book eventually! Starting early gives you plenty of time.
I do think it will be important to questions every day. If you work too slowly there will be too long a time period between when you learn a new concept and when you next practice that concept. If you think you are working too slowly  then just revive the recently introduced concept quickly before you start each day. If you could finish a lesson in either 1 or 2 days you probably won't need review. ( unless you take a break)
If I sat with my then 6 year old she could easily do all the questions but wasn't able to work independently on it. I decided that since I am time poor, and we have plenty of time, to hold her off Saxon for a while. If you are home schooling then there is no reason to hold them back, or to make them work independently through it so starting early is a valid option. 
If you get to a point where you are finished math at age 12 then worry about that then. Opportunities will present themselves to you when you need them. I only recently started looking for options for advancing high school and found many distance education college/university options that would be safe for kids.
Personally I want my kids to be more well rounded when they start university courses. We aim to achieve that by having them read more quality literature and gradually introducing media. starting with National Geographics magazine and readers digest I was thinking. Your thoughts? My kids are very sheltered at their private school and home life. . I also considered introducing my eldest to regular movie viewing to broaden her horizons....

[nee1]

Robert,

I'm finding that most of the things you say are being confirmed elsewhere. You gave great insights on vacations, and you also said that it was important for an accelerated child to remain level-headed and humble. I was re-reading Alexandra Swann's book titled ``No Regrets: How Homeschooling Earned me a Master's Degree at Age Sixteen.'' She made comments about vacations, and about level-headedness and humility.

This is what she said about vacations:

While we were not obsessed with our studies we were, and are, conscientious about them. Mother taught us to set high standards for ourselves. The result was that we became our own toughest critics, working diligently on a project until we felt satisfied with it. This attitude not only improved the quality of our work, it also helped us emotionally. Good grades earned through hard work gave us a tremendous sense of achievement and inspired us to work still harder.....

Within two and a half months I had completed the first grade.   It was now early April, and I could have settled into an extended summer vacation. However, Mother believed that the time away from my studies would ultimately be harmful to me, since I would have a tendency to forget much of what I had already learned. Therefore, she promptly enrolled me in the second grade.  I, thus, embarked upon a twelve-month school year, another tradition which continues in our home. I was not required to study on Saturdays or Sundays, and I was given the day off on Christmas, Easter, Thanksgiving, and most federal holidays. In addition, any time I was ill, which was seldom, I had the day off.  The rule concerning sick leave was that if I were too ill to go to school I could stay in bed. When I became well enough to get up and play, I was well enough to return to my studies. These were my only vacations. I did not receive two weeks off for Christmas, or spring break, or summer vacation. In fact, I was quite old before I learned that these holidays are observed by most school children.

Though this routine may seem strict, in reality it provided me with an enormous sense of comfort and security. If my routine had been disorganized and haphazard, I might have felt that my life had no direction. Children need constants in their lives, and for many, school is a constant. Whether the school is public or private, they know that they must arrive at a certain time, remain for a certain number of hours, and leave at a certain time. The presence of the studies per se, along with the familiar faces of friends and teachers, can be depended upon.

And this is what she said about her mom (Joyce Swann) teaching her level-headedness and humility towards peers:

Because my emotional development was as important to my parents as my intellectual development, they worked hard to teach me respect for other people for their accomplishments, talents and abilities. They were aware that if I continued to progress at the accelerated rate at which I was working, I would find myself far ahead academically of other children my age, and that this might produce feelings of estrangement. Therefore, they always assured me that while it is true that most five year olds are not in the second grade, basically all people are very much alike, and I was really no different from anyone else. “With proper training, any child with average intelligence could accomplish exactly the same thing,” Mother has often said. The older I grow, the more I find this to be true, but even then I realized that while my experiences might be different, I, personally, was very much like all other five year olds.

