Cognitive Science and the
Common Core Mathematics Standards
Eric
A. Nelson1
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.
Introduction
U.S. Math and Science
Achievement
In 2012, the Organisation for Economic Co-operation and Development (OECD) tested skills in 22 developed-world nations including the United States. In "numeracy" (solving problems with mathematical content), U.S. 16-34 year olds and 16-24 year olds ranked dead last (OECD 2013a,b). Figure 1 displays an Educational Testing Service (ETS) rendering of these data from the OECD's Survey of Adult Skills (Goodman, Sands, Coley, p. 12).
Figure 1. Average scores in numeracy for age 16-34 (millennials) and age 16-24, by participating country/region: 2012
Low skills in numeracy have consequences. Between 1984 and 2011, as a percentage of U.S. bachelor's
degrees, degrees awarded in the physical sciences and engineering fell 40% (NSB
2012, NSF 2014). Nearly 60% of
first-year college students who declared their intention to major in science, technology,
engineering, or mathematics (STEM fields) in 2003 failed to gain STEM degrees
within 6 years (Olson and Riordan, 2012).
U.S. students who are U.S. citizens have disproportionate
difficulty in achieving degrees in fields that require substantial preparation
in mathematics. According to the
National Science Foundation (NSF), of the doctorates awarded by U.S.
universities between 2002 and 2006, non-U.S. citizens received 44% in
chemistry, 57% in mathematics, 58% in physics, and 65% in computer
sciences. Of U.S. PhD's awarded in
engineering, 33% went to U.S. citizens, while 44% went to citizens of China,
India, and Korea. Of U. S.
doctorates in electrical engineering, 23% went to U.S. citizens (Falkenheim
2007).
A high percentage of foreign students remain in the US
after schooling (Finn 2012) and it is well-documented that they make a
disproportionate contribution to the American economy. One wonders, however, how long these
"best and brightest" from around the globe will remain in a nation where
schools fail to prepare their children to follow in their footsteps.
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.
Cognitive studies also suggest ways in which U. S. math
standards and achievement can be repaired.
Cognitive Architecture
The study of how the human brain solves problems is a sub-discipline
in the field of cognitive science. Cognitive
research divides problems into two types. In a "well-structured" problem, experts
agree on a "right answer" and paths to the answer are well-defined. The calculation problems explained and
assigned in "traditional" math textbooks are nearly always well-structured. "Ill-structured" problems include those
in which experts may disagree on answers (as in many problems in economics or
political science) (Simon 1973).
Between 1990 and 2009, as new technologies and research
increased scientific knowledge of how the brain works, the importance of memorization
and "constructivist" strategies in instruction was debated among cognitive
experts. In the 2009 book Constructivist Instruction: Success or Failure?, methods of
teaching ill-structured problem solving continued to be contested among leading
cognitive scientists, but substantial agreement was noted on methods to teach well-structured
problem solving (Tobias and Duffy, 2009).
Thomas Kuhn proposed in The Structure of Scientific Revolutions (1962) that the progress of
science is a history of models and assumptions that change when they are found
to be in conflict with new measurements and discoveries. Formerly debated assertions by experts
become "new scientific facts" and "accepted models" when they are not challenged
by other domain experts for a number of years.
Between 2009 and 2017, among leading cognitive scientists,
the agreement noted in Constructivist
Instruction for well-structured problems has continued to be uncontested.
Solving Well-Structured
Problems
Below is a simplified description of the cognitive
science model for well-structured problem solving. When available, references cited will be
relatively jargon-free reviews for educators written by cognitive experts that
include extensive citation of peer-reviewed research.
Learned knowledge can be divided into two types. "Primary" knowledge is that which humans
are biologically programmed to learn instinctively during "window periods" of
development. Examples during
childhood include learning to distinguish faces, comprehending the local spoken
dialect, and creating fluent speech in the dialect.
"Secondary" knowledge is what we are not programmed by
evolution to acquire automatically, but is beneficial to know for functioning
in our society. Examples include
learning to read, write, and solve math problems. Primary learning occurs without effort
by repeated exposure to primary knowledge, but learning secondary knowledge
generally requires cognitive effort.
The purpose of schooling is to move secondary knowledge into memory (Pinker
1994, Geary 2002, Sweller 2008).
Two types of memory relied upon in problem-solving are long-term memory (LTM) where elements of
knowledge are stored and working
memory (WM) where elements are processed.
