Math: Spiral vs. Mastery

[MunRoLy]

After reading tons of math threads today, I think I figured out why math has been so hard for dd10. We have only ever used a spiral curriculum (Everyday Math and LOF elementary) and she is having so much trouble! I don't know if LOF is spiral, but the elementary series does seem to jump around to much for her.

 

So I would love it if the popular homeschooling math curriculums could be divided into spiral and mastery. I'm specifically wondering about Math Mammoth and MEP. I was looking into Horizons for her, but now that I see it's spiral I'm sure it wouldn't be a good fit.

 

Also, how can I find out if spiral or mastery is best for my ds7? He picks up math concepts very easily, but I'd love to know now while he's young instead of when he's heading into 6th grade like dd10.

 

Thanks!

 

Thanks!

[mom2bee]

Math Mammoth is mastery based, Math-U-See is, Primary Mathematics (Singapore) too, I don't know about MEP, I think it is a blend of mastery and spiral.

Math mammoth is also availalbe in topical units though, so you can really work on mastery based stuff by just working every part of that topic.

Saxon is spiral. I can't think of any other spiral math programs. Rod and Staff? CLE? I don't know much about those last two.

[Ellie]

After reading tons of math threads today, I think I figured out why math has been so hard for dd10. We have only ever used a spiral curriculum (Everyday Math and LOF elementary) and she is having so much trouble! I don't know if LOF is spiral, but the elementary series does seem to jump around to much for her.

 

So I would love it if the popular homeschooling math curriculums could be divided into spiral and mastery. I'm specifically wondering about Math Mammoth and MEP. I was looking into Horizons for her, but now that I see it's spiral I'm sure it wouldn't be a good fit.

 

Also, how can I find out if spiral or mastery is best for my ds7? He picks up math concepts very easily, but I'd love to know now while he's young instead of when he's heading into 6th grade like dd10.

 

Thanks!

 

Thanks!

 

 

I'm not sure there is a consensus on that. :-)

 

Something to consider, and probably before spiral vs mastery, is process math versus traditional math.

 

A process math uses/depends on manipulatives such as base 10 blocks, cuisenaire rods, and so on; examples would be MUS, Miquon, and Making Math Meaningful.

 

A traditional math...doesn't, although there may be some usage of manipulatives (or visuals). ABeka, BJUP, Modern Curriculum Press, Ray's Arithmetics, Strayer Upton, and Rod and Staff are all traditional math.

 

If you decide on a process math because it seems to be spiral, and your child does NOT need manipulatives, you'll be going down a rocky path. Ditto with traditional; some children really need those manipulatives, and if you choose a traditional math because it seems to be spiral, you'll be going down a rocky path.

[junepep]

I had once tried to put together just such a list, and then I realized how utterly futile it really was. Few programs are entirely one thing or another. I'll put my list here for people to pick through / argue over and add more information about each program -- keep in mind I haven't seen most of the programs myself and when I made the list I was just going off how people described them on the forums.

 

Math Programs by Classification

Abeka - Spiral / Incremental (Drill & Review)

Christian Light Education / Publications (CLE) - Spiral / Incremental (Drill & Review)

Everyday Math (EM) - Spiral / Fast

Horizons - Spiral / Slow moving

Math on Level - Spiral

Saxon - Spiral / Incremental (Drill & Review)

Teaching Textbooks (TT) - Spiral(?) / fast (little review)

Bob Jones - Mastery (?)

Centre for Innovation in Mathematics Teaching (MEP) - Mastery(???) (based off hungarian prog) -- This was where I classified it before I used it, however, in practice I've found MEP to be a spiral program.

Beast Academy - Mastery -- In my very limited experience, (we're in book 1, but I did all of the problems through the end of year 3 before starting with my daughter) at first glance this is a Mastery program, but each chapter really builds off of the ones before it, so it is far more mixed than it appears.

K12 - Mastery (has review, not enough for dc who need an incremental approach)

Math Mamoth (MM) - Mastery (Singapore based but w/ far more review - if not bought in indiv modules it is like a spiral program)

Progress in Mathmatics - Mastery (?)

Right Start (RS) - Mastery (some said spiral - long enough gap in between to need 'refreshers')

Rod & Staff (R&S) - Mastery (?)

Singapore - Mastery / Fast (little review, can be slowed down with WB, IP, CWP - however, does spiral yr over yr)

Math U See (MUS) - Mastery (hands on w/ worksheets)

Miquon - Mastery (hands on) -- In my experience this program can actually be done as spiral or mastery depending entirely on how you want to implement it (doesn't go all the way to 6th)

Chalk-Board - DVD program

Video Text - DVD program (beginning @ algebra)

CMSP - ?

Professor B - ?

Life of Fred - Sometimes used as a full curricula, other times used as a break or supplement (Kit didn't care for Fred, I wouldn't call it a stand alone program at all).

Living Math - supplemental reading (lots of ideas / lists on the forums - but here's a link to math books: http://www.livingmat...US/Default.aspx

 

I suspect that one way to tell whether your child would benefit more from mastery or spiral would be how many reminders does he need. If he remembers how to do problems that he hasn't seen in 3 / 6 / 12 mos (or doesn't need a refresher after taking a summer break) then perhaps he would be a good fit for some version of a "mastery" program. In the end, your best bet is to evaluate each program on it's merits. MEP is free, you can go look at the scope and sequence and Math Mammoth has (if I'm remembering correctly) lots of samples. Sit down with your child in mind and just run through the sections to see if either would work. I will say that I've found it harder to hop forward in MEP when Kit has a leap in math and doesn't need the review. We wind up having to go through each page and cross off the non-challenging / interesting problems and then cobble multiple lessons together, but she's been really happy with the program over all (even though a mastery program might be a better fit) and so it's worth the extra time and trouble for us.

