Phoenix Rising
Bringing
the Common Core State
Mathematics Standards to Life
BY HUNG-HSI WU
Hung-Hsi
Wu is a professor emeritus of mathematics at the University of California,
Berkeley. He served on the National Mathematics Advisory Panel and has written
extensively on mathematics curriculum, textbooks, and teacher preparation. Since
2000, he has conducted professional development institutes for elementary and
middle school teachers. He has worked extensively with the state of California in mathematics
education, and was a member of the Mathematics Steering Committee that
contributed to revising the National Assessment of Educational Progress
Framework. In recent years, he has focused on writing textbooks for the
professional development of K-12 mathematics teachers.
Many sets of state and national mathematics
standards have come and gone in the part two decades. The Common Core State
Mathematics Standards (CCSMS), which were released in June of 2010[1], have
been adopted by almost all states and will be phased in across the nation in
2014. Will this be another forgettable standards document like the
overwhelming majority of the others? Perhaps. But unlike the others, it will be
a travesty if this one is forgotten. The main difference between these standards
and most of the others is that the CCSMS are mathematically very sound overall.
They could serve—at long last—as the foundation for creating proper school
mathematics textbooks and dramatically better teacher preparation.
Before the CCSMS came along, America
long resisted the idea of commonality of standards and curriculum—but it did not
resist such commonality in actual classrooms. Despite some politicians'
rhetoric extolling the virtues of local control, there has been a de facto
national mathematics curriculum for decades: the curriculum defined by the
school mathematics textbooks. There are several widely used textbooks, but
mathematically they are very much alike. Let's call this de facto mathematics
curriculum Textbook School Mathematics (TSM)[i].
In TSM, precise definitions usually are not given and logical reasoning is
hardly ever provided (except in high school geometry texts) because the
publishers mistakenly believe that intuitive arguments and analogies suffice.
Thus, fractions are simultaneously (and incomprehensibly) parts of a whole, a
division, and a ratio; decimals are taught independently from fractions by
appealing to the analogy with whole numbers; negative numbers are taught by using
patterns and metaphors; the central idea of beginning algebra is the
introduction of the concept of a variable
(which implies, wrongly, that something is going to vary), when it ought to be
becoming fluent in using symbols so as to do generalized arithmetic; solving
equations is explained by the use of a balance to weigh variables on the
weighing platforms; etc.
Worse, with TSM in the background, the
prevailing dogma in mathematics education is that the main purpose of a set of
standards is either to pick and chose from a collection of tried-and-true
topics (from TSM, of course) and organize the selected items judiciously, or to
vary the pedagogical approaches to these topics. For example, when California's Number
Sense Standards ask that, in grade 5, "Students perform calculations and
solve problems involving addition, subtraction, and simple multiplication and
division of fractions and decimals," it is understood that all of the
classrooms will do these arithmetic operations on fractions in accordance with
TSM. From this perspective, the main point of this standard is that these
calculations with fractions are taught in
the fifth grade. Indeed, the very purpose of mathematics standards (prior
to the CCSMS) seems to be to establish in which grade topics are to be taught.
Often, standards are then judged by how early topics are introduced; Thus,
getting addition and subtraction of fractions done in the fifth grade is taken
as a good sign. By the same ridiculous token, if a set of standards asks that
the multiplication table be memorized at the beginning of the third grade or
that Algebra I be taught in the eighth grade, then it is considered to be rigorous.
The CCSMS challenge this dogma. Importantly,
the CCSMS do not engage in the senseless game of acceleration—to teach every
topic as early as possible—even though refusing to do so has been a source of
consternation in some quarters. For example, the CCSMS do not complete all the
topics of Algebra I in grade 8
because much of the time in that grade is devoted to the geometry that is
needed for understanding the algebra of linear equations.[ii]
But the real contribution of the CCSMS lies in their insistence on righting the
many wrongs in TSM. As opposed to
the standards of years past, the CCSMS
are aware of the chasm between what TSM is and what school mathematics ought to
be. They are unique in their realization that the flaws in the logical
development of most topics in TSM—not how early or how late each topic is
placed in the standards—are the real impediment to any improvement in mathematics
education. Garbage in, garbage out, as
the saying goes. If we want students to learn mathematics, we have to teach it
to them. Neither the previous mathematics standards nor the TSM on which they
rely did that, but the CCSMS do.
