A recent NPR article entitled, 5 Lessons Education Research Taught Us in 2014, seems to have a lot of definitive answers about our currently polarized educational climate. The article mentions a research paper which encourages the use of teacher-directed, explicit instruction of mathematical computation skills for procedural fluency with students with mathematical difficulties. To me this read as favoring explicit instruction and direct modeling of mathematics for students with disabilities over the project or problem-based, hands-on (manipulatives), collaborative (student-led), investigative style instruction that makes up some “reform” mathematics curriculum.
As a counterpoint to the NPR article, the National Council of Teachers of Mathematics lists procedural fluency as just one part of what is referred to as Mathematical Proficiency. In chapter 2 of the book, Achieving Fluency: Special Education and Mathematics, mathematical proficiency is discussed as including the following four components: procedural fluency, conceptual understanding, strategic and adaptive mathematical thinking, and productive disposition. Together these four components lead to mathematically proficient students which lead to mathematically proficient adults, disabilities or not.
1. Procedural fluency involves using basic skills such as facts, procedures, and formulas efficiently (i.e., quickly and accurately). It also entails knowing when to use them and, if necessary, how to adapt them. In other words, procedural fluency is skill in carrying out routines appropriately and flexibly as well as efficiently.
2. Conceptual understanding is knowledge of facts, generalizations, or principles underlying the comprehension of concepts (categories), relations (between categories), or operations (actions or events involving categories).
3. Strategic competence involves the ability to formulate, represent, and solve mathematical problems, and adaptive reasoning entails the capacity for logical thought, reflection, explanation, and justification.
4. Productive disposition entails believing that mathematics makes sense and is useful, that learning it requires diligence, and that everyone is capable of significant mathematical learning.
Since the NPR article about educational research only references one paper specifically about mathematics instruction, you only get one point of view. This NCTM book provides another viewpoint of what are important goals for mathematics lessons with struggling students.
Another aspect of the research paper mentioned in the NPR article is that the authors freely list the limitations of their research. One of the most interesting limitations is:
We were unable to measure the relative quality with which these practices were implemented. Stronger achievement gains may have been observed if teachers had used specifically structured and integrated mathematics curricula or supplemental programs (p. 16).
This is an important distinction that is not highlighted in the NPR article. When one is writing a pop culture article which makes claims about the effectiveness of one pedagogical strategy over another, all aspects of the topic should be presented. Not taking into account the effectiveness of the implementation of “reform” mathematics teaching strategies such as manipulatives and movement is not only harmful to the educational professionals who employ and research these strategies, but also to the students whose cognitive pathways and learning styles dictate the need for them.
If you’re interested in great implementation of manipulatives in elementary classrooms there are plenty of examples on Graham Fletcher‘s blog. If you’d like to integrate movement and dance into your mathematics curriculum, then check out Malke Rosenfeld‘s expert work in this area. Both of these teachers have found success using the same pedagogical strategies disparaged in the NPR article.
Also we have to consider what we are valuing in our mathematics goals for students with disabilities. Do we want them to be better test takers or do we want them to achieve their highest level of independence in our current technological society? Steve Leinwand has written extensively about the positive aspects of gearing mathematics curricula towards the latter. From his book Sensible Mathematics:
Less and less are employees called upon to execute calculations with pen and paper, but more and more are they expected and required to know when and why to perform a particular calculation. Textbooks give students pre-packaged situations and models. The real world expects workers to create and use appropriate models for complex and variable situations! (p. 10)
This idea, however, bleeds into the educational research field of transfer of learning, which is fodder for another blog post unto itself. Michael Pershan had this to say on the topic…
@TracyZager@bkdidact Andrew’s standard (transfer of learning post-school) would essentially dissolve most edu research in one fell swoop.
— Michael Pershan (@mpershan) March 3, 2015
Despite all of these seemingly contradictory ideas and conclusions about what is the best method with which to instruct students with mathematical difficulties, the focus always has to be on the success of each individual student no matter what. So I’ll let NCTM author, Arthur J. Baroody, have the last word.
Helping children with learning or behavioral difficulties achieve mathematical proficiency and adaptive expertise will require an approach different from and more sophisticated than the traditional direct-instruction-and-drill method. It will require purposeful, meaningful, and inquiry-based instruction that promotes all aspects of mathematical proficiency in an integrated manner. Planning and implementing such instruction will require considerable time, effort, and knowledge from teachers, but significantly greater student development will reward their investment.