kids need spiraled practice, not spiraled instruction

Hainish left a link to a terrific post by a math teacher:

11th grade here = 9th grade here. In fact, Algebra 2 was such a rehash of the district's Algebra 1 course that some teachers called it "Algebra T-o-o." And really, the same point could be made about math curriculum as a whole in the U.S., since most content for any given year is a review of content from previous years. (The Common Core State Standards may help change this, but I'll believe it when I see it.)

This approach, where we touch on lots of topics each year--rather than go deep with fewer topics--and then revisit them in subsequent years is often called spiraling. But what it is for many students is stifling. And this is as true for kids who've yet to master a skill as it is for those who nailed it right away. I first noticed this when I taught 9th grade Algebra classes where every student was performing at least two years below grade level.

"Meet them where they are," fellow math teachers advised me. Makes sense, I thought, since I couldn't imagine teaching Algebra to kids who didn't know basic arithmetic. But what I soon learned is that perception matters more to students than performance. For many kids, having seen something is akin to having learned something. "Man, we already know this," students said, as I presented lesson after lesson on fractions, decimals, and percents.

Other students, meanwhile, knew they didn't understand the material, but had given up hope of ever understanding it. The implication was therefore the same for all students: encore presentations on previous years' topics were pointless. And though I was able to engage a few students when I found new ways to present old topics, one group of students was always slighted: those who really did "already know this."

[snip]

The problem, of course, goes back to the disconnect between kids seeing something and actually learning--and retaining--it. But if it didn't sink in for them the first, second, or third time a teacher presented it, why should we present it again?

We shouldn't. At some point the focus needs to be on students practicing math rather than teachers presenting it.

[snip]

[W]e should provide students spiraled practice, not spiraled instruction. When I did this in 10th grade Geometry classes, students said they learned more Algebra than they had learned in their 9th grade Algebra course. And, as a result, they were ready for more advanced math--starting with Algebra T-w-o.
Spiraled Instruction, Stifled Learning
By David Ginsburg on March 5, 2012 8:35 PM

Wow!

momof4

It doesn't say anything good about ed school that only one basic science course is required. BTW, as first-graders in the mid-50s, my class learned parts of plants, plant nutrition, photosynthesis, heliotropism etc. Why am I not surprised that global warming is part of this curriculum? I also agree with the last comment about the inefficiency of this approach, but efficiency seems to be fighting with mastery for last place in the pantheon of ideas that concern the ed system.

I love that: efficiency fighting with mastery for last place.

In my experience it's no contest. Mastery is last and efficiency isn't even in the running.

Public schools are almost anti-efficiency at all levels: in hiring and spending as well as teaching and learning.

That's why public schools are happy to slow the progress of gifted students by substituting enrichment for acceleration.

chemprof on handwriting

My students' handwriting is generally abysmal. A lot of them write like elementary school kids, and it impacts their ability to take notes, to solve algebraic problems, and to do well on exams.

The whole idea that they'll all just use computers so handwriting doesn't matter is baloney.

Absolutely.

learned disability

from the Comments thread on Paul's post, Creating Learning Disabilities,

Exo wrote:

I think you are right, Paul. Learned disability - and it's almost impossible to correct in later years.

I see the same in my HS science classes. Elementary computations, numbers make them look like the deer in headlight...They ARE afraid. The ones that are not are either my ESL students who recently moved to the US or "math kids." And please, we don't do anything higher than what in Soviet schools would count as 6th grade... Maybe even 5th.

It's just that immediate "I don't get it" as soon as the numbers are involved.

I think the psych term for this phenomenon is learned helplessness.

According to this woman, it takes about 5 minutes to induce learned helplessness in a 20-year old.



Unfortunately, the teacher ends up saying girls are specific victims of learned helplessness.

I'm pretty sure the person operating the camera doesn't agree.

Creating Learning Disabilities???

This quarter our school established three cohorts in the 7th grade and three in the eighth that are based (primarily) upon academics. We also split our math blocks in half with one half dedicated to our grade level curricula and the other half dedicated to remediation of core number sense skills. The core skill blocks are further homogenized such that there are six distinct groupings that are independent of grade level. If you are in the seventh or eighth grade you are in one of three groups attending normal curricula for your grade and in one of six groups attending remediation, independent of your grade. These splits are the best we could do based upon scheduling, teacher, and room considerations. Our goal was to create the most homogeneous groupings possible.

After about five weeks with this schedule I've come to the conclusion that there are two kinds of learning disabilities. There are those that are inherent to the child and there are those that we have created. Each of my grade 7 cohorts are about a third of the class, with the highest being no more than one year below grade level, the middle group being 2 or 3 years below grade level, and the lowest group being more than 3 years below grade level. The really interesting feature of this schedule is that I see most of my kids in two entirely different academic settings.

