Backwards Mapping: Three Student Goals of the Ideal Primary Mathematics Education System

I wrote previously about my vision for secondary mathematics education, a world in which "math class" is simply students working on problems that matter to them. Now, I want to begin backwards mapping from there into the primary mathematics education that would be a necessary cornerstone to making that vision possible and successful. What must be achieved in the primary education of students so that when they are given a space to openly work on problems of their own choosing in secondary school, they will be genuinely and wholly engaged? I think it's these three things:

1) The establishment of insatiable curiosity
When students are curious, they wonder ("I wonder if...", "I wonder why...", "I wonder how..."). When students wonder, they are formulating those problems that matter to them. When students formulate their own problems out of their curiosity, they are personally invested, greatly motivated, and will have a much more impressive recall of the work that they do. Students must enter secondary school with the confidence from primary school that they are allowed to be curious. They are allowed to be curious in school, they are allowed to be curious in every class, and they are allowed to be curious for their entire lives.

2) The fearlessness of failure
In my vision, when secondary students are all working on their own self-selected problems that matter to them, they aren't given much (if any) direction, and they don't have an adult frequently breathing over their shoulders to guide them. Students may get started on their problems, and then, surely, as students work on their problems, students will get "stuck", and then it's up to them to get themselves "un-stuck". (That's the nature of authentic problem-solving! Repeated trial and error.) Knowing what to do when you don't know what to do can sound like a tall order. But actually, primary students show us time and time again that, when they are driven by curiosity, they do know what to do: they keep trying! I'm picturing a toddler building a tower of toy blocks. They might try the exact same thing over and over and over a few times. And then they'll try something a little different. And then they'll try something a little different again. And again. And again. Until it works! But somewhere between toddler age and secondary school age, students "un-learn" to keep trying, un-learn that they are allowed to fail, and un-learn to try something a little different (to experiment). When they can't do these things, they are more likely to feel confined, to feel inadequate, to get frustrated, to give up, and to hate math (or whatever it is that they are working on). Students must enter secondary school with no fear of experimenting, failing, and trying again.

3) The clear articulation of their own ideas
This is another thing that I think young students usually do really well but then tragically "un-learn" by the time they get to secondary school. Kids generally love talking, and adults generally do well talking to them, getting them to talk and to explain their understanding of things. This shouldn't stop. As kids grow older, they need to keep talking, their brains need to keep being probed, and they need to keep being pushed to explain their ideas, communicating them to other people, both adults and peers. The better students can explain problems, the better they can process them, analyze them, and collaborate with others to solve them.

To conclude this post on primary mathematics education, it may be helpful to draw a parallel between mathematics and English. In secondary English language education, mastery is generally shown when you can understand & interpret any text and construct your own text that serves whatever purpose you will it to accomplish (e.g. to inform, to entertain, to persuade). The ultimate goal for students throughout English language education is the ability to communicate, to engage in the discourse of the world. But before students can achieve that goal, they must receive the basic building blocks of the language in primary school: the alphabet, phonics, spelling, grammar. I would say that the ultimate goal for students throughout mathematics education is to authentically and meaningfully engage with problems. But like English education, there are basic building blocks that need to be in place before that goal can be achieved. Without the "soft skill" core foundations of insatiable curiosity, fearlessness of failure, and clear articulation, students will never become the brilliant, innovative, effective problem-solving adults that are so desperately needed in this day and age.

Two Thoughts For Engaging Students in Real Mathematics

I wrote in a prior blog post ("My ultimate career goal"), "Mathematics is wonderful, beautiful, expansive, powerful. But I'm afraid that students rarely leave our current American secondary mathematics education system with that conviction." As I continue thinking on this issue, I keep coming back to two broad solutions to this problem (and I realized while writing this post that the first is actually a foundation to the second, or inversely the second is a particular extension of the first). Furthermore, it should become obvious as you consider these that the broader foundation of both and the necessary catalyst throughout is that teachers have personal relationships with their students and know them (their personalities, their interests, and their goals) well.

1) Give students problems that actually matter to them.

