Kristin Umland

Math Departments and the
Mathematical Education of Teachers:
Beyond Specialized Courses
Kristin Umland
CBMS Forum
October 6, 2014
What is MET II?
• The Mathematical Education of Teachers (MET
I, 2001) made recommendations for the
mathematics that teachers should know and
how they should come to know it
• It was updated in 2012 to reflect events and
advances, including the advent of the
Common Core State Standards for
Mathematics (CCSSM)
The call
• MET II urges greater involvement of
mathematicians and statisticians in teacher
education so that teachers have the
knowledge, skills, and dispositions needed to
ensure that high school graduates are collegeand career-ready
• The mathematical community is charged with
nurturing the mathematical development of
all teachers, including elementary teachers
Students should be doing real
• At every level, there is important mathematics
content that is both intellectually demanding to
learn and widely used
• In addition to learning content, students should
learn to think like mathematicians:
– Make sense of problems and persevere in solving
– Construct viable arguments and critique the reasoning
of others
– Model with mathematics
– Attend to precision
• The CCSSM are constructed out of learning
progressions in key domains. They call for
students to have a balanced mathematical
– Conceptual understanding of important ideas
– Fluency with key algorithms and procedures
– Opportunities to apply mathematics to
mathematical and real-world contexts
The reality
• Many prospective teachers come to college
without having had a balanced mathematical
diet in their own K-12 schooling. Many do not
know the mathematics described in the
CCSSM at a student level, much less at a
teacher level.
Challenges and opportunities
• Teachers need to develop a deep, coherent
understanding of K-12 mathematics
• That knowledge is much more clearly
specified than in the past
• There is an unprecedented opportunity for the
mathematics community to work across state
lines to solve the thorny problem of how to
support teachers’ mathematical growth
How do we get there?
• Prospective teachers need mathematics courses
that help them develop a solid understanding of
the mathematics they will teach
• They need opportunities to engage in reasoning,
explaining, and making sense of the mathematics
that they will teach
 Many institutions provide these in specialized
courses for teachers. MET II provides guidance for
such courses.
Some examples
• In third grade, students should be able to
reason about equivalent fractions without
using fraction multiplication. What is a
mathematically sound way to do this?
• In seventh grade, students should be able to
explain why they can “set up a proportion.”
Most people have never thought about why
this works or how to build it up from
mathematically sound foundations.
More examples
• In high school, congruence and similarity are
defined in terms of transformations of the
plane. Few students have studies this in HS
before the CCSSM. They need to see the
connections between the way they studied
geometry and the new expectations of their
MET II recommended coursework
• K-4: At least 12 semester-hours on fundamental ideas
of elementary mathematics, their early childhood
precursors, and middle school successors.
• 5-8: At least 24 semester-hours of mathematics that
includes at least 15 semester-hours on fundamental
ideas of school mathematics appropriate for middle
grades teachers.
• 9-12: The equivalent of an undergraduate major in
mathematics that includes three courses with a
primary focus on high school mathematics from an
advanced viewpoint.
More reality
• US students typically hold unproductive
beliefs about mathematics:
– Mathematics is a set of rules you follow; it doesn’t
matter if they make sense
– If you can’t solve a math problem in 1-2 minutes,
you just don’t get it and may as well give up
– You are born with a math gene or not; talent
matters much more than effort
Changing students’ views
• Students develop their attitudes and beliefs
about what mathematics is, who can do it, and
what it takes to do well by sitting in math
classrooms for many years.
• Undoing this is like trying to unwind a spool of
thread that has been wound up over 13+ years—
it can’t happen in one or two courses. And if the
first math course they take when they get to
college reinforces these beliefs, then changing
them during college is extremely difficult.
Beyond specialized courses
• Prospective teachers need opportunities to
develop the habits of mind of a mathematical
thinker and problem-solver, such as reasoning
and explaining, modeling, seeing structure,
and generalizing
These habits of mind develop over time and
should be cultivated in all math courses. How
we teach matters as much as what we teach.