One setting, the grade level curriculum, is fairly conceptual so you get to see kids working with new concepts and from that you can assess their prowess with connecting the dots in their zone. The other setting, core skills, is not big concepts or word problems. It is simply raw calculation of rational numbers in all their various forms. This skills component lets you see more of what they bring to the table from lower grades.

Here's the nut… My highest group is making progress in both grade level curriculum and their core skills training. My middle group is making progress in their grade level curriculum (subject to the limitations inherent with their lack of core skills) and little to no progress in the core skills block. My lowest group isn't making progress in either curriculum or core skills.

My observational shock is not so much with the lowest group as they have clearly identified, documented learning disabilities. The highest group is making progress across the board so they're not a big concern either. The real conundrum is the middle group. In their grade level curriculum they appear to have no problem attacking new material (as long as the computation is simple) but their core skills are every bit as resistant to improvement as those in the lowest group. For these kids in the middle, it's like they have two personalities, one of which has a learning disability.

One more relevant point of reference is that this middle group has a normal amount of enthusiasm and energy level in the grade level work but in the core work they have all the inherent joy of a glazed doughnut. They sit in the core class with obvious boredom and do not apply themselves at all. In this class you could easily mistake them for the kids in the lowest cohort.

I would argue (perhaps foolishly) that this middle group is capable, based on my assessment of their grade level work, but disabled when it comes to computation. I would further submit that this seeming disability is induced by their prior failure, i.e. we created it. Could it be that after enough exposure to 'failure' in a particular domain, kids simply give up on it, concluding that it is a skill that is beyond them? Remember that this skills stuff is what they've been getting for the six preceding years.

My anecdotal evidence is telling me that these kids have an externally induced learning disability. It's induced by too much early indulgence towards their early lack of mastery and the school's failure to address it before it has damaged them. As a result they level out at a place that is far below their full potential. Is it possible that at some point, the failure to master becomes a built in disability that impedes further progress? Is there a threshold, beyond which a lack of progress becomes viral, thereby blocking future attempts to improve?

Has anyone experienced this?

Am I drinking too much coffee?

What is new with the science on math disabilities?

Wednesday, December 02, 2009

Atypical numerical cognition, dyscalculia, math LD: Special issue of Cognitive Development

A special issue of the journal Cognitive Development spotlights state-of-the-art research in atypical development of numerical cognition, dyscalculia, and/or math learning disabilities.

Article titles and abstracts are available at Kevin McGrew's excellent IQ's Corner blog.



-----------
Joe Elliot on dyslexia:

"Contrary to claims of ‘miracle cures’, there is no sound, widely-accepted body of scientific work that has shown that there exists any particular teaching approach more appropriate for ‘dyslexic’ children than for other poor readers."

I am in agreement with Elliot.

I wonder if the same will be found to be true for dyscalculia and kids who struggle with math.

Education Non-Myths

I couldn't resist sharing these maxims from a new blog :
www.incentiveseverywhere.com
whose author I know from a previous book he wrote entitled Power Teaching (it's in the list of books I recommended in a post a few months ago: http://kitchentablemath.blogspot.com/2009/02/recommended-reading-from-palisadesk.html



What follows is from the "Book of Right", the set of assumptions which will produce learning.

1. Although students come from different backgrounds, and some are much easier to teach than others, what education brings to the student is much more important than what the student brings to education.

2. All subjects are hierarchically arranged by logic and there is a sequence of instruction which must be followed by all but the most exceptional of high-performing students.

3. Reinforcement is a very powerful determinant of student achievement. The main reinforcer in education is the improvement the student sees in his skills. Ill-constructed curricula, the kind found in almost every government school, result in a steady diet of failure for most students.

4. Having a system of education which is not a civil servant bureaucracy is a necessary but not a sufficient condition for effective education. You can’t do it with such a bureaucracy, but just because you don’t have a bureaucracy doesn’t mean you can do it.

5. Higher order thinking skills are explicitly taught, not fondly hoped for.

6. Methods of teaching are determined by scientific research, not consensus based on experience and sincere belief.

7. Teachers use a curriculum and lesson plans which have been demonstrated to work best and are not expected to create their own.

8. Psychological assessments are used rarely, but assessment of student progress, which means assessment of the effectiveness of teaching, occurs at least daily.

