Mathematics Teaching-Research Journal (MTRJ) on line:

The editorial team of the Mathematics Teaching Research Journal on line informs with great sadness about passing away of one of its founding editors, VRUNDA PRABHU 1961- 2013. Her creative spirit will always be with us.


Hunt for Aha!Moments in Mathematics Classrooms



The Volume 8

Mathematics Teaching-Research Journal: Two Aha!Moments: Aha!Moment from Korea, Personal communication AND An elephant – or what use can be made of metonymy?



The Creative Enterprise of Mathematics Teaching Research

Fall and Winter 2016/17 Volume 8 N 3-4

List of Content

Two Aha!Moments

Aha!Moment from Korea.
Personal communication

Bronislaw Czarnocha

Aha!Moment from Poland
An elephant – or what use can be made of metonymy?

Celina Kadej, Matematyka #2, 1999

Lynn Columba and Megan Stotz
Shifting Preservice Teacher Beliefs: The Power of the Common Core State Standards in Mathematics

Bronislaw Czarnocha
Teaching-Research NY/City Model, Chapter 1

Terrence Brenner and Juan Lacay
Getting through Calculus without using the Trigonometric functions Cosecant, Secant, and Cotangent

Jarek Stelmark
Using the Analog Imaging Visual Cues to Increase Understanding of Inverse and Direct Square Laws in Digital Imaging


Colleagues, we have two more Aha!Moments submitted to our collection, one from Korea in the geometrical context, and another from Poland during the process of understanding the concept of unknown while learning linear equations. We present them bare, without yet any attempt at interpretation. Soon we will interpret the whole collection to see how Piagetian theories and Koestler theory understand them.

Next we have an interesting article from Lehigh University in Bethlehem, PA which investigates the CCMS  professional development for pre-service teachers’ impact upon teachers beliefs. Unfortunately, Common Core curriculum has been a bit compromised due to the overemphasis on testing and the inability of testing industry to guarantee technological support for that testing. Lynn Columba and Megan Stotz, in their excellent and optimistic presentation, show that such an impact indeed exists. Our own belief is that unless such a PD is closely connected to practice, even for pre-service teachers, it will not leave lasting impression. It is an appropriate moment then to introduce our “refurbished” Teaching-Research/NYCity methodology (TR/NYCity Model), which with the incorporation of Koestler’s bisociativity theory grew in the Chapter 1 of the Creative Enterprise of Mathematics Teaching Research book published recently by Sense Publishers in Netherland announced on the MTRJ website.

As immediate examples of the Teaching-Research/NYCity model we present two papers coming from technical fields at Hostos CC, Mathematics Department and Radiology Unit of the Urban Health Department. Both of them originated through the reflection on teaching mathematics at Hostos CC and propose new approaches based on that reflection, first proposes a method of integrating trigonometric integrals without the use of sec, cosec, and cotan, but solely using sin and cos functions. As the authors, Terry Brenner and Juan Lacay say By concentrating on cosine, sine and tangent rather than all six trigonometric functions, you will attain a better understanding with less clutter in your mind. The second paper, by Jarek Stelmark addresses difficulties in understanding inverse square law by students radiology. He supported the concept of the Inverse Square Law by 3 labs exercise for student showing a very direct connection between the law, the time of exposure to the radiation and involved mathematics. He noted increase of understanding by a pre-test/post-test method.

Rules of the Hunt

As the evidence of the successful hunt we will accept the description of the Aha!Moment that took place among students in the mathematics classroom of the author or the Aha!Moment author experienced while designing and participating in the Hunt or reported by students from outside. Accounts of integrated teacher/student double Aha!Moments will be of distinguished value.

The completed evidence/ the paper to be published in the Vol 8 Wisdom of Teaching-Research: Creativity will include:

  1. Description of the mathematics situation or environment when Aha!Moment took place
  2. Some assessment, craft-based or theoretical, of the depth of learning, which took place as a result of the Aha!Moment.
  3. Reflection upon the bisociative framework within which Aha!Moment took place together with the possible hidden analogy “unearthed” with its help.
  4. Post-Aha!Moment interview with the student will be raise the value of the submission.

