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!

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.

By the time most people graduate high school, I would venture to say that, whether they themselves think they had a good or bad experience with mathematics, they actually have a very small idea of what math is. And it is at this point in their lives (when they graduate high school and enter college) that many choose (some more passionately than others) to stop learning math, or at least to avoid learning math (avoid taking college math courses) as much as possible. I'm not upset that at this point they choose to stop taking math classes in pursuit of achievement in other great fields of knowledge and practice, because their desired majors of study may not coincide with more math classes -- we all have different interests and we should each pursue our own unique interests and that's great. But I am upset that at the point of high school graduation, I believe most people haven't learned enough of the right things about mathematics to understand its true nature such that they are able to make a well-informed decision about the degree to which mathematics should be a part of the rest of their lives. They don't have an accurate idea of what it is they're turning away from when they decide that math is not "for them". It's like deciding that you don't like chocolate before you've ever even tasted it. Of course, in turn, these high school graduates become the adults and the citizens of our society that continue to go and live clueless of the wonders in mathematics right under their noses. And this is simply a tragedy.

There are really fantastic supplementary materials out there to help "normal people" access the wonderful, true, beautiful expansive, powerful nature of mathematics (much especially from England comes to mind). I continue to learn and love so much from the passionate work and sharing of resources of individuals like Brady Haran (the Numberphile guy) and Marcus du Sautoy (the mastermind behind "The Story of Maths" among other projects). But why would anyone choose to spend their time looking at supplemental math stuff if they have a poor opinion or even just an inaccurate perspective of mathematics from their school experiences that actually draws them away from the grandeur of mathematics rather than towards it? The opportunity to recognize said grandeur of mathematics should not have to come from some supplemental or extracurricular thing outside of school. Rather, students' regular, mandated, daily mathematics education must be driven by helping students to understand and appreciate what real mathematicians do and what real mathematics is.

My ultimate career goal is to be the liaison connecting every child and youth to authentic mathematics in all its beauty through the schools that they attend every day.

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If you aren't familiar with Brady Haran / Numberphile or Marcus du Sautoy / "The Story of Maths", check out some of these short and engaging YouTube videos!

1) Here are a few of my favorite Numberphile videos:

• "Philosophy of Numbers"
• "The Mathematics of Crime and Terrorism"
• "Mathematical Way to Choose a Toilet"
• "Unexpected Shapes"
• "Fair Dice"
• "Was YOUR Vote Counted?"
• "Batman Equation"

2) Here's a trailer for "The Story of Maths": https://www.youtube.com/watch?v=eDwFElncCxc

The Problem With Many High School Mathematics Course Pathways

Because of the way that secondary mathematics courses are often structured and/or sequenced, many high school students (who then grow into many adults) have a very limiting linear view of the different disciplines of mathematics and furthermore of mathematics as a whole field. I say that students have a "linear" view of mathematics because they usually take a fairly linear sequence of mathematics courses in 6-12 education: Algebra 1, then Geometry, then Algebra 2,... one course after another that has been predetermined for them in a sequence that usually places AP Calculus as the "ultimate" (from the average student perspective) high school math course.

Occasionally, there may be some variation in this linear mathematics course pathway, e.g. a high school student may have the option of taking a statistics course somewhere in the mix, which is wonderful, but unfortunately I've often seen this statistics course referred to (by students and staff) as the alternative to a calculus course for those who "aren't as good at math", which, though it may end up being somehow to some degree true about the course because of the way it's taught and/or the way it's been perceived over time (e.g. it's become a self-fulfilling prophecy for Statistics to be the "easier" mathematics course), it is a huge insult to "real" statistics. But, I digress... And perhaps I'll be qualified to write a more in-depth piece specifically on statistics and statistics education a year from now after I've piloted my school's first statistics program.

In actuality, mathematics as a field is not a rigid linear sequence of courses; rather, its structure ought to be considered more like that of science or even history. In science, there are fields like biology, physics, chemistry, etc. We understand that they are distinct sub-fields, yet we also understand that they have shared natural principles, and although perhaps we must begin by studying one at a time, we can thereafter consider how one affects the other in nature, producing disciplines of study like biochemistry or biophysics, and there is even the study of string theory that seeks the elegant union of all sciences. In history, we understand the distinction of US history, European history, and Japanese history, but we also understand that they do at times interact, for how can the story of the US be told without telling the story of the land from where its founders came, or how can Japanese history be complete without mention of its military exchanges with the US? The histories are at those junctions intertwined, two sides of the same story.

The structure of mathematics must be understood the same way: a set of distinct yet interrelated sub-fields. Geometry is not "higher than" algebra, trigonometry is not "higher than" geometry, and calculus certainly is not superior over all the other high school mathematics courses, and certainly not the end-all of mathematics as a whole field (though it is so elegant and really beautiful). And don't even get me started about the so-called "Pre-Calculus" course... What the heck is that anyway? (I taught it for three years and I can't tell you.) I mean, what is it really? What is the essence of it? And does its content have any true relation with its name? (I think not, but perhaps more on that another time.)

Actually, some even consider mathematics as a whole to be a sub-field of science, at the same organizational rank as biology, physics, and chemistry. Biology is the study of life, physics is the study of matter and its motion, chemistry is the study of matter and its reactions with other substances, and mathematics is the study of patterns. And again, everything is interrelated, intertwined; understanding in one area grants enlightenment in the others.