Beyond the frequent absence of
reasoning, the disconnectedness in the presentation of mathematical topics in
TSM turns a coherent subject into nothing more than a bag of tricks. Students
are made to feel that what is learned one year can be forgotten in the next. By
contrast, the CCSMS succeed in must instances in maintaining continuity from
grade to grade. The most striking example may well be the seamless transition
from eighth-grade geometry to high school geometry. In fact, the CCSMS succeed
in integrating geometry into the overall fabric of school mathematics. The
mathematics in the CCSMS finally begins to look like rnathematics.
Unfortunately, textbook developers have
yet to accept that the CCSMS are radically different from their predecessors.
Most (and possibly all) textbook developers are only slightly revising their
texts before declaring them aligned with the CCSMS. Do not be fooled. TSM is
much too vague and has far too many errors to be aligned with the CCSMS. For
example, when the National Mathematics Advisory Panel reviewed two widely used
algebra textbooks to determine their "error density" (which was
defined as the number of errors divided by the number of pages in the book), it
found that one had an error density of 50 percent and the other was only
slightly better at 41 percent.[iii] We
must start from scratch. Since teacher education in mathematics has long been
based on TSM, both pre-service and in-service training must also be created
anew.
Let us give two examples of the kind of
change the CCSMS (if properly implemented) will bring to the mathematics
classroom.
Example 1:
Adding Fractions
How should students add 1/8 + 5/6 ?
The way it is done in TSM is to not say anything about what it means to add
fractions, but instead to prescribe the procedure of finding the least common
multiple of the denominators 8 and 6, which is 24, and note that 24 = 3 x 8 and
24 = 4 x 6. Students are then instructed to add as follows:
By all accounts, this procedure
makes no sense to fifth-graders, but many seem to memorize it and it remains
firmly entrenched in TSM. Adding is supposed to "combine things." The
concept of "combining" is so basic that it is always taught at the
beginning of arithmetic. Yet, can one detect any "combining" in the TSM
approach to 1/8 + 5/6 ? Children who have made the effort to master the addition
of whole numbers naturally expect that the addition
of fractions will be more of the same, i.e., "combining things." But
when "adding fractions" is presented as having nothing to do with
"adding whole numbers," the fear that they cannot articulate is
undoubtedly that mathematics is impossible to understand. Indeed, there are
reports that much math phobia begins with adding fractions.
In the CCSMS, adding fractions is spread
through three grades, progressing from the simple to the complex, giving
students time for complete mastery.[2]
Briefly, in grade 3, students learn to think of a fraction as a point on the
number line that is "so many copies" of its corresponding unit fraction. For example, 5/6 is 5
copies of the unit fraction 1/6 (and 1/6 is 1 copy). When we represent a
fraction as a point on the number line, we place a unit fraction such as 1/6 on
the division point to the right of 0 when the unit segment from 0 to 1 is divided into 6 equal segments. It is
natural to identify such a point with the segment between the point itself and
0. Thus, as shown below, 1/6 is identified with the red segment between 0 and
1/6, 5/6 is identified with the segment between 0 and 5/6, etc. Then, the statement
that "5/6 is 5 copies of 1/6" acquires an obvious visual meaning: the
segment from 0 to 5/6 is 5 copies of the segment from 0 to 5/6.
In third grade, students also learn
about simple cases of equivalent fractions: 2/5 is the same point as — i.e., is equal to - (3 x 2)/(3 x 5), or 6/15.
This is because 2/5 is the second division point to the right of 0 when the
unit segment from 0 to 1 is divided into 5 equal segments. When each of these 5
segments is divided into 3 equal segments, it creates a division of the unit
segment into 3 x 5 = 15 equal segments. It is then obvious that the point 2/3
is exactly the same point as 6/15, which is (3 x 2)/ (3 x 5), as shown below.
In grade 4, the CCSMS call for students to
learn about adding two fractions as joining
two parts of the same whole. Think of the two fractions as segments, put
them together end-to-end on the same number line, and the sum is by definition
the length of the joined segment. For fractions with the same denominator,
adding these fractions yields a fraction whose numerator is the sum of the
respective numerators, as we can see clearly from an example. Let's show that
2/3 + 5/3 = (2 + 5)/3. On the number line below, 1/3 is the red segment.
