kitchen table math, the sequel: Research for The Spiral?

Tuesday, May 1, 2007

Research for The Spiral?

I'm trying to find the original research that spawned spiraling curriculum.

I can't find it. Admittedly, I haven't looked all that hard.

I've pulled out my copy of Everyday Math's Teacher Reference Manual. Surprisingly, spiral, spiraling, and spiral curriculum are not in the index or the table of contents. In fact, I can't find the word spiral anywhere in the book. It must be there, but I'm not going to read it cover to cover just to locate it.

Anyway, the introduction makes several vague statements that seem as close to promoting a spiral as I can find. For example, here's a representative quote:
Students using Everyday Mathematics are expected to master a variety of mathematical skills and concepts, but not the first time they are encountered. Mathematical content is taught in a repeated fashion, beginning with concrete experiences. It is a mistake to proceed too quicly from the concrete to the abstract or to isolate concepts and skills from one another or from problem contexts. Students also need to "double back," revisiting topics, concepts, and skills, and then relating them to each other in new and different ways. (p. 3)
Why is it a "mistake" to proceed quickly to the abstract? This is a reference manual for 4th through 6th grade teachers. This is not kindergarten where I can see abstraction might be counterproductive. But why not get to the abstract pretty quick once the foundation is set?

The problem as I see it, both on the concrete-abstract concept, and the whole spiral thing (repeated encounters?) is that EM isn't telling me how they reach these conclusions.

They toss these statements out there, with no reference, no support, no research. Where is it? They can't have imposed this spiraling drive-by exposure thing on our teachers and students with no research, could they?

The manual continues:
. [R]epeated exposures to key ideas presented in slightly different contexts are built into the EM program. (p. 5)
What makes them think that a key idea should be presented through repeated exposures?

I don't see this as intuitive and I'm more than a little annoyed that I can't find even a real kernel of research that might support the whole spiral thing.

In our experience over the past five years with EM, it is the spiral that is the most pernicious and destructive aspect of EM.

For example, the manual states that the square root symbol is introduced for the first time in 4th grade. Students need this exposure because of the symbol's "unfamiliarity and peculiar look."

By the time my daughter got to 5th grade, she had forgotten that she had ever known what the symbol meant. This is just one illustration of how the exposures are inefficient. There seems to be nothing to gain by introducing a square root symbol in the 4th grade when they won't be using it until the 5th grade. There was no time saving at the 5th grade level. A year is a long time to keep in the memory a concept you were exposed to, but never used or mastered.

So I'm left with my initial inquiry, is there some research somewhere that says kids learn better when they have repeated exposures to a concept before they are expected to master it?


SteveH said...

Look for "circling" or "repeated partial learning".

My son actually brought home something that didn't look like Everyday Math. It was a worksheet of fraction problems - add, subtract, multiply, and divide, including mixed fractions. Practice and mastery. Score one for the teacher.

I could come up with both good and bad definitions or examples of spiraling, depending on the steepness (?) of the spiral. In Everyday Math, the steepness is very shallow (circling), so that a complete topic is not properly covered each time. For my son, it's OK because he just absorbs it and requies little practice (so far). For many, EM moves right along to the next topic before some specific amount of material is understood and mastered.

The problem with EM is that they don't define the topics covered in the spiral very well. For example, they don't cover adding and subtracting fractions completely before they move on. Since they don't believe in much practice (drill and kill) then this is OK to them. The kids will see it again. But without mastery, some will have to start from scratch. One parent complained to me that she had kids in 3 different grades and they were all working on the same material!

Spiraling can be done well, but it's not an excuse for low expectations and less practice. My opinion is that the reason they use spiraling is that they wanted a method that makes social promotion look pedagogically sound. They lower expectations and practice and hope that repeated exposure to the material can take the place of mastery. It will work for some (especially with all of the supplementing going on), and I suppose that for the rest, they will just think that it's their own fault.

Anonymous said...

I have no idea about the research, but I'll note that spiraling works for some subjects.

Like history.

1st round for a young child, you go over George Washington and the cherry tree (maybe tell kid it is a myth). You read "George Washington and the General's Dog".

2nd round as the child is a bit older, you read the Landmark Book biography of George Washington. Maybe find out something about Valley Forge and the French and Indian War.

3rd round, you have the child read the Federalist Papers and a very detailed biography of George. Find out why *he* was selected to be the topmost general even though more skilled military choices were available.

For *history*, this works well.

