Mission Statement
Math Curriculum for Puente Valley Charter School
Our curriculum seeks a productive middle ground in the contentious debate in American
mathematics education between “discovery” and “skills” camps. Discovery models (which go by
names like whole math, reform math, and new-new math) advocate open-ended teaching with an
emphasis on student innovation and group work, and a corresponding de-emphasis on teacher
lecturing.1 Supporters of skills models, reacting largely to the “fuzziness” of new-new math,
argue that students are not served by the removal of traditional elements of math teaching,
including direct instruction, repetition, and priority of accuracy in computation.2 For the Puente
Valley Charter School, we seek a pedagogical approach that combines the advantages of these
two schools of thought.
To accomplish this, we urge teachers to use three stages in addressing each unit in the curriculum.
(1) The first stage is a presentation and discussion of definitions, tools, and other preliminaries
such as historical background or social context. (2) The middle stage is the exploration cycle,
flowing between student innovation and teacher guidance. Students are split into small working
groups and presented with a problem – in fact, usually several nested problems – of significant
difficulty. Groups collaboratively innovate ideas for approaching the problem and work to
implement the ideas. Meanwhile, the teacher or teachers in the classroom circulate to monitor
student progress. Critically, the teachers offer hints and suggestions, intervening in case of deadend approaches. The teachers need to be sure to nurture the discovery process by letting ideas run
their course when necessary. (3) The final stage is the vitally important period of consolidation.
This involves a summary of the group work, which the students can present to each other. Then
the teacher needs to engage in standardization, for instance introducing the usual notation, while
connecting it back to the ideas generated by the students. We believe that this is eminently
possible without undermining the value of discovery, and furthermore that it is necessary for
future applicability of the skills developed in the lesson. When the unit has social relevance, this
aspect of the discussion should be advanced all stages, including in the last stage. To complete
the consolidation process, transferring conceptual understanding to useable knowledge, there
must be opportunity for extensive practice of the core math skills in the lesson.
An organizing principle behind the arrangement of students for the exploration cycle will be to
create groups containing a range of ability levels. A cooperative atmosphere will be fostered so
that students who understand a given topic rapidly can help their peers who need more time or
explanations. Peer instruction is an excellent way to solidify the understanding of the more
advanced students while bringing the less advanced students up to speed, keeping the unity in a
non-tracked class. We strongly urge that the mathematics classes not be tracked (split up by
tested “level”) because this impedes the academic mobility of students placed at the lower levels.
Though teaching math in the non-tracked classroom poses many challenges for the instructor,
these are not insurmountable. Getting away from tracking produces an environment with much
1
The National Council of Teachers of Mathematics (NCTM, found at http://www.nctm.org) and in
particular the NSF-funded Connected Mathematics Project series of texts (http://www.mth.msu.edu/cmp/)
are strongly associated with this movement.
2
One vocal source that advocates a return to skills-based teaching is Mathematically Correct,
http://www.mathematicallycorrect.com.
greater equity—this is essential in avoiding a schooling system that calcifies early preparedness
into the major determinant of academic success.
We consider it critically important that teachers subscribe to a philosophy of honesty with the
students. This can take many forms: first, it means giving thoughtful and meaningful answers to
the question “why math?”3 that go beyond checkbook-balancing and address access to higher
education and democratic participation. We hope that students will be exposed to the idea that
mathematical literacy is a key to civic engagement, and that good instruction is their civil right.4
Honesty in instruction also means that teachers should seek real applications of the abstract ideas
in the curriculum—this means that the algebra of voting or the statistics of population
demographics would be preferable to “toy” problems about batting average, a favorite of the
California Standards.
Because we believe that algebra serves as an entry to math literacy, we have designed the
curriculum with the goal in mind that all students will have completed the equivalent of Algebra I
by the end of eighth grade. Meeting this algebra goal will also ensure that graduates of this
middle school have the background for a true college-preparatory mathematics regimen by the
time that they arrive in high school. Also, recognizing a form of mathematical reasoning that is
essential in analyzing arguments from scientific studies to policy proposals, we have built a
significant statistical component into the curriculum.
