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## Professor B Mathematics Activating the Contextual Learner

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### Harnessing the Power of Language for World Class Mathematics Education

Our article on this website entitled, "You Could Be the Difference-Mathematical Excellence or Mediocrity for Your Children," presented the vital importance of "truth-telling" or meaningful verbalizations in elementary mathematics education. It exposed traditional verbalizations of elementary mathematics education as untruthful and meaningless. We concluded that the word "nonsense", by its very definition, appropriately describes ninety-five percent of elementary and intermediate math teachers' traditional verbalizations on a daily basis.

Truthful verbalizations of the division 2,087÷3, for example, begins by stating that this problem asks a question: How many 3's can you subtract from 2,087? This question makes the contextual connection between long division and subtraction. Take note of the fact that most teachers do make the connection between multiplication and repeated addition, but they rarely go to the logical extension: division is repeated subtraction. Traditionally, the question associated with the above example is "How many 3's 'go into' 2,087?" Unfortunately, for beginners, the words 'go into' do not directly connect them to any effective context. On the other hand, repeated subtraction is so effective contextually, that any child who can subtract will quickly learn to apply it for simple examples like 25÷3. This example will require 8 subtractions of 3 until the remainder is 1. Continuing to function within the context of repeated subtraction, young learners comfortably learn that what took 8 subtractions can be replaced by 2 subtractions, if 3's are subtracted in "bunches" of 4 at a time. Since a bunch of four 3's equal 12, then two subtractions of 12 each time will leave a remainder of 1. Consequently, in two subtractions, eight 3's were subtracted (since four 3's were subtracted in each of the two subtractions). Of course, the answer to this problem can be most concisely determined by subtracting the largest bunch of 3's possible: one bunch of eight 3's or 24.

Do you see how much contextual "mileage" we get out of this one connection (division to repeated subtraction)? Let us now get more mileage using the example 2,087÷3. Young learners with this awareness will waste no time in telling you that doing this example, by repeatedly subtracting three, will take "forever." In fact, they will be quite anxious to subtract the largest possible bunches of 3. Within this context, young learners soon appreciate the effectiveness of using place values in the number 2,087 to suggest the most convenient bunches. Accordingly, they can subtract no thousands of 3. However, they can subtract a maximum of six hundred 3's, leaving 287. From 287, they can subtract a maximum of ninety 3's, leaving 17. Finally, from 17, they can subtract a maximum of five 3's, leaving 2. The total number of 3's they can subtract from 2,087 is 695, with remainder 2. By means of this approach, children develop a favorable disposition toward long division. They like the idea that what long division accomplishes in three subtractions could have taken 695 subtractions. Long division is a short cut!

The "inner voice" (the mental voice that only you can "hear") of youngsters, who have learned this method for long division, talks them through the above example as follows: "How many 3's can I subtract from 2,087?"
1. "One thousand 3's is too large, so I cannot subtract any thousands of 3."
2. "One hundred 3's is small enough, so I can subtract some hundreds of 3."
3. "Hmm, let's see, the most hundreds of 3 I can subtract is 6 hundred 3's."
4. "Six hundred 3's equal 1,800; so subtracting 1,800 from 2,087, I have 287 left."
5. "I can subtract some more 3's from 287."
6. "Ten 3's is small enough, so I can subtract some 3's in the tens' place."
7. "Hmm, let's see, the most 3's I can subtract is ninety 3's."
8. "Ninety 3's equal 270; so subtracting 270 from 287, I have 17 left."
9. "I can subtract five 3's from 17, leaving 2."
These youngsters conclude that the number of 3's they can subtract from 2,087 is 695, with remainder 2.

Clearly, every single step they take, every single statement they make, as they talk themselves through the example, is comfortably perceived as contextually connected to the initial problem: How many 3's can I subtract from 2,087? Since virtually all children are competent contextual learners, this contextual performance of long division has universal appeal.

