kitchen table math, the sequel: how long does new knowledge last?

Friday, August 31, 2007

how long does new knowledge last?

Unfortunately, this isn't directly relevant to Concerned's post, but it's interesting:

Translated into practice, the spiral curriculum is a series of different, unrelated topics that parade past the kids year after year. Kids dabble in measurement for a while before moving on to the next unit, which may be geometry, which is followed by whole-number operations, which is followed by fractions, ... and so forth. Typically, about 60 school days pass before any topic is revisited. Stated differently, the spiral curriculum is exposure, not teaching. You don't "teach" something and then put it on the shelf for 60 days. It doesn't have a shelf-life of more than a few days. It would be outrageous enough to do that with one topic--let alone all of them.

Bruner's endorsement of the spiral curriculum suggests the extent to which cognitivists lack a comprehensive schema of a kid's brain. Don't they know that if something is just taught, it will atrophy the fast way if it is not reinforced, kindled, and used? Don't they know that the suggested "revisiting of topics" requires putting stuff that had been recently taught on the shelf where it will shrivel up? Don't they know that the constant "reteaching" and "relearning" of topics that have gone stale from three months of disuse is so inefficient and impractical that it will lead not to learning but to mere exposure? And don't they know that when the "teaching" becomes reduced to exposure, kids will understandably figure out that they are not expected to learn and that they'll develop adaptive attitudes like, "We're doing that ugly geometry again, but don't worry. It'll soon go away and we won't see it for a long time"? Apparently not, even though it would take very little time working in a classroom to document all of the above.

War Against the Schools' Academic Child Abuse by Siegfried Engelmann, p. 108

Early in the book he has this to say about practice:

In addition to having a good program, we have a great deal of knowledge about how kids learn and how to teach well. We know how much practice it takes for the kids to master the various details of our sequence. Oddly enough, the amount of practice that we've had to provide to meet our goal is possibly five times the amount provided in other published programs that teach the same subject. We have also learned thast kids tend to "lose" information if we don't keep it "alive" in the program." This observation had led us to activities that require kids to use all the important skills and concepts they've been taught.

War Against the Schools' Academic Child Abuse by Siegfried Engelmann, p. 17

Saxon Math is built on this principle. At the end of the year students are still practicing skills and procedures they learned at the beginning of the year.

Saxon gets content into long-term memory.

I've decided to use a variant of the Saxon "recursive" structure with C. & Singapore Math.*

We're going straight through the Primary Mathematics textbook & workbook starting with 3A. But I'm going to assign daily problems from the Intensive Practice book out of sequence, making them last throughout the entire year.

* The principal in Linda Perlstein's new book Tested describes Saxon this way (will post the passage at some point). I like it.

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary


concernedCTparent said...

Actually, I think it's very much related to the post about overlearning particularly as it applies to the "spiral". The problem Engelmann discusses as to the spiral curriculum being exposure and not teaching is exactly one of my biggest bones with Everyday Math. I only have experience with Singapore Math at this point, but all this information certainly makes a very strong case for Saxon, doesn't it?

I think that's why the article spoke to me so loudly. It's been weighing heavily on me since I decided to homeschool my fifth grader. I keep wondering if Singapore is enough...

This will require further thought, I suppose.

Catherine Johnson said...

Yes -- definitely.

It doesn't relate directly to the question of how much time, etc.

If I were you I'd adopt the Saxon approach; there's no question it gets material into long-term memory AND KEEPS IT THERE. It's amazing.

It will take some work; you'll have to create your own problem sets to some degree....but I don't think it's going to be hard.

I'm going to be doing it myself. Not sure what's the most efficient way to go about it --- draw up a list of problems, then choose one problem type from each category for daily practice sets??

Catherine Johnson said...

I don't think the 20-different-problems a day approach probably applies to building fluency, but I'm not sure. Athletes do a lot of massed practice.

We desperately need a textbook on precision teaching.

Anonymous said...

"I keep wondering if Singapore is enough... "

It is my understanding that in Singapore there is a lot of additional practice besides the workbooks.

I'm using Singapore with my son (we homeschool) and I supplement the workbooks a lot. This is working so far ...

-Mark Roulo

concernedCTparent said...

Mark... what do you supplement with? The plan was for my 9 yo (5th grade) is currently 4B/5A/5B for this year (4B is more a safety net to plug any gaps left by Everyday Math last year). She completed the Saxon placement and tested into Algebra 1/2. I'm not sure how that all plays out but I'd be interested in hearing what you're doing and how it's working for you.

I like Catherine's idea of creating problem sets but if that's what I'll be doing I've got lots of catching up to do. I hadn't originally planned to "rework" the Singapore aside from mixing in intensive practice and challenging word problems so creating a system of cumulative review would be an interesting challenge. There is some review incorporated into each Singapore unit but definitely not to the degree suggested by the study on the "spacing effect" or in place in Saxon.

Anonymous said...

I use Singapore AND Saxon AND the fact sheets I made up myself.

It is also very common for homeschool parents to use two different math textbooks. The common combinations are:

For early grades: RightStart + Singapore, or Miquon + Singapore

Most use Saxon + Singapore, or Horizon + Singapore.

You don't necessarily need to match the contents of both, just go through one alongside the other.

The other options are: Singapore + KUMON, or Singapore + ALEXS.

Tex said...

I will be examining my daughter’s new NSF math program, Think Math!, to see if it teaches to mastery and how long the interval is before it spirals back. In fact, this will be a good question on which to focus, instead of jumping on the school with all the other potential issues. (Other issues including the emphasis on group work, which I detest for my daughter’s math learning.)

