Head of Mathematics – Miles Ford

Foremostly I am a teacher of senior mathematics (A, B & C) and middle school mathematics. As the Head of Mathematics I am responsible for the strategic direction and day-to-day management of the department. At St John’s we utilise the Dimensions of Learning pedagogical framework and the International Baccalaureate Middle Years Program (MYP). We have a strong focus on the use of technology within the mathematics environment, including Moodle, Mathletics and Mathematica.

I am particularly interested in computer-based education, especially in mathematics. This is more than providing resources or exercises online as it also involves utilising computers to undertake the computational grunt-work, we use Mathematica, which often makes up the majority of time in mathematics classes. By taking advantage of the computer’s strengths we create the time for greater understanding of the concepts being examined and improve successful problem engagement.

Miles Ford

Using Mathematica for computational learning

For the past five years at St John’s Anglican College we have used Mathematica with our senior mathematics students. We have gradually expanded its use from an initial trial with Year 11 Mathematics B and C classes to the point where it is now used, to varying degrees, across all year levels (7 – 12). Mathematica is a symbolic mathematical computation program, or computer algebra program, developed by Wolfram Research and is available at https://www.wolfram.com/mathematica/

The advantages of using Mathematica on a computer over a graphics calculator include, being able to use a keyboard, mouse and data projector to demonstrate and share work. You can see what you’re doing quite clearly due to screen size and superior graphics quality. There is also the facility to cut and paste graphics and code straight into another application, such as a word processor.

Mathematica also has a superior library of instructions to that of a graphics calculator and the scope to allow students to develop complex solutions to computational problems. Mathematica enables the development of your own resources within its Notebook environment, so both content and the computations occur in the same screen; i.e. the Notebooks are interactive and not just readable documents. Mathematica allows format and style text and images within the Notebook and has executable code ready to go, or functions can be added as users progress.

Students at St John’s use Mathematica across the full range of complexity, from solving basic computations any calculator could do right through to writing their own programs that are capable of modelling and solving more complex real-life situations.

The following task demonstrates a simple modelling application that can be done using Mathematica to determine a line of best fit for a set of data; taken from a larger assignment that examines the relationship between logistic and exponential functions.

Developing a logistic model in Mathematica

The table below shows the height of a sunflower changing with time. In this task we will model this data using a logistic model:

Week

1 2 3 4 5 6 7 8 9 10 11 12
Height (cm) 17.93 36.36 67.76 98.10 131.00 169.50 205.50 228.30 247.10 250.50 253.80 254.50

Source:  http://www.seattlecentral.org/qelp/sets/009/009.html

 

Using Mathematica students are first able to enter the data into a variable, called data:

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The text that follows the “In[#]:=” is what is entered by the user and the “Out[#]:=” is followed by the output that Mathematica produces in response.

The data can then be modelled by any function, as specified by the function FindFit. This example below generates values for the parameters ,  and  with the independent variable being.

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These values can then be substituted into the function and plotted, but one handy feature of Mathematica is that it provides some likely next steps. In this case we can select plot fit from the context menu that appears below the output line, as circled above.

This produces the relevant code to show the line of best fit and the data on the same graph:mathematica1-3

We can also use a different function called NonlinearModelFit if we’re interested in doing some further analysis. In this example the model is stored to a variable simply called model:

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We can present the function as either the equation or the parameters determined:

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We can examine the accuracy of our model to the original data:

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These are three examples of over forty properties that can be examined directly.

We can use these results to then work out other values, such as the percentage errors:

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This takes the “FitResiduals” values of model and divides each by the original value from data, then multiplies by 100. The instruction – data[[All, 2]] – tells Mathematica to get all the values from the second column of data.

In this demonstration I’ve focused primarily on the calculation side of Mathematica, but as mentioned in the beginning it also has graphical and presentation features that can be utilised to produce a more visually pleasing product.

Over the past five years we have gradually increased the use of Mathematica within the College and continue to see it as a fundamental element of our STEM program. As computational thinking and coding become more and more significant aspects of the curriculum we believe that being proficient users of Mathematica well positions our students for their future.