9. Teachers are taught how to teach in detail rather than being expected to apply vague philosophical maundering.

10. Special education is rarely needed because students are taught well on the first go round.

11. If a student does not learn, the blame is not placed on neurological impairment, but on faulty teaching methods.

12. Self-esteem is not taught because it does not have to be.

13. Students are not given "projects" until component skills have been mastered and rarely thereafter.

14. No attention is paid to individual "learning styles" because these hypothetical entities have no effect on learning.

15. Academic success can be measured by reliable and valid standardized tests, although many of these tests are too simple.

16. Students are expected to perform correctly in spelling, writing, reading, and mathematics and it does not stifle creativity.

17. The precepts of Whole Language are not used to teach reading because these precepts are wrong.

18. Students are not expected to create their own reality because this leads to frustration and slow learning.

19. Students are not expected to learn when it is developmentally appropriate but when they are taught.

20. The concept of multiple intelligences is ignored because it has no positive effect on learning.

21. The teacher is a teacher and not a facilitator.

22. The spiral curriculum is not used because things are taught properly the first time.

23. The customer is the parent and the customer must have the economic power to move his child to another teaching situation when unsatisfied.

24. In private education, the cost of education is known. In public education, the cost can never be known because there is no motivation to tell the truth and every motivation not to.

25. The curriculum must be tested on children and provision must be made for mastery learning. Passage of time or exposure does not guarantee learning.

26. Students are not tortured by "creative problem solving" because this is just another crude IQ test and has no value aside from categorizing students yet again. http://incentiveseverywhere.com/2009/10/09/education-non-myths/


I'm not sure I agree that "special education will rarely be needed," because I have observed that students with certain exceptionalities (autism, some LDs, some language impairments) need the same effective instruction but can't benefit from it in an inclusive setting, at least not initially. However, I agree with the general case, that much "special education" is simply ineffective general education, watered down in in a smaller group. As Lloyd Dunne (I think) observed, "It's not special, and it's not education."

All students deserve better.

you are here

from Paul B:

Kids can't do 6 times 7 because they are 'taught' multiple ways to multiply. Then they discover the one they like best (usually involves fingers) and stick to it until somewhere along the way they are given a calculator to expedite things.

They are assessed subjectively. They are promoted without regard to those assessments. And finally, they're never given targeted remediation or practice that is up to the task of keeping them on track.

Other than that there's not much wrong.

I have already confessed my own problems with 6 times 7.

upstate New York

from lgm:

Here (upstate), the norm is that homework or classwork is checked by the teacher, but if it shows lack of concept/mastery understanding, there is no reteach unless student qualifies for specialist help. If the parent doesn't and the child is unclassified, the child will be dropped in to a lower section at the end of the unit (gr. 4-5) or the year (K-3). When he falls behind far enough he'll qualify for specialist help via RtI. The in-the-know parents just hire the tutor or do it themselves rather than let the student fall that far behind.

Also, is Scarsdale teaching for excellence or 'for the pass'? I notice here that the info needed in math to perform at the '4' level on the grade level state exams is not taught in class. One has to afterschool with a quality curriculum.

Here, 3rd graders do write a 'research paper' in class. The topic is 3rd grade friendly and most can do it. It reminds me more of a five paragraph essay than a paper.

I would like schools to teach kids how to write book reports.

Then I'd like schools to assign book reports.

Thank you very much.

Steve H: mastery and understanding are tightly linked

[U]nderstanding and mastery are tightly linked. Curricula like Everyday Math try their best to unlink them. They think that understanding concepts with just a little bit of practice is all that you need. This conveniently fits their dislike of drill and kill and how they believe that there is no one way to solve a problem.

Mastery is not just about speed. It's directly related to understanding.

My friend K. came up with an apt definition just the other day.

She said that understanding without mastery is a case of "That makes sense."

It's grasping the logic.

Being able to grasp the logic in a number of different realms is an important goal of a liberal education; it's my liberal education that allows me to read articles about the economic crisis and follow the argument reasonably well.

But following the logic of an article about the economy is completely different from understanding economics with any degree of sophistication.

It was for this reason that, a few years ago, I stopped reading books explaining "math for non-mathematicians" and started working my way through math textbooks.

the middle 95%

While we're on the subject of college-for-all.... I found all kinds of interesting books while trawling Amazon today:

“After forty years of intensive research on school learning in the United States as well as abroad, my major conclusion is: What any person in the world can learn, almost all persons can learn if provided with appropriate prior and current conditions of learning. This generalization does not appear to apply to the 2% or 3% of individuals who have severe emotional and physical difficulties that impair their learning. At the other extreme there are about 1% or 2% of individuals who appear to learn in such unusually capable ways that they may be exceptions to the theory. At this stage of the research it applies most clearly to the middle 95% of a school population.

