The Benefits of Enriched Mathematics Instruction

2nd ed. June 2023; 1st ed. April 2010

The term “mathematical maturity” is sometimes used as short-hand to refer to a blend of elements that distinguish students likely to be successful in mathematics. It is a mixture of mathematical interest, curiousity, creativity, persistence, adventurousness, intuition, confidence, and useful knowledge.[1],[2],[3]

With advances in machine learning, computer science, robotics, nano-materials, and many other quantitative, fascinating subjects, students today have increasingly more choice in technical studies besides mathematics. To attract and retain mathematics students, it is important that mathematics instruction be experienced as both intellectually and culturally rewarding in addition to being technically empowering. Losing students from mathematics who are otherwise capable, engaged and hard-working is tragic when it could have been avoided.

In this article, building on observations gained over the years teaching and coaching students in mathematics, we consider how enriched mathematics instruction (inquiry-based/discovery learning, historiography, great ideas/survey approaches, and philosophical/humanist) can help (1) develop mathematical maturity in students from at-risk backgrounds and prevent their untimely departure from quantitative studies, (2) strengthen the understanding of those that are already mathematically inclined, (3) expand mathematical and scientific literacy in the wider population.

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For context, here is the first article in the Teaching Mathematics series: Teaching Enriched Mathematics

1. Motivating Observations

During my experience teaching university mathematics (freshman/sophomore students), and coaching junior and high school students, and inquiring into the background of my students, some correlation have emerged. Corroborating encounters over the years prompted conversations with industry and academic colleagues on their own journeys in mathematics, conversations with successful students, and reflections on my own path. In what follows below I will share these observations, and some of the directional results of the various discussions.

Observation 1: Students with “mathematical maturity” are more likely to have come from backgrounds (parents, siblings, relatives, peers, mentors) that are quantitative or that encourage engagement with mathematics, science, and technology.

Observation 2: Conversely, students coming from backgrounds that are disinterested in and disengaged with these ideas are more likely to pass through their formal mathematical education without having gained these elements, and so are less likely to make the kind of rewarding intellectual and emotional connections with the material as their more mathematically mature peers. As a consequence, they are less likely to continue mathematical/scientific/technical studies. If they do continue, they remain at a comparative disadvantage until, somehow, they manage to strengthen within themselves this mix of elements.

Focusing specifically on students that develop a sense of alienation from mathematics, there seems to be a common thread running through their educational experience that may help to identify how their disempowered situation can be turned around.

Observation 3: When good students claim to be turned off by mathematics, it has often turned out that their aversion to mathematics and the source of their difficulties stems in part from unaddressed meta questions about mathematics itself rather than particularly about its content. Questions such as “What is mathematics really about?” “What is the use of learning more mathematics?” “What’s the point of topic X?” “How is all of this connected?”. Such questions are rarely addressed within the typical school curriculum, and if they are, the answers are usually provided in side conversations with a concerned teacher rather than using a structured approach.

Observation 4: In schools, the notion seems to be that such meta-questions are neither essential nor relevant. By the time students get to advanced studies in university or graduate, it is assumed that they must already have answered these questions for themselves or that answers to any remaining questions should by now be self-evident. In my experience, none of these assumptions are correct. The majority of high school students, and indeed many undergraduate students, don’t yet have well formulated answers to these questions. They may be able to perform specific computations on demand, and yet have failed to grasp the essential ideas underpinning the mathematics.

Observation 5: Where interventions were made in time and with persistence and patience, it was possible, even in some difficult cases, to effect a change in appreciation of the kinds of questions mathematicians ask, and renewed willingness to study and eventually master the technical content. I.e. in our experience, mathematical maturity can, and indeed should, be strengthened through appropriate teaching. If on the other hand, the epiphanies did not not occur, then over time a growing sense of alienation with advanced maths developed, ultimately leading to their departure from further mathematical studies. If these students could have been satisfied earlier in their careers on their particular meta questions about the essence of mathematics and how it works, their engagement might have been recovered, the ability strengthened, and their eventual departure from math/science/tech prevented.

