**4.1 Calculus AHA! Moment: mild**

During my Calculus 1, the teacher gave us an example to solve: lim*<sup>X</sup>*�!<sup>0</sup> ffiffiffiffiffiffiffiffi <sup>1</sup>þ*<sup>X</sup>* <sup>p</sup> *<sup>X</sup>* � ffiffiffiffiffiffiffi <sup>1</sup>�*<sup>X</sup>* <sup>p</sup> *<sup>X</sup>* =


I remember in my previews class math 150 when the teacher gave us a rational fraction to solve, he said that we must eliminate the radical in the denominator by multiplicated by the conjugate. But for this equation we do not have radical in the denominator but in the numerator.

I'm a little bit struggling. What can I do?

3. I was looking at the limit and said to myself why not apply the same rule for the fraction when we have the radical in the denominator.

$$\lim\_{X \to 0} \frac{\sqrt{1+X}}{X} - \frac{\sqrt{1-X}}{X} = \lim\_{x \to 0} \left( \frac{(\sqrt{1+x} - \sqrt{1-x})(\sqrt{1+x} + \sqrt{1-x})}{x(\sqrt{1+x} + \sqrt{1-x})} \right) \tag{1}$$

$$=\lim\_{\mathbf{x}\to\mathbf{0}}\left(\frac{\left(\sqrt{\mathbf{1}+\mathbf{x}}\right)^2-\left(\sqrt{\mathbf{1}-\mathbf{x}}\right)^2}{\mathbf{x}\left(\sqrt{\mathbf{1}+\mathbf{x}}+\sqrt{\mathbf{1}-\mathbf{x}}\right)}\right) \tag{2}$$

$$=\lim\_{\mathbf{x}\to\mathbf{0}}\left(\frac{\mathbf{1}+\mathbf{x}-\mathbf{1}+\mathbf{x}}{\mathbf{x}\left(\sqrt{\mathbf{1}+\mathbf{x}}+\sqrt{\mathbf{1}-\mathbf{x}}\right)}\right)\tag{3}$$

$$=\lim\_{\mathbf{x}\to\mathbf{0}}\left(\frac{\mathbf{1}+\mathbf{x}-\mathbf{1}+\mathbf{x}}{\mathbf{x}\left(\sqrt{\mathbf{1}+\mathbf{x}}+\sqrt{\mathbf{1}-\mathbf{x}}\right)}\right)\tag{4}$$

*Bisociation: Creativity of an Aha! Moment DOI: http://dx.doi.org/10.5772/intechopen.110694*

$$=\lim\_{\mathbf{x}\to\mathbf{0}}\left(\frac{2\mathbf{x}}{\varkappa\left(\sqrt{1+\mathbf{x}}+\sqrt{1-\mathbf{x}}\right)}\right)\tag{5}$$

$$=\lim\_{\mathbf{x}\to\mathbf{0}}\left(\frac{2}{\left(\sqrt{1+\mathbf{x}}+\sqrt{1-\mathbf{x}}\right)}\right)\tag{6}$$
 
$$=\mathbf{1}$$

4.Now the limit is defined. I can solve it and finish.

#### *Analysis*

The student reports the experience of an Aha! Moment in the Calculus class while solving a limit problem. The content of the student insight is the discovery of the analogy (hidden analogy) between the algebraic expression involved in the limit problem with the algebraic expression she learned in the previous algebra class topic on rationalization of algebraic fractions. The similarity suggests to her the method of conjugates used in the rationalization case as the method of solution for the limit problem.

Thus, this single-connection schema element is constructed from two very close yet separate (in the student's mind) representations of the pre-conjugated algebraic expression: one positioned in the denominator of the fraction, and the other in the numerator of the fraction. By applying conjugates method the student learned in the past for the rationalization purposes to the calculation of the limit she has understood (and verbalized) that the application of the method does not depend on the position of the pre-conjugated algebraic expression within a fraction.

We assess the DoK of this insight as Mild for two reasons: (1) the increase of understanding was of just one new connection and (2) the two unconnected initially frames of reference were conceptually very close to each other. This classification is based on the local nature of the search process leading to the moment of insight. This moment of insight takes place in second level of P&G Triad, as the moment of bisociation connects two existing matrices one introduced in calculus the other earlier in algebra. Finally, the concept formed was a new understanding of the limit process.

This moment of insight occurred as the student reflected on one scheme she was learning in calculus class M1 with another she had learned previously in college algebra course M2. Thus, it was mostly constructivist in nature, or a moment of untutored bisociation as Koestler would phrase it. However, it contains distinct relationships to insights within the internalization process, as the student begins with replicating a matrix M1 modeled in class: internalization.

#### **4.2 Fir tree AHA! Moment: normal**

Consider the following function that generates the geometric pattern of a reverse growing fir tree (**Table 1**):

Instruction: Draw and describe the Stage 5 of the patterns in terms of its shape and the number of unit squares.

ANSWERS of the student


*The pattern is growing by increasing 1 unit square to the left, 1 unit square to the left, one additional unit in the center and 1 unit down compared to stage 4*.

3.How many unit squares are needed to build Stage 10 of the fir tree?