I briefly mentioned this idea in a prior blog post ("My Vision for the Future of Secondary Mathematics Education"), and this is a driving point throughout the texts of Paul Lockhart (which I also talked about in that same prior blog post). If mathematics is an art (as Lockhart argues), then it must be understood and practiced as an art. It is enough for mathematics to be done for its own sake, for its own inherent beauty and pleasure.

This doesn't mean that there are no benefits to it. Far from it! Are there benefits to the arts of music, of painting, of poetry? You know that there is plentiful research and there are countless more writings on the benefits of the arts for the holistic development and thriving of our students and our society. As students are assisted to find problems that actually matter to them (which you can do when you know them well) and given space to work on them at their own pace, they are naturally and implicitly developing their skills in investigation, critical thinking, and problem-solving. It is OK for students (in fact, it is OK for people, of ALL ages) to work on problems that they just find fun, without any identifiable applications. (Interestingly enough, historically, the applications are actually usually discovered later!) The political pressures might tell you it's not OK: we're wasting time, kids aren't learning or growing fast enough. But how much of what students have learned in math-class-as-it's been-done can we say are contributing factors to their growth as human beings 10 or 20 years later, and how much of it just felt like hoops to jump through and motions to go through to graduate high school and move on with their lives? Real, lasting growth takes time. And so does real mathematics. Our teachers needs to believe this so that our students can believe this. It must be built into the fabric of the classroom culture.

I strive to model continuing to work on new mathematics for fun. I've been thinking lately about the famously unsolved (unproven) Collatz Conjecture. If you're not familiar with the Collatz Conjecture, here is a great video tutorial on it (by Numberphile) that you should stop and watch at least the first few minutes of: https://www.youtube.com/watch?v=5mFpVDpKX70. "Erdos actually said this is a problem for which mathematics is perhaps not ready. Turns out all this fuss is about a problem that any fourth grader can understand." This is an utterly fascinating problem! Are there any practical applications to this? Not that mankind is currently aware of. But people keep working on it. Is it because they're hoping to find a practical application of it? Doubtful. They're just working on it because it has sparked their curiosity and their wonder, and in this way has made itself matter to them. This is just one of many unsolved, mysterious, "pure" mathematics problems that have captivated people just by what it is, by its inherent being.

2) Give students problems that they will actually have to solve outside of math class immediately or in their foreseeable futures.

Sometimes (though I would argue that it's not as often as you might think) what makes a problem "matter" to a student is its authentic applicability to him/her. That's been somewhat recognized for years, and so this is a repeated outcry we in mathematics education have all heard: give students "real-world" math problems! But unfortunately, the good intent there has often led to the creation of problems that, while they may include real-world elements, are contrived and impractical -- they're not problems that any sane person would actually encounter in the "real world". Here are two examples:

Example #1:
Maria is two years older than twice her age seven years ago. How old is Maria?

In what situation would you ever know that someone is two years older than twice her age seven years ago and not know her actual age!?

Example #2:
Summer Nail Salon is having a special this month on services. Over the weekend, they performed 33 manicures and 40 pedicures, bringing in a total of $1509 in receipts. So far this week, they have administered 13 manicures and 43 pedicures, with receipts totaling $1330. How much does the salon charge for each service?

In what situation would you ever know all the information given in this problem and not know the how much was charged for each pedicure and each manicure!?

Again, the authors of these problems have thrown "real-world" elements (age, manicures, pedicures) together into a fake situation to lure students into using a mathematical idea (e.g. systems of equations) in an impractical way. And, intrinsically, students know this, even when adults deceive themselves of it.

Now, does that mean that there is no value to students solving problems like these? Not necessarily! Refer back to my first wider-reaching thought/solution: "Give students problems that actually matter to them." What matters to students is highly subjective, relative, variable (which, again, is why teachers need to know their students well). If a student finds these problems fun and/or interesting and want to do them, let them do that! And they will garner the aforementioned benefits of developing their mathematical thinking. Just don't pretend that this is a practical application of mathematics. That will only send students down a line of thinking that mathematics is anything but practical.

If students want practical mathematics, then instead of these contrived "real world" problems, we must give students authentic problems that are applicable to students immediately or in the foreseeable future -- that is, the students must be able to foresee and believe that they will have to apply this in the future. When asked, students, especially older students, should be able to state with clarity and conviction the applications of their mathematics. The students need to believe that they will encounter problems like these outside of math class. Students knowing why what they're doing matters is the hinge factor for engagement and learning.