(emphasis mine)


By the way, Alexandra Swann is the oldest  of the 10 Swann children. All 10 Swann children got their Masters degrees by age 16. PokerDad started a review thread on the Swann children here - http://forum.brillkids.com/teaching-your-older-child/swann-family-10-children-with-ma-at-age-16!-book-review-and-discussion-thread/.
Joyce Swann (their mom and educator) authored a series of education-related articles here - http://www.home-school.com/Articles/columnists/joyce-swann.php

[PokerDad]

I know we keep moving back and forth on topics here, but I'm currently working through the short book Homeschooling for Excellence by David & Micki Colfax. It's an older book, but so far cogent none the less. On page 46 under the heading "Efficiency," David Colfax makes the argument that school, public in particular, is woefully inefficient. This speaks to Robert's curiosity of what in the world all that time is being spent on...

... there is no question that homeschooling is dramatically more efficient than public schooling.

   The numbers are straight forward and irrefutable. The child that attends public school typically spends 1,100 hours a year there, but only twenty percent of these - 220 - are spent, as the educators say, "on task." Nearly 900 hours, or eighty percent, are squandered on what are essentially organizational matters.

  In contrast, the homeschooled child who only spends two hours per day , seven days per week, year-round, on basics alone, logs over three times as many hours "on-task" in a given year than does his public school counterpart.

Colfax goes onto say that this leaves a substantial amount of time for other interests.

I'm not sure if it's been pointed out or not, but school suffers from constant distraction. Anyone that has studied efficiency can tell you that distractions are a large percentage of waste because it takes time to get back focused. In a typical classroom, the class might be distracted several times per hour - and therefore, Colfax's 220 hours might be overstated.

[nee1]

Article on Lee Binz's website  with interesting insights from a University professor on the importance of Math. (I also noticed the mention of Saxon Math in the professor's bio.).  Link - http://www.thehomescholar.com/blog/homeschooling-curriculum-why-is-math-important/

[Wolfwind]

Kerri, I know this is a late reply,  but I just read what you said about Alex.  I love reading about her math progress!  I think it's wonderful, and honestly, I don't think you need to worry about slowing her down.  Not that I have any experience in the subject :-).  But look at the Robinson Curriculum: when you finish Calculus, you start physics and chemistry, which both keep the math coming.  Life of Fred goes through Linear Algebra, and then she can do professional-level computer programming.  OK, yes, she'll be doing all of this at, what, age 7?  But then she can decide which one(s) spark her interest and start going into depth in that.  With your access to university professors, she can meet mentors who can help her go deeper into whichever one(s) she picks.  It's not like she'll be unable to have a fun time playing with other kids just because, when they're doing their multiplication tables, she's programming AIs.  And once you're into that professional level, there is no upper limit to math.  (Assuming, of course, that you have those mentors available.)  I mean, people out there make a living doing math all day, so there must be more left to do!

[PokerDad]

Thought I'd bump one of the all-time great BrillKids threads to point out how constructivist math has hit the news this week in a BIG way (constructivist math is the arch enemy of Robert Levy).

http://hechingerreport.org/content/common-core-math-problem-hard-supporters-common-core-respond-problematic-math-quiz-went-viral_15361/

I saw the problem on Facebook many days ago and immediately recognized it as a constructivist approach, while the ignorant parents (on facebook) could only blame "Common Core". I'm not here to defend Common Core, but to point out just how frustrating constructivist math is for parents. They rightly see it as confusing and adding steps.

My wife is a big constructivist believer and yet even she admitted to me the other night (when I told her this was a big problem with constructivist math) that low IQ kids have a really difficult time with the approach. She says "true."

If the approach is inferior for someone with lower capabilities, you can ask yourself why. The reason is because it builds artificial complexity that is both superfluous and harder to grasp. Additionally, those with lower working memory can quickly exhaust their stores and get lost (the same problem happens when trying to do traditional algorithms in one's head). The problem with the lower capable student is simply that the approach surpasses their mental ability. I would argue that this presents an inefficiency in the teaching method.

When working with our kids, we should be looking for the most efficient ways to teach - not the most complex.

