It took Julia Robinson and her mathematical team of four, 22 years to crack Hilbert’s Tenth problem, one of 23 he passionately and optimistically issued at the International Congress of Mathematicians in Paris in the summer of 1900.

His clarion call to mathematicians to penetrate the future of mathematics has reverberated ever since.

“Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose? “

David Hilbert, August 1900

A recent one hour movie based on Julia Robinson and her burning desire to resolve this problem in her lifetime has just been released. Below is the trailer.

Mathematics, quite simply, is about problems. Some are easy. Some are so challenging that the mathematics needed to solve them might not be discovered yet. This was the fate of how Andrew Wiles finally cracked Fermat’s Last Theorem, where he cleverly fused ideas of the 20th century. So, as much as Fermat hinted at a possible solution, it is highly unlikely that he had the proof. In an interview with NOVA, Wiles gave a very realistic insight into his poorly lit immersion into Fermat’s Last Theorem.

Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of — and couldn’t exist without — the many months of stumbling around in the dark that proceed them.

I counted the use of “dark” four times. Mathematics is a journey. That is something that we all agree upon. However, the narrative which has been adopted by math education has been one that makes that journey seem unrealistically lighted and linear. While there is more dialogue now around productive struggle — all struggle is good — and failure, math education continues to be obsessed with discrete packets of success. As such, both students and teachers feel anguish/anxiety if correct answers are not found — quickly.

If by the time students get to high school and have not been given problem(s) that take days or even weeks to solve, then this journey through mathematics has been more tourist in nature than traveler.

My title at Buzzmath, the digital company I work for, is Mathematics Learning Specialist. I am not a teaching specialist. I am not an expert. I excel at devouring knowledge to reassure myself that when it comes to mathematics, I will always be a novice.

We all will be. We are all wonderfully lost together. But, we can only truly believe that if we believe that with mathematics we all start in the dark. And while of course we are curiously driven by our collective desire to know, we should not measure ourselves or mathematics for any conditional reveal. Did Julia Robinson feel less of mathematics in her 20th year with Hilbert’s Tenth Problem than when she solved it? Sure, there was jubilation, but her attitude towards the inherited darkness of mathematics probably did not flinch one bit.

How deep do I feel about this communal celebration of seeing mathematics as an endeavor of finding light? It will be the introduction to my next book, Chasing Rabbits: A Curious Guide to a Lifetime of Mathematical Wellness(2021)

In many ways, mathematics is like those Escape Rooms — which by the way, I have never successfully completed! You might have a goal, but really, you are really enjoying being in the moment of the perplexities that confront you and your team. If you crack it, you crack it. If you don’t, you don’t. Pressure should be fun, not a weight to negate the experience.

Let’s teach mathematics with more darkness. The light? That is provided by us…

Mathematics: We Are Lost Together was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

A D minus…

I made sure that I saw that “minus” after the D. Not saying I wouldn’t have written this article without it, but the inclusion of that seemed petty, filled with an almost cartoonish malice a la some dark authoritarian figure in Pink Floyd’s nightmarish state of education.

The real gatekeeper in math education is not algebra. The real gatekeeper of math education is fractions. It’s where alienation begins…

So, like with Newton’s Third Law of Motion — for every action there is an equal and opposite reaction — this article is my reaction.

She failed a test about improper and mixed fractions. You know, converting between the two and such. Frankly, I hope she failed them because she, like thousands before her, just absolutely hated fractions. The fact that in 2019 my daughter received such a mark — a mark that still has that negative of ring of “D is for Dummies” — is more of an indictment on the failure of what is still generally accepted in some math communities than my daughter’s interest/abilities in mathematics.

As some of you know, my daughter has, in spite of that permanent stain of fraction ineptitude, a positive and joyful interest in mathematics.

She knows her primes up until 223(thank you Prime Climb). She loves figuring out operations to create numbers(thank you Albert’s Insomnia). She loves probability/strategy(thank you Yahtzee). She loves drawing geometric configurations for social connections of 3 and 4 people(thank you Ramsey Numbers).

We not only jettison whole numbers for a contrived fascination with fractions, we do it freakishly early. This dumping is a bloody slap in the face to an actual branch of mathematics — number theory — which consumed the likes of Gauss, Germain, Ramanujan(who incidentally did cool work with continued fractions).

But, that is why I am not worried about her mark in 5th grade fractions. I mean, I am not worried about marks in general — it’s a sorting tool for adults packaged off as something for students to get simultaneously excited and worried about.

