Mathematical Practices

The Common Core State Standards list Standards for Mathematical Practice at each grade level.  These practices are

Mathematical Practices

1.     Make sense of problems and persevere in solving them.

2.     Reason abstractly and quantitatively.

3.     Construct viable arguments and critique the reasoning of others.

4.     Model with mathematics.

5.     Use appropriate tools strategically.

6.     Attend to precision.

7.     Look for and make use of structure.

8.     Look for and express regularity in repeated reasoning.

These same eight expectations are listed in the descriptions for every grade level and for every advanced (a.k.a. high school) course.  But carrying out the Mathematical Practices will look different at different grade levels.

The practices that may require the most clarification are probably #4 (Model with mathematics), #7 (Look for and make use of structure) and #8 (Look for and express regularity in repeated reasoning).

Modeling with mathematics typically involves using and perhaps even creating mathematical objects (such as algebraic expressions, equations, inequalities, graphs, etc.) to capture key aspects of a situation to be explored.  A kindergartner might use 2+3 to represent the number of people involved if two people are joined by 3 more; a sixth grader might describe  a relationship between the numbers of tables to chairs in a room by the ratio 1:4; an Algebra I student might use the expression 10x to represent the value (in cents) of x dimes.

But not all word problems involve mathematical modeling.  It is not mathematical modeling to use a contrived algebraic expression such as a quadratic expression obtained by curve-fitting bi-variate data without any plausible a priori reason for believing that the two variables should be related quadratically.

Mathematical Practice #7, seeking  and using structure, is key to both pure and applied mathematics.  A first-grader begins to recognize that an addition fact such as 2+3=5 carries with it a family of related arithmetic facts, e.g., 3+2=5, 5-2=3, 5-3=2, etc. ; a seventh-grader can see that because a+0.05a = 1.05a, increasing a quantity by 5% is equivalent to scaling the quantity by 1.05; a geometry student recognizes and introduces structure by adding an auxiliary line to a geometric diagram.  Mathematical Practice #7 is definitely not about memorizing or plugging into formulas--both practices,  when applied inappropriately, can allow students to ignore the underlying structure .

I personally need further explanation of Mathematical Practice #8.  Here is how it's first described in the CCSS:

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation \( \frac {y – 2}{x – 1} = 3\) . Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

The phrase "repeated reasoning" presumably refers to the "R" in Guershon Harel's DNR. Some of Harel's work is listed in the mathematics CCSS references, but there do not appear to be any direct attributions cited.

CCSS and Community College Math Programs

We may need a complete redesign of the developmental math program in US two-year colleges.

My campus currently uses a placement test (Mathematics Diagnostic Test Project) to determine if students are ready for transfer level courses (math for elementary school teachers, stats, trig, precalculus, calculus)  or what remedial course (arithmetic, prealgebra, elementary algebra, intermediate algebra) they should take.

But the Common Core State Standards for mathematics will have high school students studying mathematics organized in a fashion that does not align with our existing math courses.

California is one of the 45 states that have formally adopted the CCSS for mathematics, and I am on a recently appointed state committee whose charge is to align California’s math standards (a.k.a. the California Framework) with the CCSS.

One of the main reasons that I applied to be on the Mathematics Curriculum Framework and Evaluation Criteria Committee (MCFCC) was to better familiarize myself with what is to be taught in California's K-12 schools.  (Another reason was to lose myself in abbreviations:  SBE for State Board of Education, CDE for California Department of Education, IQC for Instructional Quality Commission, the body that forwarded my name to the SBE for approval to serve on the MFCC to align the CF with the CCSS.)

The CCSS specify a consensus of what math is required for students to be college or career ready.  The standards are grouped into six conceptual categories:  Number and Quantity, Algebra, Functions, Modeling, Geometry, and Statistics and Probability.  (There are separately eight standards for mathematical practice that go across all grade levels.)

The CCSS differ significantly from what is typically required for graduation in most American high schools today.  For example, the treatment of statistics and probability includes not only descriptive statistics but also conditional probability, inference, decisions based on probability, and rules of  probability. 