Answer #3. It needs 110 unit squares. I developed this answer from [the discussion] next page

4.Given any stage **n** determine the close form of equation to determine the number of unit squares needed to build the tree.

*The formula is: n(n + 1).*

*Any stage number multiply by its following number equals the number of unit square, for instance: stage number: 3 multiplied by its following number meaning 4 equals 12 (unit squares.)*

5.Your mate tells you that exactly 274 unit squares make up a fir tree. He is wrong. Explain to him why he is wrong.

Answer #5. *It is really wrong because:* *Bisociation: Creativity of an Aha! Moment DOI: http://dx.doi.org/10.5772/intechopen.110694*

*Stage 16 = 16(16 + 1) = 272. Stage 17 = 17(17 + 1) = 306. The fir tree would never have 274 unit squares.*

*I had a tremendous Aha! Moment. I just realized that the formula I got from the patterns was a factorized expression and if I multiply it "n(n+1)= units square" I would have something like an algebraic expression exactly a trinomial expression that can be factorized as well, and it equals real numbers for example: n(n+1)= 12 n^2+n= 12 n^2+n-12=0 n^2+n-20=0 n^2+n-56=0 n^2+n-90 (n+4)(n-3)=0 (n+5)(n-4)=0 (n+8)(n-7)=0 (n+10)(n-9) And when it comes to n^2+n-274 it cannot be factorized.*

#### *Analysis*

The Aha! Moment took place after the student solved the assigned problem. From that solution, we know about the student's cognitive capabilities: She obtained the general formula using the variable n by generalization from the table of results for each term. She solved the question 5 by trial and error and demonstrated from the logic of these calculations that proposed result is incorrect. These are the components of the knowledge schema before the insight. The insight provided new conceptual solution to the same problem (**Table 2**).

It connected the factorized quadratic algebraic expression, a binomial with the corresponding quadratic trinomial by incorporating data of the problem. The second constructed connection is between quadratic trinomial and solutions of quadratic equation via its factorization. It is interesting that both connections are made through the process of factorization, and the concepts they connect are close to each other conceptually. However, the concept of quadratic equation is significantly further apart conceptually from factorization of trinomials, so much that the student does not recognize it within the mathematical situation. Yet the fact that student recognizes that constant terms in the binomials obtained through factorization are the solutions of unrecognized quadratic equation and have bearing upon the solution of the whole given problem indicates a larger conceptual distance within the second connection of the constructed schema. What also adds to the depth of the new schema is that from


#### **Table 2.**

*Numerical relationships discovered at each stage of the fir tree Aha! Moment.*

the language used it is clear the student (despite some holes in the overall schema of the situation) has a control over the "gestalt" of the schema—the student owns it that is that it has been interiorized as well as internalized, with evidence for both contained in the student's description. The evidence of internalization is in the ability to explain the logical connections verbally, and interiorization is evidenced by the ability to provide the second solution, hence independent action upon the problem. We assess this progress of understanding at the second level of schema construction and name it as normal.

#### **4.3 The domain AHA! Moment: strong**

The problem starts with the function f(x)= ffiffiffiffiffiffiffiffiffiffiffiffi *<sup>X</sup>* <sup>þ</sup> <sup>3</sup> <sup>p</sup> . The teacher asked the students during the review: "Can all real values of x be used for the domain of the function f(x) = ffiffiffiffiffiffiffiffiffiffiffiffi *<sup>X</sup>* <sup>þ</sup> <sup>3</sup> <sup>p</sup> ?"


Student, after a minute of thought:


A moment later the student adds:


Student, after a minute of thought:


#### *Analysis*

This Aha! Moment clarifies student misconception concerning the domain of the ffiffiffiffiffiffiffiffiffiffiffiffi *<sup>X</sup>* <sup>þ</sup> <sup>3</sup> <sup>p</sup> . The original and habitual student answer in line 1 represents student misconception. The intent of the teacher's question was to direct the student's attention to the contradiction of her answer with the situation at hand, what led the student into cognitive conflict. The Aha! Moment takes place as a resolution of that conflict. The student's short reflection on the previous verbal interaction results in the correct reorganization of her approach.

Note that student's insight engages in her the domain of the function f(x)= ffiffiffiffiffiffiffiffiffiffiffiffi *<sup>X</sup>* <sup>þ</sup> <sup>3</sup> <sup>p</sup> and order on the real number line. The bisociative frame of this bisociation were the axes of the cognitive conflict: the habit of well-established misconception and the data brought to the student's attention by focusing her attention on the relevant details. The conceptual distance between these components is not very large – just the change of the parameter but to traverse it the student had to engage her schema of addition for integers, which makes student thinking a bit more complex and places this insight on the second level.

Her language is straightforward, does not convey any doubts and places understanding on Normal level of DoK. However, follow-up questions of the instructor reveal deeper levels of understanding. For instance, the student can easily transfer her understanding to a different related example (line 11). More important, she can abstract and generalize the answer for arbitrary parameter a (line 14). This is the second-order reflection on the "family type" expressions, which leads to abstraction with generalization of the -trans stage. That indicates significantly larger conceptual distance between the components of the bisociative frame and places the Domain Aha! Moment on the third Strong level.
