Talk to me about teaching math non-sequentially

[Staceyshoe]

Ds is always pushing the envelope with math. Since age 3, he's been begging to learn multiplication and division, and I've been telling him that we will learn that after addition/subtraction. Regardless, he's memorized a surprising amount of the multiplication tables. He begged me to teach him fractions this year, and I again told him that we need to do add/subtract first. Well, he figured out fractions and just memorized those that are equivalent so he can reduce them. I learned by accident that he has a talent for skip counting. If I get him started with skip counting by 4, 6, 7, 8, he can go well past 100 with barely pausing (and he can't do the addition to get him through this:tongue_smilie:--he said that he just recognizes the pattern and continues). Now he's become interested in decimals and he's starting to figure out which fractions and decimals are equivalent. When I gave him a math placement test, he was creating number sentences that correctly used negative numbers (he also skip counts backward right past zero and into the negatives).

 

I've resisted skipping around because I learned math sequentially and don't know how to teach it another way. In my mind, it builds and you need the basics first. Now I'm reading Deborah Ruf's book, and she mentions that highly gifted often do better learning multiplication and division before addition and subtraction. So, I'm re-thinking our math plan. Should I be a stickler for learning addition and subtraction before we move on? Right now I'm planning to continue Singapore and then do Miquon 1-2 days/week so he can dabble in multiplication and such but not have gaps in his basic math skills. Does this sound like a good plan? If you teach math non-sequentially, HOW do you do it?

[MaMa2005]

I have a DS that is very good in math, too. He was chomping at the bit to do multiplication/division before he had mastered addition/subtraction. However, I am a stickler with math skills as one DOES build upon another.

 

As a retired teacher, I saw too many children who hadn't mastered the basics and it really caught up with them as they got to higher level math.

 

I use Horizons Math with DS and he has seen why I have made him learn A before B before C. The underlying reasoning for math concepts is essential to the overall learning and retention of skills.

 

Just my .02.

[MissKNG]

Now I'm reading Deborah Ruf's book, and she mentions that highly gifted often do better learning multiplication and division before addition and subtraction.

 

:lurk5:

 

The home schooling book by Lisa Rivero also mentioned that bright math students tend to get concepts easily and are more "eh" with computation.

[Violet Crown]

Not all the math in the standard ps sequence is taught at a certain stage because it requires all the math that came before it.

 

For instance, we started both our older dds on algebra, such as it was, as soon as they could add and subtract (at that point, even children who don't know how to "multiply" or "divide" can find double or halve an amount), by using scales with colorful gram cubes, and playing games where you have to figure out how to re-balance the scales after adding to, subtracting from, doubling, or halving what's in one pan. After a while you start concealing some cubes in a piece of paper, and the child has to figure out from the operation how much is hidden. Etc.

 

Our kids also start fractions very early, but my dh teaches them as a graph-paper game called "Who's the Fastest?" where fractions are first discovered as rate/slope. Later they're re-introduced as division, and finally re-re-introduced as fractional parts of a whole.

 

I do want to point out that all of this is my dh's idea; he's had a bee in his bonnet since grad school about the nonsensical order that math is taught in. He claims it's taught in the order it was discovered, not in the order that makes the most sense. I just implement it as I'm told, like a good wifey.:glare:

 

In return, I get to implement my unorthodox ideas about teaching literature. :D But anyway, the point being that there's a lot to question about the conventional order of teaching math. And the nice thing about homeschooling is we can experiment on our children with the assurance that if our theory flops completely, we just put aside that idea and try something else.

Edited May 18, 2010 by Sharon in Austin
puir speling

[LisaKinVA]

I have two who have been "eh" about computation, but grasp concepts VERY quickly. My oldest decided during fractions/decimals/percents that knowing his facts well was important and learned them very quickly. My 6yo doing long multiplication and division has decided it takes too long and will probably have them down in June.

 

In both cases, the children know how to figure their answers on their own -- but didn't concentrate on really *knowing* them fast until they felt there was a need to. Days and days of flash card "training" and drills would have been fruitless (tried this tactic with oldest ds -- miserable). Kids digging in their heels, frustration and tears. My dd took my guidance and learned them.

