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You Could Be the Difference - Mathematical Excellence or Mediocrity for Your Children?Copyright © Everard Barrett, 1998
Among the parents who read this article are those for whom the experience of arithmetic in elementary school was a near, if not complete, disaster. Even if you were among the minority who were good at arithmetic, you knew in your heart that you really did not understand it; which, in turn, meant you could not explain or teach it to anyone. In fact, many readers are totally convinced that it is impossible for them to facilitate the mathematical experiences of their children, because one of the effects of twelve years of public school education is the conviction of mathematical incompetence by the vast majority of its students. What a colossal contradiction to the mission of an educational institution! The author of this article knows that even if we "failed" math, we are all "fearfully and wonderfully made." I also know that, in spite of the feeling of woeful mathematical incompetence that burdens so many Christians, virtually all of us mastered our native language and learned (and retained) stories long before going to school. Clearly, virtually every one of us used certain mental gifts, which our heavenly Father placed in us, in order to be so competent as learners before going to school!
As members of God's Kingdom, we stand in awe of His glory as we observe the magnificence of His creation: the sun, the mountains, the forests, the animals and so much more. We must similarly recognize and awesomely observe His glory within ourselves. Those gifts that we used so competently for mastering our native language and stories are also manifestations of the Father's glory within us. As such, are they not valuable? Might they not be used throughout the course of our entire lives, while we are functioning as decision-makers and problem-solvers?
What are these gifts? Watch yourself when you are listening to someone speaking. You will find that your mind is deciphering meanings from connections among words you are hearing. This is the gift you used for teaching yourself your native language. Detectives use it to decipher the meaning of what happened at a crime scene (the solution of the crime) from the connections they perceive among the clues and statements from witnesses. Now, if I ask you to tell a story to a child that you have not told recently, you have the opportunity to observe how it comes back. Since you do not use exactly the same words when you tell it on two different occasions (as you would if you recited a poem twice from memory), you are not using memorization (of the words) to bring it back. Watch yourself. You will find you are "piecing" the story back together as you go along, and you are relying on the part you have already told to "trigger back" what comes next. So, once again, you are relying on connections. Our Creator so made the human mind that it can retain content based on the connections it perceives within it. This is another gift. For example, the connection between the words "prince" and "slippers" may suffice to bring back the whole content of the story Cinderella. Before hearing a story, children never make a determined decision that they intend to learn it. Clearly, this is such a powerful gift (which we used for learning before going to school) that children use it to learn stories without intention!
I think you are now ready to see why you were so insecure about math as a student in elementary through high school. Did you experience the content of math as all connected up and flowing like a story or did it feel as if it came to you in disconnected and fragmented bits? Virtually everyone agrees it was the latter. The gift is ready and able to absorb content when it is experienced as being all connected up. If mathematical content is disconnected and fragmented, the gift is deactivated and we are forced to memorize those bits because this is the only way to survive the demand by our teachers that we "learn" math. Sadly, however, memorization without connections results in forgetting and creates the need for remediation. Do you remember forgetting those mechanical steps (meaningless "recipes") as soon as the test was over? This assumes you remembered them in the first place. Meaning or understanding (which is the outcome of perceiving connections) is the "glue" which enables virtually all learners to retain content. Do you think children would be any more successful at learning the story of Little Red Riding Hood if you disconnected its content by (for example) telling about the wolf one day and Grandma one year later? Would children understand the story? Would they retain it?
Besides the horrible disconnection and fragmentation of math content, there is another serious problem in mathematics education: the presentation of arithmetic as a collection of rote, mechanical recipes. Look at the problem below.