2. Why the Meta Questions Matter

Why are the meta questions important?

A first reason is that meta-questions are the kind of questions that independent thinkers who value their time tend to ask before deciding whether to apply their efforts. They are the kind of questions that we expect good students to ask. When these questions about mathematics go unexpressed (and therefore not answered), or worse are asked but dismissed or ignored, this creates a mental blocker that hampers a students’ ability to absorb new mathematical ideas and engage and explore these ideas creatively and without self-doubt and anxiety. Over time, the result is a sense of alienation that leads eventually to choosing to stop pursuing further mathematics.

A second reason is that mathematics is more than the “what and how”. It is also about the “why, the so-what, and the who”. Without the latter, familiarity with mathematics is necessarily superficial. It is the only way to form a deep seated connection to the essence of the subject, its culture and methods, and its vibrant history (episodes: prehistory, uruk/susa, abstraction in algebra, rise of logic). Students who are able to engage with mathematics in this way come to appreciate the kinds of questions mathematicians ask, and show a renewed interest and diligence in studying the technical content, being rewarded by greater insight, understanding, and accomplishment.

A final reason is that we want in our students to foster a broad understanding of mathematics, not narrow specialism. Felix Klein in 1908 noted that mathematics has a depth and inner coherence “which is not brought out sufficiently in the usual lecture course”. As a remedy he launched at Gottingen a two year course of lectures on Elementary Concepts of Mathematics from an Advanced Standpoint. More recently (1989), John Stillwell has written:

“One of the disappointments experienced by most mathematics students is that they never get a course on mathematics. They get courses in calculus, algebra, topology, and so on, but the division of labor in teaching seems to prevent these different topics from being combined into a whole. In fact, some of the most important and natural questions are stifled because they fall on the wrong side of topic boundary lines. Algebraists do not discuss the fundamental theory of algebra because “that’s analysis”, and analysts do not discuss Riemann surfaces because “that’s topology,” for example. Thus, if students are to feel they really know mathematics by the time they graduate, there is a need to unify the subject.” – John Stillwell, 1989, preface to Mathematics and Its History

Meta-questions are those questions that step us back from the details, that challenge us to think bigger, seek synthesis, explore the storylines of what we have learned and make the connections of how it all fits together and above all, why it is important, meaningful.

3. Can mathematical maturity be accelerated? At its core this requires strengthening a core set of learning competencies (intellectual, behavioural, and emotional).

“The development of mathematical maturity requires a deep reflection on the subject matter for a prolonged period of time, along with a guiding spirit which encourages exploration.” [Wikipedia] It requires getting comfortable with the rather sophisticated conceptual approach which modern mathematics assumes. This is not something that one gets automatically. It is typically gained through contact, almost by osmosis, when the student’s milieu has within it peers, mentors, or teachers who already have this cultural knowledge. The absorption is most efficient when done through informal exchanges whether in conversation with mathematically or technologically literate parents, relatives or mentors.

The typical school or even university curriculum is not well suited to compensate for the lack of suitable background. For such students, there is little further opportunity for meta questions to be addressed, putting students from such backgrounds disproportionately at risk of disengagement and reduced success in their science and technology education.

Can this be turned around? The “enriched instruction” approach seeks to accelerate the natural acquisition of relevant cultural mathematical background and maturity by providing formative mathematical experiences early in the students’ education so that they gain perspective on the origins and meaning of mathematics, why mathematics has evolved in the way it has, and have been exposed to useful techniques. This ensures that the forest is not missed while they study the many, many trees that they will necessarily encounter.

Enriched mathematics material is designed to foster inter-connections which directly engage all three learning competencies. Put simply, by teaching enriched mathematics, with a guiding spirit that encourages exploration, the student has the opportunity to strengthen their intellectual, behavioural, and emotional competencies.

