The Mathematics of Faith

I subbed in a high school math classroom last week. The students were incredibly well-behaved and independent with their work, which meant that I didn’t need to hover over them to keep them working. Since I’m not one to pass up an opportunity to learn, I picked up a continuing education tome from the teacher’s desk and started to read. The book Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching by Jo Boaler from Stanford University was the initial inspiration for this post, supplemented and enhanced a day later on October 7 by a segment in the August 26, 2016, episode of Science Friday that covered the same ideas, with an emphasis on the downstream implications of entire generations of students who have been blinded to the beauty of math by gatekeeping and poor pedagogy.

Quick: What’s 18x5?

How did you get the answer?

Perhaps you simply “knew” the answer because somewhere along the line, you memorized the multiplication tables far enough to know that 18x5=90.

Maybe you solved it the “old fashioned” way:

18

x5

40

5_

90

Or, perhaps, you used one of these methods:

a) 18=10+8 (10x5)+(8x5)=50+40=90

b) 20-18=2 (20x5)-(2x5)=100-10=90

c) 18=9x2 (9x5)=45 45x2=90

d) 18=2x3x3 (5x2)(3x3)=10x9=90

e) 18+18+18+18+18=90

f) 5 groups of 18
eeeeeeeeeeeeeeeeee

eeeeeeeeeeeeeeeeee

eeeeeeeeeeeeeeeeee

eeeeeeeeeeeeeeeeee

eeeeeeeeeeeeeeeeee

g) 5 groups of 9 + 5 groups of 9
nnnnnnnnn nnnnnnnnn

nnnnnnnnn nnnnnnnnn

nnnnnnnnn nnnnnnnnn

nnnnnnnnn nnnnnnnnn

nnnnnnnnn nnnnnnnnn

h) 8 groups of 5 + 10 groups of 5
eeeee ttttt

eeeee ttttt

eeeee ttttt

eeeee ttttt

eeeee ttttt

eeeee ttttt

eeeee ttttt

eeeee ttttt

          ttttt

          ttttt

Here’s the “magic” of mathematics: ALL of these are “right” ways to arrive at the answer. And this isn’t even close to all the possible ways to arrive at the right answer; if I had any graphic design program, I could show at least a dozen more. My “ah-ha” moment was the same as that of more than one person at a corporate workshop Jo Boaler led at a Silicon Valley IT startup, which led to a change in the atmosphere of the whole company because exploring how people arrived at 90 as the answer revealed much about the way that different people think.

Boaler’s contention, and that of many other scholars based on multiple studies of attitude and aptitude across generations, is that mathematics has always been a siloed field of study. The problem with math for most of us is that if we didn’t show “natural” talent for computation at a fairly early age, we weren’t encouraged to stick with the study of math past algebra II—or maybe just geometry. Such discouragement, both subtle and overt, leads to multiple problems at the university level, where the field is overwhelmingly male, Caucasian, and Asian based on stereotypes. It also contributes to a lack of diversity in science and engineering fields (the thrust of the Science Friday segment). If you can’t “do” math quickly and accurately by middle school, the old guard insisted (and may in some circumstances still insist), you will never be “good” at math so might as well not try. As Boaler says, imagine if we gave up on writing or reading the same way…

Thus we have “common core” math in an effort to change the way we teach children and teens about mathematics. However, even if we did go through trigonometry and calculus, our children’s math is beyond us because if we graduated from high school before 2005 or so, we’re totally lost with this “common core” math. What would take us 4-6 lines eats up sheets of paper. Number lines to add and subtract large numbers? Arrays in 2nd grade? Grids (examples f, g, & h)? Real world word problems? Factoring to break the problem into smaller, easier steps (c & d, f & g in grid form) Rounding and grouping to arrive at correct answers instead of estimates (examples a & b, h in grid form)? What is this madness?

It’s numeracy, or “number sense.” One of the things that few of us who are 30 or older learned is that mathematics is about the patterns that numbers create when we play with them. Sure, we may have learned the very cool multiple of 9 trick: 09, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180…(and if you didn’t learn it then, look closely and then predict the next ten numbers in the sequence without doing the addition or multiplication, answer below). Maybe we learned about the Fibonacci Sequence before Dan Brown’s The DaVinci Code came out: 1, 1, 2, 3, 5, 8, 13, 21… I don’t remember hearing about fractals until just a few years ago, though I’m sure along the way I had reason to see and perhaps even describe them. Number sense opens up whole new vistas of understanding that can demolish the feeling of failure many of us have about math; just since February, when I started subbing, I’ve found myself doing math differently even though I’ve always been reasonably good at it (I just didn’t like it very much, though all my math teachers were good for the era). Numeracy allows us to solve 18x5 any number (get it?) of ways.

In an earlier post, The Physics of Faith, I wrote, “Faith at its best encourages questioning in the midst of doubts and has multiple answers to every problem of life.” Here I’d like to propose that math can teach us about multiple ways to solve the problems of life that get like groups to the same answer. Not only are there multiple answers to those problems, but there are also multiple ways to arrive at common answers. If physics helps to explain the number of different religious systems in the world, then mathematics helps to explain the variety of expressions within each religion: sects (or denominations, which is a softer way of noting the divisions).

