4. An even number is any number whose unit digit is 0, 2,
5. An even number is a whole number with algebra -helper/ exponent -zero.html">zero remainder
divided by 2.
6. A whole number is even if it is the sum of a whole
a) An even number is a number that can be divided into two
b) An even number is any multiple of 2.
All numbers, for example 7, 3/5, √2, π, are even!
c) An even number is any integer multiple of 2.
This is a correct definition of even number, but not usable by third
d) An even number is any number whose unit digit is 0, 2,
4, 6, or 8.
In this case, 36.7 is an even number!
e) An even number is a whole number with zero remainder
when divided by 2.
According to this, -6 is not even.
f) A whole number is even if it is the sum of a particular
whole number with itself.
This is a correct definition of evenness for whole numbers, and is
consistent with the general definition for integers that students will
Case 4: Writing a quiz and choosing
De signing a quiz on even and
odd numbers for Shea and his
classmates • Choose a list of numbers that you would
like Shea and his classmates to classify
correctly so that you can learn whether
they understand what makes a number
even or odd
– Does it matter what numbers you pose? Why or
Study of Instructional Improvement
• Study of three Comprehensive School
Reforms; teacher knowledge a key variable
• Instrument development goals:
– Usable with 5000 teachers: survey, multiple
– Measure content knowledge teachers use in
teaching – not just what they know
– Differentiate among teachers
• Multidimensionality of mathematical knowledge
– Knowledge of students and content different from
“pure” content knowledge
– Knowledge differentiable by topic domain (e.g.,
– Number and and operations mathematical knowledge
• Tentative conclusion: existence of a kind of
“professional” knowledge for teaching
Validating our measures
How do we interpret teachers’ performance on our
1. Their score reflects their mathematical thinking
– Cognitive interviews
2. Higher scores mean higher-quality mathematics
– Videotape validation study
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