Successful participation should enable students to continue more confidently and comfortably with their science and technology education, study further and with greater benefit, from both theoretical mathematics and its applications. Where the humanistic questions around mathematics are addressed, implicitly or explicitly, one often sees energy and effort, and a step change in attitude, engagement, and performance.

4. Enriched Mathematics — Examples

Over the years I have come to believe that exposure to the enduring ideas in mathematics is best achieved using Inquiry Based Learning (originated from the Moore method, Mahavier paper, IBL criteria, IBL articles) supplemented with an historical approach (teaching with original sources, e.g. TRIUMPHS, PSP, along the model of Barnett, Laubenbacher, Pengelley). These together are an ideal way to engage with, learn, and teach mathematics. They satisfy the two objectives of mastering practical methods and cultural immersion.

The Moore method of instruction was an incredibly valuable experience for those learning mathematics. It allowed students to get to the heart of discoveries and theorems and do so through building their own capability for doing mathematics, both discovering as well as justifying as well assessing and then helping to strengthen others’ justifications. A refresh of these principles is contained in “Inquiry-Based Learning” (IBL) which has developed a practitioner’s guide and created course materials. For older students, there is another strand borrowed from the teaching of the liberal arts and fused with mathematics: teaching from original sources. This has also recently benefited from extensive course materials and the proliferation of the internet. (see References)

Further proactive approaches include:

(1) a structured opportunity to philosophically examine the meta aspects of mathematical activity. Addressing this cultural prerequisite in a structured fashion can guide the student toward greater personal intellectual engagement with the subject and set students on a positive trajectory continuing successful science and technology careers.

(2) first-hand participation in an educational program that provides structured experiences in discovering mathematics or mathematically related concepts. There are programs that explore mathematics in the context of physics, technology, computer games, artificial intelligence, robotics, digital graphics, social research, monetary policy, or business.¹

There are several important pedagogical objectives to an enriched approach:

1. to expose students to the fact that mathematics is a subject that has a cultural and an aesthetic aspect that can be appreciated in the same way as can literature, art, or music, where the reader, viewer, or listener, need not themselves be writers, artists, or musicians;
2. to introduce the notion of an anthropology of mathematics, i.e. that mathematics is as much influenced by the socio-cultural context in which it is done, as it influences that context through what it creates, discovers, develops;
3. to demonstrate how mathematics develops in a dialectical dynamic akin to a conversation by individual mathematicians over their lifetimes, with mathematics from previous periods through the works they left behind, and with contemporary mathematicians through the exchange of letters and ideas, collaborations and in some cases competition;
4. to create a conceptual framework within which later technical advanced studies will find a natural and already fertilized ground in which to grow efficiently (acknowledging that in most school/university courses the time is very meagre that is spent on motivation in the ways described above);
5. to strengthen the understanding of technically important developments in a way that is memorable, i.e. that can be understood fully and reproduced at will. (For the first three objectives, see What is Mathematics?)
6. to empower students by helping them understand HOW to UNDERSTAND modern mathematics by following the WAY in which its concepts evolved, even if that construction is ahistorical.²

5. What have been the thoughts of eminent mathematicians on these equestions?

Below some quotes to provoke further discussion/consideration.

1. “Premature abstraction falls upon deaf ears, whether they belong to mathematicians or to students.” — Morris Kline, Mathematical Thought from Ancient to Modern Times (Kline/1972)

2. “Many have thought that one could, or indeed that one must, teach all mathematics deductively throughout, by starting with a definite number of axioms and deducing everything from these by means of logic. This method… does not correspond to the historical development of mathematics. Mathematics has grown like a tree, which sends its roots deeper and deeper at the same time and rate that its branches and leaves are spreading upward. It progresses according to the demands of science itself, and of prevailing interests, now in one direction toward new knowledge, now in the other through the study of fundamental principles. Our standpoint today with regard to foundations is different from that of the investigators of a few decades ago; what we today would state as ultimate principles and the latest truths will be still more meticulously analyzed and referred back to something still more general. As regards fundamental investigations in mathematics, there is no final ending, and therefore no first beginning which we could offer as an absolute basis for instruction. – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.15]

