Thanks to Jacob Mohler for sharing this video with me. This call to embody mathematics comes from Satyan Devadoss, Fletcher Jones Professor of Applied Mathematics at the University of San Diego. The video is from the 2018 Culture Care Summit, where the Fuller Theological Seminary Brehm Center director Mako Fujimura hosted a conversation on culture care in education, journalism, and more. Learn more about the Brehm Center here: www.brehmcenter.com.
This is a presentation I gave at the 2019 Society for Classical Learning Conference (an updated version of an earlier post). You can click the image above to download the slides I used in the presentation.
The title of my talk was “Math is _________ .” In introducing the talk I let that title just linger there for a while, asking the audience to consider what words or phrases come to their mind for filling in the blank. As the speaker I also enjoy soaking in the facial reactions of each member of the audience when it is announced that this next 45 minute talk will be about math. I contend that 100% of people (and as a statistics teacher, it means something when I say 100%) have a memorable, visceral experience from a math class. There are no neutral expressions on the faces of audience members. The sad thing from my perspective as a math teacher is that the majority of those memorable experiences are negative. My hope in giving this talk was to encourage people to consider some new words for the blank that they maybe had not thought of before.
I start by offering some familiar suggestions for the blank (familiar at least to our Classical Christian context where we teach). Here is the mission statement of Regents School of Austin:
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.
The bold emphasis is mine to point out a few words that might fit in the blank.
Math is TRUE. This isn’t something I need to sell people on. I mean, 2 + 2 = 4 every time, right? To take it a step further though, I encouraged people to consider some ideas I put forward in another post: God, Math, and Order.
“To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology.”
~ Hilda Phoebe Hudson
When discussing mathematics from a Christian perspective, one statement that always seem to bubble to the top of the conversation is that mathematics reveals God as a God of order. This is true. This is also way underselling the connection between God and math.
Does God use mathematics because He is a God of order or does math have order because God uses it? I would argue that order is not a characteristic God displays but a quality that He defines by His nature and math gives us a glimpse into that nature. “Our God is a God of order” – By this claim we shouldn’t merely mean that God acts in an orderly fashion. We should mean God defines what an orderly fashion is. Order is not a quality God decided to portray, rather order flows from His nature.
If this can become our perspective, then when we speak of mathematics portraying God as a God of order, that description will carry so much more meaning. Instead of just correlating our mathematical results with some quality that God displays, we can realize those results are better understood as a manifestation of God’s nature. In a way we are communing with Him in our work as mathematicians, gaining deeper insight into His character.
Math is BEAUTIFUL. This is another category that I don’t have to do much convincing on. So many people have put together so many amazing presentations on the beauty of mathematics that any rational person could be convinced of math being beautiful after a quick Google search. Here is one my favorite videos in this regard and a few quotes.
“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
~ G.H. Hardy, A Mathematician’s Apology
“The mathematical sciences particularly exhibit order, symmetry, and limitation; these are the greatest forms of the beautiful.”
Math is GOOD?
Here is where the sell gets a little harder. As I mentioned above, a lot of people associate very negative things with math class. When asked to complete the phrase “Math is _________” they may think of words like “stressful,” “confusing,” “too abstract,” “not applicable to me,” or “the exact opposite of all that is good and holy.” Here is where I focused the remainder of my talk in an attempt to get anyone who fell into this boat to start seeing math in a different way.
Whenever I am presenting at conferences I like to do the following exercise: I ask people what the number one question asked in math class is, and without fail I always hear back “when am I ever going to use this?” The reality is that this is not a question, it is a statement. It is a statement of confusion and frustration. In other words the answer to “when am I ever going to use this?” has already formed in the student’s mind as “I am never going to use this” and then they withdraw from the mental activity at hand.