 

TDLR; Experiment until you find something that your child enjoys, because that's all that really matters. I've found that the labels are fairly meaningless.

[nrself]

So is there a math program that is spiral and incremental, but that doesn't jump from subject to subject like Saxon does? I used to like Saxon but am beginning to think this is a problem for my ds.

 

Thanks,

Nicole

[Tracy]

This is huge generalization, but I have read that kids in the middle of the bell curve do better with spiral math, whereas those on the ends of the bell curve do better with mastery. The reason is that kids who struggle with math need to focus on just one concept at a time, and gifted kids can learn a single concept quickly and just move on without all the extra practice that a spiral program will provide.

 

But then you also have to take into consideration just how spiral a program is. Saxon is known to have a very tight spiral--that is, lots of frequent review. Parents of kids who are gifted in math complain that too tight a spiral is frustrating to their students. But that is not a hard and fast rule.

 

We use CSMP, which is definitely spiral. But most days, the student only works on 1-2 concepts in a single lesson. But when we used Horizons, we were doing 4-5 concepts per lesson.

[Pen]

After reading tons of math threads today, I think I figured out why math has been so hard for dd10. We have only ever used a spiral curriculum (Everyday Math and LOF elementary) and she is having so much trouble! I don't know if LOF is spiral, but the elementary series does seem to jump around to much for her.

 

So I would love it if the popular homeschooling math curriculums could be divided into spiral and mastery. I'm specifically wondering about Math Mammoth and MEP. I was looking into Horizons for her, but now that I see it's spiral I'm sure it wouldn't be a good fit.

 

Also, how can I find out if spiral or mastery is best for my ds7? He picks up math concepts very easily, but I'd love to know now while he's young instead of when he's heading into 6th grade like dd10.

 

Thanks!

 

Thanks!

 

 

MathUSee is mastery and is easy to accelerate through by only doing as much as needed to learn and then moving to next thing. If you used other paper instead of working in book, conceivably dd could use it to review and then ds7 could use it.

[Kathie in VA]

Also keep in mind that some define the terms differently.

Mastery is sometimes knocked for not reviewing and spiral for moving too fast.

 

I see them differently. I see the main difference as how they use their review. Mastery based approach uses review after a concept is learned and practiced till mastered. Spiral and incremental approaches use review to master a new concept.

 

I consider a mastery approach to teach a concept, then practice that concept. Then intention of this practice is to understand the concept and be fully able to utilize it. So if it is learning the 4 times table, you practice the concept of groups of 4, do all the facts and their reverse both horizontally and vertically, and you work on word problems. Then you practice it some more by mixing it up with review. Again you do facts and word problems but here you include the new and the review... so as to make sure you know when to use the new concepts. The point of reviewing is to keep your skills fresh and to sharpen your new skills by mixing them up.

 

The spiral and incremental approaches also teach and then practice. However the amount of practice is smaller. They also move on to review quicker. It is in the reviews that mastery is approached. Here you would learn a new concept almost every day and spend some time practicing it but more time reviewing older concepts so as to build mastery of them. Thus if you don't really get a new concept it is considered okay because you should be able to 'get it' when it is reviewed over and over.

 

hth

[birchbark]

Mastering Mathematics, Strayer Upton, and Systematic Mathematics are also mastery.

[MunRoLy]

Also keep in mind that some define the terms differently.

Mastery is sometimes knocked for not reviewing and spiral for moving too fast.

 

I see them differently. I see the main difference as how they use their review. Mastery based approach uses review after a concept is learned and practiced till mastered. Spiral and incremental approaches use review to master a new concept.

 

I consider a mastery approach to teach a concept, then practice that concept. Then intention of this practice is to understand the concept and be fully able to utilize it. So if it is learning the 4 times table, you practice the concept of groups of 4, do all the facts and their reverse both horizontally and vertically, and you work on word problems. Then you practice it some more by mixing it up with review. Again you do facts and word problems but here you include the new and the review... so as to make sure you know when to use the new concepts. The point of reviewing is to keep your skills fresh and to sharpen your new skills by mixing them up.

 

The spiral and incremental approaches also teach and then practice. However the amount of practice is smaller. They also move on to review quicker. It is in the reviews that mastery is approached. Here you would learn a new concept almost every day and spend some time practicing it but more time reviewing older concepts so as to build mastery of them. Thus if you don't really get a new concept it is considered okay because you should be able to 'get it' when it is reviewed over and over.

 

hth

Thank you. That does help. I don't think spiral works for my kids. I think they need more time to master a subject. Unfortunately all I've ever used is spiral so their math knowledge has suffered. I'm excited to make a change!

[NASDAQ]

Well, you've chosen two pretty eclectic choices there. Everyday Math is one of those curricula that huge, huge numbers of kids have problems with. Entire movements have been formed by people who suffered with Everyday Math. Teachers often seem to like it, but many kids and many parents just hate it with the passion of a thousand suns.

 

LOF is rarely considered a standalone curriculum.

 

A mastery curriculum can and should include regular review. I think that the main concepts in arithmetic actually have spiral built in. Right now my dd is starting long division in SM 3A. She's doing, say, 83 divided by two. Doing the problem requires multiplication, division, subtraction, and base ten concepts.

[MunRoLy]

Well, you've chosen two pretty eclectic choices there. Everyday Math is one of those curricula that huge, huge numbers of kids have problems with. Entire movements have been formed by people who suffered with Everyday Math. Teachers often seem to like it, but many kids and many parents just hate it with the passion of a thousand suns.

 

LOF is rarely considered a standalone curriculum.

