Reboot Mathematics.

More on Schoenfeld “Good teaching, Bad results”
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. 

Teaching understanding or algorithms?

I recently read Schoenfeld’s “When Good Teaching Leads to Bad Results: The Disasters of “Well Taught” Mathematic Courses”. 

This paper reports the results of a year-long intensive study of the teaching and learning that took place in a tenth-grade geometry class, which will be called the target class. The class took place during the 1983-84 academic year in a highly regarded suburban school district in upstate New York. The study included periodic observations of the target class and of eleven other mathematics classes, interviews with students and teachers, and questionnaire analyses of students' perspectives regarding the nature of mathematics. The target class was observed at least once a week, and was videotaped periodically for subsequent detailed analysis. Two weeks of instruction near the end of the course, dealing with locus and construction problems in geometry, were videotaped in their entirety. The analyses focused both on the mathematics that was learned, and on what the students learned about the mathematics -- including how and when they would use, or fail to use, the mathematics that they had studied.

This paper explored the difference between performing mathematical algorithms and truly understanding the underlying fundamental ideas of mathematics.  Due to the structure of the study, they were able to quantify certain classroom behaviors; time spent in questioning, active learning time, amount of praise, and amount of feedback.  The study also included the type of grouping, the size of grouping and so on.  Originally, they defined “learning” as how well the students performed on achievement tests.  I say originally, because through the article it illustrates how these tests fail in significant ways to measure subject matter understanding.

The article also suggests that providing students with repetitive routine exercises that can be solved out of context and no significance provided, actually causes the subject matter to seem frivolous to students.  This monotonous work actually deprives the students of the opportunity to apply their learning in a context that is meaningful to them.

Brown and Burton (1978) developed a diagnostic test that could predict, about 50% of the time, the incorrect answers that a particular student would obtain to a subtraction problem -- before the student worked the problem!  Teachers are still currently doing this, when we provide students with “distractors” on a multiple choice exam.  If we already know what mistakes students are going to make, before they make them, we need to start changing our instructional models.

The predominant model of current instruction is based on what Romberg and Carpenter (1985) calls the “absorption theory of learning”: “The traditional classroom focuses on competition, management, and group aptitudes; the mathematics taught is assumed to be a fixed body of knowledge, and it is taught under the assumption that learners absorb what has been covered” (p. 26) This view is essentially implying that the “good” math teacher has multiple methods of covering the same outcome.  Through these multiple methods the students will eventually “get it”.  Unfortunately, what most math teachers fail to realize, is that through these multiple methods we are forcing the students to “get” something entirely different; resistance to change.

Math teachers need to stop testing students on algorithms and more on the understanding of math concepts.  Below is an example of two problems Werthemier (1959) gave to various elementary school students.

Many of the students, who were deemed as high achievers, added the terms in the numerator and then performed the indicated division.  They followed the conventional order of operations of BEDMAS, or to some PEDMAS.  An idea which should not be taught to students out of context, which I have wrote about here.  Even though these students calculated the correct answer, I would argue that these students do not demonstrate any depth of understanding of mathematics.  To truly understand the underlying substance students should recognize that repeated addition is equivalent to multiplication and division is the inverse of multiplication. 

This example truly illustrates that being able to perform the appropriate algorithmic procedures, does not necessarily indicate any depth of understanding.  Also, the sad truth; virtually all standardized exams for arithmetic competency focuses primarily on algorithmic mastery, and not deep understanding of the math concepts. 

This is the first part of the article summarized, more to come….