Doesn't this wider perspective of mathematics make it so much more curious and fascinating? College math majors will understand this far-reaching, diverse view of the field of mathematics as they have to take a variety of math courses in their programs, many of which are not prerequisites to others (e.g. discrete mathematics, linear algebra, real analysis, game theory; but calculus is a prerequisite to many of them, demonstrating what I said before about calculus certainly not being the superior, ultimate epitome of mathematics). But of course, these individuals are not the ones who need to be convinced of the beauty of mathematics since they must have already glimpsed it to commit thousands of dollars to higher education in advancing their study of mathematics. Most others, however, come to this realization of mathematics too late, or not ever at all, their traumatic past experiences in the high school mathematics classes they've taken having made them callused to the very thought of every being intellectually stimulated and amazed by anything with the word "math" in it. How students view mathematics in school will 90% of the time be how they view mathematics beyond school. So it is my hope that secondary mathematics education can shift in ways that will steer students away from the rigid, linear perspective of mathematics as a field to a perspective of the truer, broader picture of mathematics as many distinct but wonderfully interrelated sub-fields before those students grow into adults that will perpetuate the former perspective.

I think one way to make this shift is to throw in as many forks as possible into the secondary mathematics course sequence, leading in to as many different diverse types of mathematics courses as possible. Here's one diagram for a diversification of mathematics pathways that I sketched over a year ago:

For starters at my school, less than 10 years old and still very much in its formative stages, I'll be teaching the very first statistics course of this school. I'm very excited to lead this first step at my school towards broadening students' perspectives of the wide, vast, beautiful world of mathematics.

a snippet of an email

A friend of mine who does ministry work in Oakland recently got her laptop robbed right before her eyes. This is a snippet of an email I wrote her in response to the news:

"...I'm so sorry to hear about your laptop getting stolen, but I'm glad that you're OK and I'm encouraged by your heart within your response to this incident. There was a recent event at my school that helps me to relate to your still choosing to be in Oakland despite the crime. A student brought a knife to school a few weeks ago. Praise the Lord that it was discovered and safely confiscated before anyone was hurt, but certainly the incident was and is cause for students, staff, and parents alike to question the delicate environment and culture of safety that we strive to create and maintain at our school. Several staff members shared how their own friends and family didn't understand why we choose to continue to work in an environment where our physical safety is possibly threatened. Headlines like "Student Brings Knife to School" invoke certain knee-jerk reactions (e.g. fear, blame, avoidance), and understandably so, but this is why context is important. Why did the student feel the need to have a knife with him? Why did the men (or women? don't mean to be sexist) who robbed you feel the need to do so? There are bigger systemic themes and issues and oppressive patterns at play here that you and I are in the work to disrupt, and it's really difficult if not impossible to explain that to those who aren't in the work..."

you can't understand... but i love you trying

I came across this post in which a high school teacher in southern Louisiana, Alice Trosclair, states a few reasons why teachers cannot live "normal lives" that non-teachers don't usually realize: http://www.washingtonpost.com/blogs/answer-sheet/wp/2015/02/05/why-teachers-cant-have-normal-lives/. This spurred me to share some of my own ideas.

I do believe, and find myself somewhat frequently saying that non-teachers simply cannot truly, completely understand the difficulties of being a full-time classroom teacher, especially in under-served, low-income communities. I don't mean to say that in an arrogant or a condescending way; simply in a factual way. There's just no way that you can know without actually experiencing the flurry of real, nonstop day-to-day highs, lows, joys, sorrows, busyness, ambiguity, confusion, craziness, wonders, etc. of teaching. Basically, there's no way that you can claim to completely understand something that you haven't actually experienced yourself. Certainly by the same token, I readily admit that I cannot (at this moment in time) completely understand what it means to work in the corporate world, or in the medical field. Those sectors have their own unique set of difficulties that I won't pretend to be able to fully relate to. There is just no way that I could completely understand, having never worked in those sectors. However, sometimes people don't apply that same principle when it comes to the education sector because most people have spent over a decade of their lives in classrooms as students, growing up and getting an education. So, they do have an extensive amount of experience in the classroom, and so it's understandable to me (but not agreeable) why they think they know something about teaching. But I strongly encourage those people to reconsider because the student experience and the teacher experience are very different. It's not a perfect analogy, but it's kind of like saying that you know how to play basketball really well because you've watched a lot of games. Sure, watching games does help, analyzing them helps more, but if you've never been on the court, you are some combination of arrogant, foolish, and ludicrous to claim that you're a good basketball player. Such things can also be said of non-teachers who think they know what it means to be a teacher.

Again, I say this with the intention of being informative and not condescending. I don't want this to become an excuse for me or other teachers to be moody or to complain all the time, to be inconsiderate of others, or to have to be accommodated all the time, like a man who uses having had a bad day at work to exasperate his wife and children when he gets home. I also don't want to brush aside non-teachers who genuinely want to understand. Everyone has their difficulties, and it's often part of relationships to empathize with them. Just because you can't completely understand, doesn't mean that you can't show your love and support. And in fact, I feel very blessed to be surrounded by many non-teachers in my life who get this, and I dearly appreciate when they lovingly empathize with the struggles that I face as a teacher without pretending like they completely understand them. It only irks me when people talk as if teaching is easy and nothing more than glorified babysitting, and/or a back-up plan because I couldn't do anything "better" with myself (e.g. "You know what they say; those who can't do, teach!") Boy, that really riles me up. Don't get me started. But I would venture to say that Trosclair, the author of the aforementioned article that motivated this post, has heard that, or things like it, from one too many persons, resulting in her article coming off as perhaps a bit -- abrasive? (in my humble opinion). I hope that, while holding the views that I've stated in this post, I can be gracious in my lifetime in trying to correct the misperceptions that people have about teaching. Nonetheless, Trosclair brings up some honest considerations for non-teachers, and I hope that you can read them informatively to construct for yourself a more accurate (though incomplete) picture of what it means to be a teacher.