Thus, 2/3 is 2 copies of the red segment
and 5/3 is 5 copies of the red segment, so "combining" 2/3 and 5/3
yields (2+5) copies of the red segment, which is 7/3. Therefore, adding
fractions is "combining things" in this case.[3]
In grade 4, students also go beyond the
simple cases to learn about equivalent fractions in general. Then in the fifth
grade of the CCSMS, students handle the sum of any two fractions. Of course, it
is still obtained by joining parts:
putting two segments together so that the sum is the total length. This they
are ready for because, by use of equivalent fractions, any two fractions may be
regarded as two fractions with the same denominator. For example, 1/8 and 5/6
are equal to (6 x 1)/(6 x 8) and (8 x 5)/(8 x 6), which now have the same
denominator, 48. So these fifth-graders can easily address our original
question — How should students add 1/8 + 5/6? — in a mathematically sound manner.
With their strong foundation from the third and fourth grades of the CCSMS,
they know that this addition problem is the same as asking how long the
following combined segment is.
This is the same answer as before
because, by equivalent fractions, 46/48 = 23/24. Therefore, students get to see
that adding fractions is
"combining things." Incidentally, there has been no mention of the
least common multiple of 8 and 6, and this is as it should be. (My pointing out
that 46/48 = 23/44 should not be interpreted as affirming the common practice
of insisting that every fraction be reduced to the simplest form. There is no
mathematical justification for this practice; I did it merely to show that we
got the same answer either way.)
I hope this example begins to clarify
the vast differences between TSM and the CCSMS. Adding fractions is a foundational
topic: TSM gives students (and teachers) a gimmick; the CCSMS require that
students actually learn mathematics.
Example 2:
Multiplying Negative Numbers
Why is (-2)(-3) = 2 x 3 ? This is
quite possibly the most frequently asked question in K-12 mathematics: why is
negative times negative positive? The answer, according to TSM, can be given
in terms of patterns. For the specific case of (-2)(-3), we observe that the
values of 4(-3), 3(-3), 2(-3), 1 (-3), and 0(-3) are as follows:
There is an unmistakable pattern:
the answer on each line is obtained by adding 3 to the answer from the line
above. Thus, starting with the last line, 0 = 3 + (-3), -3 = 3 + (-6), -6 = 3 +
(-9), -9 = 3 + (-12), and of course the pattern persists if we also take into
account 5(-3), 6(-3), etc. But if we now continue the sequence of multiplications
of 4(-3), 3(-3), 2(-3), 1(-3), and 0(-3), then the next couple of items in line
will be
(-1)(-3) = ?
(-2)(-3) = ?
Encouraged by the pattern we just
observed, we are confident that the number (-1)(-3) should be one that is
obtained from the number 0(-3) (which is 0) by adding 3:
(-1)(-3) = 3 + 0 =3.
Similarly, (-2)(-3) should be one
obtained from (-1)(-3) by adding 3:
(-2)(-3) = 3 + 3 = 2 x 3.
Is this a good explanation? No.
There are two problems. First, if instead of dealing with the product of
integers, we consider a product such as (-5/11)(-4/3), then a little thought would
reveal that this reasoning by patterns breaks down completely. Second, we must
convince ourselves that the pattern should
persist all the way to (-1)(-3), (-2)(-3), (-3)(-3), etc. In greater detail,
this pattern asks students to believe that
Of these, the critical one is the
first: (-1)(-3) = 3. If we know that, then, with or without a patent, we will
have the remaining equalities for the following reason. The distributive law, which is a statement
about how multiplication behaves with respect to addition, says if x, y,
and z are any three numbers, we
always have [y + z] x = yx + zx.
Thus, for example, [2 + (-1/3)](-4) = 2(-4) + (-1/3)(-4). The fact that all
numbers positive or negative obey the distributive law is a fundamental
assumption in mathematics. Now if y =
z = (-1) and x = (-3), then we have [(-1) + (-1)](-3) = (-1)(-3) + (-1)(-3).
Making use of this fact and assuming (-1)(-3) = 3, we now get :
(-2)(-3) = [(-1) + (-1)](-3) = (-1)(-3)
+ (-1)(-3) = 3 + 3 = 2 x 3.