The availability of out-of-class study time, after-school resources, and tutoring is an important
component of this model. These structural elements at the charter school will allow
underprepared students to catch up and will provide a safe space for students to complete their
classwork.
Finally, we believe that lack of physical movement in the mathematics classroom has an adverse
effect on students’ attention spans. Teachers should seek to incorporate activities that bring
movement into the classroom, which may be as simple as experimentation with physical models,
or as involved as outdoor activities and field trips.
Mark Behrens, Christopher Degni, Moon Duchin, and Daniel Reuman
April 2004
EXAMPLE: Circles, radius, circumference, area—Sixth-grade lesson
The three approaches (skills; discovery; “Puente”) are contrasted.
Skills
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Introduction of π =3.1415926…
Presentation of other estimates to π, such as 22/7
Definition of radius, diameter, circumference, area
Statement of formulas: A=πr², C=2πr = πd
Presentation of examples of computing A and C from different values of r and d
Skill-building: students work dozens of similar examples
As in “why do I need to know this formula?” or “what is this good for?” or “I’m never going to use this.”
The analogy of mathematical literacy to voting rights has been persuasively argued by Bob Moses, in
particular in Radical Equations. Moses argues that algebra, in particular, is a gatekeeper for higher
education and for access to skilled jobs.
3
4

More challenging problems ask students to find C in terms of A, to find areas of semicircles and
annular regions, etc
Strengths: lesson is compact in terms of time taken; offers plentiful practice.
Weaknesses: formulas are unmotivated and must be memorized; lesson lacks variety; conceptual
understanding may never be attained.
Discovery
 Distribute manipulatives (physical objects) which contain circles of many sizes
 Ask students to measure (approximate) circumference, radius, diameter
 Have students graph data, notice relationship of C to r (constant ratio, no matter size of circle)
 For given physical disk, use squares whose sidelength is circle’s radius; Cut up and rearrange
radius-squares to try to determine how many cover the disk
 Notice relationship (constant number of radius-squares needed, no matter size of disk)
 Discussion: “Did you know? You have discovered that the area of the circle is a little more than 3
times the radius squared. You have also found that the distance around a circle is a little more
than 3 times the diameter. There is a special name given to this number that is a little more than
three. In 1706, William Jones used π (pronounced “pi”), the Greek letter for p, to represent this
number. …”5
 Go on to give application: tree replacement policies in cities
Strengths: offers students opportunity for physical manipulation; creativity; data plotting.
Weaknesses: vagueness of idea of π (is it any number a little more than 3?); scantiness of practice.
Puente
 Definition of radius, diameter, circumference, and area – and explanation of what kinds of figures
the terms can apply to, connecting circumference to perimeter
 Estimating area of arbitrary regions as built up from area of small squares – this is explored with
regions on graph paper, showing that finer graph paper gives greater accuracy. (Note: this plants
the seed for limiting processes, an idea that will be key in calculus.)
 Problem: how does C relate to r? How does A relate to r?
 Hint in the exploration cycle: since C is a length, it can be measured in terms of r. Since A is an
area, it might be best measured in terms of r². If students have trouble with the notion of r²,
teacher suggests that radius-squares, as above, be built to demonstrate.
 Consolidation: teacher explains that findings (of a similar multiplicative factor in both problems)
are not coincidence, but an occurrence of a special constant called π, which has a decimal
expansion that never repeats or terminates. Common approximations of π are 3.14 and 22/7. π is
important because it appears in almost every context relating straight lines to circles and circular
motion, including arcs, spheres, and spirals.
 Skill-building: students work problems relating A and C to r and d for circles and disks, and for
other shapes like semicircles and annular regions. Students find π on the number line.
 Challenge: using the rule A= π r², students create visual demonstrations that π>2 and that π<4
Strengths: plants seeds for future concepts; puts practice after motivation of ideas; connected to broader
ideas in geometry, as well as the algebraic notion of squaring.
Weaknesses: no applications offered in this unit.
5
This is a quotation from a sample lesson of the Connected Mathematics Project.