Category Archives: Object oriented concepts

Fractions as operators to fractions as numbers

Steps from fractions as operators to fractions as numbers. Natural numbers are developed with all their logical structure including successor function and functions as ordered pairs.  Students are made familiar with proofs involving functions as ordered pairs including the recursion … Continue reading

Posted in Fractions as Operators, Number classes, Object Oriented Rational Number | Leave a comment

Fractions same or equal?

Is the fraction 2/4 equal to or the same as 1/2? 2/4 and 1/2 are Equal but not the same. Equal and the same. 2/4 and 1/2 are Equal and different Equal and different. Equal and not different. 2/4 and … Continue reading

Posted in Difficulties Learning Fractions, Fractions, Fractions same or equal, Object Oriented Rational Number | Leave a comment

Common Core Standards Grade 4 Number and Operations Fractions

Given a denominator n, such as 8, we can consider fractions as having a successor type relation.  So (m,n)’ = (m’,n) where ‘ is successor, so m+1 = m’. So (3,8) = (2′,8) = (2,8)’ = (1′,8)’ = (1,8)” = … Continue reading

Posted in Common Core Number and Operations Grade 4, Common Core Standards Grade 4, Common Core Standards Math, Common Core Standards Number and Operations, Object Oriented Rational Number, Uncategorized | Leave a comment

How to teach fractions

We should teach fractions by letting the whole truth hang out.  We teach that fractions were developed before Euclid to deal with practical problems.   These are similar to what is taught in school. Then Euclid tried to come up with … Continue reading

Posted in Difficulties Learning Fractions, Fractions, Number classes, Object Oriented Rational Number | Leave a comment

The dead hand of Euclid on teaching fractions

The history of the Euclid concept of proportions as taught in Euclid’s Elements is sketched below.  This is based on sources cited in the previous post and additional references below. Late ancient world they couldn’t follow Euclid on proportions so … Continue reading

Posted in Center for Improving Learning of Fractions, Difficulties Learning Fractions, Euclid's Elements, Euclis's Concepts of Number, Fractions, Object Oriented Rational Number, Oxford Department of Education, Proportion and Magnitude, Uncategorized, University Education Departments | 2 Comments

Teach Explicitly and Distinctly Algebra on Naturals and Rationals

Algebra is typically taught using a letter x with no distinction as to whether x is natural, rational or real.   Keith Devlin illustrates this thinking by advocating teaching the field axioms instead of the progression from naturals to rationals to … Continue reading

Posted in Algebra functions restricted to naturals, Algebra functions restricted to rationals, Algebra on Natural Numbers., Algebra on Rational Numbers, Algebraic slope and rational numbers, Functions and ordered pairs, Object Oriented Rational Number, Slope concept on rationals, Slope Concepts | Leave a comment

Fractions first then definition of length

Suppose we have defined fractions algebraically and worked out their rules including for adding fractions with different denominators.  Now we are ready to apply this algebraic construct to length.   We consider a function from point sets on the fraction number … Continue reading

Posted in Confusing area and rational numbers, Confusing length and rational numbers, Number classes, Object Oriented Rational Number, PCK dependence on math foundations | Leave a comment