When I was a youngster, my teacher told my class that to find the missing number I should say, "Five into twenty is four; so four times three equals twelve." The missing number is 12. I did not know why this recipe worked; but I knew that as long as I remembered and used it, I could get that check mark from the teacher indicating my answer was correct. Furthermore, to the extent that I could remember and execute the other rote (meaningless) recipes for a particular test, I was assured of a high score. But that was the problem! There were so many different recipes and steps, it was very difficult to remember them and execute each one in the right place. What kind of mental understanding (meaning) could we use as the "glue" for retaining this recipe and using it in the right place? Let me now activate one of your mental gifts for learning that our Lord gave to virtually every one of us. It is the gift we use when we can voluntarily and immediately put the image of an elephant on our mental screens (in our minds) and even make it dance. Put a rectangle in your mind. Draw four vertical lines in your rectangle, which divide it into five equal parts. This is all mental! Color the first three parts (on the left of the rectangle) yellow. We will henceforth refer to this as the "yellow part." Do you see the yellow part in your mind? Color the other two parts green. Can you "see" (in your "mind's eye") that three out of five equal parts (also known as three fifths) of the rectangle are yellow? Consequently, "three fifths" is a description of the yellow part. Now let us get another description for the yellow part. Cut every one of the five parts into four equal parts. What was five has now become how many equal parts? If what was five has now become twenty, then what was three has now become how many? So what was three out of five has now become what? Yes, it is now twelve out of twenty. Do you agree that twelve out of twenty or twelve twentieths is also a description of the yellow part? Since the size of the yellow part has not changed, you can clearly see that the two measures of the yellow part are equal.
Now here is another problem.
Can you now reason that if seven equal parts become twenty-one equal parts, then each of the seven was cut into three? Hence, if seven becomes twenty-one, the five of them become what? So five out of seven becomes what out of twenty-one?
Here is one more example.
If ten equal parts become fifty, then seven of them become how many? So seven out of ten becomes how many out of fifty? Does this now have meaning? Is there now substance behind the recipe? This is the glue that will enable you to retain that recipe forever and never use it in the wrong place.
The author of this article knows, from considerable previous experience that the vast majority of adults (including those who were "poor" math students in elementary school) can understand the arithmetic described above. That became possible for you because some of your learning gifts were activated. You put imagery on your mental screen and (mentally) manipulated it. You connected the recipe to the mental activity. Consequently, the recipe became anchored in meaningful mental activity. It really does not matter how bad you think you are at math; any of the other recipes in arithmetic or algebra can be similarly anchored in meaningful mental activity by virtually all the minds of all those who can read this article.
If we accept the reality that we are "fearfully and wonderfully made", then it is very important to understand how the schools of this world have convinced the vast majority of us that we are mathematically incompetent. As we investigate this further, you might wish to keep I Corinthians 3:19 in mind: "For the wisdom of this world is foolishness in God's sight."
Traditionally, in the public school systems of the United States, first graders are exposed to one-digit numbers; second graders to two-digit numbers; third graders to three-digit numbers; and so on. Traditional math education requires at least five years to have children experience numbers up to the millions. Let us now activate God's gifts for learning to read large numbers and see what happens.
Get a child of at least six years of age. Tell him or her that you are going to play a game. Write the following on a sheet of paper.
Tell the child that when you touch the number on the left of each set of three numbers with a pointer (a pencil, for example) he or she must say "hundred" with it. So when you tap the "first numbers," which are underlined (6, 7, 8, 4 and 5), the child must say "six hundred," "seven hundred," "eight hundred," "four hundred" and "five hundred." Make it "gamey" by tapping the underlined numbers randomly (not in sequence-jump around) and quickly. When children are nudged to respond quickly, they experience this as a game. Practice until the child has facility. This will take less than five minutes.
Now rewrite the numbers on the paper with those in the middle underlined. See below.
Tell the child that when you touch the number in the middle of each set of three numbers with a pointer, he or she must say "ty" with it. So when you tap the middle numbers that are underlined (8, 6, 7, 9 and 4), the child must say "eighty," "sixty," "seventy," "ninety" and "forty." Remember to tap quickly and out of sequence. The child must not know which one you will tap next. Practice until the child has facility.
Rewrite the numbers on the paper with those at the "end" underlined. See below.
Tell the child that when you touch the number to the right of each set of three numbers with a pointer, he or she must only say that number. So when you tap the end numbers which are underlined (7, 8, 6, 5 and 9), the child must say "seven," "eight," "six," "five" and "nine." Tap out of sequence and practice to facility.
Now rewrite the same set of numbers without underlining any of them and mix up the tapping until the child is consistently correct in his or her response. This means that when you tap a middle number, a first number or an end number, you always get a rapid, accurate response. You now have readiness for the next step.