If a stimulus from the senses becomes a focus of attention, the brain decomposes
this newly acquired knowledge into small "elements" such as a simple image, sound,
or relationship, and moves the elements into working memory (Anderson et al.
2000).
Long-term memory
is composed of tens of billions of "neurons" (one type of brain cell) and their
connections. Each neuron can store
a knowledge element. Elements
processed in working memory that are not yet stored in long-term memory tend to
be added to LTM.
Human LTM has
enormous capacity. Most U. S. 18-year olds can generally define about 60,000
"dictionary words" (Biemiller 2001).
In addition, entering adulthood, humans can identify tens of thousands
of images, odors, places and faces, apply math facts and procedures, and
fluently apply many thousand rules and procedures in areas including motor
skills and creating speech.
Each LTM neuron
can grow hundreds of physiological connections to other neurons. This "wiring" can carry electrical
impulses to and from other neurons, with linkages between cells connecting at
"synapses."
If an element
entering WM has previously been stored in LTM, the element can serve as a "cue"
that causes the LTM neuron storing the matching element to "activate" and
"fire," sending out an electrical impulse.
The neurons connected to a firing neuron may also fire and cause other
linked neurons to fire (Anderson et al. 2004, Willingham 2008, Koedinger et al.
2012). A large network of linked
elements is said to form a conceptual framework or "schema" (plural
"schemata").
Connections
between neurons tend to form, or to speed and/or strengthen, when neurons fire
at close to the same time during problem solving (Koedinger
et al. 2012). In general, "memory is the residue of
thought" (Willingham (2003, 2008). Relationships
between the elements in the neurons are more likely to be stored and
retrievable if the elements are repeatedly processed together in WM. Repeated thought that involves a relationship
between elements tends to strengthen their wired connection so that when one is
activated, the other is more likely to activate.
Strengths and Limitations of Working Memory
Working memory
is where the brain solves problems.
Problem solving begins with a goal and initial data. Data are processed in WM in steps that may
include storing data from the senses, recalling a relationship from LTM, or calculating
a math result.
If a cue is part
of a relationship that has been "memorized to automaticity" (is easily and
quickly recalled), the cue can cause activation of related elements in its LTM network. Working memory has an essentially
unlimited ability to recall elements from relationships that have been substantially
activated and apply those elements to solve a problem (Ericsson and Kintsch
1995, Anderson et al. 2004, Kirschner et
al. 2006).
Elements
entering WM that have not previously been stored in LTM or do not tend to be substantially
activated by cues in the problem are termed "novel" elements. These may include novel input from the
senses (such as problem data), or a "middle step answer" determined during the
steps of problem solving, or a "looked up" or a calculator answer (Kirschner et
al. 2006).
For these novel
elements, working memory is limited in both duration and capacity. Unless novel elements are rehearsed
(repeated to keep in memory), WM can generally retain an element for only 3 to
30 seconds (Peterson and Peterson 1959, Cowan 2010). For most adults, WM is generally
estimated to be able to hold and process only 3 to 5 novel elements at each step
during problem solving (Cowan 2001, 2010).
Working Memory Implications for Learning
Perhaps the most
significant discovery of cognitive research on learning in the past two decades
is that working memory has an essentially unlimited ability to hold and process
elements of knowledge that can be activated by cues and quickly recalled from
long-term memory, but can hold only a few elements at each point in processing
which cannot quickly be recalled. An
individual's problem solving ability depends on how much information about the
topic is stored in the individual's LTM and the quantity and quality of the
linkages that identify the relationships among those memorized elements.
If information must
be calculated on a calculator or looked up on the internet, instead of recalled
from memory, the resulting novel element or elements are difficult to retain in
memory for any length of time during subsequent steps of problem solving.
Further, at each
step of problem solving, if too much information from any source is novel, the
limited slots in novel WM may be full, and a new novel element moved into WM
will bump out a stored novel element.
If the bumped-out element is needed to solve the problem, mental
confusion tends to result (Kirschner et al. 2006).
Cognitive
scientists have identified three major ways to "work around" the bottleneck in
working memory: Automaticity,
algorithms, and chunking.
• If the facts, rules, and procedures
governing a topic are "automated" (stored in LTM and well linked), information can
be recalled via cues into WM and processed when needed. Extensive practice using "automated" knowledge
to solve a variety of problems gradually tends to construct fluency: an intuitive sense of what
facts and rules to apply when (Clark 2006).
