Effective Math Instruction: Hiding in Plain Sight


This article originally appeared in Education News in 2015. It received about 80 comments, most of them indignantly negative, which told me I was probably on the right track. I revised the article to bring it up to date and it is now included in my book “Math Education in the US: Still Crazy After All These Years”, which I heartily encourage you to purchase–in fact get several copies and send them to people who disagree with you on educational matters. They’ll never forget you.

I include this article here because there has been much talk lately on Twitter about how reading is not taught properly (due in large part to a great article written by Emily Hanford in the New York Times). The first line of my article echoes some concerns being discussed now. I extend the argument to math.


In a well-publicized paper that addressed why some students were not learning to read, Reid Lyon (2001) concluded that children from disadvantaged backgrounds where early childhood education was not available failed to read because they did not receive effective instruction in the early grades. Many of these children then required special education services to make up for this early failure in reading instruction, which were by and large instruction in phonics as the means of decoding. Some of these students had no specific learning disability other than lack of access to effective instruction.

This phenomenon has been observed by others, which was documented in a blog which recounts a British teenager’s achieving reading fluency over 18 months (after having struggled for years to read). The incident demonstrated that “correct methods” can result in reading success and in the case documented in the blog, greatly reduced the teenager’s special education needs.

These findings are significant because a similar dynamic is at play in math education: the effective treatment for many students who would otherwise be labeled learning disabled is also the effective preventative measure.

In 2013 approximately 2.3 million students were identified with learning disabilities — about three times as many as were identified in 1976-1977. (National Center for Educational Statistics, 2015)  These numbers raise several questions:  1) How many of the students identified with learning disabilities were related to math?  2) Of those, how many students were so classified because of poor or ineffective instruction?  and (perhaps most importantly) 3) How many could have kept up  with classmates if they had been taught using the more traditional math teaching methods that had once prevailed? 

In my opinion, what is offered as treatment for learning disabilities in mathematics is what we could have done—and need to be doing—in the first place. While there has been a good amount of research and effort into early interventions in reading and decoding instruction, extremely little research of equivalent quality on the learning of math in the United States exists. Given the education establishment’s resistance to the idea that traditional math teaching methods are effective, this research is very much needed to draw such a definitive conclusion about the effect of instruction on the diagnosis of learning disabilities.

Some Background

Over the past several decades, math education in the United States has shifted from the traditional model of math instruction to “reform math”. Although the shift has not been a uniform one, evidence of such transition is indicated by perennial articles in newspapers and the internet featuring parents who question and protest the methods being used to teach their children math.[ii] The traditional model has been criticized for relying on rote memorization rather than conceptual understanding. Calling the traditional approach “skills based”, math reformers deride it and claim that it teaches students only how to follow the teacher’s direction in solving routine problems, but does not teach students how to think critically or to solve non-routine problems. Traditional/skills-based teaching, the argument goes, doesn’t meet the demands of our 21st century world.

The criticism of traditional math teaching is based largely on a mischaracterization of how it is being and has been taught: rote memorization and procedures being the main focus of instruction, with little or no conceptual understanding. It is often described as having failed thousands of students in math education despite evidence of its effectiveness. Reacting to this characterization of the traditional model, math reformers promote a teaching approach in which understanding and process dominate over content.  Such emphasis is often represented by statements made by teachers, or school administrators such as “In the past students were taught by rote; we teach understanding.”

In order to ensure that students have understanding rather than performing a procedure by “rote”, students in lower grades may be required to provide written explanations for problems that often are so simple as to defy explanations. Also, they may be asked to solve a problem in more than one way, either by pictures as well as numerically, or by different methods. Failure to do so may result in the student judged to be operating via rote procedures and not possessing true mathematical understanding.  In lower grades, mental math and number sense are often emphasized before students are fluent with the procedures and number facts to allow such facility.

In lieu of the standard methods (algorithms) for adding, subtracting, multiplying and dividing, in some programs students are generally required to use inefficient procedures for several years before they are exposed to or even allowed to use the standard algorithm). This is done in the belief that the alternative approaches confer understanding to the standard algorithm.[iii]  In reformers’ minds, to teach the standard algorithm first would, once again, be considered “rote learning” thus eclipsing the conceptual underpinning of the procedure. 

