Triad of Piaget and Garcia, Fairy Tales and Learning Trajectories- -on the Trail of the Dialectic Triple of Learning.

Bronislaw Czarnocha,

Hostos Community College, CUNY, NYC.

Triples discovered teaching by the Discovery Method

Fairy Tales of the Heroic Quest (For a Hero thrice is enough!) Triad of Piaget and Garcia (PG, 1989)

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From Jenna Hirsch, Jessica Pfeil (2012) On Teaching Logarithms using

the Socratic Pedagogy in Mathematics Teaching-Research Journal online,

vol.5 N 4. (Contains two more examples.)

Review of the domain of 𝑿 + 𝟑 0 The teacher asked the students during the review: “Can all

real values of be used for the domain of the function 𝑿 + 𝟑 ?” 1 Student: “No, negative ’s can not be used.” (The student is confusing here the general rule which states that for the

function 𝑋 only positive-valued can be used as the domain of definition, with the particular application of this rule to

𝑋 + 3.) 2 Teacher: “How about X= -5?” 3 Student: “No good.” 4 Teacher: “How about X = -4 ?” 5 Student: “No good either.” 6 Teacher: “How about X= -3 ? ” 7 Student, after a minute of thought: “It works here.” 8 Teacher: “How about X= -2?” 9 Student: “It works here too.” A moment later she adds:” Those X’s which are smaller than -3 can’t be used here.”

10 Teacher: “How about 𝑋 − 1?” 11 Student, after a minute of thought: “Smaller than 1 can’t be used.”

12 Teacher: “In that case, how about 𝑿 − 𝒂?” 13 Student: “Smaller than a can’t be used.” ACHILLES and TORTOISE

Blackboard Text Log28 = ?

Instructors questions Log base 2 of 8 is equal to what number?

Student responses 4, 28,16, 2

Log216=? Lets’ try another, what is the log base 2 of 16?

4, 32, 14,

etc.

*Log232=?

(at this point most students should catch on). Raise your hand if you think you know what the log base 2 of 32 is equal to?

5

*Log39 =? Raise your hand if you think you know the answer to log base 3 of 9

2

**Logba=? raise your hand if you can tell me how I could rewrite log base a of x equals b using an exponential equation

x

QUOTE 1: Achilles a1-------a2-------a3-------a4-----a5------a6

5miles 5miles 5miles 5miles 5miles Tortoise h1-------h2-------h3-----h4------h5 Let us imagine that the race is 25 miles long with a check point at every 5 miles, so that a1 a2 = 5 miles., h1 h2 = 5 miles, and so on. Let’s also imagine Achilles runs 10 mph and the Tortoise runs 5 mph, since Achilles is faster. Knowing this, we can determine that it will take Achilles 30 minutes to run between each point… Therefore, 30 minutes into the race, Achilles would be at point a2 and the Tortoise would be between h1 and point h2. An hour into the race, Achilles would be at point a3 and the Tortoise would be at point h2. Note a3 = h2. An hour and a half into the race, Achilles would have passed the Tortoise and be at point a4, while the Tortoise is between the point h2 and the point h3. Though Zeno’s conclusion that the quicker will never pass the slower (if given a head start) may have been valid during the fifth century B.C., it does not stand true to today’s experiences. In the first essay that determines students’ spontaneous understanding this student has determined that the race will be won by Achilles. QUOTE 2 As you can see, Zeno’s conditions would never be satisfied because it is impossible for the Tortoise to win the race. Achilles would always win because for every 10 seconds, he increase his position by 100 m. On the other hand, the Tortoise only increases his position by 10m every 10 seconds. Here, the students’ reasoning is not correct inspite of the correct conclusion, since the conditions of the paradox do not imply that measurements are made at equal time intervals. QUOTE 3 The sequence of distances is an increasing sequence and is bounded above since once Achilles passes the Tortoise the race is over. The smallest number greater than every term in the sequence is 11.2 s. The limit of the sequence is the point at which Achilles and the Tortoise intersect. The slope of the line for Achilles is y = 10x and the slope for tortoise is y = x+100. To figure out the limit 10x = x + 1009x = 100x = 100/9x = 11.11 The proof implies Achilles passes the Tortoise. Note that in spite of his use of the word proof, he has not used the geometric or Weierstrassian or Least Upper Bound Axiom (LUB) for the proof. He begins the use of the LUB axiom, however, does not continue that line of reasoning and reverts to the intersection of the two graphs. QUOTE 4 However, the sequence in this particular problem is {…, 111.1, 111.11, 111.111, …}; which can also be written as {…, 111.1}. In situations like this we can implicitly use infinite sums. 111 1/10 + 111 1/100 + 111 1/1000 + 111 1/10000 + … = 111 Observe that as you add more and more terms, the partial sums become closer and closer and closer to 111 1/9. Therefore, we can say 111 1/9 is the smallest number greater than every term of the sequence (and also the limit of the sequence). To prove that 111 1/9 is the limit of the sequence, we can apply the geometric definition of convergence of a sequence. In the last and final essay he is able to provide a proof using the material covered in class and his use of the geometric definition in the text and the drawings illustrate the successful coordination of the A & T generated sequence with the definition of the limit based on the LUB axiom.

