In May
this year, the National Center on Education and the Economy (NCEE) released its
report, "What Does It Really Mean to Be College and Work Ready?" [1]. The report is in two parts: Mathematics and English Literacy. It is based on a
national study of the proficiencies required and actually used for the most
popular associate’s degree programs at two-year colleges. Just a few weeks
earlier, Jordan Weissman published a piece in

*The Atlantic*, "Here’s How Little Math Americans Actually Use at Work" [2]. That was based on a 2010 report written by Michael Handel at Northeastern University, "What Do People Do at Work? A Profile of U.S. Jobs from the Survey of Workplace Skills, Technology, and Management Practices" [3]. This large-scale survey includes an assessment of what mathematics is actually used in the workplace.
I hope
that it will come as a surprise to no one that not everyone actually uses the
contents of Algebra II in their work or that College Algebra taught in
two-year colleges is essentially high school Algebra II. Weissman highlights
Handel’s data that less than a quarter of all workers use any mathematics that
is more advanced than fractions, ratios, and percentages. He raises the
question whether requiring Algebra II for high school graduation is placing an
unnecessary roadblock in the way of too many students. NCEE poses more nuanced
questions. Why are so many students being hurried through the critical early
mathematics that they will need to be work and college ready, especially
fractions, ratios, and percentages, just so that they can get to Algebra II?
Isn’t there a better way to prepare them for what they will need?

I will
begin with the data. Handel divided the workforce into five categories:

- Upper White Collar (management, professional, technical occupations)
- Lower White Collar (clerical, sales)
- Upper Blue Collar (craft and repair workers, construction trades, mechanics)
- Lower Blue Collar (factory workers, truck drivers)
- Service (food service, home health care, child care, janitors)

Generally,
the best paying and most desirable jobs are Upper White Collar (UWC) and Upper Blue
Collar (UBC). We should be equipping our students so that they can aspire to
such jobs. It’s still not true that everyone needs Algebra II, but 35% of UWC
workers reported using basic algebra, geometry, and/or statistics in their
work. This level of mathematics is even more important for UBC workers, with 41%
reporting using mathematics at the level of basic algebra, geometry, and/or
statistics. This is still not Algebra II (which Handel lists as “complex
Algebra” as opposed to “basic Algebra”), which was reported being used by 14%
of UWC workers and 16% of UBC workers. Much less is it Calculus, which was
reported by 8% of both UWC and UBC workers. But about 40% of those working in
UWC or UBC jobs need a working knowledge of some high school mathematics, a
higher bar than simply having passed the relevant courses. It is interesting to
observe that UBC workers are more likely to use mathematics than UWC workers.

The NCEE
report looked at the mathematics required for the nine most popular associate’s
degree programs at two-year colleges: Accounting, Automotive Technology,
Biotech/Electrical Technology, Business, Computer Programming, Criminal
Justice, Early Childhood Education, Information Technology, and Nursing, as
well as the General Track. This ties nicely to the Handel study because the
nine are generally seen as preparation for UWC or UBC careers. NCEE selected
seven two-year colleges in seven states and examined the texts, assignments,
and exams in the introductory courses for these disciplines as well as for the
mathematics courses required for these fields. There were three notable insights:

**First**, except for some work on geometric visualization, NCEE found no content in either College Algebra or Statistics, two college-credit bearing courses, that goes beyond the high school curriculum described in the Common Core State Standards in Mathematics (CCSS-M). They found that College Algebra had a large component of middle school topics, especially CCSS-M for grades 6–8 in Expressions and Equations, Functions, Number Systems, Geometry, and Ratios and Proportions. Statistics was a mix of CCSS-M middle and high school level statistics, with a significant component of grades 6–8 Ratios and Proportions and Expressions and Equations.

**Second,**the introductory textbooks in the disciplinary fields used nothing beyond Algebra I. Ratios and proportions are important as well as interpreting quantitative relationship expressed in tables, graphs, and formulae, but, as the report says,

When mathematics is present in the texts, equations are not solved, quadratics are absent, and functions are present but not named or analyzed, just treated as formulae. […] Students do not have to perform algebraic manipulations nor construct graphs or tables. […] The area of high school content with the highest representation in the texts, Number Systems, is found in six percent of the text chapters. [p. 16]

**Third,**the mathematical knowledge that was tested in these introductory courses in the disciplinary fields was far lower than what was in the textbooks. Not only was there nothing requiring Algebra II on the exams, the NCEE team could find nothing, or almost nothing, that reflected knowledge of Algebra I. Furthermore, the questions that were asked on examinations were of low difficulty. The NCEE team used the PISA (Program for International Student Assessment) Item-Difficulty Coding Framework with four levels. Examples of what is expected at each level include

- Level 0: perform simple calculations and make direct inferences;
- Level 1: use simple functional relationships and formal mathematical symbols, interpret models;
- Level 2: use multiple relationships, manipulate mathematical symbols, modify existing models; and
- Level 3: solve multi-step application of formal procedures, evaluate arguments, create models.

The team found that over 60% of
the mathematical questions on the examinations given in introductory courses were
at Level 0. Few rose to Level 2, much less Level 3. (This was not the case in
College Algebra and Statistics where most of the examination items were at
Level 1 or 2 and some attained Level 3. This suggests that even though the
material of College Algebra and Statistics does not go beyond topics covered in
CCSS-M, the level of expected proficiency may be higher than what is typically
encountered in high school.)

