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 college- and career-ready • The mathematical community is charged with nurturing the mathematical development of all teachers, including elementary teachers
Students should be doing real mathematics • 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 them • Construct viable arguments and critique the reasoning of others • Model with mathematics • Attend to precision
CCSSM • The CCSSM are constructed out of learning progressions in key domains. They call for students to have a balanced mathematical diet: • 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 students.
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(x2)=ax2^2+bx2+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 justabout "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 course-takers. 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 perseverance. • All students should expect math to make sense. • 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