Dan Meyer has written a handful of insightful blog posts over the years about "real-world math" (http://blog.mrmeyer.com/category/fake-world-math/), so I'll close by quoting him on "relevant" math problems:

The real test of whether a math problem is “relevant” is not “do you use this in ‘real life’,” whatever that means, but “do you want to solve it?” It’s not that you want to solve it because it’s relevant; wanting to solve it is what it means to be relevant.

My Vision for the Future of Secondary Mathematics Education

I first read Paul Lockhart's "A Mathematician's Lament" when I was earning my undergraduate degree in Mathematics For Teaching. (If you haven't read it yet, you should click on the link and read it right now, at least the first two pages.) It left an impression on me then, but I didn't think until in the middle of my third year of teaching to dig it back up and read it again. Strongly compelled and contemplative after re-reading, I soon discovered that there was a second part to "A Mathematician's Lament" not on the free on-line PDF ("Exultation") and a whole (second?) book written by Lockhart (Measurement). (I think most people would consider "A Mathematician's Lament" to be an essay instead of a book, even though it was published together with "Exultation" as a book, so Measurement would be his first book, not his second book. But please correct me if you find/think otherwise.) I immediately ordered these texts and spent my summer break reading them. (As a result, I was also shamelessly inspired to construct my own proof from scratch that two triangles constructed by taking any triangle and then constructing the line segment from any vertex of that triangle to the midpoint of its opposite side will always have equal area, while on vacation in Germany. I'm quite sure that it wasn't the most efficient proof of this fact that's ever been done, but it was my proof. I owned the proof as my own.)

(This is not the whole proof, just a snapshot of part of it.)

Months later, Lockhart's ideas and passions continue to resonate in my heart. Lockhart has drastically changed my ultimate vision of what I hope mathematics education will one day look like. I don't know if said vision will be attainable in my lifetime (probably not), but I think that that makes it all the more a worthwhile goal to have, a goal that stretches beyond my lifetime, a life-long striving that makes life worth living. Here's my vision...

Imagine a mathematics secondary education in which every student is working on a math problem of his/her own choosing that matters to him/her. Imagine every student getting a weekly one-on-one check-in or mentoring session with his/her instructor, or perhaps a more appropriate name in this world would be a coach. Just as sports coaches mentor their athletes to improve their athletic abilities and reach their potentials, so this mathematics coach would mentor his/her students to improve their problem-solving and critical thinking capacities. At the end of every week, students write a report to explain the progress that they have made in their individual problems. The purpose of this report is at least three-fold: (1) to hold the student accountable to himself/herself as each student innately desires to write a good report that shows significant progress; (2) to hold the student accountable to his/her math coach or whoever might serve as the student's academic evaluator/grader; (3) to create a written record of the student's weekly progress that can act as a portfolio of the student's work and can be shared with other students to spark curiosity and incite opportunities for student collaboration (just as many sports are team sports that require collaboration). These reports will be carefully reviewed by the coaches and discussed with students in their weekly one-on-one check-ins when coaches might provide guiding questions, thoughts, or resources to help students continue to move forward in their inquiries and progress on their problems...

That's just the tip of the iceberg. I originally wrote more (and I am definitely thinking more that I'm still trying to put into words), but I'll stop my description there and save the rest for later (in the hopes of decreasing unnecessary long-windedness and increasing coherency and readership).

Of course this seems fluffy and strange and maybe even impossible or at least impractical to you now. And I'd love to talk about that! I don't have all the answers -- not by a long shot! There are many, many questions this vision brings up. (And as much as I wanted to have all the answers before posting this... that just wasn't going to be possible. I just needed to start somewhere before I could get anywhere.) There are many reasons why this kind of system would not be successful if we suddenly started doing this in our schools tomorrow. But if I believe in this vision, then to make it happen some day, I need to ask:
(1) What conditions would need to exist for this kind of system to be successful?
(2) How can we gradually build bridges to transition from all that is "today" to this vision of math education in the future?

I want to hear your every thought on this and think through every conceivable problem with this system, so please share!