The CCSS include not only right-triangle trigonometry but also trig functions of a real variable, to be used in modeling periodic behavior.  Thus trig spans the geometry, algebra, and function categories.

The CCSS gives math standards for high school without specifying courses or order of topics.  But evidently the introduction of functions includes an emphasis on (linear and) exponential functions with domains restricted to a subset of the integers--sequences are explicitly studied as functions.

California community colleges do not require a high school diploma for admission.  A student who masters the first CCSS high school math course will already have compared exponential functions with linear functions and solved equations both algebraically and graphically. The student will have had explicit instruction on descriptive statistics.  The student may have worked with constructions and transformations in the plane and proved simple geometric theorems algebraically but not yet worked with polynomials (and specifically not with quadratic functions or quadratic equations).

How will our placement system advise this student?

One of the recommendations  of California's Student SuccessTask Force is for better alignment between high school and college curricula.  With the CCSS adopted across states, it looks as if most community colleges will need to make adjustments to their way of placing and educating their math students.

JMM 2010 (continued)

Further points of interest...

The MAA plans to undertake  a huge study of what's going on in college Calc I classes.  They hope to survey 25% of Calc I classes across the US.  Campuses not selected in the random sample will be given the opportunity to be take the survey to learn how they compare with the national sample, but their data will not be included in the national statistics.  (From David Bressoud, speaking at the department liaisons meeting)

States have called for Common Core State Standards (CCSS) for K-12 math and English.  The math group is being headed by Bill McCallum (U. Arizona).   For the first time ever (to my knowledge), "modeling" will occur as a separate topic/strand in a math standards document.  No working drafts have been made public, and public comment will evidently be limited to a few weeks in February 2010.  Although the initiative came from states, the feds will support it by tying Race to the Top funds to compliance with CCSS.  (AMS Committee on Education Panel Discussion, "The common core State Standards: Will they become our national K--12 math curriculum?")

Web 3.0 is the "Semantic Web"--searches  will not be simply over key words but key concepts.  The meaning of each item will be considered in the search.  For example, your search can include the word "red" to refer only to the color and to ignore the name "Red" and the meaning of "communist".  Or perhaps more enticingly, you can search for information about some algebraic expression (independent of Wolfram|Alpha).  (From Tom Leathrum, speaking at the MathDL advisory board meeting)

The Committee on Technologies in Mathematics Education (CTiME) submitted a proposal to sponsor (jointly with the Web SIGMAA) a panel session about the tablet pc and similar mobile stylus-type devices for the 2010 MathFest in Pittsburg.  This session will be dedicated in memory of Howard Penn, a long time CTiME member and supporter.  The committee will attempt to hold an online meeting during the spring 2010.  (Lang Moore, CTiME committee meeting)

Davide Cervone's newly unveiled MathJax looks like an exciting way to present math on the web.  It's still undergoing final tweaking.  (David Cervone speaking at a session of Publishing Math on the Web)  Davide's design also will  provide an improvement over programs like Beamer, which creates presentation slides with editable TeX-like math expressions.  (Mike Gage chatting between sessions)

The Committee on Two-Year Colleges (CTYC) voted to co-sponsor (with CRAFTY) a College Algebra contributed paper session for JMM 2011.  Rob Kimball talked about The Right Stuff, AMATYC's project about revamping college algebra.  AMATYC president Rob Farinelli spoke about AMATYC's Project ACCCESS (a professional development program newly hired in tenure-track position community college math positions) and rapidly developing work on a grant proposal involving the Hewlett Foundation and the Dana Center:  Mathway.  (CTiME committee meeting)

CRAFTY's  Curriculum Foundations II report (revamping college algebra)  is in production.  New committee chair Andy Bennett will investigate acquiring funds so that expenses could be paid to send  CRAFTY/Curriculum Foundation  II gurus to speak at MAA section meetings about best practices for a revised College Algebra.  (CRAFTY committee meeting)

CRAFTY's next big project will be to investigate the course commonly called "Precalculus."    The committee is providing the CBMS suggestions on possible questions for the next CBM survey.  (CRAFTY committee meeting)

The MAA's new VMware and servers will allow the MAA to host websites for all the MAA SIGMAAs and sections.  (John Wyatt, department liaison meeting)

Solving quadratic equations via geometric construction

We can solve a quadratic equation of the form x² - sx + p = 0, , using the standard construction tools of compass and straightedge.  The method has been attributed to critic Thomas Carlyle.