 

It's one thing to not have mastered the concepts and moving on to new concepts -- but to hold some children back because they can't write their facts down in a minute can be more akin to torture.

[Guest]

Here's another who had a child doing multiplication before addition, and algebraic equations before she could properly count the number of candles on her cake. One of the great secrets of math instruction is that its sequential arrangement is artificial; any number of kids do not learn that way.

 

I read once that in France children do not learn to multiply and divide fractions until seventh grade; at that point is comes easily to them and is not the sticking point it often is in US classrooms. My daughter learned this later than the norm; but at the same time she was learning about slope and working with infinite series.

 

As others have said, if your child is actively and eagerly grabbing for higher level math and abstract concepts, let him go at it. Don't hold him back to master basic addition first. He'll need it to work out some of those higher level problems and then it will have a context and a purpose.

 

You can also work on boring old facts through games, which may make them more palatable than flashcards or other methods.

[catz]

t

It's one thing to not have mastered the concepts and moving on to new concepts -- but to hold some children back because they can't write their facts down in a minute can be more akin to torture.

 

I totally agree on this! I treat "math facts" and conceptual math as 2 different things. For the record I have a math degree and a 3rd grader comfortably at a pre-algebra level. My 3rd grader knows his facts pretty well now, but might be hard pressed to pass some of the timed tests they give on that kind of thing. For some kids, the conceptual leaps might come at the same time as knowing your math facts. But for other kids, it might not. I went on to do a lot of higher level college math at the undergrad and grad level. But I struggled in elementary math because it was very drill based and not at all conceptual. I knew my math facts well enough to get through, but I don't think I really knew them until later. When I hit algebra, everything clicked for me. I'm sure I would have been much happier as a child using a curriculum like Singapore. I literally hated math until I got to Algebra.

 

My 3rd grader has been making leaps and connections in math for as long as I can remember. And we regularly discuss things he's not working on formally yet. But I put him all the way through the Singapore curriculum. I don't want any holes as we approach higher level math. I just know that at certain points, he won't need a lot of practice. I also like how Singapore might covers several points at a time. For example, introducing order of operations instead of doing problems that look like this ...

 

1 + 2 x 6 - 3

 

Which would be pretty mindless for a mathy kid once they knew the rules. They introduce the concept and then have problems like this .... (we recently did some of this in 6B)

 

1/2 + 1/3 X 1/4 + 2

 

So your having to think not just about order of operations, but also about several fraction operations as part of the same problem. Their word problems are really good too. I've looked at lots of math curriculum, and Singapore is really good for mathy kids that need a lot of depth.

 

Anyway - after all that, I think your plan sounds great for your mathy kid! :) Letting them do some exploration where they're interested while continuing to give them a good base for later sounds ideal.

[LisaDSB]

Your child may be a visual-spatial learner and may need to do math in this fashion. Sometimes, if the concepts are too simple, their brains basically shut down. It's not until they get into more challenging stuff that the neurons fire up.

 

Also, VSL's tend to be big picture thinkers. They get complex stuff, but fall down on the details.

 

My DS9 used to struggle with computation, and his school held him back from the higher math he really wanted to do. Now, as a homeschooler, he does algebra and whatever else he feels like teaching himself and his computation has improved dramatically.

 

I still make sure he's covering the basics by making him work through his Singapore math (doing a bare minimum of exercises, if he establishes competency), but I don't hold him back. I think it would do him a great disservice and probably turn him off of math.

[LisaDSB]

I guess I didn't answer your specific question. HOW do you teach math non-sequentially? Basically, I don't really teach -- I let him explore what he wants to learn himself. I've introduced lots of math books along the way that inspire an interest in various concepts (books like The Number Devil, The Great Number Rumble, G is for Googol, etc.) and he gets excited and goes off to learn more.