Place commas between the sets of numbers and label them with the letters t (for thousand), M (for million), B (for billion) and T (for trillion). See below.
First, tap the commas rapidly and randomly while the child responds with the appropriate name (for each comma). Now tell the child to say only what you touch and not to go ahead of the pointer. Starting on the left and moving in sequence, touch the 6, then the 8, then the 7, then the comma, then the 7, and so on, continuing in sequence (be sure you also touch the commas) until you touch the 9 at the right end. As you proceed in this manner, you will hear the child reading this 15-digit number: "six hundred eighty seven trillion seven hundred sixty eight billion eight hundred seventy six million four hundred ninety five thousand five hundred forty nine."
You have just taken about fifteen minutes to teach a six-year-old child to read a 15-digit number! It would take another ten minutes to teach the child to read very large (or small) numbers containing zeros and to know place value up to hundreds of trillions (if you know the simple strategies). Contrast this with the fact that traditional math education takes five years to get children up to seven-digit numbers!
Once again, the methods used for achieving this result activated the child's gifts for learning. They enabled the child to connect (there's that word again) the sound "hundred" to the first numbers, the sound "ty" to those in the middle and no sound to those at the end. They also activated the child's gift for labeling things.
Whereas these two lessons carefully and deliberately activate God's gifts for learning in students, tradition (in math education) virtually ignores them. As you may now clearly see, there is an awesome difference in the two outcomes.
Many years ago, a public school teacher told me that my third grade son was a slow learner. I was concerned enough to take complete responsibility for his math education. Three years later, when he was a sixth grader, he was so advanced that his school allowed him to take the New York State Ninth Year Algebra Regents Examination. This exam is traditionally reserved for the brightest ninth graders in the state. He scored 100%. The following year, as a seventh grader, he scored 97% on the New York State Eleventh Year Mathematics Regents Examination (Algebra II and Trigonometry). As an eighth grader, he scored 95% on the New York State Tenth Year Mathematics Examination (Geometry).
Following these achievements by my son, I wanted to see if it was possible to deliver advanced mathematical achievement to disadvantaged fifth and sixth graders by training their teachers. By this time I had become an Associate Professor in the math department at State University of New York, College at Old Westbury, New York and in 1973 I approached Mr. Earl Moseley, then principal of the Theodore Roosevelt Elementary School (in Roosevelt, Long Island, New York), with the proposal to voluntarily train some of his teachers. He replied, "Let me see your stuff." Over the next three years, I trained four of Mr. Moseley's teachers after school every Wednesday in the strategies I had used to activate my son's gifts for the learning of mathematics. Mr. Moseley and all four of his teachers were also voluntary participants over the three years.
The outcome was wonderful. In 1976, 51 fifth and sixth graders, attending the Theodore Roosevelt Elementary School, took the New York State Ninth Year Algebra Regents Examination. Never before, in the history of this nation, had whole classes of elementary school children taken the same state wide math exam, traditionally reserved for the brightest ninth graders in the state! Thirty-four out of fifty-one passed (67%). The lowest score was 40%. Of the selected ninth graders in the same school system who took the same exam, at the same time of day and under the same circumstances, only 25% passed! Their lowest score was less than 20%. Similar results were accomplished in PS44, an elementary school in Bedford-Stuyvesant, Brooklyn, New York over the years 1981 to 1983. Bedford-Stuyvesant is a socially and economically depressed section of New York. The Wicomico County Public School System in Maryland has used the author's methodology over the past eight years. Currently, every class in every grade, kindergarten through sixth grade, uses a textbook at least one year ahead of itself. The author believes that this is also unprecedented (in a public school system) nationwide. These results confirm that there is an awesome difference between outcomes that are a consequence of traditional math education strategies, and those achieved when God's gifts are activated!
We have identified two serious problems in traditional mathematics education:
- the disconnected, fragmented presentation of mathematical content; and
- the experience of math as a collection of meaningless, rote, mechanical recipes.
There is another. Consider all the arithmetical statements made by all your math teachers from first through seventh grade. It will probably surprise you to learn that almost every arithmetical statement made by all those teachers, through all those years, was either untrue or meaningless! Let us look at long division, for example. Consider the problem below:
The vast majority of teachers have been "explaining" this problem by saying, "Four into two, you can't." This is untrue for two reasons:
- that is not two; it is two hundred; and
- lots of fours "go into" two hundred.
