Math Games in an Educational Context (Part 1 of 3)

As a pupil at an English prep school in the 1970s, the curriculum was surprisingly predictable. English, Maths, Science, History and Geography were interspersed with languages, both ancient and modern, and a weekly period of divinity to keep us on the straight and narrow. However, the first year pupils – those aged seven and eight – were timetabled to have one lesson each week with the headmaster. In modern parlance, these periods would be described as ‘logic and verbal reasoning’, but in reality they comprised a strange mixture of riddles, brainteasers and mathematical games. While for the remainder of the week we were taught, in these sessions with the headmaster we were encouraged to think, to apply logic and deduction, to sharpen our mental faculties. In turn, it afforded the headmaster an opportunity to identify those with real academic potential, rather than simply the ability to learning by rote.

Each week, the pupils were presented with one brainteaser to solve. These puzzles were usually wrapped up in some form of a story, provided as much to obscure as to enlighten, but in essence they were mathematical in nature. However, the most satisfying puzzles offered a logical short cut that allowed one to reach the correct conclusion without resorting to algebra, geometry or quadratic equations. The example of the water lily and the lily pond will illustrate their equal mix of beguiling simplicity and frustrating complexity:

The Water Lily and the Lily Pond
The Squire had recently constructed a lily pond at the Manor House which was 15 feet long and 8 feet wide. He ordered his gardener to plant a water lily in one corner of the pond. The chosen water lily was a newly introduced cultivar renowned for its rapid growth. Indeed, the water lily grew at such a prodigious rate that the surface area of the pond it covered doubled each day. Seven days after it was planted, the water lily was covering half the surface area of the lily pond. How long would it take until the pond was completely covered?

There is an immediately obvious and wrong answer, which is fourteen days. There is also a long-winded mathematical approach, which involves calculating the area of the pond and thus the speed of growth of the water lily, which ultimately yields the correct answer. Much more satisfying is the logical approach, which quickly discerns that the dimensions of the pond are as immaterial as whether the water lily was planted by the gardener or the Squire himself.

If none of the pupils arrived at the correct answer in the course of the thirty-minute lesson, we were left to puzzle until the following week when the answer was revealed. In the same spirit, rather than simply reveal the solution for those who didn’t reach it instantaneously, I’ll leave you with little thinking time. After all, there’s nothing quite as satisfying as labouring over a puzzle until the moment when, quite unexpectedly, the light dawns.

An Overview of a Classical Education

One method of formal instruction uses the ‘classic’ approach. A classical education includes a three-part system to lay the foundation in a student’s mind.

The Grammar Stage

The grammar stage encompasses the first five years of school, from kindergarten through the end of the fourth grade. During this period, scholars receive basic instruction that lays the groundwork for additional learning that will occur later. During this period, children are typically eager to learn, and their minds are receptive to absorbing information. The grammar stage, therefore, involves rote learning of facts. Students will learn how to read, phonics, grammar, and spelling rules. Children also begin learning foreign languages, mathematics, history, literature, and science fundamentals.

The Logic or Dialectic Stage

By the time students reach the fifth grade, their minds are capable of a different kind of thought process. Scholars are able to use the information they have learned to approach knowledge in a more analytical way. This logic stage involves students considering cause and effect, and thinking about how facts fit together. Abstract thought enables students to use their foundation of academics to explore additional areas. For example, reading skills enable them to absorb information. They can then process the information, apply logic to it, form hypotheses, and present their final conclusions. Academic study during the logic stage includes algebra, advanced writing, and learning the tenets of scientific method.

The Rhetoric Stage

Once students enter high school, they are ready to begin the rhetoric stage. This final phase of classical education involves building upon and advancing from the grammar and logic phases. Scholars learn advanced writing and speaking skills that enable them to communicate their ideas and knowledge in a concise and understandable way. Students also learn how to apply logic and communicate eloquently. Scholars in this stage begin to focus their studies on the areas that interest or attract them personally. This initial focus enables students to consider career ideas, which is a necessary process while preparing for college and other types of specialized training.

