Several years ago, in the early years of my teaching career, I had a “definitive moment” while teaching a mandatory non-honors calculus course for 12^th graders.  At the time, I required students to take tests without calculators and arranged the computation and algebra of the tests to be easily handled with basic math skills.  Nevertheless, if a student could not remember or figure out a computation, he/she could ask for my help.  At one point, I found myself staring at a student who, with all honesty, asked me, “What is two plus negative one?”  12 years of grade school had brought this young man to the seat of calculus without endowing him with the knowledge of elementary math: 2 + (-1) = 1.

Over two decades of teaching and conversing with other teachers, my understanding of this problem has deepened.  There are many layers to it, but one fundamental part of the problem has to do with an aspect of education that hears complaint in all fields (reading, writing mechanics, arithmetic, historical facts, etc.).   Teachers lament that young people simply do not have core skills to adequately participate in higher levels of the intellectual life.  This is most evident in mathematics, where high school math classes depend so heavily on students having a firm foundation in basic skills to progress to more advanced topics.  Often math teachers have to open up courses of minimal rigor just to get students to be able to succeed in a third or fourth math credit.  Perhaps even worse, teachers attempting advanced classes must slow down or gloss over algebraic or arithmetic details in order to reach the traditional goals demanded by those courses.  

This problem, among others, remains a glaring symptom of the crisis that is easily called a crisis of elementary math education in our schools today.  If studied seriously, this crisis will manifest its ultimate origin in the central crisis of modernity: a loss of the use of common sense.  By “common sense” is meant the perennial set of principles that are “commonly” held by ordinary human beings.  In philosophy, common sense means the perennial ideas starting from principles such as non-contradiction, causality, and finality.  In education, the perennial principle (known by common sense across cultures) is that teachers must meet students “where they are at.”  Proper educational praxis follows accurate assessment of a student’s capacity to learn and his/her mode of learning.[1]

In this essay, I will demonstrate how the loss of common sense in elementary math education has led to a false understanding of the nature of the elementary student and thus a failure to understand the proper mode of learning that this stage of life presents.  As a result, curricular regulations governing elementary education often take math pedagogy into areas it should never go at that stage, and well-meaning and qualified elementary school teachers are led into a quagmire that adds unnecessary stress to their lives and harms their efficiency in helping young people learn math. Toward the conclusion of this essay, I will offer a possible solution to this problem that reorients the approach to mathematics in elementary school and points the way toward a positive reform of K-12 math, in general. 

Where, then, are elementary school children in the process of learning?  That is to say, what is their mode of learning?  In her famous essay, “The Lost Tools of Learning,” Dorothy Sayers describes the elementary stage of learning as the “poll-parrot” stage of learning.   “The Poll-Parrot stage is the one in which learning by heart is easy and, on the whole, pleasurable; whereas reasoning is difficult and, on the whole, little relished. At this age, one readily memorizes the shapes and appearances of things; one likes to recite the number-plates of cars; one rejoices in the chanting of rhymes and the rumble and thunder of unintelligible polysyllables; one enjoys the mere accumulation of things.” Sayers explains that this first stage is one of three stages of learning that can be likened to the medieval use of the trivium: grammar, logic, and rhetoric.   As many “classical” educators today would say, the poll-parrot stage is the “grammar” stage of learning.

In mathematics education, the “poll-parrot” stage (or grammar stage) of learning corresponds to an accumulation of facts and basic skills.  It is through rote memorization and frequent repetition that this mode of learning best grows into knowledge.  It is not through discussion or debate that this mode prospers.   Moreover, personal reflection or inquiry into abstract concepts and connections is not an effective process for educational flourishing at this age.  Elementary school children are “hardwired” to memorize the key facts of addition, subtraction, multiplication and division while cultivating habits of the intellect in working basic computations within these four key areas.

The objection arises that children will find math dull and unrewarding if they are just presented facts over and over.  But this is not in line with the common experience of little children.  They love facts and the rehearsal of facts. They love the challenge of learning more and more facts and mastering the ones they know.  It is the grownups -- not the little children -- who find fact regurgitation less interesting than penetrating into the deeper meaning of why things are the way they are. 

Unfortunately, this basic nature of the elementary student is being systematically ignored or abandoned in curricular design.  This has led to three prominent symptoms of the crisis. The first symptom of the crisis is that curricula often encourage discussion, dialogue, conversation in exploring why this or that fact is what it is.  

As a case study, consider the state of Texas math standards for 3rd grade.