For exactly the same reason, we
would get (-3)(-3) = 3 x 3, (-4)(-3) = 4 x 3, etc., provided we assume (-1)(-3)
=3. But how do we now (-1)(-3) =3? In TSM, there is no answer. This is the
nature of TSM: it often half-satisfies students' appetite for knowledge — but
given the precise nature of mathematics, this is almost the same as no
knowledge at all.
Let us now look at what the CCSMS say on
this matter. In die broader context of understanding negative numbers, it is
important that students have a clear conception of what a negative number is.
It should be a specific object rather than some ineffable philosophical idea.
For this, the CCSMS go back to the number line just as in the case of
fractions.[4]
One standard in the CCSMS for grade 6 has this to say:
Recognize opposite signs of numbers as
indicating locations on opposite sides of 0 on the number line; recognize that
the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3,
and that 0 is its own opposite.
Negative numbers are points on the
number line to the left of 0. More precisely, for each fraction that is a point
to the right of 0, its negative is the point to the left of 0 that is
equidistant from 0. We can think of a fraction such as 3.4 (which is 34/10, by
definition) and its negative -3.4 as mirror
images of each other with respect to 0, as shown below.
Jumping ahead to multiplying
negative numbers, the CCSMS for grade 7 say the following:
Understand that multiplication is
extended from fractions to rational numbers by requiring that operations
continue to satisfy the properties of operations, particularly the distributive
property, leading to products such as (-1)(-1) = 1 and the rules for
multiplying signed numbers. Interpret products of national numbers by
describing real-world contexts.
This standard needs amplification,
which I will provide in the process of giving a correct explanation of (-2)(-3)
= 2 x 3. This explanation will be valid also for the general case of (-m)(-n)=
mn for any integers m and n. When m and n are fractions (which is what this
standard calls for), a slightly more sophisticated explanation will be
necessary (and thus should be provided by any decent textbook), but we will
settle for the simpler case here.
The key step in the correct explanation
lies in the proof of (-1)(-1) = 1 (as asserted in the grade 7 standard).
Pictorially, what this equality says is that multiplying (-1) by (-1) flips
(-1) to its mirror image 1 on the right side of 0. A more expansive treatment
of this topic in accordance with the CCSMS would show that, more generally,
multiplying any number by (-1) flips it to its mirror image on the other side
of 0.
Now, how to find out if (-1)(-1) is the
number 1 or not? For students in grades 6 or 7, the most desirable way to do so
is by performing a direct computation that starts with (-1)(-1) and ends with
1. However, since there is no known way of doing this, we'll take an indirect
approach by anticipating the right answer (which is 1, of course) and asking:
is (-1)(-1) + (-1) equal to 0? If so, then we will see that (-1)(-1) is equal
to 1 and we are done. The key difference between (-1)(-1) and the longer
expression (-1)(-1) + (-1) is that we can actually do a computation on the
latter! We appeal to the distributive law in the second equal sign below.
(-1)(-1) + (-1) = (-1)(-1)+ 1(-1) =
[(-1) + 1] (-1) = 0 (-1) = 0
Notice that it is only when we get
to [(-1)+1] (-1) that we can begin to "compute" in the usual sense of
arithmetic: (-1) + 1 is equal to 0, and 0(-1) is also 0. In any case, we have
finally demonstrated - using familiar arithmetic — that (-1)(-1) =1.
Now we can prove (-2)(-3) = 2 x 3. We
first show (-1)(-3) = 3. We have (-1)(-3) = (-1)[(-1)+(-1)(-1)] which, by the
distributive law again, is equal to (-1)(-1) + (-1)(-1) + (-1)(-1) = 1 + 1 + 1
= 3. Thus (-1)(-3) = 3. Having taken care of our earlier concern as to why
(-1)(-3) is equal to 3, we can now easily complete our reasoning about (-2)(-3)
= 2 x 3, namely: (-2)(-3) = [(-1)+(-1)](-3) = (-1)(-3) + (-1)(-3), by the distributive
law (yet again!). And, by what we just proved, the latter is 3 + 3 = 2 x 3. So
(-2)(-3) = 2 x 3 after all.