How Classical Education Differs

A fundamental difference between classical education and typical education rests in the focus on language. With a classical approach, learning occurs with an in-depth focus on language. Scholars learn a diverse vocabulary to enable them to express their thoughts and opinions. This form of instruction also links all different types of knowledge. Sometimes connections between academic disciplines are obvious, and other times they can be elusive and ambiguous. Classically educated children learn to look at the world with a different perspective.

While classical education can vary among institutions, most schools of this type utilize the three different phases in their academic approach to teaching. Scholars learn their basic skills and then use them to dig deeper into the fount of knowledge.

Multiple Intelligences and Its Importance in Education

Learning theories in general are derived from the way theorists interpret human nature and how human beings learn.

Among the theories of learning proposed in the second half of the 20th century, I would like to highlight the theory of Multiple Intelligences developed by Howard Gardner. Initially proposed as a theory of human intelligence, that is, as a cognitive model, MI attracted the attention of educators around the world due to its description of cognitive competence in terms of a set of skills, talents, or even intellectual competences, which Gardner called “intelligences”. Gardner’s intelligences are relatively autonomous, although they are not completely independent. It seems that the importance of MI for educators is in their recognition that each child has a different set of different skills, or a spectrum of intelligences.

In reality, Gardner’s theory of learning is an alternative view to the theory of traditional intelligence (Binet and Simon’s IQ). It is a pluralistic theory of intelligence. According to Gardner, the MI model has used, in part, knowledge that was not available at the time of Binet and Simon (1908): cognitive science (study of the mind) and neuroscience (study of the brain). In MI, intelligence comes to be understood as multiple skills. These categories (or intelligences) represent elements that can be found in all cultures, namely: music, words, logic, paintings, social interaction, physical expression, interior reflection and appreciation of nature. In fact, MI theory is being used, with excellent results, in diverse educational environments, so demonstrating how cultural contexts can shape educational practice. Furthermore, MI represent eight ways to learn content. IM theory, therefore, does not privilege only language and logic as vehicles for learning. MI theory provides a kind of context in which educators can address any skill, topic, area, or instructional objective, and develop it in at least eight ways of teaching it. Used not only in the classroom, but also as a conceptual model in a science park, MI are proving to be a way of ensuring that learning takes place and is fun.

At first, the set of intelligences proposed by Gardner presented seven basic intelligences. In a later work, the author added an eighth intelligence (naturalist), leaving open the discussion about the possibility of adopting a ninth intelligence (spiritual). To arrive at this model, Gardner reports that he studied a wide and unrelated group of sources: prodigy studies, gifted individuals, brain-damaged patients, idiots savants, normal children, normal adults, experts in different fields of study and individuals from different cultures. The eight intelligences proposed by Gardner are defined as abilities to: 1) use language in a competent (linguistic) way; 2) reasoning logically in mathematics and science (logical-mathematics); 3) note details of what is seen and visualize and manipulate objects in the (spatial) mind; 4) understand, create and appreciate music and musical concepts (musical); 5) use one’s own body skillfully (bodily-kinesthetic); 6) recognize subtle aspects of other people’s (interpersonal) behavior; 7) having an understanding of the self (intrapersonal); and 8) recognizing patterns and differences in nature (naturalistic). As Gardner believes, intelligence is a human capacity that is linked to specific world content (for example, musical sounds or spatial patterns). Gardner notes, too, that these different intellectual forces, or competencies, each have their own historical development. For this very reason, they are valued differently by the different cultures of the world.

Finally, according to Gardner, certain domains or skills, such as the logical-mathematical one, which was deeply studied by J. Piaget, are universal. In a nutshell, Piaget investigates the minds of children to glimpse what is unique and generic about intelligence. However, there are other domains that are restricted to certain cultures. For example, the ability to read or to make maps is important in certain cultures, but minimally valued or even unknown in others.