 

A mastery curriculum can and should include regular review. I think that the main concepts in arithmetic actually have spiral built in. Right now my dd is starting long division in SM 3A. She's doing, say, 83 divided by two. Doing the problem requires multiplication, division, subtraction, and base ten concepts.

I wish I would have know this when I started... A new question, if I begin using Math Mammoth would I need to supplement with regular review or is it built in?

[ErinE]

. 

 

Edited February 19, 2018 by ErinE

[MunRoLy]

Thank you. I'm really leaning toward MM next year for ds13, dd10, and ds7.

[Spy Car]

I really think the question of "mastery" vs "spiral" is overblown, and takes away from the more important initial question one should ask and that is does the math program in question have a "procedure only" (or almost only) approach where children mechanically learn "how" to do math, or does the math program teach both procedural competence along with a through understanding of the underlying mathematics?

 

If you don't teach "why" but only "how" the spiral review become more of a factor as kids tend to forget what they don't understand in the first place. Hence the need for constant review.

 

Not to mention that so-called "mastery" programs are constantly working the acquired skills as new ones are added. Just because a new topic involves area, or quantity, or metric measurement, or some other "topic" doesn't mean that addition, subtraction, multiplication, division, fractions, decimals, etc. (depending on the level) are not being worked constantly under the new "topic."

 

Further, if a program teaches deeply, the math operations are understood by children to be interrelated. Addition and subtraction are interrelated (subtraction is adding negative numbers and is the inverse of addition). Multiplication is serial audition. Division is inverse of multiplication. Fractions are another form of division, etc.

 

If a child really unders the math, the order of topics becomes a secondary concern.

 

While "spiral" vs "master" deserves some consideration it is far better to ask the most important questions first. Is the math program aiming for deep mathematical understanding with procedural competence, or is it limited to teaching "how" to do math? Or is it failing on both? Answer these questions first and one will be a lot better off in the long run.

 

Bill

[Farrar]

I think it's a red herring. I think because so many public school programs were so uber-spiral and also bad that spiral came to be thought of as bad, at least in homeschooling circles. And because there are so many quality mastery programs, like Singapore and Math Mammoth, that mastery came to be thought of as good. But I agree with Bill that it dodges the important question of whether the program aims for a deeper understanding in the end.

 

For one of my twins, mastery style has worked better. He wants to work through something, get it, then move on. For my other boy, he needs to jump around. If he doesn't, then it touches on his anxieties or he gets bored - he just needs new challenges. So I think a lot of it is about the personality of the learner. But even within that, obviously just being spiral or mastery wouldn't make something the right program. There are lots of other divisions to look at with math programs.

[lifesadream83]

Math in Focus is Mastery and Spiral. It teaches a concept and has you master it at one level for example it will teach the 2 times tables until it is mastered along with how to divide it. Then it will cover other topics in a similar fashion in the next level they review what you should know the two times tables and cooridinating division and then shows you how to implement the same concepts for the 3 and 4 times tables. We use it and love it.

[mathwonk]

I am not an expert on "spiral" technique as used in elementary textbooks, but I know that review has always been crucial in my teaching of college age kids. This is a form of spiraling, but in subsequent coverage one moves faster over material that has been seen before. In advanced courses one also covers the same material but from a higher point of view, or in greater depth, and this too is a form of spiraling. My impression is that Saxon uses the term to mean just repeating the same stuff from before at the same level, but repeating in a way that assumes prior familiarity is a ubiquitous standard teaching method.

 

In teaching calculus e.g., first one teaches calculus computation methods in one variable, usually without full justification.. then one teaches computations in several variables, again without full justification. Then one uses more theory to justify the methods learned earlier in one variable, then one learns the theory in several variables. After that one studies the same ideas but more abstractly and even in infinite dimensions. It's all calculus, but more and more advanced. In each subsequent course it is advisable to recall quickly the material from the earlier course, from the new perspective, in order to motivate the desired generalization.

 

The same holds in algebra, first one learns to manipulate and solve formulas in one variable, of degree one. Then one raises the degree of the formulas to two, (quadratic equations) and learns to solve these by completing the square. Then one learns some manipulations of polynomials of higher degree, like multiplying and dividing them, but not solving them usually, unless one is reading very high quality books like the algebra book of Euler.

 

Hopefully, but not always, one learns to create new number systems algebraically, like complex numbers, and manipulate with them analogously to usual numbers. One then increases the number of variables first to two, then more, but this subject is called algebraic geometry, because the solutions sets are no longer finite but have geometric structure.

 

In college one learns to study algebra more abstractly, by focusing on properties of the operations, like whether the product of two non zero "numbers" can ever be zero, and what the consequences are. One also studies new finite systems of numbers like "clock arithmetic" and learns applications of this to studying usual numbers. At every stage one compares the behavior of integers versus polynomials versus finite number systems, to unify and distinguish these various algebraic systems. Ideally every new course adds to the understanding of the previous more elementary courses.

 

Then one studies more abstract operations like geometric motions, from an algebraic viewpoint, by combining motions as if it were a sort of multiplication, and tries to apply methods of algebra to geometry. One needs to give up some familiar properties like commutativity.

 

the same holds for geometry, as one passes from one dimension to two, to three and more, then relaxing requirements to se what flexibility is gained. I.e. one changes the requirements for geometric objects to be equivalent, from rigid motions carrying one to another, to motions that allow some distortion, such as similar triangles versus congruent ones, and later "continuous" or differentiable transformations, in topology and differential geometry.

 

So there are in traditional math only three topics, algebra, geometry, and calculus, and all the myriad courses of math are spiraling versions of these subjects getting ever more complex and abstract. Of course there are also somewhat more modern topics like statistics and finite math.