NCEE did
find three mathematical topics required for the introductory courses that are
not covered in CCSS-M nor in the College Algebra or Statistics classes: complex
applications of measurements, schematic diagrams (2-D schematics of 3-D objects
and flow charts), and geometric visualization. They also found a much greater
demand for knowledge of statistics, probability, and modeling (“how to frame a
real-world problem in mathematical terms”) than is commonly taught in most
mainstream high school mathematics programs today.

What
makes the NCEE report even more depressing is that it restricted its attention
to college-credit bearing courses. Most of the mathematics taught at two-year
colleges is below the level of College Algebra (see Figure 1). The mathematical
requirements for UWC and UBC jobs may not be high, but we do not seem to be
doing a very good job of preparing students even for what they will need.

Figure 1.
Fall term Mathematics course enrollments (thousands). “Introductory” includes College Algebra, Trigonometry, and Precalculus. |

All of
this raises serious questions about whether Algebra II should be expected of
all graduating high school students. This parallels the situation that has been
my primary concern: Should Calculus be expected of all graduating high school
students who are going directly into a four-year undergraduate program,
especially those who may need to take Calculus in college? I would far prefer a
student who can operate at PISA Level 3 in Algebra I over a student who cannot
handle problems above Level 1 in Algebra II. I would prefer Level 3 in
Precalculus over Level 1 in Calculus. When students are short-changed in their
mathematical preparation simply so that Algebra II or Calculus appears on the
high school transcript, with little regard to what that actually means, then
neither they nor society as a whole are well served.

It also
raises questions about what mathematics should be required for an associate’s
degree. College Algebra constitutes a significant hurdle for most two-year
college students. Should there be alternatives? In this case, I believe that
most two-year college students would be better served with a program that
combines demanding use of the topics of Algebra I with a college-level
introduction to Statistics.

We are
not at the point where we can demand Algebra II for high school graduation. To
do so would either create unacceptable rates of high school failure or force us
to change what we mean by “understanding Algebra II.” But I worry that if we
simply lower our sights and decide that, since few of our students actually
will use anything from Algebra II once they have graduated, it should not be
expected for graduation, then that will actually weaken the preparation that
occurs in the earlier grades. Elementary and middle school mathematics should
be laying the foundation for a student to succeed in Algebra II. If we want our
students to have a strong working knowledge of the high school mathematics that

*is*needed for 40% of the UWC and UBC jobs, then we want them to have the mathematical preparation that would enable them to succeed in Algebra II.
References:

[1]
National Center on Education and the Economy. 2013. What does it really mean to
be college and work ready? The mathematics requirements of first year community
college students. Washington, DC. Available at http://www.ncee.org/college-and-work-ready/

[2] Jordan
Weissman. April 24, 2013. Here’s how little Math Americans actually use at
work.

*The Atlantic*. Available at http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/
[3]
Michael Handel. 2010. What do people do at work? A Profile of U.S. jobs from
the Survey of Workplace Skills, Technology, and Management Practices. OECD
(forthcoming). Available at http://www.northeastern.edu/socant/?page_id=366

There are a number of practical problems with Professor Bressoud's recommendations in the real world of the high school student. I will focus on just two:

ReplyDelete(1) Omit the second year algebra requirement and what do you have left: Nothing is said in his essay about geometry but, assuming (hopefully) that subject remains, you require just two years of math content. That would be followed for many students by two years with no math at all.

(2) Most of algebra 2 is simply algebra 1 again. That may seem to a college teachers (and other outside curriculum critics) to represent a waste of time, but in the real world of the high school classroom it upgrades the level of understanding of that content. If you want to raise that PISA level of algebra 1 understanding, teach algebra 2.

In my experience (I am a retired high school math teacher), too often the first semester of Algebra II has devolved into covering material that was once covered in the second semester of Algebra I. I am specifically thinking of quadratic functions, factoring, and solving quadratic equations. Where the goal is to get everyone through Algebra II, the courses are watered down at least going all the way back to 7th grade math.

ReplyDeleteI admire Prof. Bressoud's efforts in these matters. For years I have worked to try to help high school teachers develop materials that would show students how advanced high school math (I don't like to call it "pre-calculus") can solve real problems. Perhaps if training included that, more could be used in the workplace.

ReplyDeleteMy efforts in calculus were similarly directed:

Why do so many students take calculus?, Notices of the AMS, Sept. 2011, pp.1122-1124

Some time ago a group of us from all of Iowa's public Universities tried to develop a large project to work with high schools. Jerry Mathews and Elgin Johnston at ISU collected a number of real problems from Iowa industry that could be solved with algebra and trig. (Design a conference table, a ground silo for cattle, ...) When we tried for NSF funding we were told our problems weren't "clever."

Calculus teachers often think they should teach from Spivak's "Calculus" (that isn't calculus!, but IS clever) - perhaps great for math majors, not so much for the "workplace."

I worry about "downstream" consequences of changes. When you don't teach fractions in late grade school because calculators can add 0.25 + 0.5, students have more trouble with 1/x+1/y in algebra and in calculus. The course I took in thermodynamics in engineering school (a LONG time ago) had a lot more math than the one our students take at U of I now. (Too bad, some great math!)

We DO want more students to succeed in Algebra 2. A little time spent on real workplace problems could help.

(Thanks David for making me lose sleep another night. Keep it up! It IS important.)