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Abstract

# Abstract

We propose a roadmap to achieve three goals: (i) to educate every child in mathematics (particularly in the developing world) via a scalable, minimally language-dependent curriculum (textbook and teaching manual) that is crowdsourced by teachers and former students around the world; (ii) to optimize the math education in the anglophone world from secondary school level up to tertiary education; (iii) to create a platform for an international education system for the sciences.

# Introduction

The use of English as a major world language simplifies global cooperation on projects that increase our chances of survival: Globalization is in full swing and humans are increasingly connected over the internet, paving the way for construction of modern “Towers of Babel”. For example, a multi-billion dollar international collaboration of scientists and data clusters centered around the Large Hadron Collider enabled the discovery of the Higgs Boson. Research and development for another multi-billion dollar science experiment, the fusion experiment at ITER, is sourced by countries representing half of the world’s population. Over the next century, we want to include the other half of the world’s population in these projects by improving math literacy. Math literacy will help sow the seeds for future scientists, economists and mathematicians directly or indirectly collaborating on grand projects.

## Theory

How do we learn a language? Learning a language begins with our parents pointing out objects and saying their name, e.g. “tree”. Henceforth our little memories will recall that object whenever its label is called and we similarly attach the appropriate labels to other objects with the same properties. Sometimes, we combine labels to describe new things (e.g. German is notorious for combining words to form a new word). Over time we pick up a certain set of rules (grammar) and can combine objects (nouns), properties (adjectives) and actions (verbs) to form sentences. With a little practice we will have learnt a craft that makes us unique on this planet, namely the ability to record and efficiently transfer information among ourselves [Robert Sapolsky, Mark Pagel].

Mathematics could be seen as just another language. Its grammar rules are those of logic. Its alphabet is made up of individual components that can be combined to make an equation just like syllables and words are combined to make sentences. It is not surprising, therefore, that a recent study found that brains think about physics using regions trained for common tasks such as language processing [Jordana Cepelewicz, Robert Mason]. We rarely need mathematics in everyday life, but it has been essential for scientific and engineering breakthroughs that have lead to the high living standards of today. It is not a spoken language, possibly due to its heavy reliance on our working memory. For example, a relatively simple equation “the sum of terms of one over n squared, with n taking integer values between one and infinity, equals pi squared over six” is opaque to the listener, but clear to the reader, who sees $\sum_{n=1}^\infty \left(\frac{1}{n}\right)^2 = \frac{\pi^2}{6}.$ More complicated equations form too long sentences to be understood when heard, but make for concise bits of information on paper. Its unique notation could be a strength when one wants to educate people of many backgrounds in mathematics. We can do math using hardly any elements from another language, strengthening the case for math as its own self-contained language built upon logic.

## Why and how to collaborate?

According to Adam Smith, wherever individuals are acting in “enlightened self-interest” an “invisible hand” leads to unexpected social benefits. An enlightenend or altruistic individual’s total utility ($$U_t$$) can be defined as the sum of individual utility one generates for oneself without collaboration ($$U_i$$) plus the sum of the average altruistic utilities ($$\bar{U_j}$$) one generates for N other individuals. $U_{t}= U_{i} + N \bar{U_j}$ It becomes apparent very quickly that one’s total utility can be increased very easily by generating utility for other individuals. This could be a call for communism if it had not been empirically shown in the last century that self-interest is the driving force for the greatest wealth generating mechanism in the world, the free market. Therefore, it seems that for most people, most of the time, $$U_i >>\bar{U_j}$$. However, our society is a collaborative one [Daniel Abrams], so helping others seems to give benefits to individuals. It could be that collaboration is most beneficial for an individual where another person is able and likely to return a similarly sized favor in the near future. For example, family members, co-workers and friends are more likely to return favors than strangers. To capture this effect, we get $\label{utils} U_{t}= U_{i} + \sum_{i\neq j}^{M} m_j U_i+ (N-M)\bar{U_j}$ , where the second term represents the sum of utilities generated for individual $$i$$ as a result of a multiplier effect of generating utilities for $$M$$ others. The total coupling depends on the number of impacted others and is given as some multiplier $$m_j$$. This multiplier is written as some sizeable fraction $$m_j \leq 1$$ of non-collaborative utility $$U_{i}$$, adding $$m_j U_{i}$$ per group member of $$M$$. Therefore, as we strive to increase our individual utility ($$U_i$$), we increase the utility of ourselves and society at large. Utility is maximized where individuals are able to pursue their enlightened self-interest, particularly when in a strongly interacting group of enlightened individuals (a favor economy).

