Foundation Notation app for iPhone and iPad

Foundation is an important mathematical discovery. It offers a concrete way of introducing Mathematics concepts that could make a considerable difference for all students – including young children before they expand their capability in mathematical abstraction - and high school students having difficulty with mathematical abstraction. Success was demonstrated in an 18 month Grades 1/2 pilot project (see "Unlocking Mathematics for All Students" at http://www.FoundationNotation.com).

In the pilot project Foundation was motivating, engaging and easy to understand and learn. It worked with existing lesson plans and helped prepare students for mathematical exercises on paper. It was a powerful introduction to the physical manipulatives students used as well as the paperwork they did.

For high school students, Nicolas Balacheff (2001) emphasizes the importance of a link between Arithmetic and Algebra as a way for students to deal with abstraction. Because Foundation is a common base for both Arithmetic and Algebra, it provides a strong link. It is a stepping stone that eases the transition from Arithmetic to Algebra and enables mathematical generalization to be better understood in the early years.

Foundation uses a virtual manipulative that is a powerful partner - enabling rules to be demonstrated by counting coloured squares. Movement across the screen is of critical importance and understanding requires this movement to be experienced. The on-line interactive Teacher Guide has short videos followed by an exercise (with instructions on “how”) for this purpose.

Without purchasing the app, teachers can read an off-line version of the Teacher Guide at http://www.FoundationNotation.com and watch the videos to see if they wish to use Foundation. Future versions of Foundation will update the apps with expanded capability.
The focus on notation as the “spine” of Mathematics can help students perceive the subject as a developing science rather than a series of disconnected algorithms. They can also be helped to understand what mathematical expressions are by watching them be both composed and calculated one step at a time.

Foundation provides a solution to other pedagogical problems identified in the literature on the teaching of Algebra. For example, it has been fairly recently realized that it would be highly desirable to teach Mathematics through functions (rather than abstract operators). Functions are the building blocks of Foundation. Since the building blocks of Computer Science are also functions, it provides a simple and elegant way of uniting Mathematics and Computer Science.
Foundation calculates any school mathematical expression directly, making Mathematics notation a computer language. Neither computer programming nor coding is required. As a mathematical expression is built through Foundation’s function composition, the computer builds its calculation program. This makes Foundation a model of school Mathematics, enabling the subject to be taught as scientific experimentation.

The origins of Foundation are to be found in the 1960s, when computer input devices were modified typewriters using only one dimension of the paper. For example, the expression for calculating the hypotenuse of a right angled triangle would be input as something like SQRT((x*2)+y*2). This rendered mathematical computing too tedious and confusing to be of much use for teaching purposes.

When these input problems were overcome with new technology, the capability was used for digitizing, editing and printing Mathematics papers, but not for calculation. This surprising outcome may have originated from the fact that two dimensions were not necessary for the original link between Mathematics and Computer Science - Alan Turing’s first machine design -which was also a modified typewriter. However, school Mathematics pedagogy requires a second link between Arithmetic and Algebra - for which two dimensions are necessary.