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I'm back from presenting work in the "23rd Conference on Applications of Computer Algebra - 2017" . It was a very interesting event. This first presentation, about "Active Learning in High-School Mathematics using Interactive Interfaces", describes a project I started working 23 years ago, which I believe will be part of the future in one or another form. This is work actually not related to my work at Maplesoft.

At the end, there is a link to the presentation worksheet, with which one could open the sections and reproduce the presentation examples.
 

 

Active learning in High-School mathematics using Interactive Interfaces

 

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Abstract:


The key idea in this project is to learn through exploration using a web of user-friendly Highly Interactive Graphical Interfaces (HIGI). The HIGIs, structured as trees of interlinked windows, present concepts using a minimal amount of text while maximizing the possibility of visual and analytic exploration. These interfaces run computer algebra software in the background. Assessment tools are integrated into the learning experience within the general conceptual map, the Navigator. This Navigator offers students self-assessment tools and full access to the logical sequencing of course concepts, helping them to identify any gaps in their knowledge and to launch the corresponding learning interfaces. An interactive online set of HIGIS of this kind can be used at school, at home, in distance education, and both individually and in a group.

 

 

Computer algebra interfaces for High-School students of "Colegio de Aplicação"  (UERJ/1994)

   

Motivation

 

 

When we are the average high-school student facing mathematics, we tend to feel

 

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Bored, fragmentarily taking notes, listening to a teacher for 50 or more minutes

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Anguished because we do not understand some math topics (too many gaps accumulated)

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Powerless because we don't know what to do to understand (don't have any instant-tutor to ask questions and without being judged for having accumulated gaps)

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Stressed by the upcoming exams where the lack of understanding may become evident

 

Computer algebra environments can help in addressing these issues.

 

 

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Be as active as it can get while learning at our own pace.

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Explore at high speed and without feeling judged. There is space for curiosity with no computational cost.

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Feel empowered by success. That leads to understanding.

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Possibility for making of learning a social experience.

 

Interactive interfaces

 

 

 

Interactive interfaces do not replace the teacher - human learning is an emotional process. A good teacher leading good active learning is a positive experience a student will never forget

 

 

Not every computer interface is a valuable resource, at all. It is the set of pedagogical ideas implemented that makes an interface valuable (the same happens with textbooks)

 

 

A course on high school mathematics using interactive interfaces - the Edukanet project

 

 

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Brazilian and Canadian students/programmers were invited to participate - 7 people worked in the project.

 

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Some funding provided by the Brazilian Research agency CNPq.

Tasks:

-Develop a framework to develop the interfaces covering the last 3 years of high school mathematics (following the main math textbook used in public schools in Brazil)

- Design documents for the interfaces according to given pedagogical guidelines.

- Create prototypes of Interactive interfaces, running Maple on background, according to design document and specified layout (allow for everybody's input/changes).

 

The pedagogical guidelines for interactive interfaces

   

The Math-contents design documents for each chapter

 

Example: complex numbers

   

Each math topic:  a interactive interrelated interfaces (windows)

 

 

For each topic of high-school mathematics (chapter of a textbook), develop a tree of interactive interfaces (applets) related to the topic (main) and subtopics

 

Example: Functions

 

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Main window

 

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Analysis window

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Parity window

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Visualization of function's parity

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Step-by-Step solution window

The Navigator: a window with a tile per math topic

 

 

 

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Click the topic-tile to launch a smaller window, topic-specific, map of interrelated sub-topic tiles, that indicates the logical sequence for the sub-topics, and from where one could launch the corresponding sub-topic interactive interface.

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This topic-specific smaller window allows for identifying the pre-requisites and gaps in understanding, launching the corresponding interfaces to fill the gaps, and tracking the level of familiarity with a topic.

 

 

 

 

 

The framework to create the interfaces: a version of NetBeans on steroids ...