 

 

The point is that every new course should benefit from and reinforce the lessons from earlier ones. This is what spiraling means to me. But just mindless repetition of operations, on the assumption that the student did not remember it the first time, may be tiresome to students who do. To keep their attention it seems wise to include something new when reviewing, i.e. my understanding of a spiral is that it moves constantly upwards.

[Spy Car]

The point is that every new course should benefit from and reinforce the lessons from earlier ones. This is what spiraling means to me. But just mindless repetition of operations, on the assumption that the student did not remember it the first time, may be tiresome to students who do. To keep their attention it seems wise to include something new when reviewing, i.e. my understanding of a spiral is that it moves constantly upwards.

 

 

What "spiraling" means to you is exactly the approach many so-called "mastery" programs take (and there are some so-called "spiral" math programs that do the same). When you teach for understanding you get this sort of progression of skills building on skills any way you slice it.

 

Unfortunately, this leaves "spiral" and "mastery" somewhat nebulous terms. Some "spiral" programs do include mindless repetition of operations, and they are boring and tiresome, and it is because they don't teach for understanding from the outset, and adopt the "drill and kill" strategy as a replacement for building a strong conceptual foundation from the start.

 

But a so-called "mastery" program could suffer the same problem. Teaching "procedure" without developing mathematical thinking and deep understanding is the critical issue. Then skill building can take place on a solid foundation.

 

To often the mastery vs spiral debate ignores the more important question.

 

Bill

[Spy Car]

Another import question to ask of a math program is: does it make the children *think*.

 

And here, to define my terms, by *think* I am differentiating between "recall" of memorized facts (as "not thinking") and appropriately challenging work that requires deductive reasoning and logic to arrive at the solution (which is thinking).

 

It is beyond scientific doubt that stimulating the mind with cognitively challenging work builds the neural pathways in the brain. Children's minds are incredibly plastic, and especially able to build neural networks though use.

 

Unfortunately too many have fallen sway to Dorothy Sayers' flawed characterization of "the Trivium" and believe grammar-staged children's mind are only good for memorizing "facts." This is tragically wrong.

 

Great math programs for kids include developmentally appropriate mental challenges as part of the math mix. This sort of work makes math interesting, and it (literally) helps to create a better brain.

 

With the mind it is a "use it or lose it" proposition. If the math work is mind-numbing one loses out on a huge area of opportunity to make a better (and more densly-wired) brain.

 

Again, this is a more important question than "spiral" vs "mastery."

 

Bill

[ktgrok]

What "spiraling" means to you is exactly the approach many so-called "mastery" programs take (and there are some so-called "spiral" math programs that do the same). When you teach for understanding you get this sort of progression of skills building on skills any way you slice it.

 

Unfortunately, this leaves "spiral" and "mastery" somewhat nebulous terms. Some "spiral" programs do include mindless repetition of operations, and they are boring and tiresome, and it is because they don't teach for understanding from the outset, and adopt the "drill and kill" strategy as a replacement for building a strong conceptual foundation frm the start.

 

But a so-called "mastery" program could suffer the same problem. Teaching "procedure" without developing mathematical thinking and deep understanding is the critical issue. Then skill building can take place on a solid foundation.

 

To often the mastery vs spiral debate ignores the more important question.

 

Bill

 

 

I'll add another issue into this muddle. One can have a good understanding of the why and the concepts, but if the "drill" wasn't done one will then be left figuring out each problem from scratch. My son had the concepts down. He could figure out the problems. He got nearly a perfect score on the problem solving section of the ITBS. But was slow as dirt on the computation stuff. And that held him back when it came to algebra. He knew HOW stuff worked, but because it wasn't second nature and wasn't practiced he had to do it all mentally anew each time. And that is a real drain on the brain. We switched to something with more review and now that, coupled with the conceptual understanding he had before, has finally got him doing really well. .

 

I'd love a program that had good conceptual meat to it, but did lots of repetition on a daily basis of the old stuff...some kind of meld between Math Mammoth and TT. Anyone find that and I'll buy them a drink.

[Tiramisu]

I will disagree with some others. Because a student understand the concepts does not necessarily mean they it's enough to ensure retained mastery, if that makes sense. I have four kids, each very different. I feel very comfortable with saying lots of practice in rote memorization can be extremely helpful for some kids. I had one who was very conceptually-minded and used EM in the early years, but needed the rote practice to consistently do well in higher math. I had another who was very conceptual but got bogged down with a weakness for fact memorization. It came, but it took A LOT of consistent review. She thrived with CLE. Another is not conceptual at all and gets stuck. She used EM and MIF. She would probably benefit from a more traditional approach. My last one is using a traditional program, CLE, because it gets done right now. She will do the problems with a Singapore-type conceptual approach, not because she was taught that way but because she came up with it herself.

[Spy Car]

 

 

I'll add another issue into this muddle. One can have a good understanding of the why and the concepts, but if the "drill" wasn't done one will then be left figuring out each problem from scratch. My son had the concepts down. He could figure out the problems. He got nearly a perfect score on the problem solving section of the ITBS. But was slow as dirt on the computation stuff. And that held him back when it came to algebra. He knew HOW stuff worked, but because it wasn't second nature and wasn't practiced he had to do it all mentally anew each time. And that is a real drain on the brain. We switched to something with more review and now that, coupled with the conceptual understanding he had before, has finally got him doing really well. .

 

I'd love a program that had good conceptual meat to it, but did lots of repetition on a daily basis of the old stuff...some kind of meld between Math Mammoth and TT. Anyone find that and I'll buy them a drink.