Thanks for this post. As usual, KTM is helping me out.

concernedCTparent said...

Although my fifth grader tested into Saxon algebra 1/2, I'm thinking 8/7 may be a better supplement/enrichment alongside Singapore 5A/B. Or would Algebra 1/2 provide sufficient revisting of previously learned concepts? Any thoughts?

Catherine Johnson said...

I like Catherine's idea of creating problem sets but if that's what I'll be doing I've got lots of catching up to do.

I'll get some more things posted, but anyone who wants copies of the original articles should just email me at

I've got both (including the one I found this morning on "shuffled problems.")

Once you read them, you realize it's not especially hard.

The basic principles (off the top of my head) are:

* "mastery" means 100% correct answers on a problem set - no more studying after that until later

* there's an optimal spacing ratio (will get that posted)

* you can create "shuffled problem sets" simply by distributing the problems in the book across weeks & months

I'll probably try to be semi-scientific about this, seeing as how C. will be taking algebra at school & doing 3rd grade math here at home....which means I will, again, be trying to keep his head above water at school, etc., etc.


I'm going to read these two articles side-by-side & figure out THE SINGLE MOST EFFICIENT WAY to incorporate both the "shuffled" principle AND the "efficient study/spacing" principle.

At the moment, my plan to use the "Extra practice" workbooks as a source of problems to distribute seems good.

I'm also going to take a close look at the two mixed practice books I have & see if they might work.

I also think you could simply take any standard workbook and simply assign one problem from each page in the book (or from 10 pages in the book - whatever works) & do a different 10 pages on different days.

Catherine Johnson said...

You could also use edhelper for this.

Say you have a list of all the problem-types your child knows to date....just print them all out and keep distributing them as days go by.

PaulaV said...

I used edhelper quite a bit last year with my third grader for math and spelling. While he was busy learning his addition and subtraction math facts in Kumon, his class was forging ahead with multiplication, measurement, geometry, and other topics. I would print corresponding worksheets from edhelper and he would do these first and then his Kumon lesson.

The one thing I love about Kumon is the mixed review. Today he worked on multiplication, divison and subtraction problems in one set.

I had wanted to use Saxon this summer, but chose not to because I felt it would be too much for him while doing multiplication and division in Kumon. Saxon will have to be done on the weekends during the school year.

Anonymous said...

"Mark... what do you supplement with?"

My wife and I supplement with a number of different things.

*) We use flashcards for basic plus,minus,times,divide drill. Also "telling time".

*) SM has "extra" worksheet books with more (and, I think, usually more difficult) problems. We use those sometimes.

*) My wife keeps an eye out for math worksheet books (like from scholastic) and we sometimes use those if they seem reasonable.

*) I make up my own problems. This is not as hard as it sounds because I often make them up with him next to me. I can tune the problems to how he is doing *THAT DAY* (e.g. if I give him a two-part word problem with three digit subtraction using borrowing and he nails it, we don't need to do five more of those that day). I also do this because I want to get multi-part word problems in sooner than Singapore Math does. So far this has not been a problem.

For example, we've been spending some time the last month working on "place value". To do this, I mostly just ask him to write out numbers for me. E.g. 2000, then 2020, then 4006, etc. By paying attention to the mistakes he makes, I have an idea what to focus on.

My wife and I (mostly me!) may be outliers for a few reasons:

1) We are homeschooling.
2) We are overemphasizing math
3) I don't think my child is particularly gifted at math
4) I am determined that he be *very* competent at math

The result is that we spend a disproportionate amount of time on math, even though we don't think he is gifted at it. Of course, "disproportionate time" is probably 30-45 minutes per day ... not hours.

I don't know how relevant what we are doing will be to you (but I am/was willing to answer your question ... :-))

-Mark Roulo

concernedCTparent said...
This comment has been removed by the author.
concernedCTparent said...

Thank you, you've been most helpful. I find much of the input at KTM be so. It's most always worth pondering, evaluating and researching further.

My daughter is not necessarily gifted in math either, but she has certainly demonstrated talent and more importantly, enthusiasm for mathematics. She often says it is her favorite subject-- I want to keep it that way.

Finding an efficient way to do that without spending inordinate (and wasteful) amounts of time doing so is of extreme interest to me (and likely to her as well).

Catherine Johnson said...

This is funny:

1) We are homeschooling.
2) We are overemphasizing math
3) I don't think my child is particularly gifted at math
4) I am determined that he be *very* competent at math

If you substitute "afterschooling" for "homeschooling," that's me.

I am absolutely determined that C. be very competent at math.

Of course, at this point I've scaled back my goal to "very competent at arithmetic."

Catherine Johnson said...

That last was a semi-joke; I'm still determined that C. be VERY competent at K-12 math PERIOD.

The whole thing is so frustrating (now there's a novel remark!)

C.'s natural abilities lie in the verbal arena. We could easily teach him everything he needs to know about reading, writing, and history -- and we could do an OK job on literature.

What we need from our school is solid and accountable instruciton in math & science.

We've had terrific instruction, at the middle school level, in science.

But math is a dead loss.

The one thing we desperately need from our middle school is the one thing we don't have.

The whole district seems to be run this way.

Find the kids who are "naturals," put them in advanced classes where the kids are heavily self-teaching; consign everyone else to the slow track.

Catherine Johnson said...


The other thing that is SOOOOOO frustrating is that C. is your classic "good student." He's plenty smart enough to learn this material & learn it well, he likes school, he doesn't have emotional issues/ADHD/behavior problems ---

We're lucky enough to be bringing all this "to the table," and the school still can't teach him math.