Whole class and teacher-led explicit instruction (and even teacher-led discovery) has given way to what the education establishment believes is superior: students working in groups in a collaborative learning environment reducing the teacher to a mere facilitator of holistic “inquiry-based” or discovery learning experiences. Students teach themselves. Providing information directly is done sparingly or in combination with a group activity. 

The grouping of students by ability has almost entirely disappeared in the lower grades—full inclusion has become the norm. Reformers dismiss the possibility that understanding and discovery can be achieved by students working on sets of math problems individually and that procedural fluency is a prerequisite to understanding. Much of the education establishment now believes it is the other way around; if students have the understanding, then the need to work many problems (which they term “drill and kill”) can be avoided.

The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, Manitoba, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of some of the other panelists, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.” This sentiment was also echoed in an article written by Keith Devlin (2006). Such opposition has had limited success, however, in turning the tide away from reform approaches.

The Growth of Learning Disabilities

Students struggling in math may not have an actual learning disability but may be in the category termed “low achieving” (LA). Recent studies have begun to distinguish between students who are LA and those who have mathematical learning disabilities (MLD). Geary (2004) states that LA students don’t have any serious cognitive deficits that would prevent them from learning math with appropriate instruction. Students with MLD, however, (about 5-6 percent of students) do appear to have both general (working memory) and specific (fact retrieval) deficits that result in a real learning disability. Among other reasons, ineffective instruction may account for the subset of LA students struggling in mathematics.

A popular textbook on special education (Rosenberg, et. al, 2008), notes that up to 50 percent of students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction. What Works Clearinghouse finds strong evidence that explicit instruction is an effective intervention, stating: “Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review”.

Also, the final report of the President’s National Math Advisory Panel (2008) states: “Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities, as well as other students who perform in the lowest third of a typical class.” (p. xxiii). The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes math memorization and the other traditional methods for teaching the subject that have been decried by reformers as having failed millions of students.

The Stealth Growth of Effective Instruction

The Individuals with Disabilities Education Act (IDEA) initially established the criteria by which students are designated as “learning disabled”. IDEA was reauthorized in 2004 and renamed the Individuals with Disabilities Education Improvement Act (IDEIA). The reauthorized act changed the criteria by which learning disabilities are defined and removed the requirements of the “significant discrepancy” formula. That formula identified students as learning disabled if they performed significantly worse in school than indicated by their cognitive potential as measured by IQ. IDEIA required instead that states must permit districts to adopt alternative models including the “Response to Intervention” (RtI) model in which struggling students are pulled out of class and given alternative instruction.  If they improve under RtI, then the student is presumed to not have a learning disability and is returned to the normal class.

The RtI process provides a strong indication that LD diagnoses may in fact be caused by other factors such as poor instruction.  Although the number of students classified as learning disabled has grown since 1976, the number of students classified as LD since the passage of IDEIA has decreased. Why the decrease has occurred is not clear. A number of factors may be at play. One may be a provision of No Child Left Behind that allows schools with low numbers of special-education students to avoid reporting the academic progress of those students. Other factors include more charter schools, expanded access to preschools, improved technologies, and greater understanding of which students need specialized services. Of particular note, the decrease may also be due to targeted RtI programs. Since students who improve under RtI are presumed to not have a learning disability, the RtI program itself may have reduced the identification of struggling and/or low achieving students as learning disabled.

Having seen the results of ineffective math curricula and pedagogy as well as having worked with the casualties of such educational experiments, I have no difficulty assuming that RtI plays a significant role in reducing the identification of students with learning disabilities. The problem remains, however, that after a student shows improvement under RtI, he/she is then returned to the type of teaching that caused the student to be referred to RtI in the first place. In my opinion it is only a matter of time before high-quality research and the best professional judgment and experience of accomplished classroom teachers identify effective and non-effective teaching methods. Such research should include:

1) The effect of collaborative/group work compared to individual work, including the effect of grouping on students who may have difficulty socially;

2) The degree to which students on the autistic spectrum (as well as those with other learning disabilities) may depend on direct, structured, systematic instruction;

3) The effect of explicit and systematic instruction of procedures, skills and problem solving, compared with inquiry and other reform-based approaches;

4) The effect of sequential and logical presentation of topics that require mastery of specific skills, compared with a spiral approaches to topics that do not lead to closure, 

5) How students who improved under RtI fared when returned to a reform-based classroom and

6) The extent to which students who are doing well in a reform-based classroom are receiving outside help via parents, tutors or learning centers. 

