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Designing Communities of Learning: Certain Non-Negotiables


Wrong Way!


By Clarence Thompson

As I wrote in the last post, the focus and mission of the Flying University Project is to teach basic arithmetic, algebra and college-prep math to children of color and other marginalized populations who would otherwise be failed by the public school system.  This mission requires the design of learning communities that maximize our effectiveness in fulfilling this mission.  The design of such communities must be guided by a wise and accurate overall strategy, implemented by teams whose members are united in their strategic goals.  Obviously, the strategic goals of such teams arise out of their mutual agreement on the overall end result they are trying to accomplish.  This means that such teams must agree on what teaching basic arithmetic, algebra and college-prep math looks like.  It also means that such teams must agree on what people who have mastered these subjects look like - specifically, what they should be able to do as a result of their mastery.  Teams who have an accurate idea of these things should also easily be able to tell what people who know basic arithmetic, algebra and college-prep math don't look like, and should also be able to tell what educational malpractice looks like.

But where should we turn for help in forming such an accurate picture?  Where especially should we look if we want to see an accurate picture of effective math instruction?  To me, the best place to look is to look at people who have been recognized for their expertise in mathematics, and especially to those people whose jobs and livelihood depend on math expertise.  Thus it was that a few years ago I was very glad to find a paper titled "Elementary School Mathematics Priorities" by W. Stephen Wilson, who is a professor of mathematics at Johns Hopkins University.  His paper gave an objective justification and confirmation of many realizations that I had come to on my own, and that I found to be shared with many adults who had gone through elementary school before the first decade of this century.

Dr. Wilson's premise is that mastering basic arithmetic is of foundational importance for people who want to go on to algebra, calculus and post-calculus mathematics.  As he says, "Ultimately, solving problems is what mathematics is all about.  Our basic mathematics is fundamental to this enterprise because all other mathematics is built on it."  (Emphasis added.)  This statement implies that if a person can't solve numerical problems, they don't know math.  It is also true that if they are taught solution methods that are cumbersome, time-consuming, and don't apply to the widest, most general kinds of problems, they don't know math.  

For Dr. Wilson, there are five basic building blocks for mastering elementary school mathematics, as listed below: 
  • Numbers.  This includes learning to count, learning to express counted quantities numerically, and most importantly, memorizing single-digit number facts for all four basic operations (adding, subtracting, multiplying and dividing).  "...Students must learn counting and acquire instant recall of the single digit number facts...Instant recall allows the student to concentrate on new concepts and problem solving."  (Emphasis added.)
  • Place value system.  This is foundational for understanding the standard arithmetic algorithms, especially when solving problems with multiple-digit numbers.  It also provides a basis for intuitive understanding of manipulating polynomials in algebra and calculus.  Those students who want to learn computer engineering must also learn the place value system as applied to the base 10 numbering system so that they can properly use numbers expressed in numbering systems other than base 10.
  • Whole number operations.  To quote Dr. Wilson, "Addition, subtraction, multiplication and division of whole numbers represent the basic operations of mathematics.  Much of mathematics is a generalization of these operations and rests on an understanding of these procedures...The standard algorithms are among the few deep mathematical theorems that can be taught to elementary school students."  Once the whole number operations are learned by means of the standard algorithms, it is possible to use these algorithms to work with negative integers, rational and irrational numbers, and complex numbers.  The standard algorithms are also the building blocks for the more complex algorithms found in algebra and calculus.
  • Fractions and decimals.  As noted before, mastery of the use of the four basic operations on fractions and decimals is a fundamental building block because it shows students that the same rules apply both to whole numbers and to the extension of the number system beyond whole numbers.
  • Problem solving.  "The translation of words into mathematics and the skill of solving multi-step problems are crucial, elementary forms of critical thinking."  This involves solving problems which involve quantities in physical situations.  It also involves learning to express the numerical elements of a physical situation in the form of equations or inequalities in order to predict the physical behavior of the situation as these quantities are varied. These skills are essential for people who are interested in the technical professions, such as engineering or science.  Engineers, doctors, scientists, and inventors are solvers of word problems - it's what they do.
Dr. Wilson makes an important point toward the end of his paper when he says, "Mathematics is an activity.  It is not enough to believe you understand something in mathematics.  You must be able to do something with it." (Emphasis in original.)  But even more important is this: "[The] necessary skills and concepts for the engineering student begin with the foundation discussed in this paper in early elementary school.  There is a tendency to suggest that most students do not need all of these skills because most students will not become engineers.    Even if this were true...we would be in a serious quandary.  Would this mean that we should not teach them to all students?  Students who don't get these skills will definitely not become engineers...Any attempt to separate elementary school children into two groups, one group that will never have the option of becoming an engineer and another group that will be given that option, would seem grossly unfair."

