Grant Wiggins was the keynote speaker last night at the annual “Anja S. Greer Conference on Mathematics, Science and Technology” hosted by Phillips Exeter Academy in Exeter, NH. The focus of his talk was mathematics education, and the points that were noteworthy to me included the following:
Increasingly, schools and standards bodies are setting their goal for mathematics education to be the development of good problem solvers. Yet,
– few schools focus their curriculum on problem solving
– nationally, dismal percentages of students can successfully solve problems of types they have been taught to solve, let alone problems they are not familiar with
– a significant percentage of students hate their mathematics courses
We face some big questions that are challenging to answer:
– What is the problem with mathematics education today?
– What are we going to do to address it?
This is the problem that math teachers and curriculum designers must solve.
If students are to be able to solve problems of types they have not necessarily seen before, they need the ability to transfer their knowledge and skills to new domains. Yet, most of mathematics education today focuses on using existing skills within existing domains. Homework and items on tests are generally “exercises” in demonstrating an acquired skill, not so much “problems” which require novel thinking on the part of the student. In fact, students and parents often complain if assessment is based on anything other than variations on what students have already seen.
An analogy can be found in sports: the drills which have been run repeatedly in practice often do not reflect the situations which arise in competition. Yet, students thrive on the challenges presented by competition, and parents do not complain when the opposing team does not line up in the way the coach anticipated yesterday in practice. The current state of high school mathematics education is like four years of practicing for a sport, without ever playing in a real game. The sum of all practice drills does not equal the game – but do we even know what “the game” looks like in mathematics education?
So what are the attributes of the “real problems” that we wish to prepare our students to tackle? Among other potential attributes, they:
– are novel or unfamiliar to the solver
– do not provide sufficient information – some research or assumptions will be needed
– sometimes lead to dead ends – typical approaches may not solve the problem
– often present some ambiguity
All of this leads to the notion that we, as teachers, often do not know what a “real problem” is anymore, as we already know how to solve most (if not all) of the problems we give to our students.
Other current issues include:
– We teach many skills which students are not learning (as shown by test results)
– Students often dislike mathematics because they feel they are bad at it
– Students do not see a purpose to what they are learning in math class
– If we show a student how to do something, we deprive them of an opportunity to learn to “solve a problem” that is before them… we often “teach too much”
– How much time going down a dead end is a good thing? At what point should we re-direct a student, and how?
– What strategies are built into mathematics curricula to help students develop autonomy in problem solving? Many curricula seem to build dependence on the teacher.
– Have schools audited their assessments for rigor and complexity? If they do, they will probably be startled by how low the levels of rigor and complexity turn out to be.
A potential goal for secondary mathematics curricula could be something like: develop students’ abilities to answer rigorous and complex questions of types that they have not seen before, without teacher assistance.
- “The nature of proof” by Harold P. Fawcett
- “How to Solve It” by George Polya
- “The Art and Craft of Problem Solving” by Paul Zeitz