Is it the journey or the arrival?

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Photo by @przemekklos: https://www.twenty20.com/photos

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:

Calculus Is the Peak of High School Math. Maybe It’s Time to Change That. ~ Education Week

Should We Stop Teaching Calculus in High School? ~ Forbes

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.

Some, most notably Andrew Hacker, go so far as to suggest that the “I’m not a math person” crowd doesn’t even need algebra much less calculus.

Is Algebra Necessary? ~ NY Times

However, upon closer inspection one might come to realize that the argument is not one of course sequencing or graduation requirements. Here is a quote from Hacker in the above article:

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.

This issue seems to be one of “usefulness.” Calculus isn’t useful for most people. Statistics is useful for most people. Algebra isn’t useful for most people. Programming (or some type of computer course) is useful for most people.

However, things start to get muddled when you define the end goal or purpose of education as one of utility. Picking up right where the above quote leaves off:

It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.

So we should teach how a certain applicable function is influenced by variable inputs but we shouldn’t teach Algebra? Isn’t that the definition of Algebra? How can we jump to the “useful” application without grounding students’ understanding enough for the application to mean anything?

I believe what is really happening is that Hacker would rather see “useful” algebra rather than something like, say, factoring trinomials. Nobody (well, almost nobody) factors trinomials for their profession or to get through their daily life. One could make the counter argument that nobody (well, almost nobody) diagrams sentences for their profession or to get through their daily life – but an understanding of grammar and syntax lays a foundation for language development that allows people to craft a blog post (or something more substantial).

Perhaps the sentence diagramming argument isn’t really a counter argument but just the same argument Hacker and others are making but applied to English rather than mathematics. 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.

If you try to convince students that the value in learning to factor trinomials is in its usefulness then it should be no surprise that we see articles like Hacker’s today – eventually those students grow up and realize the farce they endured in math class (actually, they recognized it as a farce instantly but now as adults they are able to make their voices heard more easily).

How we teach math certainly matters. I’ve written about that numerous times (see: Cultivating Mathematical Affections). The problem with these articles is that they don’t really address the how question but rather focus on changing the what. Their focus is on altering the end goal and not about altering the methods of the journey. It saddens me that this argument has gained such wide popularity (as seen in the number of these types of articles).

Let’s try a different approach and let’s start by defining terms.

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Notice the term “course” in the definition above. Curriculum, from Latin, literally refers to the act of running or a race track. It is the same concept used in Hebrews 12:1-2 and in 2 Timothy 4:7. It is a reference to being active, to engaging in the struggle of the race, to enduring the distance, to competing. It is not an arrival or a finish line – it is the journey through the race itself.  Discussions about the “peak” of the math curriculum miss this.

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.

Page 17

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?

I think there is.

I can’t offer a prescription of how to do this in every class so I conclude by simply challenging teachers to consider how their classrooms and their curriculum are focused on getting students to a destination versus equipping students for a transformative journey.

“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 (from the beginning of Paul Lockhart’s “A Mathematican’s Lament” book)

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Christianity: Always More Than Thinking, Never Less

A student stopped me in the hallway the other day and posed what he thought was a nice brainteaser:

A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?

After doing the basic algebra:

Bat + Ball = 1.10

Bat = 1 + Ball

Substituting:

1 + Ball + Ball = 1.10

2Ball = .10

Ball = .05

I responded that the ball costs $0.05.

The student seemed disappointed that I had solved the puzzle using math of all things. He objected that I shouldn’t have thought about it but just answered with the first thing that came to my mind (…a little glimpse into the thought process of my students).

This response was expected. What he said next was not. “I guess you are less likely to believe in God.”

Wait…WHAT?!

“It’s true,” he said. “I read it in an article.”

I thought that couldn’t possibly be accurate…but it is. Thank you internet.