The triple of Golrokh One day, he [the Padishah] sat on his throne, flanked on either side by his two vizirs and surrounded by his courtiers and summoned his daughters. Presently they were ushered in, dazzling in their fineries and jewelry, and the audience gasp at their beauty and grace. “The moon had divided into three!”: they whispered. The nightingale would not know which rose to choose.!” And other such complements. The Padishah turned to his eldest daughter and said: “Tell me Shahrock, is it the lining that protects the coat, or the coat that protects the lining?” “The coat protects its lining, Crowned Father” – Shahrock replied. “very well answered” – Said the Shah. “I see that your education has not been in vein. You are thoughtful and discerning, and you deserve a good husband. I will give you to my Right Hand Visir.” Now the Padishah turned to his second daughter “Does the coat protect its lining or vice versa?” The coat protects its lining, Sire, naturally.” Responded Makrokh without hesitation. Well done my dear. Your worthy of no less a good man than my Left Hand Visier. Finally the king called forward Golrokh, the youngest and the loveliest of his daughters, and put the same question to her. “I believe it’s lining protects the coat” answered Golrokh. “Surely you don’t mean that!” Frowned her father. “When it rains or snows, if there is a fight or some other accident, it is the coat that takes the brunt and gets damaged, while the lining But Golrokh was adamant “I still think beloved Crowned Father that it is the lining that supports and safeguards the coat not the other way around.” The Shah said that Golrokh was not only ignorant, but stubborn and opinionated, that her education had been wasted. “Never let me see your face again” he roared, and banished his beloved daughter forever. He ordered his servants to search the city and find the lowest, poorest, most unworthy man, and let him marry and take away his youngest daughter. End of the Fairy Tale: Golrokh laughed and kissed her father, saying: ”Didn’t I tell you, Beloved father, that the lining protects the coat, not the other way around. Woman is the lining, man is the coat, and its is woman that makes the man, not other way around. The Shah gladly admitted that his daughter was right…He ordered the whole town to be illuminated…for Golrokh and Hassan official wedding. The Triple of Hassan, her husband: her lowly husband Hassan, guided by Golrokh, matures through another triple to become the inheritor of his mentor – the merchant Ahmed’s wealth, respect and leadership. He is invited to join the trade caravan to Damascus and Aleppo:

Padishah and his Three Daughters

On the way back to Persia, Hassan trebled his investment while crossing the sea; he divided the profit with his boss. In the mountains Hassan saved the caravan by guiding it to the top of the mountains encampment. Haji Ahmed was astonished by the foresight of his protégé. He wrote his will stating his wish that in the event of his death Hassan would inherit his business, the title of Grand Merchant and all the wealth. That night the black scorpion crawled into Haji Ahmed bed and stung him – within minutes he was dead.

The main effort is to understand how theories can be connected in a successful

manner while respecting their underlying assumptions, which we call

‘networking theories’.

Networking “can be observed by starting with different theories and focusing on

the relationships of their principles, methodologies and paradigmatic questions.

In such a case, there is a possible need for a theory and methodology for

“networking theories”. Working Group #16 Cerme 8

Triad of Piaget and Garcia, Fairy Tales and Learning Trajectories-

-on the Trail of the Dialectic Triple of Learning.