Next, they say, "Four into twenty-seven is six," and they write the "six" above the seventy. This is again untrue for two reasons:
- that is not twenty-seven, it is two hundred seventy; and
- that is not six, it is sixty.
Next, they say, "Four times six is twenty-four." Untrue again, because that is not twenty-four, it is two hundred forty. Next, they say, "Twenty-four from twenty-seven is three." Untrue again, because that is not three, it is thirty. Next, they say, "Bring down the five" (next to the three). When young truth seekers asked, "Why do you bring down?" They said, "Just do it. That's how you get the answer." Whereas the other statements are untrue, this one is simply meaningless! The few remaining statements happen to be truthful and are among the relatively rare occasions when you hear truth being spoken in arithmetic. Just think of all the untruths and meaningless statements you heard through all the years you labored over fractions, decimals and percents. Do you remember "invert and multiply," "cancel," "multiply across," "count the decimal places" and lots of other rote mechanical recipes you encountered from grades one through seven? Such meaningless responses to truth seekers' questions do not provide the "ring of truth" that they earnestly seek.
It is important to note that prior to this awful encounter with long division, you may have been taught place value. Those who have attempted to teach third through sixth graders this concept by traditional means will testify that it is rather difficult. So having labored to teach it before long division, a homeschooling parent or teacher virtually forces the child to unlearn it by making all those untrue statements. In fact, if a parent or teacher insists that a child call the 2 in 275 "two hundred" in the past and in the future calls it "two," the child has no choice but to disconnect, rather than make the connections necessary for retention of content. This is the very opposite of how we were designed by the Creator to function as learners.
Now let us tell the truth in the above long division example. Note, as we proceed, how place value is maintained and reinforced. Begin by telling children that this problem asks the question, "How many fours can we 'take out of' two hundred seventy-five?" Explain the work as follows. Because the highest place value in 275 is hundred's let us ask if we can take 100 fours out of 275. Can we? Since the value of 100 fours is 400, we cannot. Because the next place value in 275 is ten's, let us ask if we can take 10 fours out of 275. Can we? Since the value of 10 fours is 40, we can. If we can take 10 fours out, maybe we can take out 20 fours, or 30 fours, or 40 fours, and so on, up to 90 fours. Among these possibilities, what are the maximum fours we can take out in the ten's place? Yes, it is 60 fours (if 6 fours equal 24, then 60 fours equal 240, which is less than 275), because 70 fours (280) is greater than 275. After we take the 60 fours (240) away from 275, we have 35 left. Show the work as follows:
Now that we have taken 60 fours away from 275, can we take some more fours away from what is left (35)? In what place shall we take away some more fours? Yes, in the one's place. How many fours can we take away from 35? Yes, we can take away 8 fours (32) and have a remainder of 3. Since we cannot take any more fours away from what is left, the problem is solved. We have taken 68 fours away from 275 and there is a remainder of 3.
Do you see that all the statements we made, throughout the second way of doing the long division above, were true (and therefore meaningful)? Do you see how the place value concepts, that one may have labored to teach (in the past), have been maintained and reinforced? As a math teacher then, one either reinforces and maintains concepts learned by students in the past by consistently telling truth or one destroys them by telling untruths. There is no gray between the two extremes! By this "truth-telling" approach, seven-year-old children, who have experienced the activation of their (God-given) learning gifts for approximately two years, have been mastering long division. A huge number of statistical analyses and other forms of documentation accumulated over the past twenty-five years attest to the power unleashed for learning by this approach to mathematics education.
The real reason for the huge amount of remediation associated with the teaching of arithmetic is the enormous incidence of untruths and meaningless statements by parents and teachers. Since such statements cause children to disconnect from understandings in previously taught content; it becomes necessary to teach that content over and over again.
