Why do standards matter?
•
•
•
•
•
goal posts for teaching and learning
coherence across grade levels
determine the content and emphasis of tests
influence the selection of textbooks
form the core of teacher education programs
"It is critical that the curriculum, instruction,
and assessment deeply align."-Mathematics, K-10 Grade Level
Expectations: A New Level of Specificity
The State of State Math
Standards 2005
Fordham Foundation
Co-authors of the Fordham Foundation Report:
•
•
•
•
•
Bastiaan Braams, Emory University
Thomas Parker, Michigan State University
William Quirk, Ph.D. in Mathematics
Wilfried Schmid, Harvard University
W. Stephen Wilson, Johns Hopkins University
What’s Wrong with
Washington's Standards?
Fordham Foundation grade: F
Excessive use of calculators, standard algorithms
missing, poor development of fractions and
decimals, weak algebra standards (little more than
linear equations), very little geometric reasoning
and proofs, weak problem solving standards, too
many standards unrelated to math
Definition
“What is Mathematics? - Mathematics is a language and science of
patterns.”
“As a language of patterns, mathematics is a means for describing the world
in which we live. In its symbols and vocabulary, the language of mathematics
is a universal means of communication about relationships and patterns.”
“As a science of patterns, mathematics is a mode of inquiry that reveals
fundamental understandings about order in our world. This mode of inquiry
relies on logic and employs observation, simulation, and experimentation
as means of challenging and extending our current understanding.”
-- Office of the Superintendent of Public Instruction
www.k12.wa.us/curriculumInstruct/mathematics/default.aspx
Patterns: 6th Grade
• Recognize or extend patterns and sequences using
operations that alternate between terms.
• Create, explain, or extend number patterns involving two
related sets of numbers and two operations including
addition, subtraction, multiplication, or division.
• Use rules for generating number patterns (e.g., Fibonacci
sequence, bouncing ball) to model real-life situations.
• Use technology to generate patterns based on two
arithmetic operations. Supply missing elements in a pattern
based on two operations.
More Patterns, 6th Grade
• Select or create a pattern that is equivalent to a given
pattern.
• Describe the rule for a pattern with combinations of two
arithmetic operations in the rule.
• Represent a situation with a rule involving a single
operation (e.g., presidential elections occur every four
years; when will the next three elections occur after a
given year).
• Create a pattern involving two operations using a given
rule.
• Identify patterns involving combinations of operations in
the rule, including exponents (e.g., 2, 5, 11, 23).*
*Note: 3 x 2n – 1 and 1/2 (4 + 5n + n3) both give these values starting with n = 0
6th Grade WASL
Karen made a triangle out
of number tiles. She used
a rule to create the pattern
in the number tiles.
•
Extend the pattern to
complete the next row of
the triangle.
•
Describe the rule you
used to extend the pattern.
Calculators
“Technology should be available and used throughout the K–12
mathematics curriculum. In the early years, students can use
basic calculators to examine and create patterns of numbers.”
1st grade:
Use strategies and appropriate tools from among mental math,
paper and pencil, manipulatives, or calculator to compute in a
problem situation.
2nd grade:
Solve problems involving addition and subtraction with two or three
digit numbers using a calculator and explaining procedures used.
4th Grade:
Use calculators to compute with large numbers (e.g.,
multiplying two digits times three digits; dividing three or
four digits by two digits without remainders).
No requirement to learn the standard algorithms of
arithmetic.
NSF-Funded Connected Math (CMP)
Question from a unit quiz for grade 7:
Find the slope and the y-intercept for the
equation 10 = x − 2.5.
Middle School Mathematics Comparisons for Singapore
Mathematics, Connected Mathematics Program, and Mathematics
in Context: A Summary (Including Comparisons with the NCTM
Principles and Standards 2000)
by Loyce Adams, K. K. Tung, Virginia Warfield and others (2001)
“We also expect that in the next edition [of CMP] some of the
typographical errors, on which much has been written and which are
usually interpreted by critics of these curricula as mathematically
incorrect reasoning, will also be corrected (e.g. Complaints have
been raised about ‘Find the slope and y-intercept for the equation 10
= x – 2.5’, CMP 7th grade Moving Straight Ahead, where y was
mistakenly printed as 10.)” – page 49
Connected Math (CMP)
The answer given in the CMP Teacher’s Guide:
“The equation 10 = x − 2.5 is a specific case
of the equation y = x − 2.5, which has a
slope of 1 and a y-intercept of −2.5.”
In addition to the answer, the Teacher’s Guide
contains two student papers and teacher’s comments
on them. First, the work of two pairs of students.
Next,in the Teachers Guide, what a teacher wrote about
this student work:
“Beth and Kim’s work for question 12 makes it clear how they
found the slope for the given equation. Their work even suggests
that they may have learned something from doing this problem.
By constructing and finding a couple of values for a table related
to the equation, they found the rise and run between two points
and thus the slope. It appears that they could not just use the
equation to give slope. The question I have as a teacher is, after
finding slope as they did, do the students now see how they could
have found the slope for the given equation?”
“Susy and Jeff received 1 point for the correct y-intercept.”
From Good intentions are not enough (2001)
by Richard Askey, Dept. of Mathematics, University of
Wisconsin at Madison
Regarding this example, Askey writes:
“If the students have learned anything they have learned that pattern
matching of a simple type will give you a good grade in a math
quiz. They will have also have learned some incorrect mathematics.
The first pair of students did a correct calculation for a different
problem.”
“I was told about this problem by a parent whose child took this
quiz. The marking was exactly as in the text. This is far from
the only error in these books.”
Quote from TERC manuals
If you have students who have already memorized
the traditional right-to-left algorithm (of addition)
and believe that this is how they are “supposed” to do
addition, you will have to work hard to instill some
new values – that estimating the result is critical, that
having more than one strategy is a necessary part of
doing computation, and that using what you know
about the numbers to simplify the problem leads to
procedures that make more sense, and are therefore
used more accurately
From 5th grade manual
(fractions)
Teacher: Now let’s use the clock face to add fractions. Say the
hand moved one third of the way around the clock and then it
moved one sixth more. Where will it end up?
Write the problem on the board: 1/3 + 1/6 =
Encourage students to talk together and find more than one way to
think about the problem. Some might find it helpful to look at the
clock faces on their (student work sheet).
From 5th grade manual
(fractions)
Suggested problems for the students:
1/5 + 1/4 =
3/8 + 3/4 =
5/6 - 1/3 =
3 - 11/4 =
“These are the most difficult addition/subtraction problems for
fractions I could find in the TERC 5th grade curriculum (which is
described as ‘also suitable for 6th grade’)”--Wilfried Schmid, Dept.
of Mathematics, Harvard University