I would argue that what a student is really asking is “why should I value this?” It is not a question of finding application but of finding meaning. Maybe another rephrasing would be “why is this worth learning?” As Christian educators this deep longing should be familiar. If we believe Augustine in the Confessions that “Thou hast made us for thyself, O Lord, and our heart is restless until it finds its rest in thee,” then that doesn’t stop when students walk into math class. The most fundamental thing that is happening in math class is that students are seeking value (something we as teachers need to address in our curriculum) and are seeking to be valued (something we as teachers need to address in our pedagogy). In other words, the foundational issue of math class is an affective one as opposed to a cognitive one.
Affective issues are just present for students but for teachers as well. Another exercise I do at conference presentations: I ask people to close their eyes an imagine their best/ideal teaching moments (the O Captain My Captain moments). I then ask volunteers to share a word or phrase that describes that moment. Not once in all my years of doing this has a teacher mentioned anything about content. The language that is used is always affective – “engaging,” “curious,” “joyful.” Don’t get me wrong – I know the content was still there in the lesson and probably operating at a high level to produce those affective moments. The point of the exercise is simply to illuminate how central issues of affect are to the math classroom.
This is not just an anecdotal observation, but it is also affirmed in educational research:
When teachers talk about their mathematics classes, they seem just as likely to mention their students’ enthusiasm or hostility toward mathematics as to report their cognitive achievements.
Similarly, inquiries of students are just as likely to produce affective as cognitive responses, comments about liking (or hating) mathematics are as common as reports of instructional activities.
Affective issues play a central role in mathematics learning and instruction.
~ Douglas McLeod in Handbook of research on mathematics teaching and learning (1992)
It is also affirmed in national policy documents on math education (even though those documents never really develop how to go about obtaining these stated results – hence the motivation for my dissertation).
“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.”
~ NCTM Standards for Teaching Mathematics
“Mathematical proficiency has five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Productive disposition is the habitual inclination to see mathematics as sensible, useful, and worthwhile.”
~ Adding it Up: Helping Children Learn Mathematics (National Research Council)
OK – so students and teachers both would admit that affect plays an important role in math education, this is supported by research, and it is affirmed in national policy documents and recommendations. With all of this motivation how are we (math teachers) doing?
As it stands our current methods of teaching mathematics are producing untold numbers of students who see mathematics more about natural ability rather than effort, who are willing to accept poor performance in mathematics, who often openly proclaim their ignorance of math without embarrassment, and who treat their lack of accomplishment in mathematics as permanent state over which they have little control.
~ McLeod (1992)
This quote may seem a little dated as far as research goes but I think it perfectly sums up the situation. No matter how dated the quote is, I know this is still true today because… well, I’m a math teacher. Plus I introduce myself to people in social situations. Other math teachers will quickly confirm this: whenever you meet someone and they ask “what do you do?” and you respond “I teach math” the next response will typically be something like “I was never any good at math.”
Math teachers are probably second only to priests in terms of the number of confessions we take from people.
Also, through these conversations it gets revealed that what these people really didn’t like about math were more factors of the math classroom schooling environment than the discipline of math itself. To me though, these actions are very foreign to the actual discipline of mathematics. For instance, people might say “I hated memorizing all of those formulas.” No mathematician would describe math as memorizing formulas. In essence what these people are doing is gossiping about math.
It as if they are saying “My friend’s cousin’s roommate’s teacher said that math is a jerk. He saw math behind the bleachers making out with history behind science’s back. No thank you – I want no part of math.”
To which I’d have to say “First, math is everywhere so math is probably making out with all of the subjects. Second, have you actually met math? Maybe you should talk to math face to face to sort this out.”
I think people have this false perception of what mathematics is because their experience of the math classroom was through forced, awkward, artificial “relevance” of math topics. For example:
Like most word problems it successfully takes and interesting event/story and kills it dead by now making it a chore for students to slog through. The artificial relevance of this problem makes it seem as if Steph Curry does this before he shoots:
So how do we avoid this artificial relevance? How do we teach math differently at Regents? Because we teach at a Christian school, does that mean 2 + 2 = Jesus now? Here I had to share some thoughts from another previous post: 2+2=Jesus? Ultimately this type of question fails to see math as anything more than calculations. Math has calculations, but it is more than that. The question also sees Christianity as simply a new way of thinking (Jesus is now the answer to everything). As a pastor of mine would always say:
Christianity is always more than thinking, but never less.