I suspect that it works well for some other fields, too (geography comes to mind. "Science" seems to be taught in a kinda spiral with easy/fun stuff in the early grades, but no math [e.g. play with magnets] and then more rigorous versions of the same thing later).

Could this just be a case of generalizing inappropriately?

-Mark Roulo

Unknown said...

Hi, Lynn.

I would also be interested in seeing some empirical research that supports the idea of spiraling. Alas, I'm not going to hold my breath.

The genesis of the idea of spiraling curricula can be found in the work of Jerome Bruner (behold the face of evil):

In the 1960s Jerome Bruner developed a theory of cognitive growth. His approach (in contrast to Piaget) looked to environmental and experiential factors. Bruner suggested that intellectual ability developed in stages through step-by-step changes in how the mind is used. Bruner's thinking became increasingly influenced by writers like Lev Vygotsky.

Unfortunately, Mr. Bruner and/or those who sought to make use of his ideas misinterpreted Vygotsky's as a prescriptive theory, arguing that if intellectual ability develops "in stages through step-by-step changes in how the mind is used" (probably true, and certainly consistent with Vygotsky's social development theory), then, by gum, that's the way we have to TEACH kids every concept—"in stages through step-by-step changes."

A classic load of baloney that tells us that because it's natural, it must be better. It's also a complete misread of Vygotsky, who saw schooling as another social institution which required a change in "how the mind is used."

Anonymous said...

Mr. Person,

Have you read Vygotsky? I haven't, his long list of books is a bit overwhelming. Where would one start?

LynnG said...

Steve, I have absolutely seen an increase in quantity of problems to be solved. When my son was a 5th grader in EM, none of these fraction worksheets existed. Four years later, I have to say that EM has responded to the criticism that the program lacked sufficient practice. 5th grade EM is better now than it was 4 years ago.

OTOH, think of where they started from.

LynnG said...

Wait a minute, did you say divide fractions?

That's the one thing EM left out. At least, it was left out of the EM fraction unit we are doing here.

We've seen add, subtract, and multiple fractions, but no division.

For that we turned to Singapore Math afterschool.

LynnG said...

Ultimately, I think the biggest selling point on the spiral is purely political.

The school administrators and teachers love to point out to parents the early exposure to higher level thinking. Our first grader's teacher positively beamed when she said they were introducing "statistics" in the first grade.

And what parent can really complain about the square root being introduced in 4th grade?

Isn't this proof that we have "raised expectations"?

Whether or not it is inefficient is not seriously considered.

Barry Garelick said...


On the Everyday Math site, there's a document called "The Research Basis of the EM Curriculum" by Andy Isaacs and William Carroll. In it, (p. 6), the following statement appears:

"An investigation by UCSMP of U.S. mathematics textbooks found that from first through eighth grade, more than half of each year’s program was typically devoted to a review of topics from previous years (Flanders, 1987). In those textbooks, a topic was typically introduced and practiced for several weeks and then largely ignored until the following year, when it was reviewed, practiced, and perhaps slightly extended. This cycle of annual repetition with little substantive development was severely criticized by researchers who studied U.S. and foreign textbooks (McKnight et al., 1987; Schmidt, McKnight, & Raizen, 1997). Texts that were essentially medleys of disconnected topics arranged in a flat “spiral” were identified as a prime reason for U.S. students’ poor performance on international tests."

WHAT? Are they saying the spiral is bad? No. There is a footnote right after that passage, and the footnote says:

"Note that these findings do not imply that all spiral curricula are necessarily flawed, only that the traditional U.S. ones are. [Ed. OH OF COURSE, HOW SILLY OF US TO ASSUME OTHERWISE] Indeed, Thomas Romberg, the general editor of the NCTM Standards, wrote as his first “principle of curriculum engineering” that “The main generic schemata (i.e. measurement, mappings, proportionality) that we wish to develop in school children must be identified and a spiral curriculum built around those conceptual strands” (Romberg & Tufte, 1987)."

WHEW. They had me worried there for a second.

This document can be found at:

LynnG said...

So, Barry, when the EM Reference Manual says:

"The program has been created so that it is consistent with the ways students actually learn mathematics, building understanding over a period of time, first through informal exposure and then through more formal and directed instruction."

They are basing that assertion of how "most" students learn on a review of U.S. v. foreign textbooks? Not on research of how students actually learn?

What a shock. [said with sarcasm]

Catherine Johnson said...

I'm not going to take the time to find it now, but Willingham reports that if you study the same material 3 years in a row you remember it forever.