The middle 95% of school students become very similar in terms of their measured achievement, learning ability, rate of learning, and motivation for further learning when provided with favorable learning conditions. One example of such favorable learning conditions is mastery learning where the students are helped to master each learning unit before proceeding to a more advanced learning task. In general, the average student taught under mastery-learning procedures achieves at a level above 85% of students taught under conventional instructional conditions. An even more extreme result has been obtained when tutoring was used as the primary method of instruction. Under tutoring, the average student performs better than 98% of students taught by conventional group instruction, even though both groups of students performed at similar levels in terms of relevant aptitude and achievement before the instruction began.

The central thesis of Human Characteristics and School Learning (Bloom, 1976) is the potential equality of most human beings for school learning. We believe that the same thesis is likely to apply to all learning, whether in schools or outside of schools. At least, it leads us to speculate that there must be an enormous potential pool of talent available in the United States. It is likely that some combinations of the home, the teachers, the schools, and the society may in large part determine what portions of this potential pool of talent become developed. It is also likely that these same forces ma, in part, be responsible for much of the great wastage of human potentiality.

Developing Talent in Young People
by Benjamin Bloom
pp. 4-5

Just to mix things up a bit!

And see: how to build a fast learner.

pop quiz

Have just stumbled across what appears to be a test of professional knowledge for aspirant teachers.

This item will amuse and delight:

24. A heterogeneous group of three to six children working together in support of their learning is called:

1. Mastery learning
2. An authentic learning group
3. A cooperative learning group
4. A think-pair-share learning group

It will come as no surprise to learn that the answer is not number 1: mastery learning.

notes on Bloom & teaching to mastery

Another Concerned Parent find:

Mastery Learning

Notes from Benjamin Bloom lecture [ACSA, April, 1987]:

With traditional instruction, the correlation of pupil performance from grade-to-grade is 80%+. The variation within each grade is greater each year. The range is double the second grade in the fourth grade, triple in grade 6. Rank order is already fixed by third grade for the next 8 years for 90% of kids. Similarly, self-concept drops grade-by-grade for the bottom 20% while it rises year-by-year for the top 20%. This is true of most countries, not just the US.

[snip]

The mid-point for conventional instruction is the 50th percentile. For individual tutoring, it is the 98th percentile. For whole class mastery learning instruction, it is the 84th percentile.

[snip]

CORE IDEA OF MASTERY LEARNING: aptitude is the length of time it takes a person to learn not how "bright" a person is, i.e., everyone can learn given the right circumstances.


How to instruct for mastery:

1. Major objectives representing the purposes of the course or unit define mastery of the subject.

2. The substance is divided into relatively small learning units, each with their own objectives and assessment.

3. Learning materials and instructional strategies are identified; teaching, modeling, practice, formative evaluation, reteaching, and reinforcement, etc., and summative evaluation
are included.

4. Each unit is preceded by brief diagnostic tests.

5. The results of diagnostic tests are used to provide supplementary instruction to help student(s) overcome problems.

Time to learn must be adjusted to fit aptitude. NO STUDENT IS TO PROCEED TO NEW MATERIAL UNTIL BASIC PREREQUISITE MATERIAL IS MASTERED.

[There is a difference between "80% of students will master the material" and "each student will master at least 80% of the material" before proceeding.]

This explains a lot.

One of the mysteries here & elsewhere is: why are administrators so unconcerned about the amount of tutoring going on in affluent communities?

I've mentioned before the tutor here who estimates that 50% of Scarsdale kids are being tutored. That tells me that Scarsdale schools are overrated.

But that's not what it says to educators.

The 98% figure may explain this. Middle-aged educators are old enough to have studied Bloom in ed school (teachers younger than 35 may never have heard of him - not sure....).

If they're thinking "98%" when they hear tutoring, no wonder they're dismissive.

In fact, I'm 98% positive (that's a joke!) Bloom was talking about tutorials, not tutoring. He was talking about what homeschoolers are doing, which is direct, one-on-one instruction in the whole of a course, start to finish.

Tutors are trying to put out fires, get kids ready for the Big Test, keep the kid from giving up, etc. Hiring a tutor to try to teach a student who is struggling in a class at school is by no means ideal.

In fact, the one family I know whose kids are doing brilliantly well with tutors simply takes it as a given that all schools, public and private, are fatally flawed. They have a "regular" tutor on the tab who works with their kids almost year round.

Preemptive tutoring.


extra credit

Was mastery teaching ever widely adopted in public schools?

"Overlearning" Overrated?