If we reflect on the reasoning above, we
see clearly that the critical step was the application of the distributive
law; without that it would have been impossible to conclude that (-1)(-1) +
(-1) = 0, that (-1)(-3) = 3, or that (-2)(-3) = 2 x 3. This is exactly the main
emphasis in the preceding standard from the CCSMS. The proof of (-m)(-n)
= mn, for whole numbers m and n, is entirely similar. Thus, a teacher guided by the CCSMS, unlike
a teacher guided by TSM, would provide a correct and complete mathematical
explanation of why a negative times a negative equals a positive. There is no
need to look for patterns that do not hold true and no excuse for providing a
half-satisfactory explanation.
It takes no real knowledge of
mathematics to see from these two examples that the leap from TSM to the
mathematical demands of the CCSMS is a gigantic one. With more space, I could
provide many more examples: most of the time, the distance between TSM and the
CCSMS is vast. We cannot expect the nation's teachers to implement the CCSMS on
their own. So far, textbook developers are not rising to the challenge of the
CCSMS. Our only hope, therefore, lies in providing professional development to
help our teachers acquire the mathematical knowledge necessary to see the flaws
in TSM.
"Start
Selling What They Need"
For in-service teachers, professional
development is hardly synonymous with learning content knowledge. Far too
often, "professional development" is filled with games, fun new
manipulatives, the latest pedagogical strategies, and classroom projects that
supposedly make mathematics easy. The more serious kind of professional
development, which some small percentage of teachers are lucky enough to participate
in, addresses topics such as children's mathematical thinking, appropriate use
of technology, teacher-student communication, and refined teaching practices.
While these are important issues for teaching, they are not sufficient for
transitioning from TSM to the CCSMS. Right now, professional development that
replaces TSM with correct, coherent, precise, and logical K-12 mathematics is
urgently needed.
A natural reaction to the last point
would be disbelief: don't colleges and universities teach future teachers the
mathematics they need for teaching? Some may, but the vast majority do not (if
they did, teachers would be continuously complaining about the errors in their
students' textbooks, and our international ranking on mathematics assessments
would be much higher).
In courses for future high school math
teachers, colleges and universities usually teach university-level mathematics.
The idea is that the "Intellectual Trickle-Down Theory» should work: learn
advanced mathematics and you would automatically be knowledgeable about school
mathematics. But it doesn't work, not in theory and not in practice. What
colleges and universities should do is erase the damage done by TSM and revamp
future high school teachers' knowledge of the algebra, geometry, trigonometry,
etc., that they will be teaching.
In courses for future elementary
teachers, who have to master a whole range of subjects, colleges and universities
often teach pedagogy-focused "math methods" that merely embellish TSM.[5]
These courses are usually taught by mathematics education professors, not
mathematicians (who avoid teaching such courses because they wrongly see
elementary mathematics as trivial); so it may well be that in most of these
math methods courses no one — not even the professor — is aware of the flaws in
TSM.
Perhaps we can better expose the
absurdity of the way we prepare mathematics teachers if we consider the analogous
situation of producing good high school French teachers: should we require them
to learn Latin in college but not French? After all, Latin is the mother language
of French and is linguistically more complex than French. Surely mastering a
more complex language would enhance teachers' understanding of the French they
already know from their school days. Is teaching future French teachers Latin
any different from teaching future geometry teachers university-level
mathematics? I don't think it is. In the same way, if we want to produce good
elementary French teachers, wouldn't we ensure that they are fluent and
literate in French before they begin courses on methods for teaching French? We
would — and we should expect no less of our higher education institution &
approach to preparing elementary math teachers.
The failure of institutions of higher
learning to take seriously their obligation to properly prepare mathematics
teachers is a main reason why TSM has become entrenched in K-12.[6]
The failure occurs on two fronts: content knowledge and pedagogy. My main concern
is with content knowledge, as I believe that mastering the mathematics is the
hardest part of becoming a good math teacher, but I appreciate that others are
focused on pedagogy. For example, an article in the fast issue of American
Educator by Deborah Loewenberg Ball and Francesca M. Forzani addresses the
inadequacy of teachers' pedagogical preparation across all subjects.[iv] I
encourage readers who are interested in improving mathematics pedagogy to read
their article. However, because Ball and Forzani are talking about teacher
preparation in general, not just in mathematics, I would like to call attention
to the following passage,[v]
which may be misinterpreted in the context of mathematics teachers:
"High-leverage content” comprises
those texts, topics, ideas, and skills in each school subject area that are
essential for a beginning teacher to know well. High-leverage content is
foundational to the ideas and skills of the K-12 curricula in this country, is
taught in some form or another across most published textbooks and curricula,
and appears frequently. In addition, high-leverage content is fundamental to
students' learning and often causes difficulty if not taught well. It also is
often known only superficially by prospective teachers, or is entirely new to
them.
