Wednesday, January 21, 2015

Mathematization and the CCSSM

Some time ago when I was learning about the Math Recovery program and reading the wonderful books co-authored by Bob Wright I came across the verb 'mathematize' and its noun counterpart "mathematization". I always thought it was a wonderful way of describing what math education at the elementary school is all about.

In his book, Developing Number Knowledge,  Wright defines the term (p15) this way;
                  Mathematization means bringing a more mathematical approach to some activity.
                  For example, when a student pushes some counters aside and solves an addition
                  task without them, we say they are mathematizing, since it is mathematically
                  important to reason about relations independent of concrete materials.

Others define it as "reduction to mathematical form" (Merriam Webster), "to treat or regard mathematically" (The Free Dictionary) and "explaining mathematically" the Collins dictionary.

The really, really interesting thing about all these definitions is the idea of reduction or movement from real life, concrete situations such as that described by Wright, to the symbolic form of symbols and algorithms typically used in math. This is completely opposite to the way math has traditionally been taught and  how it is sadly still taught in poorly taught math classes.

A classic example occurred in my math class yesterday when I asked student what 1/2 ÷ 1/4 meant. No one knew. My hunch is that if you randomly asked 100 people on the street only a handful would be able to tell you that this meant how many quarters are in a half. What really makes this intriguing is that most people would tell you to change the sign, flip the second fraction, multiply and get the answer 2. 

In other words people have not gone through the process of  mathematization when they have learned this procedure. A real world, concrete idea has not been "reduced to a mathematical form". They learned the mathematical procedure without any sense of what it meant or connection to any concept or relationship. There was no derivation, if you like from, of a mathematical relationship from an idea or concept. This happens all the time in math.

Students are taught a square number is the result of "a number times itself" instead of a number that makes a square.

They are taught a prime number is "a number divisible only by 1 and itself" instead of a number that can only make one rectangle (e.g 1 x 7 or 1 x 13).

Students are taught the symbolic mathematics first and not the idea so they cannot be mathematized. This is what the Common Core State Standards for Mathematics is trying to achieve.

Thursday, January 8, 2015

Happy New Year!


How might you contribute to our organization?

What resources do you find valuable and could share with our members? You are welcome to share links as well as book titles. Please send them to for posting on this blog.

What can our organization do to support you?

Monday, December 15, 2014

NCSM Coaching Corner

Welcome to the NCSM Coaching Corner, the "go-to" destination for mathematics specialists, coaches, and leaders!

The purpose of the Coaching Corner is to support specialists, coaches, and leaders of coaching programs as they progress through the stages of leadership growth outlined in The PRIME Leadership Framework: Principles and Indicators for Mathematics Education Leaders and the new It's TIME: Themes and Imperatives in Mathematics Education.

Wednesday, December 3, 2014

Winter 2-14-2015 NCSM Newsletter

This season's newsletter features infomation about the spring conference in Boston (Mary Fitzgerald, Lara White, and Tracy Watterson will be presenting this year), as well as articles about It's TimePrinciples to Actions, Standards of Mathematical Practice, and SBAC Rubrics Aligned to Desired Evidnce.

Friday, October 31, 2014

Upcoming NCSM Webinar: Featuring Formative Assessment

National Council of Supervisors of Mathematics

 Upcoming NCSM Webinar: Featuring Formative Assessment

Thursday, November 6, 2014 3:00-4:00PM EST
Description: NCSM has created a new resource for leaders - supervisors, coaches, lead teachers, faculty teaching mathematics methods courses - that provides PowerPoint slides and leader notes on strategies for formative assessment. Appropriate for pre-service students as well as K-16 teachers, the JUMP START series includes discussion points and activities for each strategy related to mathematics classrooms. Join the webinar for a tour through the resource and discussions about formative assessment with the educators who created JUMP START.

Presented by: Ana Floyd, Wendy Rich

Sponsored by: Carnegie Learning, Inc.

Registration Link:

Tuesday, October 28, 2014

Advice for Math Teachers Gearing Up for Rigorous Standards

From the Marshall Memo 558 (courtesy of NCSM)

        “Many of us chose mathematics teaching because it was always so neat and clean,” says math consultant Steven Leinwand in this Mathematics Teacher article. “Almost always, we arrived at only one numerical answer by using one right procedure that could be easily graded either right or wrong… But, oh, how things have changed!” He offers the following postulates for math teachers adjusting to ambitious new standards:
            • We are being asked to teach in distinctly different ways from how we were taught. Parents tend to parent the way they were parented, and teachers tend to teach as they were taught. “We build on what is familiar because the familiar ‘feels right,’” says Leinwand. But the new expectations are unfamiliar territory for many teachers. “We need to increase opportunities for collegial classroom visits,” he advises, “and we need to increase our reliance on videotapes of what the distinctly different forms of pedagogy look like.”
            • The traditional curriculum was designed to meet societal needs that no longer exist. New math standards were developed because “society’s needs and expectations for schools have shifted radically,” says Leinwand. “Schools cannot remain perpetuators of the bell curve, where only some were expected to survive and even fewer to truly thrive; education must be a springboard from which all must attain higher levels.”
            • It is unreasonable to ask a professional to change much more than 10 percent a year, but it is unprofessional to change by much less than 10 percent a year. Changing one-tenth of one’s practice is about the right amount to ask of ourselves, says Leinwand – “large enough to represent real and significant change but small enough to be manageable.” This might be revamping one curriculum unit a year, changing questioning techniques, or introducing math journals. “Even the most radical proponent of reform should be satisfied with a change of this magnitude in our mathematics classes,” he contends, “and our most cautious and tradition-bound colleagues should be able to retain a real sense of control over such a rate of change.”
            • If you don’t feel inadequate, you’re probably not doing the job. Just think what math teachers are being asked to do, says Leinwand:
-    Use manipulatives and pictures much more frequently.
-    Get students regularly working in groups.
-    Work with heterogeneous groups.
-    Focus on problems, communication, applications, and interdisciplinary work.
-    Put more emphasis on statistics, geometry, and discrete mathematics.
-    Use assessments that are more authentic and complex.
“Feeling overwhelmed by this torrent of change is neither a weakness nor a lack of professionalism,” he says. “It is an entirely rational response… We must select a few areas of focus and balance the fear and worries we understandably have in some areas with the pride and accomplishment and success we find in other areas. We must accept the inevitability of a sense of inadequacy and use it to stimulate the ongoing growth and learning that characterize the true professional.”
            [Note that this article was published before the Common Core, referencing the NCTM standards, but the ideas are still relevant today.  K.M.]

“Four Teacher-Friendly Postulates for Thriving in a Sea of Change” by Steven Leinwand in Mathematics Teacher, May 2007 (Vol. 100, #9, p. 582-583),

Friday, October 24, 2014

Minutes of the October 16, 2014, Membership Meeting

Happy Anniversary to the Vermont Math Leadership Council!

We have just marked the end of our first year together.
I would like to:
1) Congratulations to our with new Board Members: Mary Fitzgerald (President), Julie Conrad (President-Elect), Patty Kelly (Treasurer), and Member-At-Large Elaine Watson, Mary Calder, Sue Abrams, and Sandi Stanhope,
2) Welcome to our two new members: Fran Huntoon and Mary Perkins,
3) Share the energy from the meeting to those unable to attend.

Please take a few minutes to read through the meeting's Minutes, and consider how you might join in networking, collaborating, and sharing with others in math leadership.

Thank you for a wonderful first year,

Request a copy of the Minutes, Attendance and Ballot, and Treasurer's Report at