For example, let us examine the benefits in a peer instruction environment, where a teacher imparts problems and knowledge about how to solve them to a group of N peers. The teacher derives utility from the third term in equation (\ref{utils}). Engaging in peer instruction within smaller groups of $$M$$ peers benefits the instructing student, as well as his or her peers. Therefore the students in a peer instruction environment derive large utility from the second term in the above equation (\ref{utils}).

The internet is already proving to be a platform that adds value to helping strangers by allowing an exchange of favors directly ($$1\leftrightarrow 2$$) and, more commonly, indirectly ($$1\rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1$$), or rewarding contributions with upvotes and other digital currency. Therefore, the internet forms a useful tool to maximize multipliers ($$m_j$$), number of strongly interacting strangers ($$M$$), the number of different strongly-interacting groups of strangers $$\binom {N} {M}$$, the number of total people reached ($$N$$) and average utility generated for weakly-interacting members ($$\bar{U_j}$$).

## Mission Statement

The first goal of project “Babel” is to massively educate every child in mathematics. Education in itself is very important, particularly primary education, where children learn the basic knowledge used to navigate life and prepare for secondary education. Not surprisingly, primary education for every child has been set as the second millennium development goal by the United Nations. Math education is particularly important, as it forms the basis of every education in the hard sciences. It will be an important tool in bringing students from physical work in a pre-industrialized setting to mental work in large value-adding manufacturing companies, thus raising the local living standard. Parallel goals of optimizing math education for secondary and tertiary education in the anglophone world and offering an international, free platform for education in science can be tackled.

# Method

Math education is almost an “embarassingly parallel problem”. In other words, it is a problem that scales very well: It is largely independent of the lingual and cultural environment of the student and therefore applies to every student alive today and in the future. This problem could be solved by an optimum course, accompanied by an optimum mathbook, whereby most students in the world can learn mathematics and practice numerical problem solving tools with real world applications.

We define an optimum course that teaches a certain skill to an audience with the most efficient use of time and maximal amount of fun and retention for the student. How do our brains learn best? This is a question that cannot be answered without looking into the educational science and psychology literature (e.g. Physics Education Research). There are indications that a successful course should involve elements of team work and peer instruction [Heller et al., I, Heller et al., II]. Heuristically, we are also very good at learning from other solutions to a problem. Therefore, diversity in solutions to problems should be encouraged and solutions could be shared and rated. It is important to attach meaning to learning a new skill or bit of information. Therefore it is important to have real world examples that are relatable and motivate the student to do the course in the first place (e.g. Concept Rich Problems). For tertiary education, these examples can often be offered by employers who encounter problems in everyday operation.

How do we create an optimum mathbook? The authors hope to teach math using a predominantly graphic approach and logic, which has the appeal of eliminating language-dependence and therefore reaching every student in the world with one math book. However, as with learning another language, one has to learn some new vocabulary first. Fortunately, math vocabulary is not very large and translators can be crowdsourced. In fact, the entire mathbook could be crowdsourced by offering a pedagogical hardware and software platform that converges on a locally optimum mathbook from crowdsourced versions [Luis V. Ahn]. Having learnt the few necessary vocabularies from an independently sourced vocabul