To me, the key to correctly
applying this to mathematics education is to focus on their statement that “High-leverage
content ... is taught in some form or another across most published textbooks."
Having worked with Ball, I believe she is well aware of the flaws in students'
math textbooks. TSM does touch on all the important mathematics in some form or another, but almost
never in a correct form. So while I would agree that high leverage topics can be found in today's most
widely used math textbooks, I would not agree that high-leverage content can be found in them.
That, of course, brings me back to my
main concern. Because of the teacher preparation programs' failure to teach
content knowledge relevant to K-12 classrooms, the vast majority of preservice
teachers do not acquire a correct understanding of K-12 mathematics while in
college. Because the flawed TSM they learned as K-12 students is not exposed,
much less corrected, they unwittingly inflict TSM on their own students when
they become teachers. So it comes to pass that TSM is recycled in K-12 from
generation to generation. Today, this vicious cycle is so well ingrained that
many current and future mathematics educators also are victimized by TSM, and
their vision of K-12 mathematics is impaired. They have been led to equate TSM
with "mathematics," so their educational commentaries on the school
mathematics curriculum, by their implicit or explicit reference to TSM, become
an unwitting affirmation of TSM. And so TSM lives on.
As a mathematician surveying this
catastrophic education mess, I have to admit that, when all is said and done,
the mathematics community has to take the bulk of the blame. We think school
mathematics is too trivial,[vi]
and we think the politics of education is a bottomless pit not worthy of our
attention. So we take the easy way out by ignoring all the goings-on in the
schools and simply declare that if we teach high school teachers good
mathematics, the rest is up to them. In other words, we hide behind the
Intellectual Trickle-Down Theory, even though we are daily confronted with
evidence that it is not working.
Of course, some mathematicians have
tried to make a contribution to school mathematics. But most of them have not
devoted enough time to investigating the problem. They tend to be unaware of
the sorry state of TSM and end up writing books that encourage teachers to build on their knowledge of TSM to solve
problems or learn new mathematics.[7] This
is akin to helping a starving person by buying him new clothes to make him look
better without trying to address the malnutrition problem. With the
opportunity provided by the CCSMS hovering over us, it is time that we
mathematicians make amends.
In March of 2008, I was passing through
Landon's Heathrow
Airport and happened to
catch sight of an ad by IBM:
Stop selling what you have.
Start selling what they need.
If we let "they" be our
math teachers and math education professors, then this would be a pointed directive
on what mathematicians need to do for school mathematics education:
Get to know what they need, and teach it.
The advent of the CCSMS sends out the
signal, for the first time from within the education community, that TSM has no
place in the school curriculum. TSM is incompatible with the CCSMS, and now
colleges and universities are duty-bound to provide future mathematics teachers
with a replacement of TSM. Would that those institutions were aware of their duties! Teachers of
all levels now must learn to teach mathematics, not just with analogies and
metaphors, and not just with incomprehensible pseudo-explanations and decrees,
but with precision, reasoning, and coherence.
Mathematical
Engineering
It will not be enough for institutions
of higher learning to teach future teachers rigorous advanced mathematics,
because the topics in school mathematics are not part of advanced mathematics.
Nor will it suffice to pass off pedagogy-laden courses as mathematics courses,
because the mathematical difficulties that lead to nonlearning cannot be cured
with pedagogical techniques. That said, the pressing need now is to provide all
future mathematics teachers with content knowledge that satisfies both of the
following requirements:
A. It is relevant to teaching — i.e., does not stray far from the material
they teach in school.
B. It is consistent with the following five fundamental principles of
mathematics: precise definitions are the basis for logical deductions; precise
statements clarify what is known and what is not known; every assertion can be
backed by logical reasoning; all the concepts and skills are woven together
like a tapestry; and each concept and skill has a purpose. (I briefly explain
each of these in the box on pages 8-9.)
Currently, TSM satisfies requirement A,
at least in the sense that it attempts to "cover" all of the
mathematics needed in K-12 (however, it is also riddled with unnecessary topics
— but that is another article). But TSM does not satisfy requirement B at all.