(A) apply mathematics to problems arising in everyday life, society, and the workplace; (B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution; (C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems; (D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate; (E) create and use representations to organize, record, and communicate mathematical ideas; (F) analyze mathematical relationships to connect and communicate mathematical ideas; and (G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.[2] 

In this example, note that none of these standards or principles are appropriate for the grammar stage of learning.  It is not that this type of thinking is inappropriate for math teachers in general.   In fact, middle school and, definitely, high school math classes can benefit from these guidelines, as they apply somewhat to the logic stage of learning and certainly to the rhetoric stage.    However, this type of math teaching is completely inappropriate for the grammar stage which has no interest -- and often no capacity -- for this type of analytical work. 

The second symptom is the failure to teach math to the appropriate grade.  We often see that certain individual standards go above and beyond what children should be learning at a certain age.  Consider another math approach to third grade.  After positing many multiplication and division standards (themselves together perhaps beyond where 3^rd graders need to be), the standards go on to begin discussing fractions.  Consider these fraction standards (among many): “a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.”[3] How can this be?  Are there really third graders who have such mastery of multiplication and division that they are ready for fractions at this level? Is this a necessary benchmark if there are still students who cannot add or subtract with mastery?

This leads to the last point: there are simply too many standards.  For example, there are many robust grade 3 math standards in the state of Hawaii; when I copied them to a Word file in 12-point Times New Roman font, the list went over 11 pages![4] One of them is as follows: “Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.” Assuming that, in a 180-day school year, a 30+ standard approach would allow 5-6 days per standard, can it reasonably be said that the multiplication and division facts of 100 are learned in about a week?   Isn’t it more reasonable to have a month (or even a term) at least to allow children the chance to absorb to the deepest level the most important facts of elementary math?

Having identified some of the aspects of what elementary math is not, we must then ask: What should these elementary math standards be?  In short, they should be simple basic skills that are taught in rote/repetition style.   For instance, they can be reduced to one core concept per year: number recognition and counting in kindergarten, addition in 1st grade, subtraction in 2nd grade, multiplication in 3rd grade, division in 4th grade, and fractions in 5th grade.  

Since math at this level is naturally cumulative, this arrangement does not raise the concern that students learning subtraction will forget addition; for example, just have them check every answer to an a – b = c problem by adding c + b to show that it’s a.  They’ll get all the addition they need. 

For those who really want to have kids study geometry shapes, clock and time, or some bar graphs, there is ample opportunity to use these as examples in the addition, subtraction, multiplication as they surface. Moreover, for those who need advanced addition, there are larger digit problems to do, maybe word problems, and maybe peer teaching.  The same holds for all the elementary grades.

But the core teaching in, say, 3^rd grade, is multiplication tables and a slow but certain growth toward mastery of multiplication in several digits.  As long as this is the core standard, a good teacher can be creative in occasionally finding this or that new way of presenting the same math calculation.  A square has 4 sides; let’s look at the number of sides when we have 5 squares.  4x5=20 . . . so:  20 sides!   And so on.  But this should be one of many ways to rehearse and repeat the same times table and multiplication practice that cement the one standard for the 3^rd grade year.

Finally, the only hope is if we as a culture return to common sense.  To make such a move requires a humble appreciation of how things have been done well before the onslaught of modern thought.  From Rousseau’s Emile to the present day elementary school child, Romanticism has formed a pedagogy that reduces the role of the parent or teacher in guiding the child to the necessary repetitions he/she needs to do to acquire physical and intellectual good habits.

Meanwhile, the ever-present lure of the fashionable (math-chic) makes traipsing after the latest “curricular innovation” and “technological enhancement” enviable and attractive to the naive educator or educational bureaucrat.  Various textbooks and “programs” are often heavily marketed as having the “latest” breakthroughs in math pedagogy; often, though, it seems that these programs are just preying on the devastating collapse in math formation in both teachers and students and they end up perpetuating rather than solving the problem.[5] Thus, these developments usually lead us nowhere closer to the goal that is the same year after year: teaching addition, subtraction, multiplication, division.  

Above all, as modern man finds himself lost in more and more educational theories, he has less and less confidence that his in common sense can steer the course for teaching.  The tools for learning get lost behind the theories of un-learning, as adults try to shove children into a framework that meets theoretical ideals but is cut off from the reality of who children are.

And so, math curriculum gets forever batted around in the maelstrom of modern thought swirling into a chaos of incomprehensibility so that, like Orwell’s 1984 nightmare, Winston really can’t tell anymore that 2+2=4.  Similar problems can be seen in the education of reading, writing, and basic knowledge of facts about the real world.  Suffice it to say that a return to an honest appraisal of who elementary children are will best help us return to how best to educate them.