Now it is your turn to do the problem 2,087÷3. However, you must let your mental voice talk you through it using the verbalizations you learned as a youngster. Do that before you read any further. Have you noticed that even if you got the right answer, your verbalizations were disconnected and meaningless? Did you perceive them as contextually connected to an initial problem? Do you have the same contextual perspective when your mental voice says "Three into two, you can't," as you have when we say, "You cannot subtract any thousands of 3 from 2,087"? Do you have a contextual perspective on "bring down"? This absence of contextual perspective was the reason why long division was more a traumatic, than a comfortable experience during your young and impressionable years.

Truthful, meaningful and contextual verbalizations can similarly be applied to all of the algorithms of arithmetic and algebra; thereby activating, rather than deactivating children's natural and universal aptitudes for learning mathematics.

In view of the fact that elementary and intermediate teachers so consistently make untrue, meaningless and noncontextual verbalizations, any national effort to redeem mathematics education that does not effectively address this problem is doomed to failure.

We have all had the experience of struggling to find accurate verbal expressions for our ideas or for providing explanations. The urgency of such tasks is often due to a need for communicating to other people. A requirement to articulate an idea or explanation forces our mind to clarify its structure. This is certainly a desirable prerequisite (or co-requisite) for its successful communication by verbal or written expression. The degree of difficulty experienced in the performance of these tasks depends on the availability of the structures that the verbalization must express. After hearing the story of Cinderella, a child's task of telling it to someone (without reading it) is rather effortless, due to the availability of its structure (the irresistible connection and flow of it). However, even if young learners perform the relatively simple task of telling a story, we should notice, as educators, that they must immediately mobilize their minds in the effort to recall its structure.

Consequently, anyone who communicates verbally is involved in the exercise of
(a) constructing (or reconstructing) structures in his or her own mind; and
(b) verbally expressing those structures.
The communication is successful to the degree that speakers convey structures within their minds to the minds of their listeners. When this is done well, speakers receive visible evidence of their effectiveness in activating the listeners' gift for assimilating contextually connected information: composed facial expressions, frequent nodding to express concurrence and expressions of delight like, "Ah, I see."

Language, therefore, has the awesome capacity for structuring both the mind of the speaker and the mind of the listener. All mathematics teachers know the requirement that they "stand and deliver" forces them to restructure mathematical content in their minds, so they can explain a topic. This is the reason why virtually all teachers make the claim: "It was not until I started teaching that I understood some math topics I was taught in public school."

However, language has another, equally awesome, capacity. Expressed truthfully and meaningfully, it can structure the mind of the listener with remarkable speed. Notice how rapidly a friend can structure your mind when she tells you about her vacation. You are so rapidly and effectively structured, you can immediately and accurately convey both the events and their sequence to anyone else, without any effort to memorize them. Understandably, if it was a sufficiently long vacation, you might need to clarify its structure by hearing it more than once and asking a few questions, in order to convey its content to another person. However, compared to the huge effort required for memorizing, hearing about the vacation a few times is far more efficient for retaining it. Most of you, who read the meaningful "languagization" of long division above, could do 2,087÷3 the traditional way and get the correct answer. However, you may never have understood, nor have been able to explain, the various "steps" intelligently. The very best you could offer as an explanation for any step was, "Just do it, that's how you get the answer." Now, however, you are a personal witness to the harnessing of the power of language (expressed by means of truthful and meaningful verbalizations) for mathematics education: what you were unable to understand through twelve years of public school, you now understand after a few minutes of reading. How is that for speed?

Another powerful outcome of harnessing language for mathematics education is that, following the transformation of a mathematical topic to an internal contextual experience by means of truthful and meaningful articulation, learners only need to use the same verbalizations to further transform themselves into master teachers of others. The eight year old child, who uses the above verbalizations for teaching long division to a seven year old, is as good as this author (who relies on truth for effective communication with learners) is reputed to be. We can make the same inference concerning any mathematical topic. This has very strong implications for cooperative learning and represents an awesome resource for homeschooling and other educators.