The first two years
• K-4: Most departments offer math for
elementary teachers at the 100 or 200-level
• 5-8: At least 9 of these hours are going to be
“typical” 100 or 200-level math/stat courses
(college algebra, pre-calc, calc, intro stats)
• 9-12: Whatever courses a typical
undergraduate math major takes in the first
two years
Email from a colleague in chem
• My Biochemistry students are studying for a quiz over
enzymes (including kinetics) tomorrow and I have been
engaged with several students by e-mail and at office
hours who have very serious basic math deficiencies,
even though they got through Calculus 1. Simple high
school algebra, including the slope-intercept concepts
for a linear equation of the form y = mx+b, is kicking
their butts. Negative numbers and extrapolation of
lines onto the negative side of the x-axis seems to be
another set of concepts that are poorly understood.
Does anyone have any insights into why this seems
such a formidable subject for so many students?
Email conversation in my dept.
I had an interesting phenomenon occur on my recent Calculus Exam 1.
I asked them to compute the average rate of change of f(x) = x^2 + 5x
on the interval [1,4]. Some of the students used the "wrong" formula,
[f''(1) + f''(4)]/2, thus finding the average value of the two derivatives
at the endpoints of the interval, instead of using the "correct" formula,
[f(4) - f(1)]/(4-1). As I was grading this problem, I noticed that several
students used the same "wrong" formula, but they all came up with
the "right" answer. One of my students questioned my grading ... why
did I deduct half credit, if she got the right answer? I told her that it
was just a lucky coincidence that she got the right answer using the
wrong method.
So, my question to you all is this: was it really a coincidence, or would
her method ALWAYS work on quadratic functions?
One response
ok. Let f(x)=ax^2+bx+c
f(x1)=ax1^2+bx1+c .
Do (f(x2)-f(x1))/(x2-x1) = (ax2^2+bx2-ax1^2-bx1)/(x2-x1) *
f'(x2)=2ax2+b and f'(x1)=2ax1+b
The average of the two derivatives is ((2ax2+2ax1+2b)/2 )** .
When you set * equals to ** and solve for x, you get an
identity. I interpret that to mean it will work for any x1 and x2
when the function is quadratic. I just randomly made up
f(x)=2x^2+3x-1 and evaluated the average rate of change from
2 to 5 using ( f(x2)-f(x1))/(x2-x1) and compared it to
(f'(x1)+f'(x2))/2 and I ended up with so answer (17). So the
student's answer was not a lucky coincidence.
Another response
I think mathematics and calculus in particular should not be just
about "getting the right answer". Proper justification is as
important as getting the right answer. In this particular case the
student used a formula which is not in the book (and is not even
correct for polynomials of degree 3 and higher), so she should
have provided a justification for it.
BTW, the "wrong formula" works for quadratic polynomials
because the derivative of a quadratic polynomial is linear
function and for linear functions the average over an interval is
the same as the average of values at the end points.
What is good for teachers…
• Attending to the needs of prospective
teachers by focusing on reasoning and proof
across the spectrum of undergraduate
mathematics courses that they take helps
students to make sense of mathematics—and
makes it easier to understand, easier to teach,
and intellectually satisfying for all coursetakers. Thus, attending to the needs of future
teachers in this way benefits all
undergraduates. (MET II)
…is good for all students
• All students need help developing
• All students should expect math to make
• All students should understand that
mathematics can be applied to help solve
problems outside of mathematics and have
opportunities to do that.
The take-away
• What we teach in the first two years of college
and how we teach it may not serve our
students well
• The impact it has on prospective teachers is
particularly problematic, especially because
instructors may not even realize they have
such students in the class
What we should do
• Help our colleagues who teach these courses
understand that the goals of these courses
should be much broader than simply
developing procedural content knowledge
• Look for concrete ways to reach this expanded
goal set
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