Why Math Works

by John D. Mays

Back in 1999 when I began teaching in a classical Christian school, one of the first books I heard about was James Nickel’s little jewel, Mathematics: Is God Silent? Must reading for every Christian math and science teacher, the book introduced me to a serious problem faced by unbelieving scientists and mathematicians. Stated succinctly, the problem is this: Mathematics, as a formal system, is an abstraction that resides in human minds. Outside our minds is the world out there, the objectively real world of planets, forests, diamonds, tomatoes and llamas. The world out there possesses such a deeply structured order that it can be modeled mathematically. So how is it that an abstract system of thought that resides in our minds can be used so successfully to model the behaviors of complex physical systems that reside outside of our minds?

For over a decade now this problem, and the answer to it provided by Christian theology, has been the subject of my lesson on the first day of school in my Advanced Precalculus class. But before jumping to resolving the problem we need to examine this mystery – which is actually three-fold – more closely.

In his book Nickel quotes several prominent scientists and mathematicians on this issue. In 1960, Eugene Wigner, winner of the 1963 Nobel Prize for Physics, wrote an essay entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner wrote:

The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and…there is no rational explanation for it…It is not at all natural that ‘laws of nature’ exist, much less that man is able to discern them…It is difficult to avoid the impression that a miracle confronts us here…The miracle of appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

Next Nickel quotes Albert Einstein on this subject. Einstein commented:

You find it surprising that I think of the comprehensibility of the world…as a miracle or an eternal mystery. But surely, a priori, one should expect the world to be chaotic, not to be grasped by thought in any way. One might (indeed one should) expect that the world evidence itself as lawful only so far as we grasp it in an orderly fashion. This would be a sort of order like the alphabetical order of words of a language. On the other hand, the kind of order created, for example, by Newton’s gravitational theory is of a very different character. Even if the axioms of the theory are posited by man, the success of such a procedure supposes in the objective world a high degree of order which we are in no way entitled to expect a priori.

One more key figure Nickel quotes is mathematician and author Morris Kline:

Finally, a study of mathematics and its contributions to the sciences exposes a deep question. Mathematics is man-made. The concepts, the broad ideas, the logical standards and methods of reasoning, and the ideals which have been steadfastly pursued for over two thousand years were fashioned by human beings. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass; he has predicted and shown how to control radio waves which none of our senses can perceive; and he has discovered particles too small to be seen with the most powerful microscope. Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explanation of this marvelous power is called for.

The first aspect of the problem these scientists are getting at is the fascinating fact that the natural world possesses a deep structure or order. And not just any order, mathematical order. It is sometimes difficult for people who have not considered this before to get why this is so bizarre. Simply put, the order we see in the cosmos is not what one would expect from a universe that started with a random colossal explosion blowing matter and energy everywhere.

Many commentators have written about this and professed bafflement over it. All of the above quotes from Nickel’s book and many, many more are included in Morris Kline’s important work, Mathematics: The Loss of Certainty, which explores this issue at length. In his book The Mind of God, Paul Davies, an avowed agnostic, prolific popular writer and physics professor, takes this issue as his starting point. Davies finds the order in the universe to be incontrovertible evidence that there is more “out there” than the mere physical world. There is some kind of transcendent reality that has imbued the Creation with its mathematical properties.

The second aspect to the problem or mystery we are exploring is that human beings just happen to have serious powers of mathematical thought. Now, although everyone is happy about this, I rarely find anyone who is shocked by it. Christians hold that we are made in the image of God, which explains our unique abilities such as the use of language, the production of art, the expression of love, self-awareness, and, of course, our ability to think in mathematical terms. Non-Christians don’t accept the doctrine of the imago Dei, but seem to think that our abilities can all be explained by the theory of natural selection.

But hold on here one minute. Doesn’t it seem strange that our colossal powers of mathematical imagination would have evolved by means of a mechanism that presumably helped us survive in a pre-industrial, pre-civilized environment? Our abilities seem to go orders of magnitude beyond what evolution would have granted us for survival.

I know all about the God-of-the-gaps argument, and I’m not going to fall for it here. It may be that some day the theory of common descent by natural selection will be able to explain how we became so smart. That’s fine, and I’m not threatened by it. All I’m saying is that for now Darwinism still has a lot of explaining to do. And getting back to the concerns in this essay, I for one do not take Man’s amazing intellectual powers for granted. They are wonderful.