 

 

You will get no argument for me that a good math education require both conceptual understanding and procedural competence. We might quibble about how to get to "procedural competence," and here children may differ in what mix of mental math stratagies, games, and "drill" are best. But, there is no question that procedural competence WITH conceptual understanding is the goal one should strive for.

 

And the math mix should promote *thinking* and creative problem solving on top of everything else.

 

Bll

 

 

[Spy Car]

I will disagree with some others. Because a student understand the concepts does not necessarily mean they it's enough to ensure retained mastery, if that makes sense.

 

Good math programs have plenty of practice. Some kids need more, some need less. It isn't difficult to add or subtract practice problems as needed. But if one doesn't teach for understanding hoping that endless repetition of problem-sets will create "insight" is a low probability bet. Even if it "works" it is a very inefficient (boring, and tiresome) way to learn math.

 

Bill

[Kathie in VA]

 

I really think the question of "mastery" vs "spiral" is overblown, and takes away from the more important initial question one should ask and that is does the math program in question have a "procedure only" (or almost only) approach where children mechanically learn "how" to do math, or does the math program teach both procedural competence along with a through understanding of the underlying mathematics?

 

If you don't teach "why" but only "how" the spiral review become more of a factor as kids tend to forget what they don't understand in the first place. Hence the need for constant review.

 

Not to mention that so-called "mastery" programs are constantly working the acquired skills as new ones are added. Just because a new topic involves area, or quantity, or metric measurement, or some other "topic" doesn't mean that addition, subtraction, multiplication, division, fractions, decimals, etc. (depending on the level) are not being worked constantly under the new "topic."

 

Further, if a program teachers deeply the math operations are understood by children to be interrelated. Addition and subtraction are interrelated (subtraction is adding negative numbers and is the inverse of addition). Multiplication is serial audition. Division is inverse of multiplication. Fractions are another form of division, etc.

 

If a child really unders the math, the order of topics becomes a secondary concern.

 

While "spiral" vs "master" deserves some consideration it is far better to ask the most important questions first. Is the math program aiming for deep mathematical understanding with procedural competence, or is it limited to teaching "how" to do math? Or is it failing on both? Answer these questions first and one will be a lot better off in the long run.

 

Bill

 

 

I like your thoughts. Your comment on why spiral is sometimes needed more makes sense.

 

However, I never understood why division seems to always be thought of as just the inverse of multiplication. I get how works, but it is not the complete picture.... at least not when I teach it anymore. My oldest lost it when I tried to teach division as an inverse of multiplication... especially when we moved on to long division. So I backed up to multiplication as adding by groups. Then division is nothing more then subtracting by groups. If you don't want to do the division then just do the subtraction by groups. So 200 divided by 5 becomes 200 minus 5... over and over till you can't do it evenly and then you go back and count how many times you subtracted. I had her do a few of these and then showed the long division and how they related... and very quickly long division became the short cut way to solve problems!

I'm just surprised how many programs don't teach this method.

[Spy Car]

 

 

I like your thoughts. Your comment on why spiral is sometimes needed more makes sense.

 

However, I never understood why division seems to always be thought of as just the inverse of multiplication. I get how works, but it is not the complete picture.... at least not when I teach it anymore. My oldest lost it when I tried to teach division as an inverse of multiplication... especially when we moved on to long division. So I backed up to multiplication as adding by groups. Then division is nothing more then subtracting by groups. If you don't want to do the division then just do the subtraction by groups. So 200 divided by 5 becomes 200 minus 5... over and over till you can't do it evenly and then you go back and count how many times you subtracted. I had her do a few of these and then showed the long division and how they related... and very quickly long division became the short cut way to solve problems!

I'm just surprised how many programs don't teach this method.

 

If multiplication is "serial addition" and division is the inverse of multiplication then division is also "serial subtraction." I agree with you. We did this using C Rods (via Miquon-like methods) very early my son's math adventure.

 

It does to the point that these early math operations are all interrelated. It is a good thing to build up that sence of "interrelatedness" as opposed to making them seem like unrelated discrete topics.

 

Agorithm-only math education can work in the wrong direction is building the understanding of this interrelatedness by focusing on procedure prematurely (not the the standard algorithms don't need to be learned and mastered in time).

 

Bill

 

 

[ktgrok]

You will get no argument for me that a good math education require both conceptual understanding and procedural competence. We might quibble about how to get to "procedural competence," and here children may differ in what mix of mental math stratagies, games, and "drill" are best. But, there is no question that procedural competence WITH conceptual understanding is the goal one should strive for.

 

And the math mix should promote *thinking* and creative problem solving on top of everything else.

 

Bll

 

Any thoughts on a program/curricula that promotes thinking and problem solving but also offers plenty of daily review? My son has very limited working memory, so he needs as much to be on automatic as possible so he can use that limited working memory for the problem solving part. MM gave us conceptual, but lacked enough drill/review/etc. TT gives him the daily review he desperately needed, but not enough problem solving...not to my standards, and not enough conceptual stuff. Any ideas on something that would do both? He's naturally very good at problem solving, and GETS the math pretty easily, but I'd like to play to that a bit more. Even more than that, I will have a youngster starting out in a few years and am wondering what to do with her. Thinking maybe CLE plus Miquon.

[Farrar]

Any thoughts on a program/curricula that promotes thinking and problem solving but also offers plenty of daily review? My son has very limited working memory, so he needs as much to be on automatic as possible so he can use that limited working memory for the problem solving part. MM gave us conceptual, but lacked enough drill/review/etc. TT gives him the daily review he desperately needed, but not enough problem solving...not to my standards, and not enough conceptual stuff. Any ideas on something that would do both? He's naturally very good at problem solving, and GETS the math pretty easily, but I'd like to play to that a bit more. Even more than that, I will have a youngster starting out in a few years and am wondering what to do with her. Thinking maybe CLE plus Miquon.