•
Memorized stepwise procedures (algorithms) can work
around the WM bottleneck by limiting how many elements must be
retained in novel WM at any one time during processing (Willingham 2009).
•
If elements that frequently occur together
are memorized to form a single "chunk" (such as a familiar area code or acronym),
the chunk of problem data can be remembered as a single element of unique data
during processing.
Multiple well-regarded models for problem-solving have
been proposed that vary somewhat in terminology and components from the
description above, but all recent models include a WM for processing that is
large for elements recallable with automaticity but minimal for novel elements.
Summarizing the implications for learning of these new scientific
discoveries:
• When
solving math problems of any complexity, due to 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, 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 individua's long-term memory.
Experiencing (and Working
Around) WM Limits
For those who were required to learn their times tables
and practice the "standard algorithms" of arithmetic, the strengths, limits,
and a work-around for WM limits can be experienced with an experiment (of a
type suggested by Willingham): Multiply
93 times 72 "in your head." Do not
use fingers or pencil and paper.
Take a moment to try.
Did a "middle step answer" bump out of WM an element you
needed to remember from a previous step?
Now try 93 times 72 with pencil and paper.
Did you use an algorithm automated long ago? Which method was easier? Why?
Standards
The recent findings of cognitive research suggest an
explanation in part for current levels of U.S. math achievement.
Before 1975, as a part of learning math, young people memorized
math facts and algorithms. That
knowledge served as a foundation for careers including science, engineering, building
trades, mechanics, accounting, and business.
Inexpensive calculators became available in the 1970's,
and by 1996, a majority of U. S. 8th graders reported daily use of
calculators in math classes (Waits and Demana 2000). Between 1990 and 2000, over 40 states adopted
"state math standards" aligned with 1989 recommendations by the National
Council of Teachers of Mathematics (NCTM) (Klein 2002, Loveless 2002, 2003,
Schoenfeld 2004). The NCTM advocated
calculator use "at all grade levels."
Additionally, in Grades 5–8, the 1989 NCTM
standards called for "increased attention" to "reasoning" and "decreased
attention" to "memorizing rules and algorithms," "manipulating symbols," and
"rote practice" (NCTM 1989). NCTM-type
standards remained in place in most U. S. states until 2010 (Carmichael et al.
2010).
The cognitive science of 2017 suggests that with
"decreased attention" to "memorizing rules and algorithms," skill in
calculations would decline, and between 1995 and 2002, available data from
states using NCTM-type standards show a sharp decline in student test scores in
math computation (Hartman and Nelson 2016). For the few states with standards that
emphasized "computational facility," test
scores were relatively high (Schmid 2000, Loveless 2003), but by 2012, overall
U. S. young-adult skills in numeracy ranked dead last among the 22 nations
in OECD testing.
CCMS Adoption
Federal "No Child Left Behind" legislation in 2002 incentivized
testing based on state standards, but with nearly 50 different sets of standards
and tests, comparisons between states, or to national or international norms, were
difficult at best. Despite sizable
state spending and school time devoted to testing, it was difficult to tell how
students and states were doing compared to students in other states and nations.
In 2009, the National
Governors Association and the Council
of Chief State School Officers sponsored the drafting of a single set of "Common
Core" standards in English and mathematics. Between 2010 and 2012, over 40 states adopted
the Common Core Math Standards (CCMS) (National
Governors Association 2010).
A number of states rescinded CCMS adoption or revised the standards
between 2012 and 2016, but in most, standards remain similar to the CCMS
(Heitin 2015, Norton et al. 2017). By
2015, most local districts were gradually purchasing textbooks advertised as
aligned with the CCMS and most states based annual testing on CCMS-type
standards.
The impact of new standards cannot be measured until teaching
is based upon them for a number of years, but one important role of science is to
identify rules that allow accurate prediction of the outcomes of procedures and
processes.
By applying what cognitive science knows about how the
brain solves problems, can we predict the likely outcomes of instruction based
on the standards of the CCMS?
The Common Core and Automaticity
The knowledge taught in mathematics is generally divided
into facts, procedures, and concepts (Willingham 2009, Siegler and
Lortie-Forgues 2015).