   

Complementary classroom activity on a computer algebra worksheet

 

 

This course is organized as a guided experience, 2 hours per day during five days, on learning the basics of the Maple language, and on using it to formulate algebraic computations we do with paper and pencil in high school and 1st year of undergraduate science courses.

 

Explore. Having success doesn't matter, using your curiosity as a compass does - things can be done in so many different ways. Have full permission to fail. Share your insights. All questions are valid even if to the side. Computer algebra can transform the learning of mathematics into interesting understanding, success and fun.

1. Arithmetic operations and elementary functions

   

2. Algebraic Expressions, Equations and Functions

   

3. Limits, Derivatives, Sums, Products, Integrals, Differential Equations

   

4. Algebraic manipulation: simplify, factorize, expand

   

5. Matrices (Linear Algebra)

   

 

Advanced students: guiding them to program mathematical concepts on a computer algebra worksheet

   

Status of the project

 

 

Prototypes of interfaces built cover:

 

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Natural numbers

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Functions

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Integer numbers

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Rational numbers

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Absolute value

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Logarithms

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Numerical sequences

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Trigonometry

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Matrices

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Determinants

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Linear systems

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Limits

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Derivatives

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Derivative of the inverse function

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The point in Cartesian coordinates

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The line

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The circle

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The ellipse

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The parabole

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The hyperbole

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The conics

More recent computer algebra frameworks: Maple Mobius for online courses and automated evaluation

   

 


 

Download Computer_Algebra_in_Education.mw

Download Computer_Algebra_in_Education.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Last week Michael Pisapia, Maplesoft European VP, attended the opening reception of Mathematics: The Winton Gallery at the Science Museum in London. Ahead of being open to the public on 8th December, contributors and donors were invited to take a look behind the scenes of the new gallery, which explores how mathematicians, their tools and ideas have helped to shape the modern world over the last four hundred years.

The gallery is a spectacular space, designed by the world-renowned Zaha Hadid Architects, housing over a hundred artefacts of mathematical origin or significance. It is divided up into disciplines ranging from navigation to risk assessment, and gambling to architecture. Inspired by the Handley Page aircraft, the largest object on display, and suspended as the centrepiece, the gallery is laid out using principles of mathematics and physics. It follows the lines of airflow around it in a stunning display of imagined aerodynamics, brought to life using light and sculpture. You can learn more about its design in this video.

Guests at the reception enjoyed a specially commissioned piece of music from the Royal College of Music titled ‘Gugnunc’, named after the aircraft and inspired by the rhythms of Morse code and mathematical and mechanical processes, and performed at the centre of the gallery.

Of course any exhibit celebrating all things maths is of great interest to us here at Maplesoft, but this one especially so, since Mathematics: The Winton Gallery showcases the earliest available version of Maple.

A copy of Maple V, from 1997, sits in ‘The Power of Computers’ section of the Winton Gallery, in an exhibit which tells the story of the significant role played by mathematical software in improving the quality of mathematics education and research. Other objects in the section include a Calculating Machine from the Scientific Service circa 1939, a PDP-8 minicomputer from the 1960s, and part of Charles Babbage’s mid-19th century analytical engine, intended as a high-powered mathematical calculator.

As many of you will remember, Maple V was a major milestone in the history of Maple, providing unparalleled interactivity, powerful symbolics and creative visualization in mathematical computation and modeling. For a walk down memory lane, check out Maple V: The Future of Mathematics (ca. 1994) on YouTube.

Seeing this copy of Maple finally in place in the exhibit marks the end of a long journey – and not just in the miles it travelled to arrive at the museum from its home in Canada. When we were first approached by the Science Museum for a donation of Maple, we launched a hunt to find not just the right copy of Maple with its box and manuals, but also artefacts that showcased the origin and history of Maple. It was a journey down memory lane for the inventors of Maple as well as the first few employees as they dug out old correspondences, photos, posters and other memorabilia that could be showcased. Today they can be proud of their contribution to this display at the Science Museum. 