 

For us MM had, if anything, too much drill. But there is also the golden and green series. Would adding one of those help? Or using the CWP alongside TT?

[birchbark]

If a child really unders the math, the order of topics becomes a secondary concern.

 

I think this is putting the cart before the horse. The order of topics is often key to really understanding the math. The problem with spiral programs is not really the amount of review (I think everyone would agree review is needed), but the fact that they tend to jump around. Paul Ziegler talks about this a little. I wish he had more info on it. There might be more in his Getting Started kit.

 

Daniel Willingham, whose book is being discussed on the forums now, recommends this article which also addresses the problem of unorganized topics, as well as the sheer amount of them.

 

I agree with ktgrok that drill must be included with conceptual math. Willingham wrote this article explaining how factual, procedural, and conceptual math must ALL be taught for one to be successful with mathematics.

 

ktgrok, the closest thing I've come to the perfect balance is Systematic Mathematics. This year has been a huge improvement for DS. This is after years of Miquon and Singapore. :leaving:

[ktgrok]

 

For us MM had, if anything, too much drill. But there is also the golden and green series. Would adding one of those help? Or using the CWP alongside TT?

 

 

 

I should have clarified. It had LOTS of drill on the new concept introduced. We only did about half each time. But none on older concepts...so if you were working on geometry there was nothing about converting fractions or whatever. TT has the daily review that he needs. I've thought of doing the CWP alongside.

[Spy Car]

 

 

Any thoughts on a program/curricula that promotes thinking and problem solving but also offers plenty of daily review? My son has very limited working memory, so he needs as much to be on automatic as possible so he can use that limited working memory for the problem solving part. MM gave us conceptual, but lacked enough drill/review/etc. TT gives him the daily review he desperately needed, but not enough problem solving...not to my standards, and not enough conceptual stuff. Any ideas on something that would do both? He's naturally very good at problem solving, and GETS the math pretty easily, but I'd like to play to that a bit more. Even more than that, I will have a youngster starting out in a few years and am wondering what to do with her. Thinking maybe CLE plus Miquon.

 

 

I'm not sure if it's your cup of tea (or not) but the Standards Edition of Primary Mathematics has a whole separate book called "Extra Practice" geared precisely at children who need more computational work. They also have Intensive Practice books that provide review with greater challenge and increased problem solving requirements.

 

Or there is a reasonable chance you need to use a mix of math resources and programs to fit your needs in different areas.

 

Have you tried any of the "extra math" websites that are available (people help me remember the names)?

 

There is nothing wrong with using multiple sources to get the right mix. If you find something that makes concepts clear, something else that gives good "practice," and something else that adds creative problem solving, you'll have what you need. It is not as convenient as having everything in one bag, but the big advantage of home educating is being able to tailor to our children's needs.

 

Bill

[wapiti]

Any thoughts on a program/curricula that promotes thinking and problem solving but also offers plenty of daily review? My son has very limited working memory, so he needs as much to be on automatic as possible so he can use that limited working memory for the problem solving part. MM gave us conceptual, but lacked enough drill/review/etc. TT gives him the daily review he desperately needed, but not enough problem solving...not to my standards, and not enough conceptual stuff. Any ideas on something that would do both? He's naturally very good at problem solving, and GETS the math pretty easily, but I'd like to play to that a bit more. Even more than that, I will have a youngster starting out in a few years and am wondering what to do with her. Thinking maybe CLE plus Miquon.

 

If there isn't a single program that compromises too much for your needs on one angle or the other, it seems to me that your choices are (1) supplementing your main, daily-review program with problem solving (thinking out loud, supplementing concept instruction might be trickier, if that is on your list of wants) or (2) choose a different main program and then supplement with daily review (from TT or something else). Question for further thought (not something I'd know the answer to for someone else's student): if problem solving is a significant strength, are you interested in teaching the math itself through problem solving? Or would he do better just working on problem solving on the side, after the procedures are automatic, due to the working memory issue?

 

Possible problem-solving supplements for your ds: besides CWP, perhaps he could try Alcumus (if not now, then eventually; it starts with prealgebra-level). There are other problem-solving resources as well, e.g., Zaccaro books, MOEMs and other contest books, etc.

 

At the elementary level, I was under the impression that MEP is spiral, but I have no idea how much of that is daily review... maybe someone will pipe in about that.

 

eta, this is what I was trying to say:

There is nothing wrong with using multiple sources to get the right mix. If you find something that makes concepts clear, something else that gives good "practice," and something else that adds creative problem solving, you'll have what you need. It is not as convenient as having everything in one bag, but the big advantage of home educating is being able to tailor to our children's needs.

[Tiramisu]

Good math programs have plenty of practice. Some kids need more, some need less. It isn't difficult to add or subtract practice problems as needed. But if one doesn't teach for understanding hoping that endless repetition of problem-sets will create "insight" is a low probability bet. Even if it "works" it is a very inefficient (boring, and tiresome) way to learn math.

 

Bill

 

One reason why CLE is effective is that it doesn't have problem sets of endless repetition; it provides consistent limited review of different types of problems which can, with some kids at least, keep it from being boring or tiresome while reinforcing the learning of concepts and, hopefully, long-term retention.

 

Again, my experience has shown me, with one child at least, that conceptual programs that supposedly focus on understanding have not led to long-term retention. How each child best transfers new learning to long-term memory is different. We've had the educational testing which revealed that this particular child learns best through frequent review. The results of the testing have been confirmed in real life. Most of the real life happened before the testing, so the testing only validated our experience.