This is not to say that learning place value isn't important (that's the primary purpose of this anfractuous method), only that there is likely far more streamlined methods. Heck, I never had to do this convoluted "Every Day Math" style learning and I get place value quite well.

Anyway, I post this to show that the pendulum is still swinging in the wrong direction in math education today. If you're looking for your child to gain an advantage in math, my guess is that this will get easier in the years to come but that remains to be seen.

[seastar]

Hmm...I have to say I disagree on this one, I do not feel the problem is constructivist maths - rather the problem lies with the book publishers interpretation of contructivism. I believe experiential learning is crucial and even more essential for children with lower ability, especially those who are kinaesthetic learners. Most people learn best when they get some hands-on experience the phenomenon they are studying. I feel that the constructivist V rote learning debate in maths is a little like the phonics V sight word debate in reading. Both approaches have their merits and I believe a combination of the two leads to deeper understanding.

BTW The math question in that article is not experiential learning, it is simply complete nonsense.

[PokerDad]

That's an interesting view (the parallel with reading). I think it's insightful and likely right on.

For clarity's sake, I'm talking more about grades 4+ and especially 9+. I'm not diminishing the value of using manipulatives with our EL and younger kids. My post was more about the "Everyday Math" style approaches to solving problems, which would mirror the viral math problem.

I'm also not for dogmatic approaches on any side. I'll give you an example. A few years ago, my wife had a learning challenged student with "normal IQ" but low working memory and (can't recall the exact learning in-proficiency) and this young girl could not gain any ground at all in math. None. My wife went against the norm of the school and just taught her the more traditional methods and the girl was able to make progress. The math specialist at the school couldn't believe it (because his dogmatic mind wouldn't accept it). In the end though, they had to dismiss the student from the school (it's a private school) because they lacked the resources to help her (she had issues in pretty much every subject).... my wife was in tears over the whole thing.... and I have to remind myself that "normal" IQ probably means within 1 standard deviation and she was likely below the norm but within that range, which could mean an IQ as low as say 87.

There are other examples where the approach has colossally failed. In Ravitch's Left Back, I read about a school in inner-city Philadelphia a few years back that had to completely abort the constructivist approach at the school. The school went from child-driven to military style within 8 years. The change had to be made because the students just couldn't do it. It's easy to forget sometimes that not everyone has the background, home environment, or genetics to make the learning experience what it can be. Some teachers have overcome (Rafe Esquith, Marva Collins) but neither of them use a constructivist approach and both stress(ed) systematic logical style reasoning to problem solving.

With that said, I can agree that the different methods can co-exist. Math is math. 

[seastar]

This really is an interesting area worthy of discussion. I have been mulling this over since this thread was started way back when.

Over the years, I have worked with many children with Average to Above Average IQ scores struggling with basic maths. In almost all cases, using more constructivist approaches was the key to remediation. I feel that they were moved on to abstract maths too soon without having sufficient time to explore and experience concepts hands-on. I ran a maths club last year and it was incredible to see the kids learn concepts they had been struggling with through playing games. I know if I had tried the drill-and-kill method with them, they would never have grasped what I was teaching as it would have been more of the same. But, through playing games, they became interested by and excited about maths.

I also believe that experiential learning is very valuable even at advanced levels. A watered-down version of it is drawing a picture to break down a difficult problem - I was still doing this when I was studying maths at uni!

Sophie started Singapore Maths recently and I am very interested in the way they lessons are laid out going from concrete to pictorial to abstract for each task. I do not know if they continue this throughout the levels but it is certainly there in the early levels. This is constructivist teaching at its best.

Constructivist approaches are also crucial in science. For example, many people with understand a concept simply by reading it, most people will understand it better by also doing a hands-on experiment, and a few people will ONLY understand it once they have done the experiment.

Here in Ireland, constructivist math has been around for a while but I don't feel it was ever TRULY adopted, to the children's detriment. Children still learn maths facts off-by-heart, which I believe is a good thing; however, some kids, especially those with working memory difficulties, cannot hold this information in their minds long enough to successfully manipulate the problem in their heads. I feel that if they were following the concrete-pictorial-abstract format of SM, they would be able to overcome this deficit.