Let us return to the example of detectives solving crimes. They gather bits of information (clues) and statements from witnesses. In order to solve the crime, they activate their mental gift for making connections. When all the bits connect up, they can tell you what happened at the crime scene almost as if they had seen it with their own eyes. But if witnesses tell untrue statements to detectives, their gifts might actually become deactivated to the extent that the bits of information do not connect up. In a similar manner, children's gifts for making mathematical connections are continually being deactivated by all the untrue or meaningless statements their teachers consistently make, not only with respect to long division, but also when teaching other operations on whole numbers, fractions, decimals and percents.
Now do you see how disconnection, fragmentation, rote-mechanical recipes, untruths and meaningless statements deactivate God's gifts within you for learning math? Some of you might reasonably ask, why is it then that some children did so well in math (the "math whizzes")? The author is convinced that within this group, there are three categories:
- Those who are so bright that, in spite of all the untruths, meaninglessness and rote recipes, they figure out connections, truths and meanings themselves;
- Those who are blessed with a parent, relative or exceptional teacher who helped them to have the same perceptions (as above) with respect to the content of math; and
- Those who are blessed with exceptional discipline and an unusual capacity for remembering meaningless bits of information.
The child who belongs to the first category is very rare. Children who belong to the third category are often very practical. They know the expediency of rushing through a set of problems using the rote recipes, so they can go and play soccer. They have no pangs of conscience that disturbed some of us when we did some mechanical steps with no understanding, even if we got the right answer. Some of us yearned for understanding. They do not care. Their only concern is that they get it "right." These are the math whizzes in early grades who go on to be rather average (or below) in high school and college math (unless they have been influenced by an exceptional mentor or teacher who helped to amplify this narrow perspective in mathematics).
Through all those years you experienced arithmetic, from first through seventh grade, your mind was being programmed, almost on a daily basis, to memorize meaningless, rote-mechanical recipes for getting right answers. This is why, for virtually all of us, our psychological impression of arithmetic consists of disconnected, fragmented bits. Most authorities in mathematics education would fervently argue to the effect that the purpose of children's arithmetical experience is to prepare them for problem solving. But the psychological impression of solving a relatively challenging problem consists of connecting bits of information (some are available in the statement of the problem and some are available from previous experience). A solution to the problem is a pathway of connections that yields the "answer" or the conclusion. A seven-year exposure to arithmetic, in which children are constantly and consistently bombarded with disconnected fragments, is certainly not a positive experience toward preparing them to be good problem solvers. It is not even a neutral experience. Each year that passes, as children continue to experience this conditioning, they continue to become more and more incapacitated as problem solvers. This is why, as problem solving and understanding become the focus of math in high school and college, many math whizzes become rather mediocre students. Another reason for this decline in math performance is that their overburdened memories eventually arrive at the point of "diminishing returns." Clearly, a conditioning in arithmetic that constantly and consistently permits youngsters to experience it as connected up and flowing is excellent preparation for problem solving.
Reflect, once again on the untruths and meaningless statements associated with long division. You will recall that, as a youngster these "explanations" made no sense. There is one word for explanations that do not convey the ring of truth. It is "nonsense." The stuff your teachers wrote on the board, while they spoke the traditional nonsense, looked seductively mathematical because mathematical symbols were involved. However, any activity that attempts to engage youngsters in meaningless association with mathematical symbols is devoid of mathematics. Consequently, when in exasperation you exclaimed, "I am not good at mathematics!" It was not math you were unable to do. You resisted the meaningless, rote-mechanical imposter! Consequently, instead of being ashamed, may I now urge you to feel relieved about not being good at nonsense? Real mathematical experiences engage your mind in connections, meaning and truth. When you looked at long division in this way, it took you ten minutes to understand it, as compared to not understanding it (even if you could "do" it) through twelve years of public schooling.
Knowing the vital importance of connections for learning, if someone wished to deliberately create a system for teaching math that would guarantee massive fear and failure in learners, how might he accomplish it? Would he not disconnect and fragment its content? Furthermore, if he knew the value of truths for facilitating the perception of connections and retention of content, would he not ensure that the vast majority of "mathematical" statements are either untrue or meaningless? Could he then, for the accomplishment of his ignoble purpose, create a better system than that which currently exists in the school systems of this nation? In fact, if stories were presented to children as disconnected, fragmented bits, they would be no more successful at learning them, than they are at learning math.