Questions, help and submission
To Bronislaw Czarnocha
Editor: Mathematics Teaching-Research Journal
            on line.
bczarnocha@hostos.cuny.edu or bronisuavec2@gmail.com

With this new, although delayed Vol. 8 of MTRJ (Mathematics Teaching-Research Journal on line) we start our eighth year of existence. 8 is the number of wisdom because it’s the symbol of infinity  ∞ turned 90 degrees either direction.
A natural conclusion suggests itself: let’s devote this volume to the Wisdom of Teaching-Research, of Mathematics Teaching-Research. That brings the essential question, where is the wisdom of MTR hidden? In which of its aspects? What is it in our work that brings its wisdom to fore? That is what we want to explore in this Eighth Volume.

For us in the South Bronx the wisdom of MTR is in its theoretically grounded enhancement of creativity of Aha!Moment. Therefore one of issue of Eighth volume will devoted to the creativity of Teaching-Research, possibly expressed through Aha!Moments caught during our work which in the light of Koestler theory of the Act of Creation, should and are appearing while doing teaching-research. They appear amongst the students and amongst the teachers, instructors. The pathway of development of our TR Team of the Bronx has been full of unexpected Aha!Moments. And with good reasons for it.

Balanced Teaching-Research takes place when the craft knowledge of the teacher and research knowledge of the researcher contribute, conceptually, in equal measure to the activity of Teaching-Research. Once this condition is reached, it turns out, with the help of the Koestler  bisociation theory of the Act of Creation (1964), that balanced teaching-research is the creative bisociative framework pregnant with as yet “hidden analogies”.

Koestler definition of bisociative creativity as “a spontaneous flash of insight, which…connects the previously unconnected frames of reference and makes us experience reality at several planes at once ” –an Aha!Moment, formulates  the condition, which we call a “bisociative framework” specially suitable for the facilitation of Aha!Moments: the presence of  previously unconnected frames of reference. Moreover, as Koestler (1964) describes the main mechanism of creativity in terms of  “unearthing hidden analogies” (p. 179) between two or more previously unrelated frames of reference,
we define the bisociative framework as composed of unconnected frames of reference with enhanced possibility of unearthing hidden analogies.

Teaching and Research, essentially and unfortunately unconnected professions, methodologies, goals, yet at the same, Teaching-Research, their bisociative framework time is pregnant with hidden analogies, which can facilitate the creativity of both.

That means that balanced Teaching-Research or TR/NYCity model is the creative bisociative framework ready for Aha!Moments, it is the creative approach to both Teaching and Research.

It means a lot. Teaching-Research gains through bisociation its own intrinsic identity as the bisociative framework composed of previously unconnected frames of reference with enhanced possibility of unearthing hidden analogies. Looking from this perspective, one immediately establishes contact with Stenhouse work who introduced the concept of  “an act [which is] at once an educational act and a research act” – an expression of the bisociativity of teaching-research (Rudduck and Hopkins, 1985).  That single concept allowed to classify the Discovery Method of teaching, the Teaching-Research Interviews and Concept maps methodology as characteristic instruments for Teaching-Research. The same pathway of associations leads to Margaret Eisenhart (1991) formulations of frameworks for inquiry: theoretical, practical, and conceptual. “ A conceptual framework is an argument that the concepts chosen for investigation, and any anticipated relationships among them, will be appropriate and useful given the research problem under investigation. Like theoretical frameworks, conceptual frameworks are based on previous research, but conceptual frameworks are built from an array of current and possibly far ranging sources. The framework used may be based on different theories and various aspects of practitioner knowledge.”(Lester, 2010).Therefore Teaching-Research is a conceptual framework of inquiry, which acquires this way insignias of academic discipline as much as it has acquired the bearings of the craft knowledge discipline. We see here the strength of bisociation as its integrating foundational principle.

So we, Mathematics Teacher-Researchers have quite a lot, a creative methodology, which induces creativity in the classroom. Let’s do it then!

Hunt for Aha!Moments in Mathematics Classrooms

Mathematics Teaching – Research Journal (www.hostos.cuny.edu/mtrj)

  1. Invites submissions describing, analysing moments of creativity in mathematics in general, in mathematics classroom, in particular,  to  its 8th year anniversary volume titled
    Vol 8 Wisdom of Teaching-Research : Creativity.
  1. Announces Hunt for Aha!Moments in Mathematics Classrooms
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