 

When he decided he wanted to do algebra, I bought him Thinkwell's online Beginning Algebra course and he does that on his own. Today, he wanted to do some complex graphing of formulas (complex to me, at least:tongue_smilie:), and his dad was home, so they did some problems together.

 

I'm in the background, making him do his Singapore once in awhile, but he really doesn't like it because it's not stimulating enough.

 

Maybe strew some interesting books like that around and let him play with math as much as he wants. Reward him for doing a few exercises of his required math with playing on a site like CoolMath4Kids or something else that lets him jump around a bit and try things out. You can still teach him sequentially, but don't confine him to that.

[fractalgal]

Ds is always pushing the envelope with math. Since age 3, he's been begging to learn multiplication and division, and I've been telling him that we will learn that after addition/subtraction. Regardless, he's memorized a surprising amount of the multiplication tables. He begged me to teach him fractions this year, and I again told him that we need to do add/subtract first. Well, he figured out fractions and just memorized those that are equivalent so he can reduce them. I learned by accident that he has a talent for skip counting. If I get him started with skip counting by 4, 6, 7, 8, he can go well past 100 with barely pausing (and he can't do the addition to get him through this:tongue_smilie:--he said that he just recognizes the pattern and continues). Now he's become interested in decimals and he's starting to figure out which fractions and decimals are equivalent. When I gave him a math placement test, he was creating number sentences that correctly used negative numbers (he also skip counts backward right past zero and into the negatives).

 

I've resisted skipping around because I learned math sequentially and don't know how to teach it another way. In my mind, it builds and you need the basics first. Now I'm reading Deborah Ruf's book, and she mentions that highly gifted often do better learning multiplication and division before addition and subtraction. So, I'm re-thinking our math plan. Should I be a stickler for learning addition and subtraction before we move on? Right now I'm planning to continue Singapore and then do Miquon 1-2 days/week so he can dabble in multiplication and such but not have gaps in his basic math skills. Does this sound like a good plan? If you teach math non-sequentially, HOW do you do it?

 

I see nothing wrong with letting him explore and teaching him multiplication and division when you see that he is asking about it - not worrying at all about where a certain curriculum says you should be. I would just make sure that you have your bases covered on addition and subtraction so that he has a solid base in those. Try to move forward with those concurrently if possible taking extra time if necessary.

 

With my own children I like to take "easier" subjects once I feel comfortable that they understand the concept and go off on tangents to see how far they can stick with me. For example if I am teaching about base 10 with regards to regrouping: I'll introduce 10 ones = 1 ten. After teaching that and seeing that they understand I'll go on about 10 tens = 100, then 10 hundreds = 1 thousand, 10 thousands equal 10,000 while showing them the place value until I can tell that they are frustrated. It helps me gauge how much they are actually grasping.

 

You could teach addition, for example, and then expand on it showing how multiplication is an extension of addition and show examples to go deeper. For example: 4+4+4 = 12 = "3 groups of 4" = 3 X 4. Then you are teaching multiplication and addition side by side.

Edited May 18, 2010 by fractalgal

[wapiti]

Your child may be a visual-spatial learner and may need to do math in this fashion. Sometimes, if the concepts are too simple, their brains basically shut down. It's not until they get into more challenging stuff that the neurons fire up.

 

Also, VSL's tend to be big picture thinkers. They get complex stuff, but fall down on the details.

 

My DS9 used to struggle with computation, and his school held him back from the higher math he really wanted to do. Now, as a homeschooler, he does algebra and whatever else he feels like teaching himself and his computation has improved dramatically.

 

I still make sure he's covering the basics by making him work through his Singapore math (doing a bare minimum of exercises, if he establishes competency), but I don't hold him back. I think it would do him a great disservice and probably turn him off of math.

:iagree:

[Crimson Wife]

We do a combo of sequential and non-sequential math. I do require DD to work through a standard math program (first Right Start and now Singapore) to master the basics. But I supplement with the Edward Zaccaro Challenge Math series, Hands-on Equations, a bit of CSMP, and I gave her a set of Fraction Tiles to play with in kindergarten. I keep looking at Don Cohen's Calculus for Young People but haven't tried it yet.

