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.

Supporting community college faculty across the disciplines

I recently spent 48 hours in Northfield, MN (home of Malt-O-Meal) to work with educators from different disciplines and different organizations trying to find ways to increase two-year college faculty awareness of and participation in professional development opportunities.

The workshop was hosted by Carleton College and its Science Education Research Council.  SERC (http://serc.careltoncollege.edu) has amassed an impressive collection of resources across multiple disciplines including geoscience (the first discipline), chemistry, economics, mathematics, physics, psychology,  and more. 

The Pedagogy in Action page (http://serc.carleton.edu/sp/) has links for Teaching Methods, Activities, and Research on Learning. SERC is continually seeking to improve its website to become a one-stop launching point for finding discipline-specific lesson plans, research-based pedagogical strategies, student projects, career information--essentially anything of interest to an educator seeking to improve student learning.

SERC has also been learning how to run effective workshops.  We were given pre-workshop assignments to upload essays into designated spaces on the SERC website that were visible to the other  participants but not to the rest of the world.  And during  the workshop we were constantly moving from whole group to small group activities, mixing tasks from cross-discipline to the discipline-specific. 

Each group would choose a recorder, who wirelessly entered directly into the SERC system.  The others in the small working group could see the notes on their own computers during their discussion, and the notes were available to the whole group during the "share out" session. Working across disciplines allowed us to learn of challenges and strategies that gave us fresh perspectives for our discipline-specific discussions.

The real-time recording of discussions means that our notes won't be accidentally lost among papers or luggage during our journeys home. Eventually the notes from our workshop will be organized, polished, and made publicly accessible on the SERC site.

Simpson's Rule on Cubics

Simpson's rule approximates a definite integral

by replacing the integrand f with the quadratic  function that agrees with f at the endpoints and midpoint of each sub-interval.  (For comparison, the Left- and Right-Hand Riemann sums each replace f with a constant function, the Trapezoid and Midpoint rules replace f with the linear function respectively agreeing with f at the endpoints of the interval or agreeing with both f and f' at the midpoint of the interval.)

It is remarkable that Simpson's rule gives the exact values of definite integrals not only for any quadratic but also for any cubic polynomial, using only one sub-interval.

This can be algebraically verified by using the change of variable x = a + (b - a)t and verifying that Simpson's rule with one sub- interval gives the exact value for

Here is a more geometric argument.

Let f be a cubic polynomial, and let q be the quadratic function satisfying f(a) = q(a), f(b) = q(b), and f((a+b)/2) = q((a+b)/2).

Then the error in using Simpson's rule for approximating

is

where E is the cubic polynomial defined by E(x) = q(x) - f(x).

Because  E(a) = E(b) = E((a+b)/2) = 0, the inflection point in the graph of E occurs at x = (a+b)/2.  Cubic polynomials are symmetric about their inflections points, so the region lying between the curve and the x-axis on one side of the inflection point is congruent to the region between the curve and the x-axis on the others side of the inflection point. 

Hence

That is, the approximation has no error.

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.

MathFest 2012 and Common Core State Standards in Math

Andrew Hacker’s article "Is Algebra Necessary?" in the New York Times was a hot topic last week and mentioned by several presenters at the 2012 MathFest session "What Mathematics Should Every Citizen Know?".  The panelists, Bil lMcCallum, Lynn Steen, Hyman Bass, Joseph Malkevitch, and co-organizer Sol Garfunkel, were actually reacting to the Core Curriculum State Standards in mathematics.

Mathematicians and math educators agree that we are not  currently doing the best job of teaching algebra.  But unlike Hacker, the math community believes the appropriate strategy is to improve algebra instruction, not to abandon it to all but an elite few pupils.

On the other hand, the speakers on the panel, although quite civil with each other, clearly had disagreements about the best strategy to improve math education in the US.

McCallum, who was the lead mathematician in the development of the CCSS, emphasized the benefits of having commonality across states.  Having a set of standards that could be adopted by 45 of the 50 states (so far) required compromises, but the benefits accrue not only to pupils and teachers in our mobile society, but to all who do business with textbook publishers who currently provide materials for the multitude of different curricula.

Steen gave some numbers showing the dismal success of preparing US students for STEM, but argued that we should improve rather than remove algebra from the curriculum.  He favors a modeling-based approach and avoidance of common assessments.  When asked how to accomplish his recommendations, he cheerfully remarked that he doesn't need to worry about that now that he's retired.

Bass focused on pedagogy rather than curriculum as the key to improving math education.  Student learning is increased when the instructor employs appropriate classroom strategies.

Malkevitch promotes widening the curriculum.  He argued that we need to show many ways that mathematics impinges on daily lives.  He gave combinatorial graphs and fair choice algorithms as examples of mathematical topics that are new and accessible to very young children.