My daughter is a serious gamer. Check out all the games on her phone.

They run the gamut of different kinds of puzzle/problem solving games. Remind me why one of the reasons we teach mathematics again. There is one game in particular, Gardenscapes, that symbolizes the burgeoning divide between what has been traditionally valued by teachers and what my own daughter values in terms of problem solving.

She is on level 32 now. There are over 2000 levels. This game is not easy. In fact on an earlier level, which she was stuck on for a few days, she humorlessly remarked “I hate this game. It is so frustrating!” The key thing missing, but I tried to allude to, is that my daughter was laughing her head off when she said it. This is what resilience sounds like — enthusiastic proclamation of the joys of frustration. Or, as James Tanton often says, funstration.

I have listened to her strategies when playing this game. She gets the game on both a macro and micro level. She even makes fun of the purchasing part of he game, saying “how dumb do they think they I am to purchase boosters/coins for the game?” Her examination of the entire playing surface, making connections, visualizing potential moves, etc. are exactly some of the long term skills we want kids to learn through mathematics.

The problem is that most of the mathematics is boring and has a currency that is over valued in 2019. My daughter knows the basic meat and potatoes about a fraction. Whole numbers on the other hand, her appetite is pretty boundless.

Sorry, but I would rather have my daughter engaging with the medium(digital) that she enjoys/values to build problem solving skills. How to split 42 cookies with 8 kids isn’t in her wheelhouse — and her and I don’t care if you want it to be part of a larger gate keeping structure of what is valued/important in mathematics.

Actually, it is a gate keeping mess.

Fractions are important because you need it for______________ and you need_________ for _______________. Maybe my daughter “gets” fractions next year or the year after. Maybe she gets them at the end of high school. Maybe she never gets them.

I. don’t. care. Being proficient with them and doing completely meaningless tasks with them like multiplying and dividing them with zero intuitive understanding is something that cannot even begin to compete with what she is gleaning from her playing Gardenscapes.

The skeptics will say that the resilience that kids demonstrate to games doesn’t transfer over to math. Okay. Shouldn’t we wonder why that is and why we are asking that question? That kind of inquisition hints at a hierarchy of knowledge — “yeah, games are fun and all that, but it’s not real math, and at some point kids have to learn things they don’t like…”

Martin Gardner, a beast of a mathematician and writer of over 100 books, still couldn’t shake his “recreational math” label. So, I have zero faith in Gardenscapes dislodging dividing fractions as being more relevant.

A while back there was an article written by Barbara Oakley in The New York Times, that was ode to a narrative filled with classism/elitism with regards to mathematics.

Where to start? How about with the word “Make”, a gentler word than “Force”, which is a direct violation of anything playful and joyful about learning mathematics. Whether my daughter is interested in a STEAM career or not, I sure as hell don’t want her taking this path — not so subtlety marked with seeing math as a performance pursuit.

Hard pass.

Her algebra skills, regardless of what some math teachers might think, isn’t going to be compromised by when/if she becomes fluent with doing sometimes ludicrous gymnastics with them. Primarily because fractions play a minor service task role, making “guest appearances”. If they are playing a role bigger than that, then we are teaching algebra incorrectly. Fractions in algebra usually involve “clearing them”. But, that task is part of a larger sphere of understanding equality of operations.

That question above is like Level 0 of Gardenscapes. Zero thinking. Zero intuition. Any kids thinking to themselves that the answer is “3ish”?

Nope.

Mathematics is large. There is nothing contradictory about my daughter loving mathematics, puzzling, and persisting through rich problems and NOT liking/understanding fractions. In fact, it is to be expected.

So, sorry Barbara Oakley. I am not making my daughter practice math/fractions to achieve some status of a math-oriented career.

I am letting my daughter play games so she can be happy.

Oh, and mark my word, she will become a kickass mathematician one day…probably around the same time she cracks level 2000 for Gardenscapes.

My Daughter Got a D- in Fractions: Why Neither One of Us Care… was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

No other subject parallels math in terms of the fear that students feel when faced with it. You’d be hard-pressed to find a student who thinks that they are intrinsically incapable of understanding literature or geography, yet countless children and adults alike hold the belief that they ‘just can’t do math.’

Staring at the whiteboard while symbols pass them by amidst the steady drone of vocalized arithmetic, these students are being left behind. They do not understand the applications of math or why it matters. Moreover, they are expected to grasp the subject quickly, memorizing theorems and methodologies without being taught the reasons why these attributes work the way they do.