Construct the circle in the Cartesian plane with center and passing through A(0,1).  By symmetry, the circle also passes through and .

Because the circle has center   and passes through A(0,1), the equation of the circle is

This reduces to

x² - sx + y² - (p + 1)y + p = 0

So the x-intercepts of the circle are the solutions to  x² - sx + p = 0.

Alternate justification:

The segment joining the x-intercepts has a length , hence x₁ + x₂ = s.

intercepts the arc AX₁X₂, and  intercepts the opposite arc, hence the two angles are supplementary.  But is also supplementary with , so is congruent to , which in turn impies that  .  Hence

 ,

which implies that , so x₁ x₂ = p.

Thus (x -  x₁)(x - x₂) = x² - sx + p, and the solutions to x² - sx + p = 0 are x₁ and x₂ .

The Correlation Coeffiicent as cosine theta

Mathematicians define the dot product between vectors and as

On the other hand, the alternate geometric definition for the dot product popular with physicists is

So

And statisticians define Pearson's correlation coefficient r so that

Thus if we set and , then .

The idea is to think not of n ordered pairs (x₁, y₁), (x₂, y₂), ..., (x_n, y_n), but rather to think of two vectors in n-dimensional space. When the vectors are pointing in the same direction, the angle between them is zero and the correlation coefficient is cos 0 = 1. When the vectors point in opposite directions, the correlation coefficient is the cosine of a straight angle, r = -1. And when the vectors are orthogonal, the correlation coefficient is the cosine of a right angle, r = 0.

The only tricky part is that the two n-dimensional vectors are not the vectors and  , the vectors containing all the and respectively.  Instead, the necessary two n-dimensional vectors are the and defined above.

And nicely, the least-squares regression line for the data is y = mx + b, where and .  (Notice that the variance , so m can also be written as  .


One typically derives the least-squares regression line by finding m and b that minimize .  But one can alternatively use the n-dimensional vector point of view, where the coefficients m and b correspond to the solution of the vector equation .  The vector is the vector of all 1's and the vector   is the orthogonal projection of the vector  onto the space spanned by and .

Teacher Prep at Community Colleges

Although only a small proportion of the general public or even the faculty and administrators involved seem to be aware of it, community colleges are in the business of preparing future teachers.

It has been estimated that 40% of U.S. K-12 teachers took math or science at a two-year college (http://www.nsf.gov/pubs/1999/nsf9949/nsf9949.htm ) and that 46% of baccalaureates in science and engineering have attended a two-year school (http://www.nsf.gov/statistics/nsf04302/ ).

The numbers of K-12 teachers who attended two-year colleges only grows if we include courses and degrees in non-STEM disciplines.

At California State University Northridge (CSUN) , which identifies teacher preparation as one of its primary missions, more than half of their students in the multi-subject credential program (for teaching elementary school ) took math courses at a community college. Most of CSUN's math majors are considering a career in teaching, and about two-thirds of CSUN math majors took math at a community college.

Many or most universities require only 3 to 9 units of courses taught by math departments for the students preparing to become elementary school teachers. Some of these units may be taken at a community college, and considering that many students (51% at CSUN) take developmental math classes at a two-year schools, and furthermore that some students (20% at CSUN) require as much as 20 units of remediation, our prospective elementary school teachers are probably taking more math at two-year colleges than at four-year colleges and universities combined.

A number of faculty at two-year colleges take seriously their role in the recruitment and education of future teachers. The Teacher Prep Committee ( http://amatyc.dtcc.edu/ , http://teacherprep.amatyc.org/ ) is one of only eight standing national committees of the American Mathematical Association of Two-Year Colleges. Yet there are two-year colleges where the "Math for Teachers" class either does not exist or is taught primarily by adjunct faculty because the full-time faculty do not have sufficient interest to teach the course, or the institution is not sufficiently motivated to run the course.