Cerme 8, Antalya, Turkey, Feb. 6 – 10,2013

The Learning Triple has its first two events at the same level of generality while the third one is at its the higher level, or in direct contradiction of the first two. Here are two examples of an interesting triple, for which (*) two first events are on the same level of generality, while (**) the third one is on the higher level of generality than the previous two. The first instance (lines 6-9) - the student reaches first generalization; the second instance (lines 0,10-14) – the student reaches second generalization. One may add that the lines 2-6 also contain a similar triple of teacher’s questions and student’s answers leading to the cognitive conflict in lines 6,7. This way we see here a sequence of triples joined together so that every third of the previous triple is at the same time the first of the next triple. Such a sequential structure can also be encountered within the subclass of fairy tales under discussion. The process is highly similar manner to the conclusion taken by the student after the first two intra-operational events –“ all those X’s which are smaller than -3 are not good here”. Clearly, Ahmed learned after the two initial events and their inter-operational comparison that Hassan is trustworthy and will know what to do while taking care of and developing the bequeathed inheritance. The two triples of Golrokh and of her husband are constructed to keep the fairy tale within their framework, just like two generalization triples

compose into student understanding of the domain for 𝑿 − 𝒂 . The Triad of Piaget and Garcia is a mechanism of thinking leading to concept formation formulated on the basis of the thorough comparative analysis of the development of physical and mathematical ideas in history of science on one hand, and the psychogenetic development of these concepts in a child, on the other (PG,1989). It is defined as the passage through intra-operational, inter-operational and trans-operational stages.

“Intra-operational stages are characterized by intra-operational relations, which manifest themselves in forms that can be isolated” inter-operational stage is “characterized by correspondences and transformations among the forms that can be isolated at previous levels…” “The trans-operational stages are characterized by the evolution of structures whose internal relationships correspond to inter - operational transformations.”

In the elementary situation when only two different individual cases (or two classes of cases) exist, the trans-operational transformation coalesce into one with the inter-operational observation , because in this case the defining structure of the trans-operational transformation is at the same time the structure of the inter-operational one. The trans-operational statement ” Those X’s which are smaller than -3 can’t be used here” is a generalization of the inter-operational observation that “-3” and “-2”, the elements of the domain of the given function, are larger or equal (not smaller than) to -3”. It is critical to understand why such a rather complicated structure as the schema obtained by the Triad construction can have such a simple, three step sequence. The simplicity is due to the fact that we have only two separate instances at the intra-operational level. Had we had instead three separate instances of intra-operational level, we would have to, in general, take into consideration at least 3 different relationships between them in the inter-operational level, whose composition at the trans-operational level would have be more complex than each of them separately. On the other hand, when we have exactly two different instances of intra-operational nature, they have only one relationship (or one class of relationships) between them discovered at the inter-operational level, which by its simplicity and uniqueness leads directly to the trans-operational level through generalization. It is obvious that just one case of intra-operational instance is not enough to give rise to the inter-operational level, and consequently to the trans-operational level. If one accepts the general nature of the argument presented in previous sections, then we are confronted with an interesting but not unfamiliar situation: two significantly different domains of human experience, a mathematics classroom and a fairy tale share similar developmental process, a learning triple. If we apply that learning triple structure to the situation we are confronting then we have to conclude, moving to the trans-operational stage of the argument, that the learning triple is a general process of learning- the dialectic triple of learning. “These triads are much more flexible than are the thesis, antithesis and synthesis of classical dialectics” (PG.p.134) and, possibly they can become the objective basis for learning trajectories in mathematics, among others.

Total T $ Calculations with the unit rate R = $24/hr

Number of hours

1 2 7 N

1.1 How much in total will he make in 1, 2, 7, N hours?

Thinking reflection questions:

Recall the steps of the calculations you made above, thoughtfully look into the

numbers in the table and answer following questions.

1.2 If you know the number N of hours Juan works, how would you calculate

his total pay?...............................................................

1.3 Juan got a rise to $30/hr. If you know the number N of hours Juan works,

how would you calculate his total pay now?.............................

1.4 Now look back into last two problems, compare the steps of calculations and

answer the question: If the total pay is T, $/hr is the rate R, and hours of work

are N, how would you write the correct general formula governing this

problem?.............................

Learning Progression Rates designed w/h of learning triples (a component; 2 sequential triples)

Three concepts: R – unit rate; T-total amount, N – number of units. Strategy #1. (Given R,N. Unknown T) Juan is making $24/hour as a carpenter.