[NASDAQ]

 

However, I never understood why division seems to always be thought of as just the inverse of multiplication. I get how works, but it is not the complete picture.... at least not when I teach it anymore. My oldest lost it when I tried to teach division as an inverse of multiplication... especially when we moved on to long division. So I backed up to multiplication as adding by groups. Then division is nothing more then subtracting by groups. If you don't want to do the division then just do the subtraction by groups. So 200 divided by 5 becomes 200 minus 5... over and over till you can't do it evenly and then you go back and count how many times you subtracted. I had her do a few of these and then showed the long division and how they related... and very quickly long division became the short cut way to solve problems!

I'm just surprised how many programs don't teach this method.

 

I have two textbooks on mathematics for elementary school teachers. One (Parker & Baldridge) focuses on division as the inverse multiplication, and mentions division as partition. The other (A Problem-Solving Approach to Mathematics for Elementary School Teachers) gives division as partition, as the inverse of multiplication, and as repeated subtraction.

 

So it _is_ out there. I suspect the focus on division as the inverse of multiplication is because of its application to algebra.

[Spy Car]

If multiplication is serial addition (and it is) then the inverse of serial addition is serial subtraction.

 

Division is the inverse of multiplication, so division is serial subtraction.

 

Bill

 

 

[Kathie in VA]

I have two textbooks on mathematics for elementary school teachers. One (Parker & Baldridge) focuses on division as the inverse multiplication, and mentions division as partition. The other (A Problem-Solving Approach to Mathematics for Elementary School Teachers) gives division as partition, as the inverse of multiplication, and as repeated subtraction.

 

So it _is_ out there. I suspect the focus on division as the inverse of multiplication is because of its application to algebra.

 

Well, good to know it is out there. I like the idea of being clear and teaching all three concepts: partition, inverse of mult., and repeated subtraction. However I'd pick the repeated subtraction first, then partition, then link to the inverse of multiplication. I think it is a shame that most programs don't do this.

 

I get what your saying ... that the inverse of multiplication concept is prep work for algebra. That is probably why they focus on that method.

[Tracy]

However, I never understood why division seems to always be thought of as just the inverse of multiplication. I get how works, but it is not the complete picture.... at least not when I teach it anymore. My oldest lost it when I tried to teach division as an inverse of multiplication... especially when we moved on to long division. So I backed up to multiplication as adding by groups. Then division is nothing more then subtracting by groups. If you don't want to do the division then just do the subtraction by groups. So 200 divided by 5 becomes 200 minus 5... over and over till you can't do it evenly and then you go back and count how many times you subtracted. I had her do a few of these and then showed the long division and how they related... and very quickly long division became the short cut way to solve problems!

I'm just surprised how many programs don't teach this method.

 

FWIW, my dd8 is finishing up CSMP 3rd grade, and both of these concepts are being used to introduce division.

[nrself]

Would you link to CSMP?

[nrself]

I really think the question of "mastery" vs "spiral" is overblown, and takes away from the more important initial question one should ask and that is does the math program in question have a "procedure only" (or almost only) approach where children mechanically learn "how" to do math, or does the math program teach both procedural competence along with a through understanding of the underlying mathematics?

 

 

 

So the question then is which programs are most effective at teaching mathematical understanding? At all grade levels.

[Spy Car]

 

 

 

So the question then is which programs are most effective at teaching mathematical understanding? At all grade levels.

 

 

For young children with invested parents, Miquon is amazing.

 

Primary Mathematics (Singapore) is very strong, the teaching tends to run towards "implicit" rather than "explicit" understanding of the underlying mathematical laws more than is to my liking, which can be rectified by incorporating Miquon (which teaches "explicitly" if a parent follows the notes in the Annotations book). The core PM books also lack built-in concrete teaching, again Miquon fills in that difficentcy (as do the HIGs). The Singapore Model Method is outstanding overall.

 

MEP is outstanding.

 

There are very interesting ideas in CSMP, but we got too busy to continue it beyond the 1st Level, so no expert here.

 

The SMSG (School Mathematics Study Group) materials are (like CSMP) the best of the "New Math" texts, and available from on-line archives. "New Math " is often lampooned but these materials are great.

 

I borrowed many ideas from RightStart ( games, place value ideas, etc) but I have never used the full program.

 

Many people like Math Mammoth and Math-in-Focus as alternatives to PM for teaching Singapore style math. No first hand experience with either.

 

The new Beast Academy series (so far the Third Grade level, 3A, 3B, 3C, and 3D have been published). We have two sections left in 3D. It is outstanding.

 

For higher levels I have less first hand experience so these are (mostly) based on "reputation."

 

Mathematics 6, a Soviet Text translated to English intrigues me. It has gone OoP, but is available in PDF and an ebook is coming. It will not be re-published in hard copy as I have been hoping. I really wanted a hard copy. If anyone has a copy they wish to sell, please PM me!

 

Art of Problem Solving makes outstanding series of books for math-adept students who like discovery based learning that is in depth, challenging, and can handle "wordy." I have the Prealgebra book. It is outstanding. It reminds be of Miquon for big people.

 

The Dolciani Algebra (and other level) math books are highly regarded, expecially some of the more "vintage" editions.

 

Some of the Harold Jacobs books are intriguing. I have used none.

 

Singapore had two series (at least) for upper levels. NEM has pretty much gone away, in favor of Discovering Mathematics. It is integrated math, meaning topics like algebra and geometry get mixed across years rather than being taught as separated courses.

 

I am probably leaving things out, and this should not be considered any sort of "definitive list," but are things on my radar screen.

 

Bill

[Tracy]

Would you link to CSMP?

 

CSMP Math

 

There is additional information in the files of the Yahoo Group.

 

We are finishing our 3rd year. Let me know if you have any questions, or ask them on the Yahoo Group.

