Aha!Moment Research LibraryHostos Teaching Research Team


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?


Teaching-Research NYC Model


The Creative Enterprise of Mathematics Teaching Research

Summer 2017 Volume 9 N 1-2

List of Content

Section: Teaching Practice and Action Research

Rethinking Instruction of the Tangent Line of a Circle
Jae Ki Lee and Susan Licwinko, Borough of Manhattan Community College

Active Learning in Developmental Classes of Mathematics
Malgorzata Marciniak, LaGuardia Community College

Section: Teaching-Research

Assessment of the Depth of Knowledge (DoK) Acquired During the Aha!Moment Insight
Bronislaw Czarnocha, William Baker, Hostos Community College

Appendix. Collection of Aha!Moments collected during the Teaching-Experiment

Section: Notes from the Field

Minutes from the Meeting with a Student-Researcher
Malgorzata Marciniak LaGuardia Community College


The Summer’17 issue of MTRJ brings reports from Community Colleges of CUNY, from BMCC, LCC,HCC. In the future issues of MTRJ, we are planning to focus on the faculty of CUNY’s community colleges as a very unique faculty amongst the professors of community colleges nationwide. Only approximately 8-14 % of college faculty have PhD degrees in their disciplines as accordingly to AMATYC, MA or MS degrees are sufficient for the job. However ourselves, cc faculty of CUNY, in the majority we have PhD’s in our disciplines, hence we know what is research, the research skills and methods. On the other hand, addressing myself to mathematics faculty, we also have a very deep knowledge of teaching elementary, high school and college, courses of mathematics. Together, when integrated and synthesized can create a powerful, creative method of doing research on learning while teaching that each of us can develop. Together we can create a new and unique domain of knowledge and practice based on our unique experience Teaching-Research.

We present two Action Research investigations of familiar to all of us issues, construction of the tangent to the circle and the active learning. Jae Ki Lee and Susan Licwinko from BMCC point our attention towards two different methods to obtain the equation of the tangent to the circle from a distant point, through implicit differentiation and through geometric construction. They report students’ appreciation for the geometric method as much simpler. Malgorzata Marciniak from LCC discusses her approach to learning based on the worksheets, which include corrections made on the basis of student comments and efforts. She is discovering  the  role  of  classroom  iterations  of  the  intervention  with  the  refinements  in between – the basic unit of teaching-research focused on the improvement of learning. She also is discovering common problem for our students that is absence of attention to connections between different examples, different concepts.


Broni Czarnocha from HCC presents the results of the pilot teaching-experiment focused on facilitation and assessment of the Depth of Knowledge reached of Aha!Moments in mathematics classroom as the beginning of the formulation of Aha! Pedagogy. The work will be presented at the Conference of Teaching Research for All Students (CTRAS 9) in Dalian, China. He points out that the standard tools of assessing Depth of Knowledge on the basis of instruments derived from Bloom’s taxonomy are essentially insufficient in their static descriptions of the levels. We need the ability to assess “pre” and “post” knowledge reached during the insight; it involves “jumping” between different levels rather than reaching one particular level. An interesting component of his teaching experiment was the participation of student-researchers, who together with the author participated in the facilitation process of Aha!Moments as peer mentors in the classroom. Work with student-researchers connects his work with the Notes from the Field, where Malgorzata Marciniak presents notes from her Teaching-Research diary on starting the work with student-researcher in ideals of commutative algebras. The surge in the student-researcher concept present interesting and challenging idea in terms of close TR investigations concerning the process of learning in this environment.

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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