Math Mind


Finger tracing can lift student performance in maths   …   Teachers have used finger-tracing since the early 1900s, when Montessori got young children to trace over letters of the alphabet made from sandpaper with their index fingers. While this approach was at the time based on intuition rather than evidence, later studies would confirm that finger tracing aids recognition of letters and shapes. 

Yet, researchers are only now starting to explore if finger tracing’s benefits could extend to more complex mathematical tasks that require higher levels of abstract thinking and problem solving.


On The Development of Mathematical Reasoning

I am a highly visual learner.  With a passing glance I can see, through the blur of falling snow, a raven atop a tree in the distance.  I wish that mathematical concepts were always presented just as clearly.

My observation is that really good Math Books are rare, and that poorly written Math Books are common.  I make this statement after having looked at several thousands of Math Books in my hand, or having reviewed them online.

It is my considered opinion that poorly written Math Books are a major reason why so many children and adults have such difficulty with learning Mathematics in our schools and colleges. 

In my own case, I was 28 years old before I learned how to divide.  Until this time I was unable to work with fractions, and of course could not understand Algebra.  I enjoyed Geometry, however, because I could work with visible figures.

For many years I would freeze up from anxiety when I had to do even simple calculations.  I had nightmares about going to Math Exams, and would awaken from these drenched in sweat. 

I came to believe that I was born missing a piece in my brain for doing math, in the same way that some people are born with various other kinds of limitations.   I had no hope of ever being able to understand Math, and I gave up on entering any occupation that required doing math on a regular basis.

Working as a Carpenter is where I first came to understand fractions, after thousands of measurements.  Going to Trade School is where I first came to realize that I have the ability to learn math in general. 

Today I want to understand everything there is to know about Math.  But now I also understand that there is more to Math than any one human can possibly know.  So I content myself with knowing a little bit.

My particular focus is in developing, as Liping Ma says:  “a profound understanding of fundamental mathematics.” 



  • Mathematical Expressions and Language Translations
  • Ambiguous Definitions of Mathematical Symbols and Terms
  • Understood Meaning of Mathematical Symbols and Terms
  • Mental Imagery of Mathematical Concepts
  • Verbalization of Mathematical Expressions
  • Optimum Sequence for Introduction of Concepts
  • Comprehension Differences Between Children and Adults
  • Learning Rates and Age
  • Learning Rates and Learning Styles
  • Learning Rates and Tools for Learning and Using Math
  • Learning Rates and Recreational Math Activity
  • Learning Rates and Instructor Preparation Levels
  • Learning Rates and Institution or Program Types


New Shape Opens ‘Wormhole’ Between Numbers and Geometry


Recommended Resources

When Less is More: Visualizing Basic Inequalities, by Claudi Alsina and Roger B. Nelsen.   Mathematical Association of America (2009).  Hardcover:  181 pages.  ISBN:  9780883853429

Math Made Visual, by Claudi Alsina and Roger B. Nelsen.  Mathematical Association of America (2006).  Hardbound:  190 pages.  ISBN: 0-88385-746-4

Handbook of Mathematical Cognition, by Jamie I. D. Campbell, Editor.  Psychology Press (Nov 29 2004).  Hardcover: 528 pages.  ISBN-10: 1841694118   ISBN-13: 978-1841694115

How the Other Half Thinks: Adventures in Mathematical Reasoning, by  Sherman K. Stein.  McGraw-Hill (August 1, 2002).  ISBN-10:  0071407987   ISBN-13:  978-0071407984

Strength In Numbers: Discovering the Joy and Power of Mathematics in Everyday Life, by Sherman K. Stein.  Wiley (January 29, 1999).  ISBN-10:  0471329746   ISBN-13:  978-0471329749

Number Words and Number Symbols: A Cultural History of Numbers, by Karl Menninger.  Dover Publications (May 5 1992).  ISBN-10: 0486270963   ISBN-13: 978-0486270968

Mathematics: Its Content, Methods and Meaning, by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent’ev.  Dover Publications (July 7, 1999).  ISBN-10: 0486409163    ISBN-13: 978-0486409160  

The Skeleton Key of Mathematics: A Simple Account of Complex Algebraic Theories, by D. E. Littlewood.  Dover Publications (Nov. 11, 2002).  ISBN-10: 0486425436   ISBN-13: 978-0486425436

A Profile of Mathematical Logic, by Howard DeLong.  Dover Publications (Illustrated edition.  June 17, 2004).  ISBN-10: 0486434753   ISBN-13: 978-0486434759

The Book of Numbers, by John Conway and Richard K. Guy.  Copernicus (1996)  ISBN:  0-387-97993-X 

How to Solve It, by George Polya.  Princeton University Press (1988)  ISBN 0-691-02356-5 

Prentice Hall’s Reference to Mathematics, by Cheryl Cleaves and Margie Hobbs.  Pearson Education (2003).  ISBN: 0-13-061800-4 

Access to Science research papers more than one year old.

Adult Numeracy Network

Canadian Mathematical Society    

Collection of Habits of Mind Problems for Math & Logic

Dave Moursund:  On Learning and Teaching Math

Experiential Learning Cycle  

Lists of Mathematical Habits of Mind (Kien Lim)

Mathematical Cognition Centre (mc2)

National Adult Literacy Database:  Adult Learning Theory

Slide Rules and Other Calculating Instruments   

Table of Mathematical Symbols 

Tactile Math Graphics   

Teachers Investigating Adult Literacy 

The Mathematical Brain 

Tessellation, Penrose Tilings and Infinity (Tom Rocks Maths, 14.05.2020)