Garfunkel believes that the entire K-12 mathematics curriculum should be centered on modeling. He echoed Malkevitch's suggestions that the US curriculum needs to be widened, and said that Bill Schmidt had paid an advertising agency to create the phrase "a mile wide and an inch deep" that is used to characterize the US K-12 curriculum following the disappointing ranking of the US high school students in the Third International Mathematics and Science Study.

Reviewing Grant Proposals

I recently served on a panel reading NSF grant proposals.  We were admonished not only to respect the confidentiality of the proposal contents but also not to divulge the dates we did the reviews nor the title or nature of the grant types being solicited.

I've reviewed NSF grant proposals on about a half dozen occasions, and I've enjoyed each of my experiences.   There are typically 4 or 5 members to a committee, a few dozen committees representing the different STEM disciplines, and each committee is assigned about a dozen proposals to review.  NSF tries to create panels with a diversity of geographical regions and of institution type  (research universities, state universities, liberal arts colleges, and sometimes even two-year colleges). NSF brings in both faculty and administrators.

Panelists register online using the NSF program FastLane long before they see any of the proposals. FastLane facilitates many aspects of the review process, such as tracking panelist coordinates and other information for  travel arrangements, giving access to proposals, recording panelist reviews, and arranging letters to home institutions acknowledging service.

Panelists upload to Fastlane their personal evaluations of each proposal before going to DC and physically meeting as a panel.  Project descriptions are restricted to 15 pages, but with all the additional documentation of references, biographical sketches, budget, etc., the entire proposal usually exceeds 50 pages.  So it is also common that when the panels finally meet in DC on the first morning, not all the panelists have successfully prepared reviews of all the proposals.

The panels meet essentially all of the first day discussing each proposal.  Panelists are logged on the FastLane program during discussion.  This allows each panelist the ability to see the reviews of all the panelists for each proposal.

NSF assigns each panelist to be "scribe" for two or three proposals, and the scribe's task is to record the discussion about the proposal.  NSF does not require consensus, but it is common that the discussions persuade some panelists to change their initial evaluations.  Most panelists work into the night preparing the summaries required by their duties as scribe and resubmitting their personal proposal reviews.

The second morning is spent largely in approving the scribe summaries.  Again the panelists are logged on FastLane, and they are not allowed to leave until each summary has been approved.  NSF reconvenes the panels by discipline late in the morning for a debriefing before sending the panelists back home.

The debriefing is one of the highlights of the whole experience, because each panel typically describes its two favorite proposals.  So we hear the highlights, and it's exciting to learn of innovative ideas that are being pursued.

MS Input Panel and Equation Writer

Microsoft's Equation Writer converts handwritten mathematical expressions to typeset expressions which can be inserted into MSWord documents.  I played with the Equation Writer in January 2010 and found it amusing but not usable.

Now Microsoft's Math Input Panel, which I found under Accessories in Windows 7, is definitely an improvement.

I first tried writing the same equation that had given the Equation Writer so much trouble.

Although my handwriting has not improved, the Math Input Panel recognizes my writing much better than the Equation Writer had, provides a more pleasingly formatted result, and is easy to edit.

The "Select and Correct" allows you to lasso a part of your expression and gives you a list of likely interpretations.  Sadly, the program's best guesses for my cursive "dx"were "m," omega,"max," "cos," "log," "def," or the infinity symbol.  But it was easy to erase my connected "dx" and replace it with two separate letters, which were properly interpreted.

To get the expression into a Word document, I needed to have both the Word document and  the Input Panel visible, I clicked on the point of insertion in the Word document, then clicked on "Insert" in the Input Panel.

What might make this worthwhile is that once you insert the expression into a MSWord document, you can use the Equation Tools.

For example, suppose you've input "sin x" using either the Math Input Panel or typed directly into the Equation editor (by first simultaneously pressing the Alt and = keys).  Now when you right click on the expression, you get a menu of possible actions.

If you click on Integrate on x:

It's interesting that the constant of integration is included in the result.  What's also interesting is that if you type a 1 immediately after the C, right-click to get the menu of options, then select Graph -> Plot in 2D, you get an insertable graph that can first be reformatted and/or display an animation based on varying the constant of integration.

You can also go back and right click on your antiderivative, integrate on x, and type a 2 immediately after your new constant of integration.  Now when you ask for a 2D graph, you can run an animation based on either of the two constants.

And yes, you can draw 3D graphs in your Word document. Press the Alt key and the = key simultaneously, type in an expression in two variables (such as sin x + cos y), right click and select Graph -> Plot in 3D.

You can drag the image to a preferred orientation before inserting it into your Word document.

You can also use the "Calculate" option to solve some simple equations or to evaluate/simplify algebraic expressions.