Electronics, computing, and applied mathematics are gateway subjects to modern technology.
For young learners, we believe that electronics provides an ideal entry point. It is practical, with manipulables. It is easy to see cause and effect. With the right equipment and approach, exploring electronics can begin for children as early as 3 years old.
There are many tangible benefits for young learners getting started in electronics:
- fine motor skill development,
- an intuition for how technological things work at a component level,
- the integration of technology into the palette for imagination and creativity,
- improved self-confidence,
- strengthening a growth mindset,
- building resilience,
- raising the threshold of frustration,
- better dexterity,
- stronger focus.
 A three year old wiring his first circuit and the joy at seeing the LED, which he selected, light up! Continue reading this article…
By Assad Ebrahim, on January 21st, 2018 (52,119 views) |
Topic: Maths--Technical
2nd ed. January 21, 2018; 1st ed. Feb 8th, 2010
Abstract
This three part paper explores solving the sum of powers problem  using discrete maths techniques (recurrence relations, matrix systems) to obtain a solution polynomials whose coefficients turn out to be exactly the Bernoulli numbers  .
Part 1 (this paper) solves the problem using recurrence relations in a way which a high school student could emulate for small  . In Part 2, we develop a general recursive solution that works for arbitrary , from which we can build a table of values to assist in finding the coefficients of the solution polynomial, coefficients that are precisely the Bernoulli numbers discovered in 1713. In Part 3, we show how by transforming the problem into a linear system, we may obtain a direct (non-recursive) solution which directly calculates the Bernoulli number for any power . Source code is provided for all solutions.
Readers who are interested in this topic are referred also to lovely paper by Bearden (March 1996, American Mathematical Monthly), which tells the mathematical story and fills in the history (thanks to a reader for this great reference).
Continue reading this article…
By Assad Ebrahim, on January 20th, 2018 (21,737 views) |
Topic: Maths--Technical
By Assad Ebrahim, on January 19th, 2018 (12,695 views) |
Topic: Mathematics, Maths--Technical
(Discrete Mathematics Techniques III)
1st ed. Apr 2nd, 2010
Abstract
This is the last in the 3-part series of articles on finding for oneself the solution to the sum of integer power problem, and in the process discovering the Bernoulli numbers. In Part 3 (this paper), we find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem  . Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the  independent monomials  ( ), where the  are unknown coefficients. Using the recurrence relation  , we obtain a linear combination of the monomials, which reduces to an easily solvable  -by- triangular linear system in the unknown coefficients of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.
A lovely paper by Bearden (March 1996, American Mathematical Monthly), which was shared with me by a reader, tells the mathematical story nicely, with much of the history filled in.
Continue reading this article…
*New!* (29 Aug 2020) – Turtle Logo v1.8 (portable) is available! Developer kit with source code included. Suitable from ages 3 years to adult. (970 lines of Forth code).
1. Inspiring the next generation of technology builders.
A challenge facing parents and teachers is how to help children develop ‘builder’ relationships with technology rather than being limited to the passive consumption of content created by others. The consensus on what’s important for older kids and adults is clear: coding. This enables children to participate in the creation of their own technological “micro-worlds” — environments rich in educational potential.[14]
This autumn, spurred by having our own young children (one aged 4 years, the other 16 months), we began an experiment, the result of which is a Turtle Logo program for Windows computers (freely downloadable) that is simple enough to be accessible for children from 3 years and older, while providing an extensible platform that can grow with the child.
The long-term goal is to enable children to express their creativity, artistry, and natural ‘builder’ impulses using coding, computer graphics, and robotics as readily as the previous generation could using paints, brushes, and building blocks.
 Turtle Logo – Inspiring the next generation of technology builders. Continue reading this article…
By Assad Ebrahim, on November 11th, 2014 (269 views) |
Topic: Maths--General Interest
Updated: 2014-11-23
MetaMoji Note: A worthy digital replacement to Pencil & Notebook for creating freehand diagrams on Windows and Android.
Pencil & Notebook are hard to beat when diagrams, equations, and words are desired in roughly equal measure, which is common when working as an engineer, mathematician, or product designer. To be sure, there are good digital tools for subsets of these (example LaTeX), but not when all three are combined. But now there’s a new suite of digital freehand tools that are almost good enough to set aside the notebook and pencil permanently. The core of this toolkit is a vector-based graphics canvas (software) and an ultra-precise stylus (hardware).
If you use Apple iOS, you may already be using Adobe Ideas, which does the job brilliantly and is free (see this excellent article by Simon Raper). [1] But Adobe has no plans to release this to either Windows RT or Android, so for the rest of us, that’s a show-stopper, until now, with MetaMoji…
Continue reading this article…
By Assad Ebrahim, on May 21st, 2014 (7,636 views) |
Topic: Maths--General Interest
Duelling with pistols. If you were the one issuing the challenge, your dilemma was that custom dictated that your adversary be allowed to shoot first. Only then, if you were still able to shoot, would you be permitted to seek “satisfaction”.
How much of an advantage does the first shooter really have? In this article, we build a simple probability model, and implement a numerical model in a few lines of R code.
 Two gentleman face off in the snow. Convention dictates the challenged shoots first. Continue reading this article…
THIS PAGE HAS MOVED!
The essay has been incorporated into the Software list. You can find it here!
By Assad Ebrahim, on November 15th, 2010 (11,805 views) |
Topic: Maths--Data Science
(Statistics and Data Mining II)
Automated decision problems are frequently encountered in statistical data processing and data mining. An heuristic filter or heuristic classifier typically has a limited set of input data from which to arrive at a set of conclusions and make a decision: REJECT, ACCEPT, or UNDETERMINED. In such cases, pre-processing the input data before applying the heuristic classifier can substantially enhance the performance of the decision system.
In this article, I’ll motivate the use of a radar-tracking algorithm to improve the performance of automated decision making and statistical estimation in data processing. I will illustrate using the website visitation statistics problem.
Continue reading this article…
By Assad Ebrahim, on September 3rd, 2010 (16,593 views) |
Topic: Maths--Data Science
(Statistics and Data Mining I)
For a variety of reasons, meaningful website visitation and visitor behavior statistics are an elusive data set to generate. This article introduces the visitor statistics problem, and describes seven challenges that must be overcome by statistical and data analysis techniques aiming for accurate estimates. Along the way, we’ll encounter the “Good News Cheap, Bad News Expensive” Paradox of Data Mining — or, why information is often used “as-is”.
This article is the first in a series on algorithms, statistics and data analysis techniques (using free and open source tools) using the visitor statistics problem as a vehicle for illustration.
Continue reading this article…
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, the coefficients of whose solution are the famous Bernoulli numbers (1716). In this paper we show to how obtain a 
for any given 