In the 2008 Report
of the National Mathematics Advisory Panel, a presidential commission, five
of the nation's leading cognitive scientists (David Geary, Valerie Reyna, Wade
Boykin, Susan Embretson, and Robert Siegler) advised, to avoid working memory
limits, that the "central" strategy to improve student problem solving is
"[T]he achievement of automaticity, that is,
the fast, implicit, and automatic retrieval of a fact or a procedure from
long-term memory .... [A]rithmetic facts ... should be thoroughly mastered, and
indeed, over-learned."
In cognitive studies, "overlearning" is defined as
repeated practice in recalling new knowledge.
Cognitive scientist Richard Clark (2006) notes that
researchers who use computers to model information processing by the brain
make:
"... a very compelling case that all effective
applied knowledge must be proceduralized and automated in order to circumvent
the limits on working memory.... Most
[other researchers] reach a similar conclusion about the importance of the
automaticity process."
Citing extensive studies, cognitive scientist Daniel
Willingham (2009) writes:
"{A]nswers must be well learned so that when
a simple arithmetic problem is encountered..., the answer is not calculated but simply retrieved from memory." (emphasis added)
The CCMS, however, ask students to know "from memory" only
half of the arithmetic facts, and to calculate the other half. The 1st grade CCMS includes:
1.OA.6. Add
and subtract within 20, demonstrating fluency for addition and subtraction
within 10. Use strategies such as counting on; making ten
(e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten
(e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); .....
The related 2nd grade standard is:
2.OA.2. Fluently add and subtract within 20
using mental strategies (from 1.OA.6).
By the end of Grade 2, know from memory all sums of two one-digit numbers.
In a 2016 interview (Fordham 2016) of Dr. Jason Zimba, one
of three "lead writers" of the CCMS, the Fordham Foundation asked if in the
standards, "'being fluent in' also means 'know from memory.'" Zimba answered:
JZ: They aren't the same thing, and the
language of the standards makes this clear.... [In both 2.OA.2 and 3.OA.7]: If
being fluent were the same thing as knowing from memory, the second sentence
would not have been necessary. ....
In every case, fluency pertains to an act of calculation."
By this definition of fluency, the CCMS never ask that
students "know from memory" the subtraction or division facts, calling for
calculation instead. Cognitive science
emphasizes the importance of retrieval from memory, and the disadvantages of calculation,
for all fundamental facts.
Recall vs. Calculation
Why must all arithmetic facts must be "memorized to
automaticity?" According to science:
•
When 56/8 must be done on a calculator, mentally
storing the answer for transfer to paper will occupy a novel WM slot which, if
novel WM is already full, will bump out an element that may be needed during
subsequent processing.
•
Even a quick mental calculation of 56/8 requires
information to be placed into novel WM. Automated retrieval does not.
•
Any calculation takes more time than
automated recall, and each of the other elements stored in novel WM during
prior problem steps begin drop out after only a few seconds.
•
Calculated answers are more likely to have errors
than recalled facts (Willingham 2009).
•
Automated recall helps to free slots in novel
WM for "context cues" that distinguish different types of calculations. Processing those cues is a key step in
building the conceptual frameworks that promote recall of appropriate facts and
algorithms (Willingham 2003, 2008, 2015).
Given the time and considerable effort required to
automate recall, it would seem logical to take the CCMS advice to memorize
simple addition and multiplication facts, then calculate the subtraction and
division. But math standards must
work both mathematically and cognitively. Willingham (2009) notes:
"[A]utomatic
retrieval of basic math facts is critical to solving complex problems.... [B]efore they are learned to
automaticity, calculating simple arithmetic facts does indeed require working
memory."
Arithmetic is the foundation for mathematics. By asking that only half of the fundamental
arithmetic facts be automated, the CCMS creates what science predicts will be a
"critical" barrier to subsequent learning.
Learning Facts: Sequence and Timing
For all of 1st grade and part of 2nd,
the CCMS ask pupils to "fluently" solve multi-step addition and subtraction
before they know addition facts (see 1.OA.6 and 2.OA.2 above). In several 3rd grade 3.OA
standards, the CCMS ask students to multiply and divide before knowing their
multiplication facts.
The 2010 CCMS assumes the brain can work effectively with
facts before they have been well-memorized. Science in 2017 says the student brain
cannot.
Age Appropriate?
On average, 1st graders have a smaller novel WM capacity
than 4th graders, who have a smaller capacity than adults (Cowan 2001). Gathercole (2006, 2008) notes, "Poor
working memory skills are relatively commonplace in childhood" and advises to
"avoid working memory overload in structured learning activities." Asking 1st graders to perform
multi-step addition and subtraction (see 1.OA.6) before knowing addition facts would
seem by definition to raise concerns of WM overload.