I haven't read the book Left Back. However, I do have some thoughts on the area. Some authors confuse teaching methodology with school philosophy. Constructivism/discovery learning does not mean child-led, it simply means to explore the world in a scientific manner, testing hypotheses rather than accepting what you are told as dogma. It does not mean loose discipline. In fact the opposite is true as it takes discipline to operate as a scientist, working methodically, thinking creatively etc.

The story about the math specialist in your wife's school is very sad. It shows the danger and arrogance of believing there is one right way as it blinds you to other possibilities. It's like the anti-EL people, they believe so strongly that kids are damaged by EL that they cannot take in any evidence to the contrary. In a similar vein, I was reading an old thread on the WTM forum discussing the Robinson curriculum. It quickly descended into a heated debate re whether using the RC constituted neglect simply because the child was expected to work autonomously. I found the whole thing bizarre as the posters minds were so fixed on their own position it meant there was no possibility for them to learn from one another's viewpoint. Thankfully, it is the opposite here on BK and I feel I learn most from those who hold different viewpoints to my own as it helps me to think critically about my own position and to see the merits in other ideas and methods.

[Robert Levy]

ROAR!!!!!!!!!!   Did I hear "constructivism" somewhere?

Nice to see this thread light up a bit, it's been a lot of fun.  As always, I have to qualify my comments by stating that my sample size is one person for math (and basically one person for reading).  In David's case, I tried and tried to teach him addition using drawings and objects to allow him to visualize the very basic number facts.  I'd ask him if there were 2 apples here, and 3 apples there - how many total applies are there?  SEVEN! (of course).  I even have a video of us at a Miniature Horse farm counting the horses.  He says "one", "two", "three", "four".  I ask him how many total - he says TWO!  Hopeless.

I gave up on that approach and then hit him with the raw numbers.  No dice there either.  Finally, I used a number line - so he could count his way on the line to help him get to the answer.  That actually worked.  Then I started deleting numbers from the line and leaving tick marks - he adjusted.  Eventually I told him to use the "number line in your head" - he did, and then slowly he learned the addition facts without having to count.  Multiplication tables, for whatever reason, was a breeze for him.  Anyway, that's my little story.

By the way David's doing fine.  He's living in New York City, working for a large company as a computer programmer (about the only skill I didn't teach him, at all).  Without getting too specific, they are paying him well, as he has his own 0.5 bedroom apartment (kind of a cross between a studio and one bedroom) in a really nice neighborhood in Manhattan (although it is a 4-storey walk-up), and a 5 minute walk to work - he's also saving up a bunch of money.  And he (finally) gets to see his grandma in New Jersey a lot.

He's visiting us (here in Texas) this week to work on our cars and do some plumbing repairs (my back's a bit messed up, so we put him to work).  But he is having a great time here and it's really nice to have a 19 year old come home to visit without asking for money (LOL).

[seastar]

it's great to have you back Robert - I was wondering if constructivism would be enough to bring you back!

Congratulations to David on his job, you must all be very proud. He really is a shining example of what is possible with dedication and planning.

So, back to the controversial topic at hand...I am in no way saying that experiential learning is the only or even the best way, simply that for some children it helps them to solidify concepts. I can say this with confidence as my informal sample size is in the hundreds. I fully appreciate and respect the fact that it did not work for David and I feel his case is a good example of how you need to explore different methods before finding the best way for each child. My daughter is just beginning her formal maths journey and, at this stage, she benefits from using her fingers or draw pictures to aid her computation. She is quite motivated to drop these aids so she sometimes needs a little prodding to use them if she is stuck. At some point in the near future, I expect that she will progress to visualisation along with  automaticity in maths facts.

BTW, I would consider your approach with the physical numberline moving towards the visualised numberline a little constructivist as it involves experiential learning to grasp an abstract concept! (Now I really will hear you roar)

[Robert Levy]

Naa...valid point.  Using a number line is a manipulative, at least based upon my understanding.

