There is much beauty in nature’s clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things. Mathematics is to nature as Sherlock Holmes is to evidence. When presented with a cigar butt, the great fictional detective could deduce the age, profession, and financial state of its owner. Dr. Watson, who was not as sensitive to such matters, could only look on in baffled admiration, until the master revealed his chain of impeccable logic. When presented with the evidence of hexagonal snowflakes, mathematicians can deduce the atomic geometry of ice crystals. If you are a Watson, it is just as baffling a trick, but I want to show you what it is like if you are a Sherlock Holmes. Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern, exceptions suddenly stand out.The numerical tricks we learn in arithmetic are only the tip of an iceberg. However, mathematics is not just about numbers.
Mathematics uses symbols, but it no more is those symbols than music is musical notation or language is strings of letters from an alphabet.Carl Friedrich Gauss, thought by many to be the greatest mathematician ever to have lived, once said (in Latin) that what matters in mathematics is “not notations, but notions”. The pun “non notationes, sed notiones” worked in Latin too.
My own conclusions are that, given we have this wonderful freedom, we don’t have to take a route in math that will cause this math anxiety or dislike in the children or ourselves. This encourages me to keep seeing math in its richness, as more than mere calculation, and to unjustly equate success in math to success in math tests or problems contained in workbooks. Funny, his progression as a teacher it resembles mine in the Language Arts as well. With Charlotte Mason I say, if we do not enjoy and have a passion for what we do, we are not doing it right, we can’t guide or model learning for our children.