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: Wisdom of Teaching-Research: Creativity.



Fall and Winter 2015/16
Volume 8 N 1-2

List of Content

Wisdom of Teaching Research: Creativity of Aha! Moments

Bukurie Gjoci
What if it were you?

Brian Evans
Breakthrough Moments in Problem Solving

Hannes Stoppe
Creativity ≠ Creativity

Csaba Csicos and Janos Steklas
Phases of a ten-year old studentís solution process of an insight problem as revealed by eye-tracking methodology

Teaching Research

Victor Freiman and Mark Applebaum
Engaging elementary school students in mathematical reasoning using investigations: Example of a Bachet strategy game

Dieter Scott
Mathematical Curriculum, Mathematical Competences and Critical Thinking

Sergiy Klymchuk and Zlatko Jovanosky
Analysing University Students' Abilities in Making Assumptions in a Balistics Model: A Case Study

Murray Black
Teaching and Assessment of Statistics to Employees in the NZ State Sector

Reports from the Field

Rohitha Goonatilake, Katie D. Lewis, Runchang Lin, and Celeste E. Kidd
Improving Gender Disparity in Scholarship Programs for Secondary-Level Mathematics Teachers

Kathleen Doyle
Throwback Thursday


The present Vol. 8, N 1-2 double issue of MTRJ contains three components:

  1. Wisdom of Teaching Research first collection of contributions discussing emergence of Aha!Moments in the classrooms among students and teachers. We will accept these contributions for each issue of the Volume 8, and at the end of the year, in the Winter 2016/2017 we will publish their full collection.
  2. The Teaching-Research which has four contributions addressing different methods and approaches to the development of mathematical reasoning and its application in “real life”.
  3. Reports from the Field with two interesting contributions:
    1. The report of the NSF supported project from Texas A & M International University informing about increasing female participation in mathematics careers.
    2. Throw Back Thursday is a new mtrj idea for accomplished mathematics educators to look back upon their thinking in graduate school and offering new comments from their professional experience

Editors of MTRJ are excited to have received four response to the call for the Hunt for Aha!Moments in mathematics classrooms announced recently. We start with the observations of the graduate student at the Teachers’ College, Bukurie Gjoci, who had experienced Aha!Moment as a teacher of remedial mathematics in a community college. She introduced a new approach to Word Problem Solving, which had turned the class on its head. To every problem situation she asked “What if it was you?” (in this situation). It is known that the personalization of a mathematical situation eases students into appreciation of the subject (Prabhu, 2016) and Bukurie Gjoci method is a next confirmation of that knowledge. The contribution by Brian Evans describes facilitation of Aha!Moments among student-teachers with the subsequent analysis of the components that led to such a moment of understand. This analysis, an example of which is Koestler’s based bisociative framework with its hidden analogies, is critical to enable teachers’ design of instruction which may lead to Aha!Moment. Hannes Stoppel, on the other hand, analyzes his students’ views on the role of creativity in problem solving. He demonstrates that students’ understanding of creativity depends on the type of problem they are engaged to solve.

The last paper in the series of Aha!Moments looks upon this phenomenon through the physiological angle by observing problem solver’s eye movements. The method can be quite good in precisely determining the timing of the Aha!Moment.

The Teaching-Research collection of reports focuses on the development of mathematical reasoning in different context and its impact upon reasoning in “real life”. Freiman and Applebaum report the high level of student engagement in mathematics in the context of strategy games. In particular they are interested in Bachet games, different versions of which are quite popular. They ask a standard Teaching-Research question:

  1. What is effective methodology to introduce students to the strategy game?
  2. What are the student methods of reasoning emerging during the implementation of A).

Dieter Shott, on the other hand investigates the transfer of reasoning from mathematics into every day’s life to “expose dubious arguments and interests”. His arguments lead to modelling of “real life” situations easing us into the theme of the next paper by Klymchuk and Javonoski, who are interested in the methods of student argumentations in the context of ballistic models.

The section culminates with Murray Black’s describing the knowledge assessment among the New Zealand State employees.

The Reports from the Field contain the report on the NSF-based project in Texas A&M. The interest of Goonatilake et al is in addressing the female participation in mathematics. They assert they have the approach but its successful implementation requires strong political will. The last short report of Doyle compares her contemporary experience with her own thinking as a graduate student of Teachers’ College. An important question which emerges from Doyle’s Throw Back is the role of Discovery method in teaching, especially in teaching remedial mathematics. We hope to come back to this question in the near future in mTRJ on line.

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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