Required
Time
Counting members of commutative pairs separately, there
are more than 150 math facts that involve two single digits for addition and
subtraction, and over 150 more for multiplication and division. Learning mathematics requires extensive "verbatim"
recall, which is generally more difficult to achieve than less precise "gist" (summary)
recall. Geary et al. (2008) write:
"Verbatim recall of math
knowledge is an essential feature of math education, and it requires a great
deal of time, effort, and practice."
Achieving automaticity generally involves strategies such
as overlearning spaced over days and weeks, repeated self-testing, and
interleaved practice (Willingham 2003, Brown et al. 2015). These strategies are time-consuming, but
according to science they efficiently promote the automaticity in recall that
is required for math facts and procedures.
CCMS goals include "know from memory" the addition facts at
"the end of Grade 2" and multiplication facts at "the end of Grade 3," but the
CCMS do not state when learning of these math facts is to begin. The CCMS do not call for subtraction and
division facts to be known from memory at all. Given the large number of standards at
each grade in Grades 1–3 before mastery of component facts is mentioned,
it is not clear that the CCMS provide reasonable time during instruction for a
goal that all students automate recall of all arithmetic fundamentals.
Standard Algorithms
For millennia, to address poorly understood but quite evident cognitive
limitations, students were taught to solve multi-step problems by applying standard,
stepwise procedures (algorithms). Willingham
summarizes:
"certain procedures are used again and again. Those
procedures must be learned to the point of automaticity...." (Willingham 2004).
Geary et al. (2008) write that
"fundamental algorithms should be thoroughly mastered,
and indeed, over-learned...."
The CCMS call for students to fluently apply the standard
algorithms for arithmetic operations.
But in multiple topics, science predicts the CCMS instructional sequence
is likely to interfere with applying those algorithms fluently. For multiplication, the 4th grade
standard is:
4.NBT.5. Multiply a whole number of up to four
digits by a one-digit whole number, and multiply two two-digit numbers, using
strategies based on place value and the properties of operations. Illustrate
and explain the calculation by using equations, rectangular arrays, and/or area
models.
The corresponding 5th grade standard is:
5.NBT.5. Fluently multiply
multi-digit whole numbers using the standard algorithm.
For each of the four multi-digit operations, the CCMS ask
students to practice multiple strategies for one year or two before learning
the standard algorithm. Cognitive studies have found that if two
processes are learned that are similar, steps from one can "interfere" with recalling
the other (Dewar et al. 2007). Practicing
multiple "strategies" for a year will move individual steps and sequences into
memory that will be errors in the standard algorithm.
After practicing multiple different multiplication
procedures over a two year period, will students know "with automaticity" after
Grade 5 which procedure steps to apply?
Standard algorithms
are exact (verbatim) procedures. Learning
four exact procedures takes effort over time. Is it a wise use of precious
instructional time to require students to spend a year or more practicing multiple
non-standard procedures that presumably will not be used after the point that the
standard algorithm becomes the standard that students are encouraged to use?
To align the CCMS with science, for each arithmetic
operation, each standard algorithm would be the first procedure taught, and it
would be practiced until its use is automated. The underlying concepts would then be
explored in a variety of ways to promote understanding, which could include the
"strategies" in the CCMS.
Theories
of Learning
Reasoning versus Algorithms
The 1989 NCTM standards emphasized reasoning over memorization. The CCMS, for facts and procedures, ask
students to begin each topic by practicing general reasoning strategies for a
year or more. Cognitive science has
found that for students in math, using "general reasoning strategies" in the
absence of well-memorized facts and algorithms is nearly always ineffective as
a strategy to solve problems (Sweller et al. 2010). Summarizing research, Clark et al.
(2012) explain why a "reason first, then memorize" sequence is not likely to work:
"If the learner has no relevant concepts and procedures
in long-term memory...., searching for a solution overburdens limited working
memory.... As a consequence, novices
can engage in problem-solving activities for extended periods and learn almost
nothing."
Discovery and
Misconceptions
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).
Except for a small amount of initial "discovery," science
supports learning facts and standard procedures first. Clark et al. (2012) summarize:
"Decades of research clearly demonstrate that for novices
(comprising virtually all students)..., teachers are more effective when they provide explicit guidance
accompanied by practice and feedback, not
when they require students to discover many aspects of what they must
learn...."