Spiraling with mastery is what you want; I would say that's what Singapore Math does (the series definitely spirals).

Bruner's idea, as I understand it, was that you could come back to a topic each year and learn more about it (more complicated facets of the topic, etc.)

That seems to have been lost in translation.

A mom here told Ed that her child had done some kind of graph in 2nd grade.

Then in 3rd grade she did the exact same graph, except she put labels on.

Something like that.


SteveH said...

"Wait a minute, did you say divide fractions?"

The supplementing that my son gets has nothing to do with anything EM is doing. It's something that the teacher is making up or getting elsewhere.

Catherine Johnson said...

Gosh....who left the fantastic quote from her parents' book --- heck.

I think this was on the old site.

LynnG said...

The supplementing that my son gets has nothing to do with anything EM is doing. It's something that the teacher is making up or getting elsewhere.

That explains it.

Our teacher does a tremendous amount of supplementation as well, but none related to fractions. Division of fractions is not on the CMT test, so it isn't a concern.

All supplementation here is directly related to performance on the CMT. I wish they showed such single-minded focus on getting kids ready for algebra. Then we might get somewhere.

Barry Garelick said...

In the fifth grade, my daughter's math teacher didn't supplement EM so much as "supplant" it, which was fine with me. He only relied on EM for the review pages. Still, he was scurrying, providing Xeroxed pages from an old textbook. He was definitely trying to keep one step ahead of the reaper.

In Sixth Grade, I tutured my daughter using Singapore to make up for what she learned weakly in 4th and 5th grades. (Her 5th grade teacher was good, but the 4th grade didn't provide a good basis because her 4th grade teacher relied on EM--that's the big problem here. Some teachers supplant/supplement, and some don't. So you don't necessarily get a good result). In sixth grade EM treats division by fractions and I was hell bent to teach them addition, and multiplication of fractions via Singapore and then division before they got to it in EM. I succeeded. I was losing sleep at nights. EM does a poor job with fractional division in the 6th grade workbook. For all their talk about making sure students understand, they just taught the procedure, with no build up to the "pattern" of invert and multiply, which Singapore does very carefully through several grades.

Lynne said They are basing that assertion of how "most" students learn on a review of U.S. v. foreign textbooks? Not on research of how students actually learn?

Well, if students were learning from a good textbook like Singapore or Saxon, that wouldn't be bad would it?

Pamela said...

JMO (and I haven't read the others yet, but), I think the spiral is backwards in situations like you mention.

It seems that they:
intro and do little
then intro and do more
then intro and do a little more

If they did it to mastery originally, they'd just have to tie it to other things as reasonable and review ocassionally to keep the skill, concept, tool viable.

Doing the spiral after the mastery allows for less time to be taken up re-teaching COMPLETELY each time they come back to the topic.

It also would allow the majority of students to be more confident and capable. They KNOW this and can build upon it.

Exo said...

That spiraling thing bothers me a lot. When I started teaching science in middle school, I found out that the first topic all science teachers do in 6th, 7th, and 8th grade is the scientific method and measurements. EM book for K-grade my son brought home has the HW tasks (called Home Links) that introduce ... guess what? Scientific method and measurements! And I still wonder, why EVERY science textbook used in school starts with .. Scientific Method and Measurements!
Well, I thought (a foreigner), since you studied scientific method and measurements, I can start using it without wasting my time - so, go ahead, do the lab - measure mass, volume etc, write a formal report. Oh, boy, I was mistaken! I had to teach that mass is measured in grams, and length in meters, and volume in liters to 8th graders. And scientific method... I wasted a good week of time on that - teaching it fresh.
Now, that's spiraling, I assume.
When I was a student, they taught measurements and calculations for measurements in MATH (grade 5, I believe) and our biology, chemistry, and physics teachers in later grades never taught this topic. But we were applying it a lot. Is that a spiraling? Or integradted curriculum? Or interdisciplinary connections? I don't know. But I would love if I had my 8th graders coming to me with the skills learned in math so I could have them apply those skills in science. (I will never forget teaching my 7th graders to substitute numbers for letters in physics formulas and the way to derive a formula from a given. a=b/c, then c=b/a type of things. They didn't know that a=a/1)

Anonymous said...


I'm tired of every science book for young children beginning with "Our Five Senses."

Unknown said...

Testing comments.

Ah, I see.

SteveH said...

"Doing the spiral after the mastery allows for less time to be taken up re-teaching COMPLETELY each time they come back to the topic."