The third aspect to our problem is the most provocative of all. Mathematics is a system of symbols and logic that exists inside of our heads, in our minds. But the physical world, with all of its order and structure, is an objective reality that is not inside our heads. So how is it that mathematical structures and equations that we dream up in our heads can correspond so closely to the law-like behavior of the independent physical world? There is simply no reason for there to be any correspondence at all. It’s no good saying, Well we all evolved together, so that’s why our thoughts match the behavior of reality. That doesn’t explain anything. Humans are a species confined since Creation to this planet. Why should we be able to determine the orbital rules for planets, the chemical composition of the sun, and the speed of light? I am not the only one amazed by this correspondence. All those Nobel Prize winners are amazed by it too, and they are a lot smarter than I am. This is a conundrum that cannot be dismissed. John Polkinghorne said it well in his Science and Creation: The Search for Understanding:

“We are so familiar with the fact that we can understand the world that most of the time we take it for granted. It is what makes science possible. Yet it could have been otherwise. The universe might have been a disorderly chaos rather than an orderly cosmos. Or it might have had a rationality which was inaccessible to us…There is a congruence between our minds and the universe, between the rationality experienced within and the rationality observed without. This extends not only to the mathematical articulation of fundamental theory but also to all those tacit acts of judgement, exercised with intuitive skill, which are equally indispensable to the scientific endeavor.” (Quoted in Alister McGrath, The Science of God.)

Which brings us to the striking explanatory power of Christian theology for addressing this mystery. As long as we ponder only two entities, nature and human beings, there is no resolution to the puzzle. But when we bring in a third entity, The Creator, the God who made all things, the mystery is readily explained. As the figure here indicates, God, the Creation, and Man form a triangle of interaction, each interacting in key ways with the other. God gives (present tense verb intentional) the Creation the beautiful, orderly character that lends itself so readily to mathematical description. And we should not fail to note here that the Creation responds, as Psalm 19 proclaims: “The heavens declare the glory of God.” (I have long thought that when the Pharisees told Jesus to silence his disciples at the entry to Jerusalem, and Jesus replied that if they were silent the very stones would cry out, he wasn’t speaking hyperbolically. Those stones might have cried out. They were perfectly capable of doing so had they been authorized to. But I digress.)

why math worksSimilarly, God made Man in His own image so that we have the curiosity and imagination to explore and describe the world He made. We respond by exercising the stewardship over nature God charged us with, as well as by fulfilling the cultural mandate to develop human society to the uttermost, which includes art, literature, history, music, law, mathematics, science, and every other worthy endeavor.

Finally, there is the pair of interactions that gave rise to the initial question of why math works: Nature with its properties and human beings with our mathematical imaginations. There is a perfect match here. The universe does not possess an order that is inaccessible to us, as Polkinghorne suggests it might have had. It has the kind of order that we can discover, comprehend and describe. What can we call this but a magnificent gift that defies description?

We should desire that our students would all know about this great correspondence God has set in place, and that considering it would help them grow in their faith and in their ability to defend it. Every student should be acquainted with the Christian account of why math works. I recommend that every Math Department review their curriculum and augment it where necessary to assure that their students know this story.

John D. Mays is the founder of Novare Science and Math in Austin, Texas. He also serves as Director of the Laser Optics Lab at Regents School of Austin. John entered the field of education in 1985 teaching Math in the public school system. Since then he has also taught Science and Math professionally in Episcopal schools and classical-model Christian high schools. He taught Math and 20th Century American Literature part-time at St. Edwards University for 10 years. He taught full-time at Regents School of Austin from 1999-2012, serving as Math-Science Department Chair for eight years. He continues to teach on a part time basis at Regents serving as the Director of the Laser Optics Lab. He is the author of many science textbooks that I invite you to explore further on the Novare website.

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Education: An Act of Justice or an Act of Grace?

It has been a while since my last post. Ph.D. work, conference, and family have all kept me busy throughout the summer. I should hopefully have a few posts over the next few weeks that summarize the conferences that I have attend, the progress I have made in my Ph.D. research, and some general thoughts I have considered this summer. This post falls in the latter category. Enjoy.

