An earlier version of this article appears in “Math Education in the U.S.” Footnotes appear after the list of references at the end of the article.
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, recently documented in a blog which documents a British teenager’s achieving reading fluency over 18 months (after having struggled for years to read).[i] The incident demonstrated that “correct methods” can result in reading success and in this case (as in Reid Lyon’s work), greatly reduced the teenager’s special education needs.
These findings are significant because a similar dynamic is likely to be play in math education: the effective treatment for many students who would otherwise be labeled learning disabled is also the effective preventative measure. This prospect raises several questions: 1) How many of the students identified with learning disabilities are related to math? 2) Of those, how many students are 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 an intervention for students who are either identified as having a learning disability or otherwise having difficulty in mathematics is often what should have been done—and what students need to be doing—in the first place. Given the education establishment’s resistance to the idea that traditional math teaching methods are effective, research is very much needed to draw conclusions about the effect of instruction on the diagnosis of learning disabilities and the need for interventions.
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, and although people who advocate for reform insist that the traditional method is prevalent 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. [1]
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. (“Evidence” presented for such statements are often as weak as stating most adults if asked whether they like math will say “no”—thus implying that if the traditional method were effective, the answer would be “yes”.)
Math reformers promote a teaching approach in which conceptual understanding takes precedence over procedure. 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, some programs require students to draw pictures and/or use inefficient procedures before they are exposed to or allowed to use the standard algorithm. This is done in the belief that the alternative approaches confer understanding to the standard algorithm. In reformers’ minds, teaching the standard algorithm first would be considered “rote learning”, thus eclipsing the conceptual underpinning of the procedure.[2]
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”, “problem-based” or other synonyms for discovery learning experiences. 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 for years have dismissed 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 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.” 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 Need for Further Research
The use of intervention techniques in schools for students doing poorly in math provides some indications that LD diagnoses and low ability may be caused by poor instruction or lack thereof. Having seen the results of ineffective math curricula and pedagogy as well as having worked with the casualties of such educational experiments, I believe that interventions using explicit instruction plays a significant role in helping students to improve as well as reducing the identification of students with learning disabilities. 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 verify it. 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 effect of explicit and systematic instruction of procedures, skills and problem solving, compared with inquiry and other reform-based approaches; and
3) 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 interventions 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 style of teaching used in interventions 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, 2) not providing the conceptual basis for procedures, and 3) 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 interventions are susceptible to the use of pedagogical features of reform math which may have 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
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 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
Thanks for this post and for this blog in general. Going beyond the type of intervention or remediation is the question of whether any intervention or remediation is taking place in public schools in the first place.
I volunteer at the tutoring center of my local high school -- a well funded and high performing school. The other day, for example, a student came in with some "geometry" questions. The questions involved two opposite angles of a parallelogram, one labeled 2x, and the other something like x + 40. The student had no idea how to set up the problem, nor how to solve for x. Another question involved subtracting 19 from 29 -- the student could not do this in his head nor could he set it up on paper. The student had difficulty speaking at times -- it seemed like he might have tourette's -- and it was clear to me he had some kind of learning disability, so therefore he must have an IEP. So the question I would like to ask is why in the world does his teacher think he should be working on problems designed to integrate geometric and algebraic concepts, when he cannot even set up nor solve a basic subtraction problem?
Its pretty clear to me that this student needs to be spending 100 percent of his time in math working on basic numerical proficiency, whether that involves learning the "tricks" that common-core emphasizes or the now-hated standard algorithms. An ideal progression for him would seem to be basic proficiency in addition/subtraction/multiplication/division and then hopefully later a good understanding of ratios, percents, etc. But instead, despite all of the resources and attention given to the development of this student's IEP, the school and/or teachers still seem content to pass him right on through -- in a sense, pretending he understands concepts that he clearly does not.
Another thing I noticed about the student was that, although he could not add/subtract by hand nor solve basic algebraic equations, he was able to use a calculator and an online equation solver to find the answers. So it could also be that his teacher is content with his lack of proficiency so long as he demonstrates the resourcefulness to use online tools or a calculator in order to solve problems.
Either way, it seems clear to me that, beyond the content of instruction, many students with learning disabilities simply do not receive any individualized or explicit instruction whatsoever, and are instead just passed along from class to class and grade to grade -- and perhaps teachers or the school rationalize this by assuming that so long as the student can use a calculator, he will manage fine enough.