Although the case of historic software packages is visually less impressive than many of the other items in the gallery, it certainly attracted plenty of attention as guests made their way in for the first time. 

For fans of Maple V - and there are many - it’s reassuring that the Science Museum are now entrusted with preserving not only the iconic packaging, but with telling the story of Maple’s history and marking its place in the evolution of mathematics and technology.

To learn more about Mathematics: The Winton Gallery, its highlights and architecture, visit http://www.sciencemuseum.org.uk/mathematics

To see the timeline of Maple’s evolution over the years, visit:  http://www.maplesoft.com/25anniversary/

A string is wound symmetrically around a circular rod. The string goes exactly
4 times around the rod. The circumference of the rod is 4 cm and its length is 12 cm.
Find the length of the string.
Show all your work.

(It was presented at a meeting of the European Mathematical Society in 2001,
"Reference levels in mathematics in Europe at age16").

Can you solve it? You may want to try before seing the solution.
[I sometimes train olympiad students at my university, so I like such problems].

restart;
eq:= 2/Pi*cos(t), 2/Pi*sin(t), 3/2/Pi*t; # The equations of the helix, t in 0 .. 8*Pi:
               
p:=plots[spacecurve]([eq, t=0..8*Pi],scaling=constrained,color=red, thickness=5, axes=none):
plots:-display(plottools:-cylinder([0,0,0], 2/Pi, 12, style=surface, color=yellow),
                         p, scaling=constrained,axes=none);
 

VectorCalculus:-ArcLength(<eq>, t=0..8*Pi);

                           20

 

Let's look at the first loop around the rod.
If we develop the corresponding 1/4 of the cylinder, it results a rectangle  whose sides are 4 and 12/4 = 3.
The diagonal is 5 (ask Pythagora why), so the length of the string is 4*5 = 20.

 

 

 

 

The GroupTheory package in Maple includes facilities for working with finitely presented groups - groups defined by finitely many generators and defining relations.  We now have a video tutorial that covers the basics of this aspect of the package.  As always, we appreciate feedback and suggestions regarding this feature, or new features that you would like to see in the GroupTheory package.

 

 

Here we have a very brief introduction to the use of embedded components, but effective for the study of the polynomials in operations and some products made with maple 2015 to strengthen and raise the mathematics today.

 

Operaciones_con_Polinomios.mw

(in spanish)

Atte.

L.AraujoC.

Good afternoon.

 

I request your valuable suggestion for the above cited subject.

I here by uploading the file for your kind notice.

 

Good afternoon sir.

 

I request your kind suggestion to my cited query.

 

 

With thanks & Regards

 

M.Anand

Assistant Professor in Mathematics

SR International Institute of Technology,

Hyderabad, Andhra Pradesh, INDIA.

Hi all

I want to produce following c_nm's(which are differentiation based formula) . assume that N and M are known and f(t) is arbitrary. also n=1,2,...,N and m=0,1,..,M-1

how can we do this?

regards


Mahmood   Dadkhah

Ph.D Candidate

Applied Mathematics Department

Good afternoon.

 

I request your kind suggestion to my above cited query.

 

 

With thanks & Regards

 

M.Anand

Assistant Professor in Mathematics

SR International Institute of Technology,

Hyderabad, Andhra Pradesh, INDIA.

Hi Again

Assume that we have known matrix namely, Q, of order (m+1)*(m+1) and we want to construct following matrix

where 0(bar) is zero matrix of orde (m+1)*(m+1) and New matrix should be of order {N*(m+1)}*{N*(m+1)} where N is known constant.

thanks for any guide


Mahmood   Dadkhah

Ph.D Candidate

Applied Mathematics Department

Good morning.

 

I request your kind suggestion to my query posted.

 

 

With thanks & Regards

 

M.Anand

Assistant Professor in Mathematics

SR International Institute of Technology,

Hyderabad, Andhra Pradesh, INDIA.

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