Blaise Pascal, a seventeenth century French philosopher, was one of the most eminent mathematicians and physicists of his day and one of the greatest writers in Christian literature. He once wrote, "Truth is so obscure in these times and falsehood so established, that unless we love truth, we cannot know it." The falsehood that only a few brainy people can learn math is so well established that most people believe one needs to be "gifted" to learn it. Hopefully, the reader now understands the truth: we do not need any new gift to excel at math; we only need to activate the same gifts we have used so well for learning language and stories. Virtually all children can achieve excellence in math. Some by activating their mental gifts themselves (these are the rare ones) and most by having their gifts activated by a facilitator. Clearly, if you, as a parent, are such a facilitator, you are in a wonderful position to so influence your children's development, that they become excellent math students. In fact, Pascal, as a result of early tutoring by his father, became one of the best mathematicians of the seventeenth century. On the other hand, if you are not such a facilitator, it might be important for you to contemplate the question: What perspective will have the dominant influence on my children's mathematical development? Will it be that which so appreciates how fearfully and wonderfully made are your children, that it carefully activates and cultivates their gifts for learning? Will it be that which deactivates the gifts by being totally unaware of them (we regret to inform you that your child has little or no aptitude for math)? If you do not intervene, the well-established falsehoods of traditional mathematics education may have the predominant influence. Your intervention with truth is necessary to counteract its effect.
You have read the math lessons that appear in this article. Did you follow them? Did everything make sense to you? Could you sit with your eight-year-old child and cooperatively figure each lesson out so that he or she also understands it? If all of arithmetic was similarly worked out in the form of lessons, where all the recipes were consistently anchored in meaningful mental activity through the activation of your own gifts for learning, do you think you might be able to read it, understand it and facilitate your child's comprehension of it? I am quite sure that the vast majority of those who read this article are able to accomplish these tasks (even if, out of some phobic anxiety, you deny it). My confidence in you is the outcome of my understanding that, as a reader, you have the ability to assimilate connections, meanings and content from a page of written material (assuming the way it is written permits your perception of them). Now if I write carefully, in such a way as to express truth regarding mathematical content (as was done with respect to the three preceding lessons), you will similarly make the mathematical connections, decipher the mathematical meanings and assimilate the mathematical content. You will then become the kind of facilitator who can cultivate excellent mathematical performance by your children, regardless of your own past experience.
The ability to perceive connections and use them as a means of assimilating content is a universal "learning style," because this is how all children learn. You may be aware that we possess visual, auditory and kinesthetic learning styles (modalities) and that some children may favor using one over the others as a means of facilitating the assimilation of meaning and content. Whereas some educators recommend teaching consistently to the strong (favored) modalities, the author believes this will further weaken the weaker modalities. A good teacher is aware of various learning styles and conscientiously activates all of them. Weaker modalities may, consequently, become stronger. It is certainly desirable to cultivate whole rather than handicapped learners.
As Christians we should not underestimate the power of truth. When our Lord said, "The truth shall set you free," he was not talking about math; but, make no mistake about it, the truth in math does set children free from fear and failure. As children of God (The Truth), we should seek truth in all things. We should always think truth, speak truth, value truth and, as Pascal wrote, we should love truth. Satan is "the author of lies" by means of which he sows division, confusion and misery in our lives. As teachers speak untruths, they are unconsciously using the weapons of the Enemy to inflict fear and failure in learners of math.
If God's gifts for learning are within us, should they lie dormant? Does it not glorify Him when we perceive, activate and apply them? In view of Pascal's statement, should we not, as educators of children, teach them how to think for the purpose of discerning truth? Will children not become better thinkers as they practice the deciphering of truth from the connections they diligently perceive in their academic as well as their daily lives? As better thinkers, will they not be more ably equipped to discern between truth and falsehood and, as adults, to perceive the deceitful human agents of principalities and powers; to detect real character and integrity; and, generally, to serve God better?
Let us glorify God, therefore, as we activate His awesome learning gifts throughout our entire lives, as decision makers, problem solvers and most importantly, as seekers of the Kingdom of God, builders of His church and defenders of the faith. Toward this end, it is important to train children to think within their academic subjects by activating His gifts for perceiving connections, deciphering meanings, retaining information, solving problems and making decisions.