~ Neil Tomba, Senior Pastor, Northwest Bible Church, Dallas, TX
A better understanding of how Regents approaches math differently can be summed up in the following quote:
“If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.”
~ Antoine de Saint-Exupery
I pulled this from the first page of A Mathematician’s Lament, by Paul Lockhart, and it hits on the deeply affective aspect of what we do as teachers. I also proceed to read the opening imagery of Lockhart’s initial paper to communicate to everyone in the audience how we as math teachers feel about what often is described as math teaching.
So how do we go about cultivating mathematical affections? Well, I’ve written a lot about that here, but to quickly summarize:
Education is not primarily a heady project concerned with providing information; rather, education is most fundamentally a matter of formation, a task of shaping and creating a certain kind of people…. What makes them a distinctive kind of people is what they love or desire or value….. An education, then, is a constellation of practices, rituals, and routines that inculcates a particular vision of the good life by inscribing or infusing that vision into the heart (the gut) by means of material, embodied practices…. There is no neutral, nonformative education.
~ James K.A. Smith, Desiring the Kingdom (2009)
Mathematics educators who set out to modify existing, strongly-held belief structures of their students are not likely to be successful addressing only the content of their students’ beliefs…it will be important to provide experiences that are sufficiently rich, varied, and powerful in their emotional content.
~ G.A. Goldin in Beliefs: A hidden variable in mathematics education? (2002)
In other words, it is the practices of the math classrooms that shape mathematical affections. So I challenged the teachers in the audience to consider:
- Students want to know your story…
- What are the touchstone moments you can recall from a math classroom?
- What would you say are the “thick” practices/routines/liturgies of a math classroom?
- How has your experience of those practices shaped your perspective of mathematics?
- In light of our own experience of mathematics how do we work to shape our students’ experience of mathematics? How do we cultivate their mathematical affections?
(Side note here: there was a great talk at the most recent ACMS conference on relating the problem solving aspects of mathematics to the cultivation of virtue – hopefully you’ll be able to access that soon).
To help answer these questions, I closed by offering new words to fill in the blank of “Math is ____________ .”
Math is INVITING.
Here I got to share about my role as an ambassador for the Global Math Project. First, an introductory video:
I challenged people to tell me how this math problem was different than the cow-jumping math problem above. A couple of different responses: this one makes you curious – you want to solve it. This problem has no words only images. This problem makes you ask questions rather than asking them for you.
Often our invitation into mathematics is already excluding some students. The words or terminology we use to introduce the problem may already shut people down. I’m not saying we shouldn’t use proper terminology, I am just asking to consider if it is always necessary. For instance, this example is teaching binary numbers but that term is never used. We don’t start by telling students “Ok, let’s learn the properties of binary numbers.” We have this interesting video instead. I would rather students understand the underlying concepts and connections (which this video very clearly portrays) than parrot terminology without understanding.
Math is RELATING.
There should be a relational component to every math lesson – math is not meant to be done in isolation, but in community. But here I am also intending the word “relating” to not only refer to relating to others, but also to God.
I have written elsewhere about Flatland: A project of many dimensions. I love Flatland, and occasionally you’ll even see it’s title referenced as a parable. I love taking students through the though process of what would it look like for a sphere to enter into the 2D world of Flatland. In Flatland the inhabitants would only see the sphere as a circular cross section, completely unaware of the concept of a 3D sphere outside their literal plane of existence.
