The first part can be found here.
Math teachers need to be focusing on deeper understanding and not rote memorization of procedures and algorithms. A famous example, of the difference, comes from Werthemer and his observations of classroom sessions.
He witnessed students being taught the area of a parallelogram formula, which is derived from cutting a triangle-like piece off and calculating the area of a rectangle and triangle. These students performed quite well as the lesson, and were able to answer all standard questions mathematically and in correct form. However, when these students were provided with parallelograms in non-standard position, or to find the area of a parallelogram-like the students were confused. These students had memorized the formula, steps and procedures but failed to truly understand calculating area. With these type of understanding, these students would have no problem solving certain well specified exercises but in reality had acquired only the superficial appearance of competence.
The most extensive documentation of student’s performance on word problems, without understanding, comes from the third National Assessment of Educational Progress (Carpenter, Lindquist, Matthews, and Silver, 1983). This exam gave 45 000 13-year olds the following problem:
“An army bus holds 36 soldiers. If 1 128 soldiers are being bused to their training site, how many buses are needed?”
72% answered the long division algorithm correctly, however
· 29% of them wrote the answer “31 remainder 12”
· 18% answered only 31 busses.
Therefore only 23% of the students answered the question correctly.Math educators need to put more emphasis on the analysis of the answer. Our students, who are capable of performing symbolic operations in a classroom context, demonstrating “mastery” of certain subject matter, often fail to map the results of the symbolic operations they have performed to the systems that have been described symbolically. They also fail to connect their memorized algorithms to the “real world” application needed. These two ideas are truly demonstrating a dramatic failure of math instruction.
After Schoenfeld conducted many other studies he concludes that students have created four beliefs about mathematics.
1) The processes of formal mathematics (eg. “Proof”) have little or nothing to do with discovery or invention. Corollary: Students fail to use information from formal mathematics with they are in “problem solving” mode.
2) Students who understand the subject matter can solve assigned mathematics problems in five minutes or less. Corollary: Students stop working on a problem after just a few minutes since, if they haven’t solved it, they didn’t understand the material (and therefore will not solve it)
3) Only geniuses are capable of discovering, creating, or really understanding mathematics. Corollary: Mathematics is studied passively, with students accepting what is passed down “from above” without the expectation that they can make sense of it for themselves.
4) One succeeds in school by performing the tasks, to the letter, as described by the teacher. Corollary: learning is an incidental by-product to “getting the work done”.
These beliefs sadden my heart greatly as both a math teacher and a math learner. I dream of a school where instead of informing students of how dumb they are, they are reminded of their strengths and on these strengths, knowledge and wisdom are built.
How can this be done?
We need to realize that our primary goal of instruction and teaching is not to have students do well on an exam. Instead we need to focus on students’ passions, interests, and life goals. Teachers need to stop sacrificing understanding for the sake of accuracy and speed. There is a HUGE difference between effective teaching and efficient teaching.
“Failure is an option and one that I want you to experience.” This should be a tag line for all classes. Our Education Minister Hon. Dave Hancock said “If you are not failing, then you are not taking enough risks!” We should be allowing students to fail, and show that, through failure, true success can occur.
Students need to be involved in their own learning. Direct instruction only confirms to students information which someone “smarter” has already proven to be true. When we run our math classes with “true discovery,” students must make their best guess, and then test it by trying it out and seeing if their attempt meets their own empirical standards. This behavior will then be learned as an unintended byproduct of their OWN instruction.
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