Some sects are dogmatic: the answer is X and you arrive at this answer by following one procedure and one procedure only. All other procedures are false and only give the illusion of arriving at the answer of X. Translated to religious terms (I’ll use Christianity since it is my problem set of choice), Jesus is the answer and one only has the answer right if one behaves exactly this way (e.g., calling out what the sect sees as sin and falsehood aggressively regardless of who is sinning or lying); profess exactly this set of creedal statements—nothing less, nothing more, for example, that women are not allowed to preach at all or to teach men or boys over the age of 12; and devote at least so much of one’s time, talent, and treasure to the sect (which is sometimes all of the above, of course). This is religion, not faith, and in many cases, those who don’t adhere to the one true path to the answer are considered outsiders and condemned to eternal separation from God. Mathematically, it’s as though these groups say 18x5=90 but 5x18 does not. 

Other sects are less dogmatic but still circumscribe acceptable behavior, belief, and devotion. These groups recognize that other groups get the same answer in different ways, though only certain ways of arriving at that answer are fully acceptable. For example, one simple but tremendously incomplete understanding of the complex relationship among Roman Catholic and Orthodox church bodies today lies in some common beliefs: Jesus is the answer but one only has the answer acceptably if one does not have a female pastor (though women teachers are allowed and may even be encouraged); believes that the bread and wine become the literal body and blood of Jesus Christ; and devotes time in prayer and service, talent in time and service, and treasure in generous amounts to every appeal. Yet even with these similarities, issues of supremacy (is the Pope “first among equals” or “first” or even “among equals” with the Patriarchs?) and mutual recognition of ministry still cause friction. These are less religious and allow for more personal faith, though limited in acceptance of personal experience and belief over dogma. 18x5=5x18, but 18x5 may be the better way to get the answer.

More open sects allow that there are multiple ways to arrive at the same answer but demand adherence to creedal statements and particular behavioral standards for their members, though what they demand in devotion is often less onerous that the stricter sects. They recognize other sects as Christian though they expect such sects to adhere to the traditional teachings, if not all the practices. There is much more room for personal experience and belief in concert with dogma as we move away from religion toward personal and community faith. To put it in mathematical terms, these groups endorse only equations as valid: 18x5=5x18=18+18+18+18+18=45x2=5x3x3x2=(10x5)+(8x5)=90, while viewing grids and arrays and other visual representations as suspicious because these methods have too much flexibility for comfort.

And then there are the Christian sects which invite people to use whatever way works to get them to the uniquely Christian answer. These are folks who have their own paths but who recognize and value the paths of everyone else who seeks the uniquely Christian answer. Rather than imposing their own or a community’s beliefs on others, they tend to gather in communities where personal experiences and beliefs bring people together to work for common goals. This is where I see the ideal of the United Church of Christ (UCC) as an expression of Christianity. Our tagline is “No matter who you are or where you are on life’s journey, you are welcome here.” In mathematical terms, we might say, “No matter whether you use number sentences, arrays, graphs, diagrams, or your fingers to work out that 18x5=90, you are welcome here.”

I have observed that those who are most open to multiple ways of solving a problem in life with a discrete answer are also those who are most open to engaging those who are solving different problems in life. Those open to multiple procedures often recognize that those who are solving different problems use many of the same procedures: the many ways we can solve 18x5 can also be used to solve 17x5 or 19x5 or 18x7 or 34x8 or 42/7 (and anyone who knows my love of all things absurd will understand how tempted I was to use 42 as the answer and all the various ways to arrive at that answer as the example in this post!) or the most complex of calculus problems. The answers may be different, but that doesn’t mean they’re unrelated.

If you are a parent struggling to understand your child’s math homework, set aside for a little while the idea that the way you learned to do math is the only way. Mathematics is much more playful and joyful today than most of us were ever invited to imagine it as being. Forget timed facts tests and memorized formulae. Draw pictures, shade in boxes, play with objects, whatever you need to do to see the beauty of the patterns rather than just row after row of equations. 

That applies to a life of faith, as well. If you’re stuck thinking that there’s only one way to believe but that way isn’t working for you, take a step back to take in the depth and breadth of ways to find the answer. Find a way that invites joy and play rather than tedium and rigidity. We know now what we didn’t when we were young: it’s not how you find the answer that matters; it’s that you find an answer that changes everything. What’s yours? Everyone has one, even if it’s no belief at all. Being in relationship with God through Jesus Christ is my answer and I welcome anyone who wants to explore that answer to join me…I can even tutor using multiple solution options!

PS: 189, 198, 207, 216, 225, 234, 243, 252, 261, 270. The pattern continues ad infinitum: 18909, 18918…20772, 20781…27000, 27009…36045, 36054…45027, 45036…123456789, 123456798, 123456807… There’s a pattern with 11, as well: 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220… 

PPS: If you were ever told you were bad at math, take a deep breath and know that you really aren’t bad at math, you just weren’t taught well because most of our teachers, as awesome as many of them were, were also taught badly. You really do use math every day, it’s just that no one made the connection between what we learned in the classroom and what we do to drive and park our cars, arrange furniture, plan menus, and figure out our daily and weekly schedules. The best part about common core, for all the problems it has caused in implementation, is this connection between mathematics and real life.