In this way the concept of dimension in mathematics offers a great analogy for issues of faith. How can Jesus be both fully God and fully man? Well, maybe it is kinda like the sphere still being a sphere but also a circle. I also love the illustration put forward in the chapter on dimension in Mathematics Through the Eyes of Faith about what if a hand entered Flatland. As your fingers went into Flatland the inhabitants would see cross sectional circles, none of which are connected. But if they could zoom out to 3D space then they could see fingers all connected to one hand. Maybe this can help us understand how the church is composed of many separate members but is still referred to as one body.
Finally there is Salvador Dali’s Crucifixion (Corpus Hypercubus). Just as a 3D cube can unfold into a 2D net that appears like a cross, a 4D hypercube can unfold into the 3D net seen above. Dali is speaking of the mystery of the crucifixion – of something that originated in a higher plane of existence unfolding itself into our world. These moments of insight for our students are made richer by the use of math as an analogy for faith.
Math is JOURNEYING.
I have written on this point in a post entitled “Is it the journey or the arrival?” Spoiler alert: it’s the journey.
I have probably read the same argument a dozen times. The language and the nuances of the argument very only slightly between articles. These are just the two most recent articles to cross my path:
The basic synopsis: not everyone needs calculus. Stop making calculus the end goal of the K-12 math sequence. Everyone needs a better grasp of data analysis and digital technology. Teach more statistics and computer science. Teach students the math that they really need.
Typically these articles focus on replacing AP Calculus with AP Statistics and/or AP Computer Science or some other programming course. Each article on this topic does note the exception that those interested in STEM fields do actually need Calculus. These articles are typically framed around what everyone else should take – how to offer better (framed as more useful) math for the “I’m not a math person” crowd.
Perhaps what is really underlying these articles that are making the argument of teaching more “useful” mathematics is that these articles are less a commentary on the content in the mathematics curriculum and more a commentary on (perceived) pedagogies in the mathematics curriculum. In other words, perhaps the argument is less about what is taught and it is more about how it is taught.
Disenchantment with the traditional teaching methods employed have left people looking to jump to the practical applications without journeying through foundations to get there.
I have thoroughly enjoyed reading through Teaching and Christian Imagination. In it, David Smith uses the metaphors of journeying, building, and gardening to reflect on educational practices in light of Christian teaching. Below is an extensive quote from his section on journeying that I believe will be very instructive:
The journey metaphor offers us a different picture of the learner than the passive receptacle. And yet it still leaves the nature and purpose of the journey open for debate. As educational history has walked hand in hand with cultural history, imagery associated with educational journeys has shifted from travel on foot to riding in a coach and then to driving along a highway. In older Christian appropriations of the image, the path was given by God and led (at a more deliberate and deliberative pace) towards God as its destination. In the Enlightenment, the sense of destination remained, but the goal was reframed in terms of movement towards the virtuous life of the useful citizen. As travel became more widely available, the idea of education opening up new horizons took hold. The image of the 19th century explorer offered a version of travel as deliberatively leaving the well-trodden path and collecting new experiences in exotic, uncharted territories. Later still, the rise of mass tourism tilted the image of travel towards comfort, efficiency, and consumption, evoking anxieties concerning educational tourists whose shallow gaze skims the main sights but does not linger for long enough to be changed. The educational path is now giving way to talk of an educaitonal superhighway with a powerful emphasis on speed of information. Alongside these shifts came a gradual yet momentous reversal in which the experience of journeying itself overtook the pursuit of a hallowed destination as the central emphasis; simply being in motion at increasing speed and with increasing range became an end in itself. Eventually, with the fading of a shared destination, any self-chosen destination became equally valid.
Any self-chosen destination becomes equally valid – be in calculus or statistics or whatever else appears most “useful.”
In contrast to this modern perspective, Smith goes on to discuss the concept of journeying on a pilgrimage in the Biblical narrative, particularly in the Old Testament pilgrimages to the temple.