Would such research show that the use of RtI is higher in schools that rely on programs that are low on skills and content but high on reform-based techniques that purport to build critical thinking and higher order thinking skills? If so, shouldn’t we be doing more of the RtI style of teaching in the first place instead of waiting to heal the casualties of reform math?

Until any such research has been conducted, the educational establishment will likely continue to resist recognizing the merits of traditional math teaching, based on conversations I’ve had with education professors. A statement made by James McLesky (2009), a professor at University of Florida’s College of Education, is typical of what I’ve been told:

If we provide only (or mostly) skills and drills for students with disabilities, or those who are at risk for having disabilities, this is certainly not sufficient. Students need to also have access to a rich curriculum which motivates them to learn reading, math, or whatever the content may be, in all of its complexity. Thus, a blend of systematic, direct instruction and high quality core instruction in the general education classroom seems to be what most students need and benefit from.

While Dr. McLesky recognizes the value of direct and explicit instruction, his statement carries with it the underlying mistrust and mischaracterization of traditional math teaching—a mistrust that defines such teaching as 1) consisting solely of explicit instruction with no engaging questions or challenging problems, and 2) failing to teach math in any complexity. Statements such as these imply that students who respond to more explicit instruction constitute a group who may simply learn better on a superficial level. Based on these views, I fear that RtI will incorporate the pedagogical features of reform math that has resulted in the use of RtI in the first place.

The criticism of traditional methods may have merit for those occasions when it has been taught poorly. But the fact that traditional math has been taught badly doesn’t mean we should give up on teaching it properly. Without sufficient skills, critical thinking doesn’t amount to much more than a sound bite. If in fact there is an increasing trend toward effective math instruction, it will have to be stealth enough to fly underneath the radar of the dominant edu-reformers. Unless and until this happens, the group-think of the well-intentioned educational establishment will prevail. Parents and professionals who benefitted from traditional teaching techniques and environments will remain on the outside — and the methods that can do the most good will continue to hide in plain sight.

References

Devlin, Keith. (2006). Math back in forefront, but debate lingers on how to teach it.  San Jose Mercury. Feb. 19.

Geary, David. (2004).  Mathematics and learning disabilities. J Learn Disabil 2004; 37; 4

Lyon, Reid (2001), in “Rethinking special education for a new century” (Chapter 12) by Chester Finn, et al., Thomas B. Fordham Foundation; Progressive Policy Inst., Washington, DC. Available via  http://eric.ed.gov/PDFS/ED454636.pdf

McLesky, James (2009). Personal communication via email; October 20.

National Center for Educational Statistics, (2015) Digest of Educational Statistics: 2013. Table 204.30Available via http://nces.ed.gov/programs/digest/d14/tables/dt14_204.30.asp

National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel, U.S. Department of Education: Washington, DC, 2008.

Rosenberg, Michael, D. L. Westling, J. McLesky (2008). Special Education for Today’s Teachers: An Introduction; Pearson. New York.

Zimba, Jason (2015) When the standard algorithm is the only algorithm taught. Common Core Watch; January. http://edexcellence.net/articles/when-the-standard-algorithm-is-the-only-algorithm-taught


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10 thoughts on “Effective Math Instruction: Hiding in Plain Sight

  1. BOOM! Every parent, and teacher needs to read this. And then send a copy to the school principal and insist they read it as well. Why do we settle for second rate education practices when they are so simple to fix, and yet create unimaginable horrors for a child trying to learn basic facts?

    We don’t accept homeopathic treatments for medical ailments that are easily treatable by time worn, successful medicine. Why is education any different?

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    • From looking at that website, I can’t see that their approach is an actual math program. Maybe a set of enrichment activities, and certainly a way to add a few less-goal-oriented problems into a classroom (not a bad thing sometimes). But it’s not a curriculum or a theory of instruction, not at all.

