*Will the next generation of tablets lead to new and better user interfaces for mathematical software? Will this encourage the use of tablets in education?*

Back in the 80s and 90s I was a member of the Mathematical Sciences Department in IBM Research in Yorktown Heights, NY. Under the leadership of Dick Jenks and with significant contributions from many internationally known computer algebra scientists and mathematicians, we produced mathematical software called *Axiom*. While it ultimately could not compete with more commercial offerings like *Maple* and *Mathematica*, it contained state of the art computer language features that only otherwise appeared much later in languages like Java, Python, and Ruby.

Mathematical software like *Axiom* is not the same as a spreadsheet. Rather than compute extensively with decimals and produce charts, so called *symbolic mathematical software* factors huge integers, manipulates and factors polynomials, computes derivatives and integrals, simplifies trigonometric expressions, and performs arithmetic on matrices. These systems do much more and have extensive libraries for computation. *Axiom* also allows you to define categories like `UniqueFactorizationDomain` and then instantiate it into types like univariate polynomials in *x* whose coefficients are rational numbers.

Very fancy stuff. *Axiom* has the capability to express and solve some very difficult problems. This power comes at a price, however, and we spent a lot of time trying to make the system do simple problems easily and quickly. *Maple* and *Mathematica* certainly did and do better in that regard.

There are many open source systems out there of varying capabilities that do symbolic mathematics. For example, *Sage* rolls up a lot of different math software via Python. Maxima is an open source fork of one of the grandaddies of the computer algebra world, Macsyma, originally developed at MIT.

While any one system may do some things particularly well, it is *Maple* and *Mathematica* that have the bulk of the marketshare and the breadth of related products, not to mention polish and consistency. These are desktop programs for the most part, though Wolfram|Alpha does let you do some mathematical computations via its web interface.

What will mathematical software look like when the primary device being used is a tablet like the iPad or one of the upcoming Linux-based machines? Will we be using Wolfram|Alpha to solve all our problems in a browser? I doubt it.

First of all, we should not think of simply replicating the screen-based interfaces for the tablets. If I want a screen and mouse, I’ll use a laptop. What is different about the user interfaces of the tablets that will change how we do math?

Much of the focus will naturally be on education. Imagine this scenario: a teacher in a 7th grade math class tells her students “take out your tablets and compute the height of the tree given the values shown in the picture.” What would the students do?

First of all, the picture should be live in the sense that the numbers representing the lengths of various lines are not just dots on the screen but values that have units associated with them, e.g., 7 meters. Students should be able to open a calculation area, drag the numbers out of the picture, create expressions and equations, drag and drop values with their fingers from one side of an equals sign to the other, and document how the problem was solved. All on a tablet, in an intuitive way.

An economist reading a paper in a journal comes across an expression that he needs for some work. He selects the equation, drags it with his finger to his working notebook on the tablet, and then plugs in some numbers. To get to the tiny exponents he uses his fingers to squeeze open (zoom) the expression, and then shrinks it again when he is finished. When he is done, he can save the result or send it to a colleague.

Yes, I know this sounds a lot like notebooks for you *Mathematica* users. Those are primarily for desktop or laptop computers and certainly don’t use the native gestures or user interface of the iPad or those tablets that have yet to be introduced.

Will we just see a port of existing heavy duty math software described above or will we see brand new entries that rethink how problems should be solved? Probably, eventually, both, but I’m rooting for the latter since I want to see some innovation in this space. Yes porting over twenty year old technology will work for some people, but that’s not paving the way for radically new educational or problem solving software.

I want to be able to hand a tablet to a 12 year old and have the math software be so intuitive that he or she could be solving real problems in minutes. (Yes, I know there are serious questions about when computers or calculators should be used versus direct manual calculation, but that’s not a point I am trying to debate in this entry.)

What do you think? Will we see more of the same or radically new ways of doing math on tablets like the iPad? There are already fairly simple math programs for the iPhone and hence iPad, but what will the first really serious and different math applications look like and how will they take advantage of the new devices?

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