To read Scripture is to encounter on a regular basis people leaving the security of home and setting out into the unknown. (p. 19)
The worshipers find their strength not ultimately in the place of worship, but in the one worshiped there, who is with them on the road as well as in the sanctuary… It is not a journey from where God is not to where he is, but a celebration of God’s rule over the entire land. (p. 24)
Blessing is not tied to arrival… They (the pilgrims) doggedly seek blessing, practice works of mercy, and erect signs of the kingdom. Treading a pilgrim path involves placing oneself within a tradition… The paths were not already cut into the landscape, but had to be made and maintained by walking. (p. 25)
The individual pilgrim learns the path both from elders who passed this way in previous years, and by walking, by going along it for the first time and gaining a familiarity that might lead to becoming a future guide for others. (pp. 25-26)
It is a journey not towards spring break, towards a strong grade point average, or towards employment, but towards standing in the presence of God and seeing God give new life to the world… God’s glory fills creation, and setting our faces towards God and our hearts on the highway is a celebration of God’s sovereignty over every territory through which we pass. (p. 26)
In these arguments over content and methodology, destinations and journeying, the what and the how, I find it refreshing as a Christian math teacher to recall that God is not just the author of the content but also how we come to know it. There is something the journey is meant to do to us – it is not simply meant to be endured until reaching a destination.
Is factoring trinomials presented in the classroom as a task to be endured until students can reach the more useful destination of the Consumer Price Index? Or is there a way that we as math teachers can reshape our classroom and reframe our teaching methods so that students experience the value of the journey?
Math is SERVING.
I have written a lot here about service-learning in mathematics. I won’t expand here, I’ll just summarize why I think teacher’s should consider service-learning:
- Affective learning objective is primary
- Cognitive learning objective is still present and operating at a high level
- Opportunity to communicate the value of affective learning outcomes through assessment
- “It is through our assessment that we communicate most clearly to students which activities and learning outcomes we value.”
- David J. Clarke, NCTM Assessment Standards for School Mathematics
- Reflection is key
- Moves toward inculcating a servant’s heart
Here is a quote from a student who would adamantly describe themselves as “not a math person” at the end of the year after going through a service learning project. When asked if they think their attitude towards math has become more positive:
“Yeah, definitely, much more positive. It was hard, don’t get me wrong and I’m not saying the ‘I’m no good at math thing’ didn’t change, but I do think … I am sure that I can learn it, because I am sure I can learn it. It just will take longer and when you don’t feel so completely discouraged about it … When you do feel that you do have shot to understand it and learn it, for me at least it really raises my attitude towards it. It doesn’t feel like it’s this hopeless thing that I just have to suffer through. It is kind-of just a hill you climb, right?”
I like this quote because it is honest. The point of cultivating mathematical affection is not to have every student now love math and have it be their favorite subject. The hope is that students who once saw math as this hopeless thing to be endured now see value in working hard at it. They start to see why they should value math.
Finally, returning to the Global Math Project as the inspiration for my talk in the first place:
Math is UPLIFTING.
I love that the motivation behind the Global Math Project is to change people’s experience of mathematics. I would love to see the students who would say that math is “confusing” and “stressful” now start to use words like “uplifting” to describe mathematics.
Returning to the Regents mission statement:
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 (inviting) to strive for excellence as they live purposefully and intelligently (journeying) in the service of God and man (relating).
Notice the new points of emphasis. Sometimes focusing on math as true, good, and beautiful can still be an abstract exercise. Let’s start looking for math after the comma. Let’s look at the experience students are having of mathematics. Let’s care about the practices and liturgies of the math classroom so we can impact the mathematical affections of students.
If you are interested in starting the conversation with math (and leaving gossiping about math behind) I ended but sharing two great talks given by Francis Su, former president of the Mathematical Association of America:
AP and the College Board have partnered with WE to bring WE’s internationally recognized service-based learning framework and resources to AP courses, so students can use what they’re learning to tackle real-life social issues and challenges.service-learning to the AP classroom. Microsoft hosted the inaugural AP with WE Service Teacher Summit at their headquarters in Redmond, WA, March 4-5, 2019.
