Mathematics is no more computation than typing is literature.
John Allen Paulos

Developing Reasoning and Problem Solving Skills Through Math:

Topics below give an idea of general direction I would like to take your child in our exploration of Mathematics. It’s not a quick journey by any means. Our youngest students might just get a taste of some topics on their first year, sometimes in a form of a game or hands-on activity. Returning middle-schoolers, on the other hand, will be challenged to apply the familiar concepts in serious problem solving.

Numeral systems and deeper understanding of base ten notation

  • Cuneiform and Roman numerals (positional vs non-positional numeral systems)
  • addition, multiplication and long division puzzles with same digits replaced by same letters
  • exploding dots (introduction to positional systems other than decimal)
  • operations in other bases, conversion of numbers between bases

Introduction to number theory – Parity

  • even vs odd numbers, definition and easy recognition
  • keeping track of parity in arithmetic operations
  • keeping track of parity during a process with alterations (what side of the fence is a grasshopper after 315 jumps)
  • recognizing parity in longer arithmetic expressions

Introduction to number theory – Divisibility

  • review of divisibility rules by 2, 3, 5, with justification
  • skip counting on a circle predicting result
  • prime factorization, LCM and GCF

Arithmetics

  • order of operations, parenthesis
  • recognizing order of operations in word problems
  • regrouping of long sums for efficiency, introduction to finite series

Word problems

  • review of famous arithmetics problems (heads and legs, birds landing of trees, etc.)
  • organizing data into table for certain logic problems
  • working backwards problems
  • organizing data as a chart (I have twice as much as you:), converting into algebraic expressions
  • date, time, calendar and age in word problems (organizing what’s given)
  • word problems on fractions

Coordinate systems

  • number line: negative numbers (basement floors, piles and pits in a sandbox)
  • number line: what is there between 0 and 1?
  • geometry of graph paper

Geometry

  • counting polygons on a given picture
  • split a shape into identical parts in all possible ways, construct all possible shapes from parts given
  • length, perimeter, when does perimeter stay the same
  • direction and angles, properties of angles of polygons
  • area of certain polygons, area of a part as a fraction of a whole

Solid Geometry

  • review of nets of a cube, introducing nets of other polyhedra
  • building all possible Platonic solids
  • observations on numbers of vertices, edges, sides of polyhedra
  • connections of a 3d shape and its 2d images
  • volume

Various other topics

  • graphs: relationships of “phone lines”, “islands and bridges” as graphs, unicursal lines
  • elements of topology: labyrinths, verifying if a point belongs to closed shape, 3d topology tricks
  • elements of combinatorics: counting paths, permutations
  • logic: true and false statements, opposite statement, “knights and liars” problems
  • strategy games
  • puzzles

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