3. It is by logic we prove, it is by intuition that we invent. — Henri Poincare, Mathematical definitions in education (1904)

4. “The manner of instruction [that works best in the schools] can be designated by the words intuitive and genetic, i.e. the entire structure is gradually erected on the basis of familiar, concrete things, in marked contrast to the customary logical and systematic method at the university. … To the words intuitive and genetic, we can add a third word, applications.” – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.6-7]

5. Applications should always accompany theory in the teaching of mathematics! The living thing in mathematics, its most important stimulus, its effectiveness in all directions, depends entirely upon the applications, i.e. upon the mutual relations between those purely logical things and all other domains. The needs of mathematics instruction require precisely a certain many-sidedness of the individual teacher, a comprehensive orientation in the field of pure and applied mathematics in the broadest sense. – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.15-16]

6. “The greatest mathematicians, as did Archimedes, Newton, and Gauss, always united theory and applications in equal measure.” (Felix Klein) (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908]

7. “The presentation in the schools should be psychological and not systematic. The teacher must be a diplomat. He must take account of the psychic processes in the boy in order to grip his interest, and he will only succeed if he presents things in a form intuitively comprehensible. [Indeed] one should take it to heart, that in all instruction, even in the university, mathematics should be associated with everything that is seriously interesting to the pupil at that particular stage of his development and that can in any way be brought into relation with mathematics.” – Felix Klein (mathematician), from Elementary Mathematics from an Advanced Standpoint, [Klein, 1908,p.3-4]

8. In an area [of knowledge] the number of ideas is limited. Everything else is a variation on the theme. In mathematics the number of ideas is not large. Everything that is achieved is obtained from the basic (or fundamental) concepts which are applied with some degree of variations. Mastering these basic concepts in one field of mathematics helps to distinguish and use them in other fields.” — I.M. Gelfand, from introduction to Geometry

9. Very often in mathematics the crucial problem is to recognize and to discover what are the relevant concepts; once this is accomplished the job may be more than half done. — I.N. Herstein (mathematician), Topics in Algebra

10. By producing examples and by observing the properties of special mathematical objects, one could hope to obtain clues as to the behavior of general statements which have been tested on examples. — S.M. Ulam, Adventures of a Mathematician

11. A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one. — Paul R. Halmos (mathematician)

12. There are two fundamental sources of ‘bare facts’ for the mathematician, that is, there are some real things out there to which we can confront our understanding. These are, on the one hand the physical world which is the source of geometry, and on the other hand the arithmetic of numbers which is the source of number theory. Any theory concerning either of these subjects can be tested by performing experiments either in the physical world or with numbers.” – Alain Connes (mathematician) from Non-Commutative Geometry, 2000

13. [While] mathematical ideas originate in empirical facts, once conceived the subject begins to live a peculiar life of its own and is better compared to a creative one governed by almost entirely aesthetic motivations, than to an empirical science. – John von Neumann (mathematician, computer scientist), 1947

6. What aim should one have in one’s mathematical studies?

If you are a beginning student, wondering how you should approach your mathematical studies, the best advice I can give is the following:

1. Be able to do the computations, construct the arguments rigorously, and understand the big picture and how all the pieces fit in, as well as connections with other areas of mathematics and the sciences
2. Research: look for connections of things you know and understand to other areas and try to find connections and bridges. (See Connes description of Atiyah)
3. Read the ideas and understand the storyline.³

   What motivates the inquiry?
   What are the problems that are wanted to be solved?
   Why are they a problem, i.e. what is hard about solving them?
   * Try yourself to solve them.
   Why does one care to solve them?

