John Walkup

For this blog I pay homage to Mario Iona, the gadfly professor of physics from the University of Denver that made skewering textbooks authors an art form before he died five years ago. I enjoyed his monthly columns, Would You Believe?” in The Physics Teacher.  Textbook authors didn’t.

I remember one time John Saxon, the founder of Saxon Publishing, coming to me with a letter he planned to send to The Physics Teacher in response to Iona’s latest column, which critiqued John’s physics book. At the time I worked for John on his physics textbook while living in Norman, OK.

“John, I wouldn’t bother,” I told him. “His column really wasn’t that negative.  I read his columns and, believe me, you got off lightly.”

Now it is my turn, and a more suitable subject couldn’t be found than the draft of the national common core standards for mathematics.  Once the final editing is completed, you can bank on states adopting the standards for use in their own states’ classrooms, so it is important that the federal government gets it right.

So far, they haven’t.

In this first blog on the subject, I want to present some overarching critiques of the standards writers’ approach.  Later blogs will focus on the minutiae.

First and foremost, do the standards discuss concepts and skills with sufficient detail to guide instruction, curriculum development, and assessment?”

In my opinion, no.  For example, the standards fail to provide specialty teachers (such as those teaching Algebra II or trigonometry) with well-defined guidelines on the skills and concepts required of their instruction. As such, the standards reflect learning goals more closely associated with a high school exit exam, not as a guide for high school instruction.

Will California teachers be compelled to drop their Algebra II standards in favor of the common core standards?  California already provides Algebra II teachers with well-defined standards that they can use to transform into standards-based learning objectives, with clearly identifiable concepts and skills. Yes, the skills tend to correlate far too much on the third level of Bloom’s Taxonomy (apply) and the lowest order of Norman Webb’s depth of knowledge (straightforward computation), but a tweak here and there can rectify the situation. Consider California Algebra II Standard 6.0:

Students add, subtract, multiply, and divide complex numbers.

Crafting a learning objective poses no trouble for teachers: “Students, today we will learn to add and subtract complex numbers.” (Two unpacked standards are combined into one learning objective, a perfectly reasonable thing to do.)

The independent work assigned to students will mostly reflect the Bloom’s Taxonomy level is Level 3 (apply) because teachers will likely expect their students to apply their knowledge to solve unfamiliar problems.  The depth of knowledge of the assignments will likely correlate to the lowest available level (DOK-1) because the problems will likely involve straightforward application of a procedure.

In my view, this particular combination of Bloom’s Taxonomy and depth of knowledge is killing us in mathematics, so the state of California may want to consider a revision. But at least the standard for teaching complex numbers is available for California teachers.  Oklahoma too:

1.1 Define and perform operations on real and complex numbers.

But if I am an Algebra II teacher, I am not sure how the common core standards can guide me. Few scientists and engineers can doubt the importance of complex numbers and complex analysis (I can write a book on the subject), but complex numbers are never mentioned in the common core standards.

At this point, we need to reflect on the potential purposes of the common core standards.