Science Daily had an interesting article out this week discussing the value of overlearning. Recent research has found that studying material beyond mastery may be a waste of time in the long run.

University of South Florida psychologist Doug Rohrer decided to explore this question scientifically. Working with Hal Pashler of the University of California, San Diego, he had two groups of students study new vocabulary in different ways. One group ran through the list five times; these students got a perfect score no more than once. The others kept drilling, for a total of ten trials; with this extra effort, the students had at least three perfect run-throughs. Then the psychologists tested all the students, some one week later and others four weeks later.

The results were interesting. For students who took the test a week later, those who had done the extra drilling performed better. But this benefit of overlearning completely disappeared by four weeks. In other words, if students are interested in learning that lasts, that extra effort is really a waste. They should instead spend this time looking at material from last week or last month or even last year.

Researchers concluded that once mastery was achieved it was better to leave that subject alone for a while and return to it later. They found that an optimal "study break" of about a month resulted in long-term learning-- something they refer to as the "spacing effect".

Is this "spacing effect" an argument for the spiral approach? Perhaps so, yet it does seem to be a well executed spiral in which the content is first studied to mastery and then revisted for reinforcement later. This is certainly not the haphazard "spiral" I've witnessed my children being subjected to with Everyday Math and seems to be more in keeping with Saxon or Singapore Math's idea of a spiral curriclum.

I hope they keep looking into this subject. Children have such precious little time to learn so many important things. Imagine all that could be accomplished if we started implementing teaching and study skills that were actually efficient.

Source: Back to School: Cramming Doesn't Work In The Long Term


ABSTRACT—Because people forget much of what they learn, students could benefit from learning strategies that yield long-lasting knowledge. Yet surprisingly little is known about how long-term retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions—the spacing effect—depends jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning.

Increasing Retention Without Increasing Study Time


Catherine here, diving into Concerned's post.

What a find! I've just pulled the article; will read shortly.

In the meantime, here are the Willingham articles that discuss overlearning:

• Allocating Student Study Time: "Massed" versus "Distributed" Practice

• Practice Makes Perfect, But Only If You Practice Beyond the Point of Perfection

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

Doomed By Careless Math Errors

My 5th grader brought home her recent pre and post-tests on fractions. I can't fault the schools, teachers, or the typically lousy EM curriculum on this one. The topic is appropriate -- fractions in 5th grade requiring her to find the common denominator, add or subtract or multiply, and putting fractions in order, least to greatest. She's had plenty of classroom instruction and practice (she's also had a lot of afterschooling on this as Singapore Math's 5A and 5B spend a lot of time mastering fractions). The tests themselves were not particularly challenging.

So what's the problem? On a test she could easily have gotten 100% she got a 78. Why? Careless errors. She added when she was supposed to subtract, she failed to read the question properly, one she skipped entirely -- just missed the question. The one she skipped she got right on the pre-test. The ones she got wrong on the post-test she got right on the pre-test and vice versa.

So what next? Pedagogically, instructionally, what is the best way of getting kids to take their time and be careful? Is this just one of those 5th grade things? I'm tempted to ask the teacher for a copy of the test and have her retake it. Is there anything gained by having a consequence/punishment for carelessness?

Attention to detail seems critical in math. Am I overreacting? Anyone have success with this particular aspect of teaching?

is Everyday Math Anti-Mastery?

On KTM1, we frequently discussed the level of "mastery" involved in the constructivist reform curriculum. But I was never sure if we could really say that EM is philosphically opposed to mastery. I went to the UCSMP website for Everyday Math and did a google site search for the word "mastery." It appears in exactly four documents. In two of the documents, the word "mastery" is paired pejoratively with "rote," as in EM "shifts the emphasis from rote memorization of procedures and mastery of isolated concepts." In one document, it states,

This is not to say that skill mastery is not expected in Everyday Math. Fact automaticity is expected and nurtured through various games and practice routines.

Hmm. That appears to be the extent of the expectations for mastery. I don't disagree, except I think we need mastery on much more than the basic facts.

The final document with the word "mastery" in it from EM comes from the 2001 Algorithms in Everyday Mathematics. The paper states as a justification for the non-standard algorithm the fact that in one study (not identified), only 60% of US 10-year-olds achieved mastery of the traditional subtraction "borrowing" algorithm. Unfortunately, EM doesn't tell us how many US 10-year-olds achieve mastery of the alternative algorithms EM has used to teach subtraction for the past 15 years.

Anybody have any data on whether US 10 year olds are faring any better these days?

Based on the EM website, it seems that calling EM anti-mastery is probably a fair characterization. Discussion?