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A Brief History of Numeracy

The Code Wars: Reading

Teaching academic content in schools has a dubious history, especially those important basic subjects: reading, writing, and 'rithmetic. The history of this subject matter is marred by debates, and children at risk of failure remain at risk until American public education can stop arguing and figure out what works. Characterized by numerous educators, researchers, and policy analysts, the end of the reading wars brought a more balanced understanding of the alphabetic code to schools. What was more, teachers learned to integrate prevention-based practices into their instructional approach. Such practices included assessment of critical areas of development such as phonological awareness, alphabetic understanding, and fluency in sight word reading. In the "dark ages," however, conflicts among educators raged. Phonics versus whole language? Meaning versus code emphasis? Advocates on both sides called for scrutiny of the other's views. Thanks to research and reason, more balanced and effective methods prevailed while rhetorical armchair analyses died out. While these wars may be over, they have inadvertently set the stage for the next set of rather confusing, let alone vicious and unnecessary "code wars."

The Code Wars: Math

In math, while many valid points are made across sides, there is much of "both in each." That is, you can find plenty of examples of those advocating for arithmetic and algorithms also advocating for promoting mathematical thinking. It is hard to find anyone arguing for "rote memorization" of algorithms. All educators should know that conceptual structures of thought include the names of things and the ideas behind them. It appears that despite profound synthesis such as professional consensus on early math development, as well as national standards (or, focal points), mathematics educators will face an uphill battle in learning what works.Despite the quarrels, none recommend sacrificing thinking for procedural efficiency. Do we have to? Let's hear from some experts.

Dr. Hung-Hsi Wu, in his paper "Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education" would argue that this all-too-typical polarization fails to consider merits of both procedural efficiency and thoughtful analyses of the representative structures of math. That is, the little squiggly lines we call numbers and operators actually represent something very important; the lines and the concepts are inexorably linked. Some, like Dr. Michael Battista, argue that by directly teaching only the algorithms or operations, and mnemonics to remember these (invert & multiply, parentheses, powers, MDAS), we may be failing to connect the underlying structure of quantity and number. It is also clear that mathematical thinking is critically important, and that there are a few efficient ways to use mathematical principles to solve what appear to be "standard algorithms." If numeracy truly is reasoning flexibly and efficiently about number for the purposes of solving problems mathematically, then "do this like this" has to be questioned. It is important to note, however, that direct instruction has intense merit - and is misperceived as a "do this like this" instructional approach.

Demonstrating a method for the purposes of skill development, fluency, and maintenance lead to generalization and creativity when the skill set is not taught narrowly. Behaviorists and DI proponents use effective means to teach "multiple exemplars' of well-completed procedures. Generalization, novelty, and creativity are targets of every behavior analyst I've known. Problems seem to arise, however, when teachers and schools believe, erroneously, that 'constructing your own knowledge' and unfettered dialogue is qualitatively different or more effective than a teacher directed solution. That is, the belief that finding an answer on your own is better than being taught the procedure and given opportunities to practice under novel conditions.

To fight the math wars well, at all costs, schools must guard against the adoption of student-driven approaches that result from poorly trained personnel. There is nothing inherently wrong with student-directed instruction if it produces results - mathematically thoughtful and talented individuals who do well on tests because they have confidence and flexibility. However, this is clearly not the case. If we are to revere and trust teachers as instructional experts, then the recommendations of Louisa Moats for reading and Liping Ma for math need to be followed - we need to start with strong content knowledge, research-based core instruction, and secondary supports that have the backing of the scientific method. Teaching math and teaching reading is a profoundly challenging and technical endeavor.

We can be thankful that the panel ushering in reading reform (The National Reading Panel) has a new sibling in the National Math Advisory Panel. Dr. Wu, and many other profoundly accomplished and careful researchers are heading up this panel, charged with leading the way toward mathematics reform in the United States. Among the many panel members is Dr. Liping Ma, who wrote "Knowing and Teaching Elementary Mathematics." Dr. Ma argues that the conceptual understanding inherent in high-achieving nations can be attained in the United States through a more thorough study of the subject matter of mathematics, and that real reform occurs when teachers become content area experts. Dr. Ma makes a number of important points, and among many, she holds that "arithmetic is intellectually challenging and pedagogically powerful for elementary students." Dr. Ma's work is akin to the work of Dr. Louisa Moats, who reminds us that "Teaching Reading is Rocket Science."

Another member of the panel is Dr. Russell Gersten, who has written extensively on taking what is known in mathematics education, and using these important building blocks to support children with learning problems in math. This work is critical in so many ways. First, it binds general and special education together around central concepts and big ideas. Opportunities can abound if children at risk can learn beside their low-risk peers, who can benefit from learning number sense. Second, to understand dysfunction (learning disabilities and problems), it is critical to understand function. Despite the importance of Dr. Gersten's work, he and other often indicate that a sparse body of research exists on math difficulties or disabilities. If one takes into account the idea that dysfunction is predicated on function, we would need nothing more than a series of simple translations of the immense body of research on mathematical function to better understand mathematics difficulties. Nothing that is known about learning disabilities can be understood without a working model of human abilities. Dr. Gersten's papers are featured in the research section below.

Interestingly, teaching math is probably and quite literally much more like teaching rocket science than is teaching reading. Mathematics, the study of encoded logic, is a formal area of study; school mathematics is quite different. Especially when proposing standards that will guide assessment, teaching, and learning, early school mathematics has much to learn from mathematicians, especially in understanding the basic building blocks, or "amino acids" of math. These building blocks are the foundations for numeracy - mathematical literacy - known as number sense.

Whichever code you choose - the numeric or alphabetic - the challenge of becoming an expert lies in discovering the code and its basic representational system. Typical school math, such as number operations and computation fluency, sits upon a pillar of important concepts. The most fundamental of these concepts is number sense. Research-based curricula and intervention programs have finally been developed in mathematics, and are founded upon number sense. Assessment of these building blocks in each child is critical. Below are resources for math improvement, including our contact information. Included is a list of research papers, policy pieces, assessment, and instructional links that provide a foundation for understanding early mathematics.

Research Papers in Early Math, Number Sense, and Mathematical Development

Policy Issues, Institutes, and Conferences on Improving Mathematics