[Farrar]

If multiplication is serial addition (and it is) then the inverse of serial addition is serial subtraction.

 

Division is the inverse of multiplication, so division is serial subtraction.

 

Bill

 

 

Bill, can I assume you're aware of the "It ain't no repeated addition" deal from a couple of years ago?

[Spy Car]

 

 

Bill, can I assume you're aware of the "It ain't no repeated addition" deal from a couple of years ago?

 

 

I remember the (wrongness) of the original article. I never saw letsplaymath's blog post.

 

Bill

[Farrar]

I remember the (wrongness) of the original article. I never saw letsplaymath's rebuttal blog post.

 

Bill

 

 

I only ever read it through Let's Play Math's post, which is why I linked that. I liked what she had to say about it and it helped me understand how to teach it better, though the models aren't something I'm enamored of.

[Spy Car]

 

 

I only ever read it through Let's Play Math's post, which is why I linked that. I liked what she had to say about it and it helped me understand how to teach it better, though the models aren't something I'm enamored of.

 

Well, I ain't got me no PhD in no mathematics, but I'm scratchn my head on this proposition:

 

Oh, yeah. “Repeated addition†works fine as a model for 3 • 4 , and for 3 • x , and even for 3 • 1/2. But we have to use a different model to make sense of 3/7 • 51

 

The teacher concedes: Multiplication ain’t repeated addition.

 

Now hold everything!

 

How is adding 3/7s 51 times different than muliptlying 3/7 by 51. Either way you get [21 and 6/7], yes?

 

Why is anyone conceding anything?

 

Bill

[Farrar]

I am scared to answer for fear that I'll get it wrong. My understanding of math feels limited.

 

But I'll guess... If you're doing 3/7 x 51, then you should be doing 51 3/7 times, not the other way around. Even though we know the answer is the same thanks to the communicative property. It's not actually the same if you were talking about a real world application - say 3/7 of 51 pizzas, which would be different from 51 people getting 3/7 of a pizza. (And, ack, yes, I am one of those people who can only seem to do math in food analogies.)

 

The π x r squared example made more sense to me though. Or less... How do you do something, especially a variable, π times.

 

Basically, the thing I got out of it was that it's good to teach the idea of "of" as being the essential meaning, and to teach repeated addition as being one way we see of in action, especially when the numbers are whole numbers. So 3 x 8 is 3 of the 8's, which works fine to conceptualize as repeated addition.

 

Someone please correct me if I've just embarrassed myself. :leaving:

[Spy Car]

I am scared to answer for fear that I'll get it wrong. My understanding of math feels limited.

 

But I'll guess... If you're doing 3/7 x 51, then you should be doing 51 3/7 times, not the other way around. Even though we know the answer is the same thanks to the communicative property. It's not actually the same if you were talking about a real world application - say 3/7 of 51 pizzas, which would be different from 51 people getting 3/7 of a pizza. (And, ack, yes, I am one of those people who can only seem to do math in food analogies.)

 

The π x r squared example made more sense to me though. Or less... How do you do something, especially a variable, π times.

 

Basically, the thing I got out of it was that it's good to teach the idea of "of" as being the essential meaning, and to teach repeated addition as being one way we see of in action, especially when the numbers are whole numbers. So 3 x 8 is 3 of the 8's, which works fine to conceptualize as repeated addition.

 

Someone please correct me if I've just embarrassed myself. :leaving:

 

 

As you noted the Commutive Law makes the first question mute. Either way, it the same thing mathematically. Pizzas, or no.

 

As for multiplying (or serially adding) pi, one has troubles because pi is a transcendental (and irrational) number. Which makes it difficult (impossible) to deal with (without "rounding") as a computation. This is true with variables as one of the factors, or with an positive whole number as one of the factors, yes?

 

But pi • 5 is still the same as pi+pi+pi+pi+pi.

 

All this talk of pizza and pi is making me hungry :D

 

Bill

 

[Farrar]

All this talk of pizza and pi is making me hungry :D

 

 

 

I had pizza for dinner. Throw in fraction talk and it seems inevitable that I'd be posting about pizza.

 

I keep coming back to the idea that even though we can flip it because ab = ba, it's not actually the same thing, even if only for pizzas. And anyway, what if it's pi times pi? To me, it seems like it is repeated addition, except when I read Let's Play Math's explanation, it did seem to make sense. Much more so than the snooty math guy's.

 

Anyway, I'm no expert. Quite awhile back, dh and I got into a big argument about this and at the end, we both just laughed at ourselves because neither of us ever pursued any sort of higher math so it seemed very silly that either of us should know "the answer." I think it interests me in part because it's so basic - what is multiplication - yet also made me think.

[Spy Car]

 

 

I had pizza for dinner. Throw in fraction talk and it seems inevitable that I'd be posting about pizza.

 

I keep coming back to the idea that even though we can flip it because ab = ba, it's not actually the same thing, even if only for pizzas. And anyway, what if it's pi times pi? To me, it seems like it is repeated addition, except when I read Let's Play Math's explanation, it did seem to make sense. Much more so than the snooty math guy's.

 

Anyway, I'm no expert. Quite awhile back, dh and I got into a big argument about this and at the end, we both just laughed at ourselves because neither of us ever pursued any sort of higher math so it seemed very silly that either of us should know "the answer." I think it interests me in part because it's so basic - what is multiplication - yet also made me think.

 

 

Once WTM member Myrtle (do you remember Myrtle?) shot this "multiplication ain't serial addition" article to hell. She was one smart lady. Miss her.

 

Bill

[Taz007]

Farrar,

 

Is this what you are referring to?

 

http://www.artofproblemsolving.com/Videos/external.php?video_id=21