One thing I meant to mention was that I worried that he would not understand the meaning of what he was learning...be it math or reading.  Sure, he might be able to add 2 and 3 - but what does that mean?  Likewise, maybe he could read big words - but he might have no clue as to what they mean.

Then I figured his school and social activities would fill in the gaps that I was leaving - and that did work.  In fact, it becomes a lot easier to expand vocabulary if you're not struggling to be able to simply read the word (similarly in math).

[PokerDad]

I have an old saying, "Do whatever works"

That's the message I take from the discussion. Thank you seastar & Robert. I didn't realize you were able to transition to a mental number line - that's great! I know that Pete Weatherall incorporates the number line on one of his DVDs.

Last I had from you Robert, it sounded like David was pondering the big move to the Big Apple. I'm glad he did and it's all working out; I especially liked the part about him saving money. Just think of it this way, at 19 years old most of his peers would be going into debt and foregoing wages to do it. David gets to live independently, enjoy a nice stroll to the office, gain work experience, and make connections where they count: in the real world!

I'm currently working on the phonics with my (he's now 21 months!) little boy and he's doing well - though lately he's needed a bit more work to get the patterns down than he did earlier. I think it's just because he had a lot of exposure to the easier stuff for many months but now it's getting real... words like "boast" and "grind" which are fairly complicated ideas for a kid still one year old. I mention this only to say that I'm already looking forward to the day we read Hamlet! I'll make sure to take a video and put it in this thread for you (but give me a year or two to do it)

[Mandabplus3]

Poker dad! Shame on you! Two year to hamlet! you are short changing your son     
So my two cents....
Well I think math needs to be a bit of both. The way I see it kids need to be given a good solid dose of just do these sums. The purpose being to build memory, speed, resiliency and to continue the march forward. This is where Saxon shines. It is constantly moving forward. Still reviewing but my kids get to learn something new each day as well as consolidate and improve depth in learnt skills. It is endless math questions building mathematical thinking and giving solid practice in copious amounts.
Now constructivist math is real world math. It's the type of math we face in our everyday lives. I believe that's what it's supposed to be anyway. We can see the apples we bag, we can hold the milk and cream and determine their weight. It has merit in the real world. It has less merit in the classrooms of the bigger kids. Usually because the questions are so far removed from the children's realities they could never actually experience them. It is valuable for the first couple of years of school. But shouldn't be all that is on offer. Constructivist math takes too long to do enough problems to ever become fluent in numbers. It can help a child learn to think mathematically, so they might be able to tell you how to get the answer but perhaps they don't have the skills to get there in a reasonable time. This is of course assuming it is even being taught properly...which sadly rarely happens as the kids have always got the manipulatives and step by step procedures that TELL them how to get the answer. Rather than giving them the manipulatives and having them figure it out themselves. ( which clearly has more value)
It's interesting that as soon as you mentioned the number line I realised that at is how I always did my math facts, on a number line in my head. Learning math facts by rote was never in my school curriculum so I never learnt them. I figured them out on a number line in my head until my first child was born and my brain went very soggy and I started playing math computer games to get it working again ( brain training really works on baby brain, ) now they are automatic I see the value in my kids learning them, so we do
Now as for working memory deficiencies, well unless it is a specific brain injury ( and then only in some cases) it is a developable skill. If it's working memory holding kids back then teach them that skill first! Spend the time working to increase their memory, then teach the math. It's bound to be quicker.

[Korrale4kq]

Interesting. I do math facts with my fingers when I am doing it mentally. I tap my fingers where no one can see. And I use dot formations, like those on a dice, if I am writing them. I just lightly tap my pen down, not marking, on my paper in the dot formation. Number lines are awkward for me as they are a relatively new technique, but I see the merit.  I used to have really automatic recall but over the years my brain is sloppy. Honestly, I seldom ever use math. Pretty much only when I am shopping and I round up prices so much as it is easier and if I forget something I am not too far under.  I always underestimate the final price. It feels like winning the lottery at the check out.

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