The following article was published in the past month and sheds light on why math curricula based on the 1989 National Council of Teachers of Mathematics (NCTM) Standards were so problematic and led to declines in performance on national and international tests. Though the article focuses on the deficiencies of the Common Core State Standards in Mathematics from a cognitive science perspective, it has extra relevance to 1989 NCTM aligned curricula. Unfortunately, Everyday Mathematics is one of the two most popular math curricula based on the 1989 NCTM that is still around. LCUSD used a similar math curriculum in the late 1990s and a parent revolt within our district forced changes.
“Cognitive Science and the Common Core Mathematics Standards”
Eric A. Nelson
http://www.chemreview.net/CCMS.pdf
Here is the abstract:
Between 1995 and 2010, most U.S. states adopted K‐12 math standards which discouraged memorization of math facts and procedures. Since 2010, most states have revised standards to align with the K-12 Common Core Mathematics Standards (CCMS). The CCMS do not ask students to memorize facts and procedures for some key topics and delay work with memorized fundamentals in others.
Recent research in cognitive science has found that the brain has only minimal ability to reason with knowledge that has not previously been well‐memorized. This science predicts that students taught under math standards that discouraged initial memorization for math topics will have significant difficulty solving numeric problems in mathematics, science, and engineering. As one test of this prediction, in a recent OECD assessment of numeracy skills among 22 developed‐world nations, U.S. 16‐24 year olds ranked dead last. Discussion will include steps that can be taken to align K‐12 state standards with practices supported by cognitive research.
Important excerpts:
What Went Wrong?
Comparing K‐12 math standards in most U.S. states to findings of recent cognitive research, the evidence indicates that a significant percentage of state math standards, now and for the past two decades, have asked students to solve problems in ways that a student (non‐expert) brain measurably cannot do. This suggests an explanation in part for U. S. student achievement in mathematics.
…Summarizing the implications for learning of these new scientific discoveries:
• When solving math problems of any complexity, due to working memory (WM) limits, students must rely almost entirely on well‐memorized facts and algorithms.
• Solving problems involving secondary knowledge requires study: first effort to move new information into long‐term memory (LTM), and then problem solving that builds conceptual frameworks by processing new information in a variety of distinctive contexts.
• To improve problem‐solving skills, the goal of study must be to increase the content and improve the organization of an individual’s long‐term memory.
If the science is correct, this has profound implications for math curricula like Everyday Mathematics (EM), which by design de-emphasize basic skills, standard algorithms, and developing mathematical fluency in favor of what is asserted to be encouraging deeper conceptual understanding at a very young age. The lack of basic skill development is compensated by teaching calculator usage starting in KG. Through constructivist learning pedagogy, EM attempts to have kids construct their own mathematical knowledge and their own algorithms through guided “discovery” based learning, before the teacher presents multiple non-standard algorithms, and eventually the standard algorithm for some operations far after other traditional curricula have taught them. In some critical cases, LCUSD kids will never be taught the standard algorithm (e.g. the long division algorithm is not taught in EM until 6th grade and by then LCUSD kids switch to Math In Focus, which expects it to have been taught starting in the 4th grade.) Only after concepts are discovered in EM are basic skills and rudimentary practice encouraged. But according to the findings of cognitive science, this sequence is exactly backward:
Schmidt et al. align with cognitive science, but the Common Core Mathematics Standards (CCMS) for topics noted above move in the opposite sequence: from multiple activities aimed at “deeper structures” to “simple math facts and procedures.”
Physiologically, “deeper” connections cannot grow until elements are stored in LTM neurons (memorized) and fundamental relationships are established (Anderson et al. 2000).
…
Schwartz and his colleagues (2011) report that “guided inquiry to introduce a topic can prepare students to see deeper conceptual principles.” However, if extensive “inquiry” or “discovery” is encouraged before instruction identifies what is correct, there is substantial risk students will move misconceptions into memory, and those misconceptions can be very difficult to root out (Willingham 2003, Rosenshine 2012).
The bottom line is the process used by EM is exactly the opposite of what cognitive science understanding of brain function and novice learning inform us should be done. It is also potentially harmful because not only are our kids not doing what should be done in the primary years — overlearning and mastering basic skills — but they spend significant time trying to discover algorithms on their own or with teacher guidance, and misconceptions and incorrect understanding may be committed to long term memory (LTM). If nothing else, this may explain why, as many of us have experienced with our children, EM leaves many children confused and with what many of us observe are deficient numeracy skills. The takeaway is if your child is subject to EM, you or your child’s teacher will need to heavily supplement their EM math education with basic skills practice to develop computational fluency and move basic math facts to LTM.
-Sugi-
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