I have written here before that education is inherently value-laden. If you are an educator then it is not a question of “are you teaching values?” but rather “which values are you teaching?” In this vein of thinking it can also be argued that education is inherently religious if it instills within us some sense of values and some sense of faith. Now, those values and the object(s) of that faith could vary greatly depending on the educational institution, but the fact is they are always there. This is why for centuries the work of education was undertaken by explicitly religious institutions and it is only fairly recently in society’s history that the state has taken on this endeavor (Wilson). So for everyone who feels that there is something wrong with our current state/system of education, I would argue that the root issue is primarily a religious one.

Every pedagogy assumes an anthropology (Smith). Before you can teach human beings you have to have some understanding of what human beings are. Ultimately I believe this is the reason for the existence of standardized testing (at least in America). The state approaches education from the perspective that all human beings are essentially good. It is simply a matter of providing education equally for all that will result in well-trained, productive citizens who contribute to the good of society. From this perspective, education is an act of justice. It is a citizen’s right to be educated. If the educational system is an administration of justice on the part of the state then it will inextricably be tied to universal standards that students and educators are required to meet. I mean, isn’t that how the justice system works? It sets standards in place that apply equally to everyone in a blanket approach and expects every individual to live up to those standards. There are also consequences when the standard isn’t met. Fail to meet the justice standard in society and go to jail. Fail to meet the standard in school and get remediation or don’t move to the next grade level or don’t graduate. As long as the state is in control of education, expect standardized testing to always be a part of the educational process.

What is the alternative? Maybe, just maybe, human beings aren’t inherently good. Maybe they are inherently evil and no amount of knowledge is going to save them from that. If we adopt this (Christian) anthropology then education will not be seen as an act of justice, as a right of the citizens, but rather as an act of grace. Education can be viewed as an act of grace that missionally reaches out to engage the lost mind. Grace doesn’t set a standard for you to meet. In fact, grace realizes that you can’t meet THE standard. So then the focus of education shifts from universal standards to individual and communal transformation. Results aren’t measured in knowledge gained but rather in affections formed.

My thoughts are still developing on this topic, but for now I can leave you with this fact: as a Christian educator (be it in an explicitly Christian setting or even when I was in public school), I care less about what my students know, and more about what my students love. This is the purpose of education.

Some books that I have been reading that have influenced my thinking on this issue:

Desiring the Kingdom: Worship, Worldview, and Cultural Formation, by James K.A. Smith.

The Case for Classical Christian Education, by Douglas Wilson.

Mathematical Affections: Assessing Values in the Math Classroom

Presented at

19th ACMS Conference

Bethel University, 2013

math affections(Click on the image above for the presentation slides)

I. The Need for Affective Learning

How many of you, as math educators, have heard the question “When am I ever going to use this?” be uttered by your students? If you have been teaching for more than 5 minutes then it’s safe to assume that phrase has been mentioned in your presence. Occasionally it is posed as a valid question; the student is genuinely interested in the future career application of the topic at hand. However, I believe the majority of the time the phrase “When am I ever going to use this?” is spoken it is not as a question, but as a statement. A statement which implies that the obvious answer is “I will never use this so learning it is a waste of time.” The real issue being raised by these students is not one of application, but rather one of values. If we could translate their question into what they are really trying to communicate then “When am I ever going to use this?” will become “Why should I value this?” Students express their inquiry in terms of mathematical practicality because that is the language in which their culture, including their math teachers, has conditioned them to speak.

To illustrate how we as math educators have contributed to this misconception that value equals utility, let us turn our attention to the foundational document for composing the learning objectives and outcomes of an academic course: Bloom’s Taxonomy (pictured below).

Picture1

A quick glance at this chart will reveal that ‘application’ falls under the cognitive (mental/knowledge) domain of learning while ‘valuing’ falls under the affective (heart/feeling) domain of learning. The cognitive domain is almost exclusively emphasized in the preparation of teachers within the modern educational system while the affective domain is largely ignored. So while we ‘improve’ our teaching and questioning to make mathematics less abstract and to focus on real-life applications so that we can address the question of “When am I ever going to use this?” before it is even asked, we are actually implicitly teaching students that mathematical value is to be found only in application. If we really want to help those students address the true foundational question of “Why should I value this?” then we need to do so through increasing our attention on the affective domain of learning; writing rigorous learning objectives and developing quality assessments just as we do for the cognitive domain.