Intuitive Thinking May Influence Belief in God

Here are the highlights:

In a series of studies, researchers at Harvard University found that people with a more intuitive thinking style tend to have stronger beliefs in God than those with a more reflective style…”We wanted to explain variations in belief in God in terms of more basic cognitive processes,” researcher Amitai Shenhav said. “Some say we believe in God because our intuitions about how and why things happen lead us to see a divine purpose behind ordinary events that don’t have obvious human causes. This led us to ask whether the strength of an individual’s beliefs is influenced by how much they trust their natural intuitions versus stopping to reflect on those first instincts.”

The research was published online in the Journal of Experimental Psychology: General. The study from the Harvard University Psychology Department was conducted by Shenhav, a doctoral student; post-doctoral fellow David Rand, PhD; and associate professor Joshua Greene, PhD.

In the first part of the study, 882 U.S. adults, with a mean age of 33 and consisting of 64 percent women, completed online surveys about their belief in God before taking a cognitive reflection test. The test had three math problems with incorrect answers that seemed intuitive (like the problem above which has an “intuitive” answer of $0.10)…Participants who had more incorrect answers showed a greater reliance on intuition than reflection in their thinking style.

Participants who gave intuitive answers to all three problems were 1 ½ times as likely to report they were convinced of God’s existence as those who answered all of the questions correctly.

First of all, there is clearly a problem in defining intuitive (or at least measuring intuition) as answering incorrectly to a series of math questions. Intuition, as defined by dicitonary.com: direct perception of truth, fact, etc., independent of any reasoning process (emphasis added). I am uncertain then how answering incorrectly shows a direct perception of truth.

I get the impression that the real claim of this study is that people who respond without thinking tend to accept the idea of religion more than those who analyze information that is presented to them.

This reminded of me of the following comic:

For some people this is the “religious” way of thinking: just believe, don’t analyze. What makes the comic humorous is that this is an obviously flawed approach to mathematics. I contend is an equally flawed approach to faith.

The title of this post is a favorite saying of mine from our former pastor in Dallas, “Christianity is always more than thinking, never less.”

There is great danger in putting religion a step below thinking. I am reminded of a segment on NBC Nightly News, “Show of Faith,” from June 23, 2008. Video Courtesy of NBC. While there have been numerous news segments to survey and comment on the religiosity of Americans, this one stood out to me because of one sentence by Rev. Eugene Rivers (emphasis added):

In some cases, because the American public is not terribly theologically literate, they hold contradictory views because they haven’t thought deeply, or been taught deeply, about their faith tradition.

Not examining one’s religious beliefs tends to lead toward contradictory or false beliefs.

Scripture makes this point itself. The book of Romans clearly explains how we are created with an intuition toward the things of God, but that intuition becomes marred by sin. What we need is a renewal of this intuition, a renewal of our minds, not for the purpose of blind acceptance but for testing and approving the will of God.

For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men who suppress the truth in unrighteousness, because that which is known about God is evident within them; for God made it evident to them. For since the creation of the world His invisible attributes, His eternal power and divine nature, have been clearly seen, being understood through what has been made, so that they are without excuse. For even though they knew God, they did not honor Him as God or give thanks, but they became futile in their speculations, and their foolish heart was darkened. Professing to be wise, they became fools, and exchanged the glory of the incorruptible God for an image in the form of corruptible man and of birds and four-footed animals and crawling creatures (Romans 1:18-23).

Do not be conformed to this present world, but be transformed by the renewing of your mind, so that you may test and approve what is the will of God– what is good and well-pleasing and perfect (Romans 12:2).

We are charged then to approach our faith as the Bereans. In the book of Acts, the Bereans are praised for examining the Gospel message:

The brothers sent Paul and Silas off to Berea at once, during the night. When they arrived, they went to the Jewish synagogue. These Jews were more open-minded than those in Thessalonica, for they eagerly received the message, examining the scriptures carefully every day to see if these things were so. Therefore many of them believed, along with quite a few prominent Greek women and men (Acts 17:10-12).

As a math teacher I feel that I am charged with teaching students to think well and reason correctly.

As a theologian I feel that I am charged with teaching believers to think well and reason correctly.

We need thinkers. We need teachers of thinkers.