4. Try yourself to solve them:
   Can you lay out a path to solving them. Use the text for the steps along the way.
   It is a bit like the Moore Method – just put the breadcrumbs.
   **Attack the breadcrumb path; solve each sub-problem

5. Solve each step.
   Understand thoroughly the definitions (which are fundamental to understanding the math).

   Charles Yang (2019) elaborates on the mental approach a motivated high school student should use.

   The three- and five-strand models (Benjamin Braun, 2019)

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Further reading:

1. A Course in the Philosophy and Foundations of Mathematics April, 2010, Assad Ebrahim – this collection of materials can be used for self-study or for an elective course for undergraduate mathematics majors as well as beginning graduate students in mathematics, preferably during the first year of graduate school. With appropriate modifications, an approach such as this may be useful for talented secondary students in an abbreviated, colloquium setting run in tandem with faculty from a university mathematics department interested in school partnership and outreach.
2. Teaching Enriched Mathematics, Part 1, July 2016, Assad Ebrahim
3. How Algebra became abstract: George Peacock & the birth of modern algebra (England, 1830), Assad Ebrahim 2019
4. The rise of Mathematical Logic: from Demonstration to Laws of Thought to Foundations for Mathematics, Assad Ebrahim, 2019
5. What is Mathematics?, Assad Ebrahim, 2023
6. The Prehistoric Origins of Mathematics, Assad Ebrahim, 2020
7. The Mathematics of Uruk and Susa (c.3500-3000 BCE), Assad Ebrahim, 2020

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Educational References

Mathematical Education Approaches: Inquiry Based Learning, the Moore Method, Historical Learning (Teaching using Original Sources)
1. [1] [Braun, 2019] – Precise Definitions of Mathematical Maturity, by Benjamin Braun, April 2019, American Mathematics Society Blog; Precis: explores 3-strand and 5-strand models of competence, with the view that mathematical competence can be strengthened through strengthening each strand.
2. [2] [Lew, 2018] – How do Mathematicians Describe Mathematical Maturity? Mathematics Association of America; (PDF) Precis: Literature review and analysis of results of interviews with 5 pure and 4 applied mathematicians on the term
3. [3] Wikipedia: Mathematical Maturity Flawed definition, as discussed in [Lew, 2018]
4. [Ernst, 2013]: What the heck is Inquiry Based Learning?, Dana Ernst, 2013, Mathematical Education Matters, Mathematical Association of America
5. [Mahavier, 1999]: What Is The Moore Method?, William S. Mahavier, Primus, vol. 9 (December 1999): 254-339.
6. [Barnett, et.al, 2007]: Collaborative Research: Learning Discrete Mathematics and Computer Science via Primary Historical Sources, Janet Barnett, et. al, 2007
7. [Barnett, et.al, 2013]: The Pedagogy of Primary Historical Sources in Mathematics: Classroom Practice Meets Theoretical Frameworks, by Janet Barnett, Jerry Lodder, David Pengelley, 2013 (Online)
8. [Laubenbacher/Pengelley 1996]: Mathematical Masterpieces: Teaching with Original Sources, by Reinhard Laubenbacher and David Pengelley, 1996
9. [Dunham, 1986]: A Great Theorems Course in Mathematics, by William Dunham, 1986, American Mathematical Society, Vol 93, No.10, 4 pps
10. [Dunham, 1990]: Journey Through Genius: The Great Theorems of Mathematics, by William Dunham, 1990, 315pp. (Online)
11. [Laubenbacher/Pengelley]: Resource Page for Teaching from Historical Mathematical Sources
12. [Laubenbacher/Pengelley 1995]: Historical Sources for Teaching Mathematics, 1995-2008
13. [Pengelley, et.al]: Teaching Discrete Mathematics through Historical Sources
14. [Fauvel, 2003]: History in Mathematics Education – £80 Amazon,
15. Tribute to John Fauvel and his style of teaching history of mathematics

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Footnotes