University-level mathematics satisfies B, but not A. (Those who are not convinced
should read the box on page 10.) What we are witnessing, therefore, is two
extremes in the presentation of mathematics, each one satisfying one of the
two conditions but not the other.
The middle ground — which must be both accessible
to children and mathematically correct — is a modified or customized version of university-level mathematics. Examples 1 and
2 above provide illustrations of such customization.
This brings us to a clearer conception
of what K-12 mathematics education is all about: mathematical engineering, in the sense that it is a customization
of abstract, university-level mathematics for the consumption of school students.
Let us put this in context. Engineering is the discipline of customizing
abstract scientific principles into processes and products that safely realize
a human objective or function. So, chemical engineering begins with chemistry
and results in Plexiglas tanks in aquariums, the gas you pump into your car,
shampoo, Lysol, etc. Electrical engineering transforms the abstract theory of
electromagnetism into computers, iPods, lights in your hall, hybrid motors,
etc. And in the same vein, mathematical engineering takes abstract, university-level
mathematics and customizes it into school mathematics (distinct from TSM) that
can be correctly taught, and learned, in K-12 classrooms.
My hope is that the CCSMS will usher in
mathematical engineering, drive out TSM, and replace TSM with school mathematics
proper.[vii]
But if our mathematical engineering work is limited to standards and large scale
assessments (which, sadly, seems to be where we are currently headed), then
nothing will be accomplished. Proper school mathematics textbooks for teachers
and students, model lesson plans, diagnostic assessments, and professional
development are absolutely necessary. These things are often discussed as
instructional “support” implying that only weak teachers would need them. That
is absurd. Is it only the weak chemists who need proper lab equipment or only
the weak basketball players who work with coaches?
To do all the necessary mathematical
engineering work well, mathematicians, mathematics education professors, and
mathematics teachers must work together. These groups' history of working
independently has given us inadequate TSM for students, pedagogy-focused
math-light courses for future elementary teachers, and irrelevant
university-level math courses for future high school math teachers. If these
groups came together, they would finally have the knowledge of mathematics,
children, pedagogy, and classroom realities necessary to replace TSM with proper
school mathematics, and to create rigorous and relevant math courses for future
(and current) teachers. (The federal agencies that have followed the
development of the CCSMS should take note of this need and provide financial
incentives for the reconciliation.)
There are two major impediments to this work:
a shortage of willing mathematicians, and a shortage of teachers and mathematics
education professors who realize that TSM is inadequate. As a mathematician
who has worked with K-12 teachers for more than a decade, I believe the latter
shortage will be much easier to address than the former. Most of the hundreds
of teachers I have worked with are eager to improve, and they are relieved to
discover that their own difficulties with mathematics are a result of the TSM
they have been taught. In addition, once we have made progress in our
mathematical engineering, teacher preparation can be completely overhauled, and
the vicious cycle that perpetuates TSM will be broken. But first, we must
address the shortage of willing mathematicians. I have a radical proposal:
professional mathematics organizations, especially the American Mathematical
Society, should sponsor training for a new corps of competent mathematicians
to get to know the school mathematics curriculum and then dedicate themselves
to mathematical engineering. Like chemical and electrical engineering, mathematical
engineering ought to became an established interdisciplinary discipline.
Assuming the work of mathematical
engineering gets going, we
will still face a few additional obstacles. First, district leadership will
have to comprehend that teaching this new proper school mathematics to
in-service teachers requires a long-term commitment. Learning mathematics, and
unlearning TSM, will take effort and time. Two or three half-day sessions each
semester will not be sufficient. In mathematics, the most difficult part of a
teacher's professional development is the acquisition of solid content
knowledge. Preparing to teach proper school mathematics is not about learning
a craft; rather, it is about learning a discipline that is cognitively complex
and very hierarchical. Each topic, no matter how basic, is essential to some
future topic. For example, understanding place value is essential to
understanding multidigit addition, and understanding multiplication of
fractions is essential to understanding algebra, etc.
Second, although I would like nothing
more than for all of the nation's elementary-grades teachers to be immersed in
the intensive school mathematics education that they should have received in
college, two things work against that: the fact that there are more than 1.5
million elementary teachers, and the fact that they are required to teach all subjects.