To align standards with science, "discovery" could
briefly introduce a topic and its context.
Next, students would practice the retrieval of facts, followed by problems
applying procedures and activities focused on deeper understanding.
Activities and Conceptual
Understanding
Science emphasizes that conceptual understanding is
vitally important in mathematics and it needs to be taught (Siegler and
Lortie-Forgues 2015). Geary et al. (2008)
write:
"[T]he cognitive processes that facilitate
rote retention (e.g., of over-learned arithmetic facts), such as repeated
practice, can differ from the processes that facilitate transfer and long-term
retention, such as conceptual understanding."
At the right time, Clark et al. (2012) note that a
variety of problem-solving activities can be useful:
"Independent problems and projects can be effective
– not as vehicles for making discoveries, but as a means of practicing
recently learned content and skills."
The CCMS "strategies" have the potential to help students
organize conceptual frameworks, but when should those activities should occur? The introduction section of the CCMS quotes
Schmidt, Houang, and Cogan (2002):
"[T]o be coherent, a set of content standards must evolve
from particulars (e.g., the meaning
and operations of whole numbers, including simple math facts and routine
computational procedures...) to deeper
structures inherent in the discipline...." (emphasis added)
Schmidt et al. align with cognitive science, but the 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).
Conclusion
From the viewpoint of cognitive research, while the CCMS
are superior to the "1989 NCTM-type standards" in place in most states prior to
2010, the repeated omissions and delays in initial memorization in the 2010
CCMS result in learning strategies that simply do not work in the brains of
non-experts (students). When
processing information that is not well memorized, the limits of WM are new but
verified science. Whenever new
science is confirmed, existing theories related to that science must be
re-evaluated.
Teachers do not decide K–12 state standards. When state standards at each grade level
are extensive, as in the CCMS, teachers will be unlikely to have time to add
additional topics that fill in what is missing and is needed to make the
standards work.
Given the weaknesses identified by cognitive studies in
the CCMS approach to arithmetic, the foundation for mathematics, the author
would submit that a sufficient case has been made for a detailed review of the
current CCMS and similar state standards by a panel that includes both
cognitive experts and educators.
Thanks to the progress of scientific research, most of it funded by U.S. taxpayers, we now know how to help students learn math and science far more efficiently and effectively, but U.S. policy makers whose views and power decide K-12 math standards are failing to put that knowledge to good use. Reasons for this inaction, justified and not, include:
•
The
areas of scientific consensus on how the brain solves problems are complex,
recent, and not widely disseminated in non-technical terminology.
•
Institutions
within education may require time to become familiar with the new science.
•
Traditional
policy-makers in education are not accustomed to having their philosophical
preferences constrained by science.
•
The
many education public-policy organizations that endorsed the CCMS as an
improvement over past standards may not find it easy to call for a review of
2010 standards only recently implemented, even if change is supported by new
scientific research.
•
Since
1995, standards in most states were debated and frequently changed without much
evidence of positive impact, tending to promote "reform fatigue."
Given those circumstances, unless there is substantial
pressure from U.S. economic and political leaders who respect science and
understand the importance of preparing our nation's students for mathematics,
science, and engineering, math standards that do not work are likely to remain
in place in most states for quite some time.