For most educators, language in mathematics education usually represents only the vocabulary or technical expressions they see in textbooks. As you can see, this exposition takes it far beyond the limitations of mere vocabulary.

The wonderful efficiency of truthful and meaningful verbalization of mathematical content for structuring learner's minds will significantly facilitate the efforts of the Constructivists in mathematics education. Their objective is the empowerment of children for structuring their own minds mathematically, by using various manipulative materials. This author is convinced that as children are exposed to meaningfully verbalized math content, day after day, week after week, month after month, year after year, from first through seventh grade and beyond, they will have a far greater capacity to accomplish a major Constructivist objective: verbal or written articulation of mathematical concepts they derive from their manipulative involvement. This author is also convinced that teacher's improved capacity for meaningful verbalization of math content will so improve children's assimilation of math content and accelerate their learning, that the use of manipulatives, although sometimes necessary, will be decreased to some extent.

Significant authorities in mathematics education have recognized that children learn contextually. However, they propose that children should perceive the external contextual relationship of mathematics to daily life as the greatest source of motivation for learning it. This author agrees that applications have motivational value, but they are not the primary source of motivation for young learners. The motive of children for solving a puzzle, learning a story or hopping on one leg is not the applications to daily life! Children solve puzzles for the delight that follows analyzing fragments of information: a burst of inspiration that synthesizes a solution. Children learn a story, not for the moral (the application of the story to their daily lives) of it, but for the joy of experiencing its connection and flow. Analysis, synthesis, connection and flow all involve internal (not external) contextual dynamics. Children hop on one leg because others do it. Their motivation is "I can do it too."

This author contends that untruthful and meaningless mathematical verbalizations hopelessly obscure internal contextual relationships and force hapless learners to exclaim, "What do I need this for? I will never use it in my life." On the other hand, as truthful and meaningful verbalizations reveal internal contextual connections and relationships, young learners delightfully exclaim, "Ah, I see what you mean. It's so easy!" The fact that children "see what you mean" implies that their minds have captured the structure within the mind of the teacher. This type of contextual insight is a source of pleasure to virtually all learners (it actually makes us smile). As such, it represents our primary and most important source of motivation.

Most educators accept the notion that males have a greater aptitude for learning mathematics than females. In fact, there are many statistical analyses that support this conclusion. Is it true?

When this author was a youngster, for some time he was unable to do long division problems. One day, in total exasperation, he exclaimed to a friend, "Stop trying to understand that teacher, just do what he says." Having made this decision, he experienced a breakthrough. He was finally able to get correct answers to long division problems. There was no understanding of what he was doing, but at least he got them right.

His desire for understanding was preventing him from doing the meaningless steps his teacher verbalized. Eventually and unfortunately, however, the pressure of this situation forced him to sacrifice understanding for the performance of a rote mechanical skill. His grades increased. An educational institution rewarded him for the sacrifice of his understanding!

This author believes that boys, as a whole, are far more likely than girls are to sacrifice understanding for rote mechanical performance. When faced with the choice between understanding their math homework and basketball, most boys will take the path of least resistance to the latter: rote mechanical performance of math homework. Who cares? More girls than boys pay a huge price for their tenacious desire and intention to make sense of math. They receive low grades because they refuse to make the sacrifice in favor of rote mechanical performance. As more boys than girls get higher grades in math, the public school systems deliver a majority of boys who are "prepared" for college level mathematics. Hence, traditional math education appears to work better with the psyches of more boys than girls.

Let us take note, at this point, that although most educators claim a higher aptitude for math among boys, they claim a higher aptitude for meaningful verbalization among girls. Who, then, will have the advantage for learning mathematics when teachers' verbalization of mathematics becomes consistently meaningful: the girls or the boys?