This event was an opportunity for teachers to meet fellow AP teachers, strengthen their implementation of AP with WE Service, learn best practices, and hear from Microsoft staff. A total of 50 teachers were selected to attend.
What follows below is the outline of the presentation I was asked to give at the summit on cultivating student affections through service-learning. I hope to be able to share the video of the presentation soon.
My name is Josh Wilkerson and I teach AP Statistics. I also teach other math courses and as a math teacher there is one big question that I am often asked. You are probably asked it to in your other disciplines, but it is especially prevalent in math. The question usually is accompanied by the student having his kind of expression…
Help me out, what is the big question? (Audience: when am I ever going to use this?)
When am I ever going to use this (said in an exasperated way). It is never “When? When I am going to get to use this!?” (said in an excited way). Rather it is a dismissive question. Since we are at a conference on service-learning you might think you know where this is going. However, I hope to lead you in a slightly different direction. If we are honest with ourselves, how we respond to the student varies based on our mood at the time. In the best of scenarios we can give them a meaningful future application. In other scenarios we respond with how they will need the information for their next course, or more immediately, they’ll need it for the test next Tuesday.
I would like to pose to you that none of those answers are sufficient because, if we are being even more honest with ourselves, we know that the student isn’t really asking a question. The student is making a statement – a statement that they feel validates their disengagement from the lesson. I would also like to pose to you that if you were to translate their statement to an actual question it would be this…
Why should I value this? I would argue that this is the most fundamental question to address in any classroom, even (or especially) mathematics. If we respond to the student’s surface level application question with cognitive information we will always have a disconnect. The student is actually longing for affective formation – however you want to parse that, into motivation or engagement or attitude or something else. The main thing is it is more than just being about what they know.
This is not only true for students. Close your eyes for a minute and imagine a great moment in your teaching career – something where everything was clicking and you were thinking “this is why I got into teaching.” Give me a word or short phrase to describe the mental image you came up with (solicit audience responses).
Notice that none of you told me “pythagorean theorem,” or “Great Gatsby,” or “mitosis.” None of you gave me content. Now, to be sure, the content was still there and was still operating on a high cognitive level I’m sure. My point is to not to dismiss content, but to perhaps reorient us on the primary objective of a classroom.
These aren’t just our feelings. The importance of affect in education has been documented in research.
Not only in research, but also in positional standards. I have here a few in mathematics but I’m sure that you can find something similar for any discipline. So with all this agreement and support, how are we doing?
(Read slide). This quote is from 1992 but it is not dated. I know that it is not dated because, as I mentioned at the beginning, I’m a math teacher. Whenever I introduce myself to anyone and the topic of what I do for a living comes up, these are the number one responses I receive. Math teachers may be second only to priests in the number of confessions they take.
This is what keeps me up at night. This is what I want my classroom to address. How do we do that?
We don’t do it by changing content or even focusing on student beliefs – we do it through rich experiences. The experience of the math classroom needs to change.
THIS is where I think service-learning is powerful. Not as an answer to application, but as an answer to offering rich and meaningful experiences in the math classroom.
How are the affections of the student impacted when we change the experience of math class from this…
For the past three years my stats students have partnered with a local homeless ministry in Austin for survey research. Here is how they responded to a survey I gave them at the end of the year.
Transition through graphs – point to growth in content knowledge but then also appreciation.
Yes service learning is application – but it is so much more than that. Through service learning we can shift student postures to ones of self-service (when will I use this?) to the service of others. Ultimately their education is not just about them.
Let me close with some words from a student who began the year with a very negative attitude towards mathematics and that improved over the course of the service-learning project. I asked him about it and this is what he said.
(After first bullet) I like this quote because I’m also a realist – I’m not setting out to make everyone always enthusiastic about math. But there are steps we can take and I think this student took them.
(Read rest of slide)
Why should I value this? Because it will benefit me and it will benefit others.