Expecting any one person to expertly teach reading, mathematics, and all other
subjects is just wishful thinking masquerading as national policy. A more sensible
approach would be to have mathematics teachers in elementary school.[viii]
(To read more about this idea, please see "What's Sophisticated about
Elementary Mathematics? Plenty—That's Why Elementary Schools Need Math
Teachers," which I wrote for the Fall 2009 issue of American Educator, available at www.aft.
org/pdfs/americaneducator/fall7009/w-u.pdf)
A third potential obstacle is the
assessment that comes with the CCSMS. State officials should be vigilant in
safeguarding their students from being overtested. They must remember that
while some standardized assessment is necessary and healthy, several assessments
a year would be counterproductive to learning. Another concern is about the
mathematical quality of test items.
At the moment, students' need of a
mathematically valid assessment is undercut by the presence of flawed and
mathematically marginal items in standardized tests, including those from NAEP.[ix]
To minimize such errors in the future, we need assurance from both of the
assessment consortia that they are committed to getting substantive and
continuing input from competent mathematicians.
Our nation has been known to overcome
greater obstacles than these, provided the cause is worthy. Because failure in
math education has far-reaching consequences,[x]
the worthiness of successfully implementing the CCSMS is clear. Furthermore,
the CCSMS are likely our last hope of breaking the vicious cycle of TSM for a
long time to come. Can we all contribute our share to make sure that the CCSMS
will stay the course?
Our children are waiting for an
affirmative answer.
[2] For an extended discussion of how to teach fractions in grades 3-7 in accordance with the
CCSMS, please see my guide "Teaching Fractions According to the Common
Core Standards," available at http://math.berkeley.edui-wuKCSS-Fractions.pdf.
[3] For an extended discussion of how to approach these two exarnples
from the point of view of the number line, one may consult parts 2 and 3 of my
new textbook for teachers, Understanding
Numbers in Elementary School Mathematics, published by the American Mathematical
Society. (See the box on pages 12-11)
[4] This
is a small example of the longitudinal coherence of mathematics: the fact that
fractions and rational numbers are united by the number line.
[5] Future
teachers certainly do need to learn effective pedagogy, but they also must
learn the content they will teach. This article is about building relevant and
sound mathematics content knowledge into teacher preparation; it is not about
taking pedagogical studies away from teacher preparation.
[6] This
is not the only reason. The long-standing separation between educators and
mathematicians is the other one.
[7] Unfortunately,
this statement appears to hold true for almost all education writings in which
mathematicians are involved.
[i] For further discussion, see H. Wu,
"The Impact of Common Core Standards on the Mathematics Education of
Teachers," April 29, 2011, http://math.berkeley.edu/~wu/CommonCoreIV.pdf
[ii] H. Wu, "What Is Different
about the Common Core Mathematics Standards?” June 20, 2011, http://math.berkeley.edu/~wu/CommonCoreVI.pdf
[iii] National Mathematics Advisory
Panel, "Chapter 3: Report of the Task Group on Conceptual knowledge and
Skills" (Washington, DC: U S. Department of Education, 2008),
page 3-63.
[iv] Deborah Loewenberg Ball and
Francesca M. Forzani, "Building a Common Core for Learning to Teach,"
American Educator 35, no. 2 (Summer
2011): 17-21 and 38-39.
[v] But it is not. See the preface of
Wu’s book, Understanding Numbers in
Elementary School Mathematics (Providence,
RI: American Mathematical
Society, 2011).
[vi] Ball and Forzani, "Building a
Common Core," 38.
[vii] See H. Wu, "How Mathematicians
Can Contribute to K-12 Mathematics Education," February 26, 2006, http://math.berkeley.edu/~wu/ICMtalk.pdf,
for further discussion.
[viii] See, for example, pages 5-51 to
5-58 of National Mathematics Advisory Panel, "Chapter 5 Report of the Task
Group on Teachers and Teacher Education" (Washington, DC : U.S. Department
of Education, 2008).
[ix] See pages 8-3 to 8-4 of National
Mathematics Advisory Panel, "Chapter 8: Report of the Task Group on
Assessment" (Washington,
DC: US. Department of Education,
2008).
[x] See, for example, Rising Above the
Gathering Storm, Revisited Rapidly Approaching Category 5 (Washington, DC:
National Academies Press, 2010), http://www.nap.edu/catalog. php?record_id=12999.
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