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    • Quite true. The edu-establishment in general is hard to get past; they have a toe-hold on beliefs and practices and no matter how many articles are written showing how damaging their ideology is, it keeps going on. And Boaler is part of the edu-establishment.

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    • “She [Jo Boaler] is the author of the first MOOC on mathematics teaching and learning.”

      If she really wanted to prove her ideas, she would create a MOOC (stupid name!) on actually learning mathematics – not mathematics teaching and learning. She needs to put her ideas to a reality test compared with other techniques in an open world, not an educator and school monopoly world.

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  2. # 6 especially, since it would be a simple research project. All of my son’s STEM-prepared friends received help at home and with tutors – not so much for specific high school material, but for recovering from K-8 damage. Once my son got to the high school’s traditional AP track after K-8 help from me, I didn’t have to do a thing. Unfortunately, many parents (and educators) now feel that parents should be surrogate teachers at home who need to understand their fuzzy math techniques and/or to help them “flip” the classroom. Do they ever stop to think what expected parental help implies?

    Everyone knows that traditional AP/IB math in high school won the battle, so what’s so special about K-8 that it requires something else? Nobody has EVER addressed this split. At best, we hear about some sort of Zombie Effect with no proof and no examples of ANY other method that works better. All colleges ask for GPA for AP/IB classes, SAT I and II in math, AP test scores, and any AMC test scores. Where is their better model?

    As for reading, I could go on and on about our lower grades’ push of Onset and Rime instead of phonics. I remember reading about how products like “Hooked on Phonics” were bad for kids. I ignored them. I taught my son phonics before he even got to pre-school. He loved it and learned to read. I could go on and on about all of the preemptive teacher strikes against us parents. One teacher told us that our son “had a lot of superficial knowledge” and one parent even told me, when I once questioned what they were doing, that “They are the experts.”

    My own view is that their ideas are all part of their philosophy of full inclusion and natural learning – deliberately ignoring what many of us parents do at home. They can get away with it in the lower grades because reality is not pushing back, and when kids get to high school, there are enough parent and tutor-helped students who do well to provide them with enough cover. Perhaps they are still using my son as cover for their stinking bad choices of MathLand and then Everyday Math.

    After 20 years of fighting this, I see no real avenue of change. They would have to admit that they are fundamentally wrong with their ironically rote learning. Perhaps in the lower grades with full inclusion, they will accept a trans-grade acceleration model for real acceleration rather than enrichment. Perhaps some schools will figure out a model for a full inclusion environment, but not full inclusion academic classes (like high schools). But then they would have to confront the issue that all of the kids in the accelerated groups are from families that give help at home. That model doesn’t solve the problem of bad teaching.

    It could be that the only option is to fight for more charter schools, but that battle is entwined with political demagoguery about profit, not whether or not parents who opt-in know what they are getting. My view is that it’s easy for parents to compare school ‘A’ with school ‘B’. They can always switch back. Many charter schools set higher expectations and push, but unfortunately, they often still use fuzzy math.

    Many educators complain about the 19th century school model of education, but what about their 19th century monopoly? I really don’t like the idea of charter schools, but that’s because I want public schools to get it right in the first place. My son went to a public high school that pushed at least one AP class for all students. Their math teachers know that math in the lower grades is not good, but they don’t try to fix it. There is a fundamental systemic problem here.

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  3. I mostly read your blog as confirmation that I did the right thing homeschooling my children. Once again, you reassure me. I thank God and Texas freedom that I taught my children to read using phonics and math using traditional methods! Homeschoolers have known for years that those methods are effective. Look at Classical homeschooling. It’s all about using the natural learning and developmental stages that all children go through to make learning facts and concepts EASIER! This includes rote memorization, copy work, and recitation at the younger ages. Not until they are developmentally ready do they move into logic and reasoning. Of course the public schools really do not have the goal of education anyway. The goal is babysitting and keeping children out of the work force for as long as possible and preparing them to be compliant worker bees. The strong ones who emerge prepared for life do it despite the system or sometimes thanks to a teacher who does not stick to the curriculum.

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  4. Pingback: “Progressive” Education is Stuck in the 19th Century – Mr. G Mpls

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