*Math apps for tablet devices like the iPad should teach you how to structure a problem and its solution so you become better organized and capable of learning more advanced topics.*

As I continue to ponder math software for tablet devices like the Apple iPad, I keeping coming back to the core problem of what the user interface (UI) should be. For the record, I’m not thinking about early education math apps, so no counting leaping frogs or dancing princesses in my UI.

Rather, I’m thinking about the type of homework my 8th grade son does in math, with a continuation on up into high school and college. The work is not just computation with numbers and symbols, but can also include graphs. It needs to allow text so the student can explain what it going on and what the solution is.

For many years, students have used very sophisticated calculators from TI and others. Therefore, some apps try to emulate that UI model.

Classical computer algebra systems looked much like operating system console windows where the user would type in an expression or command, press enter/return, and the result would be displayed. So the process is: read an expression, evaluate it, print the result. Repeat. And so on.

Systems like this eventually evolved into ones where popup windows contained graphs and text could be included. The most sophisticated systems included notebook interfaces where all sorts of information could be placed on pages organized into sections. One page might look like a piece of graph paper while another could resemble a legal pad.

There are also dependencies among computations and graphs. For example, I might say “take the result from step 3, square it, and add 2″. You could also build a spreadsheet-like model where these dependencies are made much more explicit and the functions available go far beyond the floating point functions most spreadsheets provide.

So my model of the ideal math UI for a tablet is closest to a notebook that allows easy and flexible formatting of text, computations, graphs, and tables. Also, the math output should be displayed beautifully with subscripts, superscripts, integral signs, matrices, and so forth. If you know TeX, that’s what I mean.

You need to have some way of saving the contents to disk or for export, and be able to load documents that you or someone else worked on previously. Here you might be a teacher and a “someone else” might be one of your students. So the interface should allow the sharing of documents with others. For extra credit, tie this into a learning management system like Moodle or Sakai.

So we can think of lots of functionality this fancy interface could or should have, but the primary purpose is to help students or those doing the work structure and organize what they are doing and how they are doing it. It’s to get the problem solved but also to learn a logical way of doing it.

Math homework should not look like chicken tracks on a page. Those pages have lines on them for a reason. Text and lettering should be neat. Students should be able to explain in English (or their own language) what is going on at each step. The environment should help and encourage the student to answer the question that is asked. Where units like feet, quarts, meters, or km/sec^{2} are needed, the UI and system should allow them and enforce their correct use in computation.

Math is not about solving one particular problem enough to get credit and then moving on to something else. It is about learning how to think and recognize patterns. There are computational skills involved and those must be mastered, perhaps with help from calculators or apps, but the goal is getting to the moment of clarity where a problem and its solution makes perfect sense. The system should not do all the work for you, but it should train you in the techniques and how to avoid stupid mistakes.

Different UIs may be better for different people, but the best ones help you master the material in an organized way and enable you to synthesize the new material with what you already know. This then becomes part of the foundation on which you will layer more math, physics, engineering, economics, or perhaps just life.

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