Now, application is certainly useful in the teaching process and it should not be ignored. I am not advocating the promotion of the affective domain over and above the cognitive. My goal is to simply bring the affective up to the same level as the cognitive. The best learning is done when both domains are utilized in conjunction with each other. In The Abolition of Man, C.S. Lewis writes “Education without values, as useful as it is, tends to make man a more clever devil.” I believe this is a fairly accurate statement of the modern day system of education. If we don’t focus on values, if we don’t focus on the affective learning of our students, then their education will still be useful – they’ll increase in cognitive ability and learn to apply their thinking. But is that really valuable in and of itself? Without a proper sense of values to guide their application, aren’t we really just making students “more clever devils”? You see, you can never actually remove values from education. Education is inherently value laden, and I believe Lewis knew this. It is not a question of “Are you teaching values?” but rather “Which values are you teaching?” Lewis’ point is that the value we instill in education should be affective – loving learning for its own sake and valuing wisdom. If you don’t focus on affections, then you still have usefulness, but is that really beneficial? In the words of the Bishop in Victor Hugo’s Les Misérables: “The beautiful is as useful as the useful…Perhaps more so.” Application is indeed useful but it should be presented in a way that promotes the development of what I’ll term mathematical affections. Learning has little meaning unless it produces a sustained and substantial influence not only on the way people think, but also on how they act and feel.

II. The Place of Affective Learning in the Math Classroom

Let me take a moment to define what I mean by mathematical affections. The title of this talk is in homage to Jonathan Edwards’ Treatise Concerning Religious Affections. Edwards’ goal was to discern the true nature of religion and in so doing dissuade his congregation from merely participating in a Christian culture (a mimicked outward expression) and motivate them to long for true Christian conversion (an inward reality of authentic Christian character). The purpose of this talk is to engage us as math educators in discerning the true nature of mathematical assessment and how we use it in the classroom: does it simply mimic the modern culture of utility by requiring outward demonstrations of knowledge retention and application, or does it aim deeper at analyzing true inward character formation? For Edwards, affections were not synonymous with emotions as they tend to be in today’s culture. Because today’s culture sees affective learning as simply an emotional state the culture classifies affective learning as a purely subjective domain and therefore not worthy of developing objective standards or assessments. But Edwards understood affections as aesthetics – a way of orienting your life via a mechanism that determines what was beautiful and worthwhile. If we see affections as character producing then Edwards’ definition leads to a more objective perspective that provides more potential for assessment rather than viewing affections as emotions.

It is Edwards’ definition of affections (orientation of life, determining worth) that actually appears in popular math education literature. According to the NCTM Standards for Teaching Mathematics, “Being mathematically literate includes having an appreciation of the value and beauty of mathematics as well as being able and inclined to appraise and use quantitative information.” Adding it Up: Helping Children Learn Mathematics, a report published by the National Research Council argues that mathematical proficiency has five strands, one of which is termed “productive disposition.” Productive disposition is defined as “the habitual inclination to see mathematics as sensible, useful, and worthwhile.” Both of these foundational documents in the area of math education plainly portray mathematics as beautiful, of value, and affecting the habits of the learner to see math as worthwhile. However, neither of these documents develops how we as teachers are to go about accomplishing this task. It is almost as if these phrases are included in these documents as a courtesy – as a way of saying “this is how we as math teachers feel about math and it would be nice for our students to feel this way too, but feeling is subjective so there is no real way for us to objectively instruct or assess students in this regard.”

This is a point of connection that we as Christian educators can make with educational system as a whole – we can answer the questions of how. We have much to contribute here and we don’t have to be overtly religious in the presentation. What we should really be emphasizing is the classical approach to mathematics. The mission statement of the classical Christian school where I teach states that: “The mission of Regents School is to provide a classical and Christian education, founded upon and informed by a Christian worldview, that equips students to know, love and practice that which is true, good and beautiful, and challenges them to strive for excellence as they live purposefully and intelligently in the service of God and man.” Non-Christians will obviously reject the beginning and end pf that statement but the key phrase here for the purposes of contributing to fixing the problem in education is “equipping students to know, love and practice that which is true, good and beautiful.” The true, The Good, The Beautiful – these are ideas from Plato who believed in a rigorous development of mathematics. Legend has it that above the door to his academy was the phrase “let no one ignorant of Geometry enter here.”

