I have an ambivalent attitude to mathematical software. On the one hand, an enthusiastic user of LaTeX both here and in documents and presentations, but, on the other hand, wary of too much reliance on calculators and computers. OneÂ superb maths teacher I know was criticised for not using a computer in his A level class.Â A Chinese student in the class remarked that they didn’t see much benefit in using one and they had only started to use a calculator when they came to this country to study A levels. They have an important point; I recoil when I ask a simple arithmetic question and the student instinctively reaches for their calculator. Worse still is when they use the calculator on their phone and don’t believe me when I tell them the calculator gets it wrong because it doesn’t obey basic mathematical rules. Similarly, I find graphics calculators are too complicated (computer programs are much easier to use to sketch graphs) and I would prefer to teach students how to sketch curves so that they get a feel for the properties of various functions.
Of course there is a place for calculators and mathematical software. Checking my tax would be a pain without a calculator and graphing software for focusing in at what happens near the origin of the graph of functions like
is fascinating. Similarly, if I am marking student work I use mathematical software to check their matrix operations rather than doing them by hand. Some mathematical software is very powerful and expensive but there are free sites like QuickMath which will solve many problems. However, they should really be used to save time or offer insights after the techniques have been taught and understood – practice, practice and practice is often the best way to learn.
Equation Wizard is a Windows only program that solves real rational equations and simplifies rational expressions (rational means ratios of polynomials). In fact it claims to solve algebraic equations but that is a misnomer as it doesn’t solve equations involving fractional powers or complex coefficients, though it will give some complexÂ roots (for some reason called imaginaryÂ roots in Help). QuickMath does this for free but Equation Wizard’s strength comes from the fact that it will show the working so the user can understand the method behind the solution.
Entry of polynomials is easy using ^ for powers or using buttons or menus or the Ctrl key and the text is previewed in mathematical form as you type, so x^2 becomes and (x^2-1)/2 becomes ; the previewer does its best to interpret ambiguous expressions such as 1/2x. It will add algebraic fractionsÂ showing theÂ working, so if you input 1/(x-1)+1/(x+1) then it is simplified to by adding using a common denominator, multiplying out and collecting the terms in the numerator to get the result, with all steps shown. It uses a similar method to solve the equation 1/(x+1)+1(x+1)=1 finding the answer to 3 decimal places (or up to 9 decimal places if required). There appears to be no limit to the degree of the polynomial equations to be solved – solving was virtually instant.
However, there are limitations. The answers given cannot give exact values so gives 1.414 and -1.414 rather than . Rational equations are solved by multiplying by the denominator but the solutions aren’t checked so gives as one of four solutions. My Norwegian students were taught to always check their answers, so would know what to do. On the other hand solutions can be missed so gives 0 and 0 as the two solutions, though I expect that’s just a bug. However, gives one correct answer and two wrong ones, which is rather worrying. There are language problems with being described as a quadric equation. Quadric surfaces are interesting and there are nice pictures here.
You can print out the calculationsÂ or save themÂ in rtf or HTML format (with the equations saved as images) but it would be nice to be able to copy the selected output to the clipboard. Having solved an equation it’s not possible to change it – you have to enter a new one, though you can copy and paste the old one or use the history button, bizarrely labelled .
Your opinions on Equation Wizard are welcome and I hope ElasticLogic will improve this early version as a result of feedback from you. Can you find other equations with missing or incorrect solutions? The software costs $29 (or Â£15.55 in real money ) and you can download a free trial version here.