The problems transcend the absence of detailed content. The relevance associated with each standard is
never stated, nor is the teacher compelled by the wording of the standards to teach the relevance.
Without relevance, students disengage and consider mathematics an esoteric exercise of little practical
value. In a nutshell, if we don’t teach students the importance of learning mathematics, why would they
want to learn it?
A large proportion of the standards do not align to what is typically taught in high school but rather
reflect middle school instruction.We suspect that much of this problem could have been alleviated if the
standards writers had concentrated on an individual standard (e.g., “Expressions”) at a time, writing
core concepts and core skills related to the individual standard as it progressed through the grade levels.
(For example, what do the core practices and core skills related to the ~Expressions” standard look like
in Algebra II, Algebra I, eighth grade, seventh grade, and so on?)
“The introductory comments beginning each standard contain useful descriptions of key
mathematical components, but it isn’t clear whether these comments constitute standards. For
example, “Coordinate geometry is a rich field for exploration. How does a geometric
transformation such as a translation or reflection affect the coordinates of points? How is the
geometric definition of a circle reflected in its equation?”
Does this statement imply that teachers should teach transformations (e.g., translation, rotation, etc.)?
The answer isn’t clear.
We now turn our attention to specific examples of standards and our commentary.
Standards
1. Mathematical Practice
1. Attend to precision.
Mathematically proficient students organize their own ideas in a way that can be communicated
precisely to others, and they analyze and evaluate others’ mathematical thinking and strategies
noting the assumptions made. They clarify definitions. They state the meaning of the symbols they
choose, are careful about specifying units of measure and labeling axes, and express their answers
with an appropriate degree of precision. Rather than saying, “let v be speed and let t be time,” they
would say “let v be the speed in meters per second and let t be the elapsed time in seconds from a
given starting time.”
Ahem. “…let v represent the speed in meters per second…” (And no, we are not overly picky. Students
need to learn and never forget that variables represent unknown values—using ambiguous verbs of the
form “to be” does little to reinforce this understanding.)
They recognize that when someone says the population of the United States in June 2008 was
304,059,724, the last few digits indicate unwarranted precision.
The precision (as opposed to accuracy) of the stated census measurement is not obvious. If the
measurements reflect raw counts, rather than interpolations and extrapolations, then the last digit is
precise and should be retained. (The census would not necessarily be accurate, but it certainly would be
precise.) A better example would be the width of the universe expressed to the nearest meter, an
obvious overestimation of our precision capabilities.
Speaking of interpolation and extrapolation, we see nothing in the standards that addresses these two
topics on the conceptual level. The fact that interpolating is typically safer than extrapolating is
important to know in mathematical and scientific reasoning.
3. Make sense of complex problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They consider analogous problems, try special cases and
work on simpler forms. They evaluate their progress and change course if necessary. They try
putting algebraic expressions into different forms or try changing the viewing window on their
calculator to get the information they need.
Changing the viewing window of a calculator does not correspond to the same level of thought and
ingenuity as re-expressing (not “putting”) an algebraic expression in an alternative form.
They look for correspondences between equations, verbal descriptions, tables, and graphs. They
draw diagrams of relationships, graph data, search for regularity and trends, and construct
mathematical models. They check their answers to problems using a different method, and they
continually ask themselves, “Does this make sense?”
The granularity of Standard 3 (above) contradicts Standard 4 (below). We suggest removing the
reference to regularity and trends in Standard 3 and placing this reference in Substandard 4, thus
evening the granularities of the two standards.
4. Look for and make use of structure.
[ ] They can see complicated things, such as some algebraic expressions, as single objects. For
example, by seeing 5 – 3(x – y)2 as 5 minus a positive number times a square, they see that it
cannot be more than 5 for any real numbers x and y.
Without explicit training in the domain and range of multivariate functions, limiting the domain of x
and y will pose students trouble in most situations. Suppose -2 < x < 2 and -1 < y < 1. Can students
analyze the range of the term 3(x – y)2 in such a situation? (This is not a contrived example, as video
game algorithms typically feature functional calls of two and three variables, depending on whether the
graphics are two-or three-dimensional. Often the domain is limited by the physical constraints of the
computer mouse or virtual object movements, limiting in turn the range of the term.)
Note that the standards only mention functions of one variable. And nowhere do the standards mention
the teaching of domain and range within the context of a single term. The stated definition “functions
model situations where one quantity determines another” reinforces this misconception. (My emphasis)
Nowhere is the teacher urged to teach the importance of learning this skill as a necessary component of
analysis. Where’s the relevance? (“The object in my video game scoots offscreen whenever I touch the
mouse in the upper left corner. What’s going on?”)
By the way, the variables x and y need italicizing.
5. Look for and express regularity in repeated reasoning.
[ ] For example, by paying attention to the calculation of slope as they repeatedly check whether
points are on the line through (1, 2) with slope 3, they might abstract the equation (y – 2)/(x – 1) = 3.
We are not sure what activity is taking place. The slope is given, so why are they calculating the slope?
We think the prose needs clarifying.
Noticing the regularity in the way terms cancel in the expansions of (x – 1)(x + 1), (x – 1)(x2 + x +
1), and (x – 1)(x3 + x2 + x + 1) leads to the general formula for the sum of a geometric series. As
they work through the solution to a problem, proficient students maintain oversight of the process,
while attending to the details.
We are not sure why the term cancel is used here. There is no division operation. Perhaps the standards
writers meant to say combine.
They continually evaluate the reasonableness of their intermediate results.
Sure, but this issue is unrelated to expressing regularity in repeated reasoning. If anything, the
evaluation of reasonableness is worthy of a standard in its own right. (This same ideal is mentioned at
the end of Substandard 3 — “Does this makes sense”? — although relegated to intermediate results.
We think it would be better to combine the two statements and embed them in Substandard 3.)
2. Number
[ ] The rules of arithmetic govern operations on numbers and extend to operations in algebra:
• Numbers can be added in any order with any grouping and multiplied in any order with any
grouping.
• Adding 0 and multiplying by 1 both leave a number unchanged.
• All numbers have additive inverses, and all numbers except zero have multiplicative inverses.
• Multiplication distributes over addition.
Subtraction and division are defined in terms of addition and multiplication, so are also governed by
these rules.
This is incorrect. The commutative property described in the first bullet does not apply to subtraction or
division. We understand what the standards writers meant to say, but as stated the statements mislead.
On another note, these concepts do not correlate to the rigor and difficulty associated with high school
mathematics.
The place value system bundles units into 10s, then 10s into 100s, and so on, providing an efficient
way to name large numbers. Subdividing in a similar way extends this to the decimal system, which
provides an address system for locating all real numbers on the number line with arbitrarily high
accuracy. Place value is the basis for efficient algorithms, reducing much computation to single-digit
arithmetic. Mental computation strategies also make opportunistic use of the rules of arithmetic, as
when the product 5×177×2 is computed at a glance to obtain 1770, rather than methodically working
from left to right.
The last sentence has nothing to do with place value, but rather the commutative property, so it should
be set apart in its own paragraph.
Rational numbers represented as fractions can be located on the number line by seeing them as
numbers expressed in different units; for example, 3/5 is 3 units, where each unit is 1/5. However,
rational numbers do not fill out the number line. There are also irrational numbers, such as π or √2.
Each point on the number line then corresponds to a real number that is either rational or irrational.
Use of the word then in the last sentence implies that the final sentence follows from the previous
sentences. Such is not the case.
The core concepts and core skills expressed in this standard do not align to what is typically highschool
level mathematics.
Core Concept A
The real numbers include the rational numbers and are in one-to-one correspondence with the points on
the number line.
The statement “real numbers include the rational numbers” begs the question: what else do the real
numbers include? We would also suggest replacing “and are in” with “form a.”
Core Concept C
A fraction can represent the result of dividing the numerator by the denominator…
Dividing the numerator by the denominator produces either an integer or decimal, not a fraction. The
standards writers meant to convey the concept that a fraction represents a quotient of one quantity with
respect to another.
Core Skill 1
Compare numbers and make sense of their magnitude.
According to this standard, noting that 4 > 2 is standards-based instruction in high school. Do the
numbers in the standard correspond to decimals, fractions, irrational numbers? Shouldn’t they also be
comparing numbers expressed in mixed representations (e.g., 4.2 > 5/4)?
We suggest replacing “make sense of” with “comprehend,” “understand,” or “conceptualize