Beyond Plato, the truth, goodness, and beauty of mathematics has been attested throughout history by famous mathematicians who were not espousing a religious view. In regards to truth, Richard Feynman said “To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.” In other words, the truths of mathematics are written into the very core of nature. In regards to goodness John Von Neumann wrote “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” I believe no one would argue the good that comes from learning mathematics. There would not be nearly the outcry that we see today over fixing the educational system and increasing our place in mathematics scores if people did not think knowing math was good. And finally, in terms of beauty, G.H. Hardy wrote “Beauty is the first test: there is no permanent place in the world for ugly mathematics.” Every mathematician will attest to the beauty of mathematics. If we can argue that mathematics is objectively true, good, and beautiful, then I believe we can argue that these qualities, which contribute so greatly to our affections, can be more objectively assessed.

III. Cultivating Mathematical Affections

According to the NCTM Assessment Standards for School Mathematics: “It is through assessment that we communicate to students what mathematics are valued.” If we are going to increase our attention on the affective domain of learning we need to do so by writing rigorous learning objectives and developing quality assessments just as we do for the cognitive domain. Assessment is always tied to our objectives; by definition objectives are quantifiable learning outcomes of a lesson. Below, I would like to offer some examples of learning objectives in a statistics course followed by a list of methods for assessing affective learning that very from traditional cognitive assessments.

The following examples of affective objectives use the current taxonomy of the affective domain of learning developed by Bloom and Krathwohl (middle column of the diagram above). I am not endorsing this taxonomy, as I believe there are improvements that can be made to it. For now though, I would like to propose how we can start working within the current system. Notice that in each objective there is an affective verb, a cognitive verb, and a vehicle for assessment.

  • Receiving: The student will (TSW) differentiate (affective verb – AV) between valid and fallacious statistical arguments and argue (cognitive verb – CV) their reasons in a written response (method of assessment – MOA). Assessment should account for the initial discernment between truth and falsehood in addition to the correctness of the argument.
  • Responding: TSW engage (AV) in class discussion (MOA) comparing (CV) and contrasting (CV) religious and statistical knowledge. Assessment should account for the level of engagement in addition to the determination of proper similarities and differences.
  • Valuing: TSW support (AV) the mission of a local non-profit organization through the design (CV) of a statistical study (MOA) done on the organization’s behalf. Assessment should require student to defend the worthiness of the cause motivating the project and not simply report valid data analysis.
  • Organizing: TSW define the limitations (AV) of statistical inference procedures and accordingly make recommendations (CV) to the acting agency in an oral presentation (MOA).Assessment should account for recognition of shortcomings (humility) in the oral presentation.
  • Characterizing: TSW be evaluated (CV) in their intellectual integrity and rated positively by their peers (AV) through a written reflection survey (MOA).Assessment allows for students to communicate their personal reflection and evaluation of others.

The key to the above objectives is the integration of affective and cognitive learning. We need to be creative in our assessment techniques to allow the affective domain an opportunity to be assessed. Some examples of assessments include (but are not limited to): Math Journals, Reflection Assignments, Personal Interviews, Class Discussion/Debate, Oral Presentations, Open-Ended Group Problems, Historical Reading and Response, and Service-Learning Projects. If you are still under the belief that textbook assignments, quizzes, and tests provide a more objective measurement of student ability, let me pose the following questions: Who chooses what problems to assign? Who writes the quiz or test? Who grades the quiz or test? How is partial credit handled? I can ask many more questions in a similar vein that will hopefully allow us to realize that even the cognitive skills that we assess have some level of subjectivity involved. I am not arguing that affective learning is completely objective, but it is at least as equally objective as the cognitive domain.

There is much work to be done in this area. I hope that I have convinced you of the need to engage whole-heartedly into this work. For now I will leave you with a student quote on the impact of affective learning: “I was more dedicated because I saw a deeper purpose.”