And yet that kind of segregation is what is happening to many children of color and children from other marginalized populations in many public school districts across the United States, and especially in the Portland Metro area.  Moreover, it is not because these children are stupid, or have limited capacity for learning, or are genetically limited, or are lazy.  It is rather because many teachers in many school districts have deliberately lowered the bar of math instruction for these kids.  I suspect that many people who hold jobs as "teachers" don't know basic arithmetic themselves and can't themselves perform the four basic operations on whole numbers.  In addition, the curricula developed for the children in these districts de-emphasize memorization of single digit math facts and mastery of the four basic operations.  (Those who want to see how important memorization is should check this out for instance.)  Instead, children are taught to develop problem-solving approaches that are ad hoc and cannot be easily applied to a wide variety of problems.  Because they are discouraged from (or forbidden from) memorizing basic math facts, they must rely on calculators (or smartphones), or they must rely on problem-solving strategies that waste an inordinate amount of time and paper, and that can't be easily explained between one person and another.  One such strategy is the "guess and check" method that has become very popular among elementary school "teachers."  Click on the link in the preceding sentence and look at their guess and check problem.  Before anyone can guess and check (or reach for their calculator), I can bust out the correct answer to that problem in nothing flat by simply using a pencil, a piece of paper, and the division algorithm!

What happens when children who have been failed by the school system in this way grow up to be young adults?  If they are high school graduates who want to go to college and think they are ready for it, they are in for a rude shock - especially for those who choose technical majors.  As Dr. Wilson notes in his paper, "The majority of students who take a [math] placement test in college fail it.  When such students find they are missing their mathematical foundation they tend not to be happy.  They also seldom recover mathematically well enough to proceed with a college level mathematics course."  Can such students recover?  The answer is yes, but it takes a lot of extra time, plus the expense of an ever-growing series of remedial mathematics courses at the community college level.  An increasing number of young adults find that they need to start by learning basic arithmetic in college, and that drawing circles and dots or using "guess and check" just doesn't cut it.   The American educational system is characterized by the scandalous explosion in remedial college math courses and the explosive growth in the number of students (especially students of color) who find themselves having to take these courses.  (See this, this, this, and this for starters.)

Therefore, when I teach, I insist on finding out whether my students have mastered the fundamentals, and if not, I want to know exactly where they are lacking.  Then the strategies I develop focus on filling in the gaps in the understanding of the fundamentals of arithmetic.  Only after this is accomplished am I willing to move on to algebra.  I do not accept the notion that I should simply focus on helping students with schoolwork beyond their skill level without regard to whether they possess the fundamentals required to actually do such schoolwork.  Students must be encouraged and helped to grasp the fundamentals.  That is a non-negotiable strategic goal, because that is the only way students will achieve mastery.  In teaching the fundamentals, our mission is to insure that students have mastered basic arithmetic using standard algorithms built on a mastery of single-digit math facts.  That also is non-negotiable.  W. Stephen Wilson's paper is my strategic map.

And based on my experience, such a map can easily be followed.  Here is a picture of a student who recently mastered addition with carrying.  He's also starting to get comfortable with subtraction with borrowing.  You go, kid!


For further information on the necessity of memorization in math instruction, please see the following:
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