# #
# # #
Citation: Nelson, E.A. (2017). Cognitive Science and the Common Core Mathematics Standards , Nonpartisan Education Review / Articles. Retrieved [date] from http://nonpartisaneducation.org/Review/Articles/v13n3.pdf
References:
Anderson, J. R., Bothell, D., Byrne, M. D.,
Douglass, S., Lebiere, C., Qin, Y.:
An integrated theory of the mind. Psychological Review 111(4) 1036 (2004)
Anderson, J., Reder, L., Simon, H.: Applications and misapplications of
cognitive psychology to mathematics education. Texas Education Review 1(2) 29–49
(2000)
Biemiller A.: Teaching vocabulary. American Educator 25: 24–28 (2001)
Brown, P., Roediger, H., McDaniel, M.: Make It Stick: The
Science of Successful Learning. Harvard
University Press, Cambridge, MA (2014)
Carmichael,
S., Wilson, W., Porter-Magee, K., Martino, G.: The
State of the State Math Standards 2010. Fordham Foundation, Washington, DC
(2010)
Clark, R.: Not knowing what we don't know:
reframing the importance of automated knowledge for educational research. In:
G. Clarebout and J. Elen (Eds.) Avoiding Simplicity, Confronting Complexity. Sense Publishers,
Rotterdam (2006)
Clark, R., Sweller, J., Kirschner, P.: Putting students on the path to
learning: The case for fully guided
instruction. American Educator, 36(1), 6–11 (2012)
Cowan, N.: The magical number 4 in short-term
memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences 24, 87–185. (2001)
Cowan, N.: The magical mystery four: How is working memory capacity limited,
and why? Current Directions in Psychological Science 19(1), 51–57 (2010)
Dewar, M., Cowan, N., Della Sala. S.: Forgetting due to retroactive
interference. Cortex 43.5 616
(2007)
Ericsson, K. A., Kintsch, W.: Long-term working memory. Psychological Review 102, 211–245
(1995)
Falkenheim, J.: NSF
Infobrief 08-301, National Science Foundation. (2007) https://www.scribd.com/document/1006889/National-Science-Foundation-nsf08301
Finn, M.: Stay
Rates of Foreign Doctorate Recipients from U.S. Universities, 2009, (2012) https://orise.orau.gov/stem/reports/stay-rates-foreign-doctorate-recipients-2009.pdf
Fordham Institute: Does
common core math expect memorization?
https://edexcellence.net/articles/does-common-core-math-expect-memorization-a-candid-conversation-with-jason-zimba (2016)
Gathercole, S., Lamont, E., Alloway, T.: Working memory in the classroom. Working
memory and education: 219–240 (2006)
Gathercole, S.: https://thepsychologist.bps.org.uk/volume-21/edition-5/working-memory-classroom (2008)
Geary D.: Principles of evolutionary educational
psychology. Learning and Individual
Differences 12: 317–345 (2002)
Geary, D., Reyna, V., Boykin, W., Embretson,
S., Siegler, R.: Chapter 4 - Report
of the Task Group on Learning Processes.
In: Foundations for Success: The
Final Report of the National Mathematics Advisory Panel, U.S. Dept. of
Education, Washington, DC (2008)
Goodman, M., Sands, A., Coley, R.: America's skills
challenge: Millennials and the future. Educational Testing Service, Princeton,
NJ (2015)
Hartman, J., Nelson E.: "Do we need to memorize that?" or
cognitive science for chemists. Foundations of Chemistry, 17(3), 263–274 (2015)
Hartman, J., Nelson E.: Automaticity
in Computation and Student Success in Introductory Physical Science Courses.
https://arxiv.org/abs/1608.05006 (2016)
Heitin, L.: Nebraska's new math standards.
Curriculum Matters Blog, Education Week, (9/15/2015)
Kirschner,
P., Sweller, J., Clark, R.: Why
minimal guidance during instruction does not work: An analysis of the failure
of constructivist, discovery, problem based, experiential and inquiry-based teaching.
Educational Psychologist 41, 75–86 (2006)
Klein, D.: Brief history of American K–12
mathematics education in the 20th century. In: J. Royer (Ed.). Mathematical Cognition:
Current Perspectives on Cognition, Learning, and Instruction. Information Age Publishing, Greenwich, CT (2002)
Koedinger, K., Corbett, A., Perfetti, C.: The Knowledge-Learning-Instruction (KLI)
framework: Bridging the science-practice chasm to enhance robust student
learning. Cognitive Science, 36(5),
757–798 (2012)
Kuhn, T. S.: The Structure of Scientific Revolutions.
University of Chicago Press: Chicago, (1962)
Loveless, T.: The 2002 Brown Center
Report on American Education. Brookings
Institution Press, Washington, DC (2002)
Loveless, T.: The regulation of teaching and learning.