My point exactly. They don't like mastery because it's too much like drill and kill, so they use the spiral to avoid the issue. They are now forced to add in more practice, but I wonder what is taken away to provide the time for this?

Anonymous said...

Singapore Math does spiral but with an emphasis on moving away from the concrete at the earliest possible point, rather than belaboring it.

As someone else here said, it's spiraling with mastery.

LynnG said...

EM's spiral feels very flat as you watch the progression from grade to grade. On the other hand, and quite by random, things seem to drop from the moon, with no foundation knowledge taught.

This is called enrichment.

Also higher order thinking skills. Probably a little self-discovery, too.

Exo, I don't think science has ever been seriously aligned in US schools. Is there a scope and sequence of topics for science in elementary through HS?

It seems like there is little effort to think it through one year after another.

Next year, science is going to be tested on the state exams.

Maybe NCLB will have some beneficial impact on science.

Dickey45 said...

A free book I picked up to read during my break time:
Jerome Bruner "The Process of Education" 1960.
pg 13 "A curriculum as it develops should revisit these basic ideas repeatedly, building upon them until the sutdnet has grasped the full formal apparatus that goes with them. ...Theres is much still to be learned about the "spiral curriculum" that turns back on itself at higher levels, and many questions still to be answered are discussed in Chapter 3."

Ugh. Reading this is torture.

Dickey45 said...

lynng - each state is supposed to have state standards for science. How well they are written is the question. Oregon's are very poorly written, fuzzy without examples of what they mean. I hate that. Makes it hard to make curriculum against something written so poorly that you have to guess at their meaning.

Anonymous said...


I've seen a user of Saxon admantly deny that Saxon was sprial. They said it was "incremental" instead.

Is there merit in making a distinction between a sprial and an incremantal approach?

Anonymous said...

I suppose Saxon "spirals," but I think for those of us who have used the curriculum we just call it "review."

Saxon introduces a subject, for instance, but expects some form of mastery on that particular level. The next year it may come back in the early chapters as a review, but it will often add to that particular concept more complicated thinking.

Again, Saxon expects the new level to then be mastered through practice and then distributed practice throughout the book to ensure that the new parts have been learned.

Anonymous said...


maybe the key issue is "mastery" more so than "spiral"? It does seem that Saxon doesn't instruct a new topic for the entertainment of the child, but rather expects the child to apply what he has learned to problems. I get the impression that some math programs are spiral in the sense of "math appreciation" , they expose the child to a concept but it really goes nowhere.

To add to this confusion, I have no idea what to categorize Singapore as. For example, they treat each topic one at a time, but they don't teach EVERYTHING about that topic that one year. However, it does require the student to master whatever the topic is. If the same topic is revisted the following year there is a speedy review. I thought that this was "spiral" then folks told me it was mastery.

And truthfully I've somewhat given up figuring out the correct definitions of these terms.

Anonymous said...

I agree, Myrtle,

These terms seem to mean what people want them to.

I will say that when I use the word "mastery" around teachers I tend to get a blank or neutral look.

Saxon is very methodical about how the concept was introduced earlier and how far to push past the last exposure.

For instance, basic division is introduced one year, but each year (and even later within the same year)it gets more complicated, but with those "complications" (for lack of a better word) being explained and practiced throughly.

Dickey45 said...

Wow, if a teacher doesn't know what mastery is that would seem to me to be a problem. If they have kids with IEPs in their classroom they have to deal with the issue of mastery. A well written IEP requires measurable objectives - usually 90% or so. Mastery is like 95% to 100% - and should be followed by being able to do the objective over a period of time like 3 consecutive times.

Anonymous said...

I think more teachers are associating the word "mastery" with things like "kill and drill."
I think they think it's an old-fashioned term from another era. I'm guessing. None will say.

We have Math Trailblazers around here and so the word "spiral" is tossed about quite a bit. My kids escaped it, one being SpEd and the other accelerated. However, all of the same EM language is there.

They don't have time for math facts, they're problem-solving--

Real Problems. None of that math stuff.

Anonymous said...

They don't have time for math facts, they're problem-solving--
To respond to this
I'll steal a quote from someone that said this on a different blog, (RightWingProf's) problem solving is all that neat "outside the box" stuff. Most of the time I'd be happy to get them thinking inside the box.

Unknown said...

The way most teachers (and publishers) I've heard use the term refer to "spiraling" as a process that occurs within, not outside, a grade level.

Anonymous said...

Mr. Person,

That makes sense.