 

Art of Problem Solving is something I've got my eye on for later.

[Storm Bay]

You can teach math non-sequentially if you treat learning math facts as a separate skill from the concepts. My dc learn their math facts by doing math rather than by drill anyway. Mine have learned the basic operations sequentially, but that is only important if dc need to see addition and subtraction to understand multiplication and division as fast adding and fast subtracting.

 

What I would do if I were in your shoes after going through this age with my three would be to handle addition and subtraction as a game, perhaps with manipulatives or addtion & subtraction war or something, but still teach whatever your dc is interested in conceptually. You could use Miquon with C-rods or simply play with C-rods. Not that your dc normally needs these, but the addition is just so natural this way.

[MissKNG]

Thank you all for this thread and all the suggestions. Though I'm not the OP, I keep getting pulled in the direction of moving on conceptually by my big girl when her math facts aren't super up to par. I read in the K-8 forum very often on how kiddos MUST KNOW math facts before moving on. That scared me into thinking I was doing something wrong by letting my big girl progess in concepts more naturally verses drilling math facts then moving on. She doesn't like drill or repetition.

 

I did buy Miquon and am putting together some math games to make math facts fun. But I'm not going to be afraid to move on when she tells me to move on or complains we are working on "baby math".

 

:auto:

[Storm Bay]

Thank you all for this thread and all the suggestions. Though I'm not the OP, I keep getting pulled in the direction of moving on conceptually by my big girl when her math facts aren't super up to par. I read in the K-8 forum very often on how kiddos MUST KNOW math facts before moving on. That scared me into thinking I was doing something wrong by letting my big girl progess in concepts more naturally verses drilling math facts then moving on. She doesn't like drill or repetition.

 

I did buy Miquon and am putting together some math games to make math facts fun. But I'm not going to be afraid to move on when she tells me to move on or complains we are working on "baby math".

 

:auto:

 

fwiw, my 15 yo has made As in both Algebra 1 & Geometry and I never held her back for not knowing her facts. She learned them as she did her math and has them down cold now (actually, she knew them before Algebra, but just.) Same with my middle one who is sailing through Algebra 1 at the moment (so much for LoG Beginning Algebra being a 1 year course ;)). I was a hard core drill and rote math facts advocate until I homeschooled and had a real wake up call. Some dc just don't do well with that.

[LisaKinVA]

fwiw, my 15 yo has made As in both Algebra 1 & Geometry and I never held her back for not knowing her facts. She learned them as she did her math and has them down cold now (actually, she knew them before Algebra, but just.) Same with my middle one who is sailing through Algebra 1 at the moment (so much for LoG Beginning Algebra being a 1 year course ;)). I was a hard core drill and rote math facts advocate until I homeschooled and had a real wake up call. Some dc just don't do well with that.

 

You mean, the daily tears and meltdowns and math time dread was too much to bear? :001_huh: (haha...just kidding). I couldn't handle it either. I'm a pretty much "structured" hsler... but couldn't justify driving my son to frustrated, angry tears. Reminded me of how I felt the year in school where teachers, administrators and my parents told me that a YEAR of review would be good for me (and it wasn't because I didn't know my math facts, it was because the SCHOOL didn't want to accommodate me.)

[Storm Bay]

You mean, the daily tears and meltdowns and math time dread was too much to bear? :001_huh: (haha...just kidding).)

 

 

Actually, this is very close to being correct. My dd HATED math prior to Algebra and had a mind like a steel trap for all things math other than the facts. I tried drills, etc, with her, my first, when I pulled her out of ps after gr 2 and finally stopped. This dd is very strong willed and I literally had to choose my battles with her or I'd have been battling all day long and neglecting my other two. Over the years she battles me less about school, but I've learned to work with her. She has to do school, of course, but when I can I give her a choice between two things I have already researched, etc.

 

As for my middle one, it wasn't tears, just pointless because she is so strongly vs & I read enough about that in time to realize that drill was a waste of time for her. I used MUS for mastery in theory, but it was doing it over & over that nailed them for her.