In: Hallinan, M., Gamoran, A., Kubitschek, W., Loveless, T. (Eds.). Stability
and Change in American Education: Structure, Process, and Outcomes, pp 171–192 Eliot Werner Publications, Clinton Corners, NY (2003)
National Council of Teachers of Mathematics: Curriculum and Evaluation
Standards for School Mathematics, NCTM,
Reston, VA (1989)
National Governors Association and the
Council of Chief State School Officers. Common Core State
Standards Initiative: Common Core
State Standards for Mathematics.,
Washington, DC (2010)
National Science Board: Science and Engineering
Indicators 2012. Appendix Table 2-12 (2012) https://www.nsf.gov/statistics/seind12/c2/c2s2.htm. Accessed 20 September 2016
National Science Foundation: Science and Engineering
Indicators 2014, Appendix Table 2-17 (2014) https://www.nsf.gov/statistics/seind14/index.cfm/appendix/tables.htm Accessed 20 September
2016
National Science Foundation: NCSES Survey of Earned
Doctorates 2015, Table 22 (2016) https://www.nsf.gov/statistics/2017/nsf17306/data.cfm Accessed 17 January 2017
Norton, J., Ash J., Ballinger, S.: Common Core revisions: What are states really changing? Abt Associates, Cambridge, MA (2017)
Organisation for Economic Co-operation and Development: OECD Skills Outlook 2013: First Results from the Survey of Adult Skills. OECD Publishing, Paris (2013a)
Organisation for Economic Co-operation and Development: Literacy, Numeracy, and Problem Solving in Technology-Rich Environments -- Framework for the OECD Survey of Adult Skills.
OECD Publishing, Paris (2013b)
Olson, S., Riordan, D.: Engage to Excel: Producing
One Million Additional College Graduates with Degrees in Science, Technology,
Engineering, and Mathematics, Table C-6.
Report to the President. Executive Office of the President, Washington, DC
(2012)
Peterson, L., Peterson, M.: Short-term retention of individual
verbal items. Journal of Experimental Psychology 58, 193–198 (1959)
Pinker, S.: The Language Instinct:
How the Mind Creates Language. William.
Morrow, New York (1994)
Rosenshine, B.: Principles of instruction: Research-based
strategies that all teachers should know. American
Educator 36.1 12 (2012)
Schmid, W.: New battles in the math wars. Harvard
Crimson, May 4 (2000)
Schmidt, W., Houang, R., Cogan, L.: A coherent curriculum. American Educator, Summer, 1 (2002)
Schoenfeld, A.: The math wars. Educational
Policy 18, 253–286 (2004)
Schwartz, D., Chase, C., Oppezzo, M., Chin,
D.: Practicing versus inventing
with contrasting cases: The effects of telling first on learning and transfer. Journal
of Educational Psychology, 103(4), 759–777
(2011)
Siegler, R., Lortie-Forgues, H.: Conceptual knowledge of fraction
arithmetic. Journal of Educational Psychology 107.3 909 (2015)
Simon, H.: The structure of ill-structured
problems. Artificial Intelligence 4, 181–201 (1973)
Sweller,
J.: Instructional implications of
David C. Geary's evolutionary educational psychology. Educational Psychologist 43, 214–216 (2008)
Sweller, J.: What human cognitive architecture tells
us about constructivism. In: Tobias
S. and Duffy T. (Eds.) Constructivist
Instruction: Success or Failure? pp. 127–143. Routledge, New York
(2009)
Sweller, J.,
Clark, R., Kirschner, P.: Teaching
general problem-solving skills is not a substitute for, or a viable addition
to, teaching mathematics. Notices of the
American Mathematical Society 57(10), 1303–1304 (2010)
Tobias, S., Duffy. T., (Eds.): Constructivist
Instruction: Success or Failure? Routledge, New York (2009)
Waits, B., Demana, F.: Calculators in mathematics teaching and
learning. In: Burke, M. and Curcio, F (Eds.) Learning mathematics for
a new century.
National Council of Teachers of Mathematics, Reston, VA (2000)
Willingham, D.: Allocating student study time: Massed
versus distributed practice. American Educator. 26(2), 37–39 (2002)
Willingham, D.: Students remember what they think about.
American Educator, 27(2), 37–41
(2003)
Willingham, D.: Practice makes perfect—but only if
you practice beyond the point of perfection. American Educator 28(1), 31–33 (2004)
Willingham, D.: How knowledge helps. American
Educator 30(1), 30–37
(2006)
Willingham, D.: What will improve a student's memory? American
Educator 32(4), 17–25
(2008)
Willingham, D.: Is it true that some people just can't
do math? American Educator 33(4), 14–19
(2009)
Willingham, D.: Do students remember what they learn in
school? American Educator (Summer) 7 (2015)
# #
# # #
1 Eric A. (Rick) Nelson is a career STEM
instructor and co-author with Dr. Donald J. Dahm of Calculations in Chemistry – An Introduction, published by
W. W. Norton.
# #
# # #
(Discussion of this paper is invited in the
comment section at www.ChemReview.Net/blog ,
Post 15.)