I interviewed five adults who have struggled with math anxiety throughout their educational upbringing. You’ll hear from a PhD candidate, a singer, a nurse, a marketer, and an operations manager. This phenomenon spans the globe, with our anecdotes drawing from Greece, the United Kingdom, and the USA. Some of these individuals had to conquer their fears in order to follow their dreams, such as the nurse whose drive to help others compelled her to learn measurement calculations for drugs and infusions. Others adapted around the anxiety, pursuing passions that circumvented math. One of them had a natural gift for math yet still struggled with the apparent meaninglessness of it, ultimately finding the joy in math and ending up in a quantitative field. In a way, none of these stories is unique — so many of us have struggled with the same barriers trying to overcome math anxiety.

Memorization versus Exploration

At heart, there is something fundamentally wrong with the way we teach math. Throughout this piece, we’ll discuss a few of the ways in which we could improve math pedagogy, starting with the focus on rote memorization.

Across every single interview, participants brought up the idea that math was something to be memorized, rather than played with and deeply understood. Memorization is not inherent to the field; and yet, math has become something to be recited on tests rather than to be understand.

The whole of high school we were told that you were learning to pass exams, nothing else. (Emily, nurse, U.K.)

Mathematics — when taught well — provides an opportune stage for students to explore, feeling for the boundaries of logical pathways by way of trial and error. If a child postulates that 2+2=5, outside information is not required to help them see why this isn’t possible. In fact, this attribute is relatively unique to the field, for most other subjects do require external input: a history lesson relies on firsthand recollection and a science class relies on direct experimentation.

Math utilizes a self-contained system, and yet it is consistently taught via memorization rather than exploration. Emily*, the speaker above, said it was understood that the purpose of her class was to memorize the correct methods. Although she would try to understand the meaning and underlying systems, there was only so much she could do on her own.

Asking for Help

Emily continued by describing how math was the only subject where she felt she couldn’t ask clarifying questions. Prior to high school, she felt that there was room for the class to move at its own pace, but that changed when the curriculum turned to passing the state-mandated GCSEs.** The teacher would put problems on the board with no further discussion or room for probing.

Ali, who was educated through high school in England and is now pursuing a Ph.D. in literature in the United States, said that teachers expected students to blindly memorize material as early as kindergarten, and this memorization did not leave room for questions. “The teacher wasn’t patient. She would explain something once and expect you to immediately get it.” This extended through her American university statistics course, in which the professor teaching her introductory precalculus course looked “frustrated” when students didn’t grasp concepts immediately.

For Erin, a Midwest-educated marketer, age-appropriate pride kept her back from asking for help. Although she felt that her math teachers practiced similar pedagogies as in other subjects, she had a weaker foundation in math and so often didn’t understand the answers when she did ask a question. “Sometimes my question would be answered,” said Erin, “and I didn’t understand it, and then I’d shut down.” Grasping the answer felt high-stakes — as if it defined whether it not she could succeed in these difficult classes — and her stymied understanding dissuaded her from future inquiry.

Helen, a Greek singer, didn’t have the opportunity to ask for help in the first place. The expectation in Greek schools was that teachers would not be available for assistance, compelling nearly all students to pay for outside tutoring. “Only the very strong students who were organized and had help at home didn’t face problems,” recalled Helen. “Gradually, [without help], I started having serious problems.”

Many teachers assumed that the students understood intuitively, which Helen posited was because “they didn’t want to give help.” In the Greek system, she said, teachers often “weren’t in the mood to actually teach,” and whether you would receive a full explanation of the material depended on the personality of the teacher. This implicit assumption of understanding was particularly difficult when you missed a day of class, said American-educated John, where each day would cover a specific skill and absent students would not be able to catch up without asking — which, of course, most students did not feel comfortable doing.

No Point of Entry

Without help from teachers, Ali felt lost in math. Since math is often not encouraged in the home or in daily life, particularly for young girls, she did not even have the benefit of consistent non-academic exposure. While other subjects appeared to have a clear point of entry — sciences tie back to your experience of the world, native language studies explore stories, history is a recollection of events — math seemed like a black box. “I never felt like I couldn’t understand a book,” Ali said.

She believed she could bootstrap her way through anything linguistically-based and accessible, but math didn’t fall into this category. With other subjects, there are elements “that are in our lives…[so] we feel more comfortable with them,” said Helen. Similarly, Emily expressed that “it’s almost like it’s in another language,” although a language that isn’t taught.

The idea of math as an impenetrable language permeated our conversations. Helen put it this way: “It was like I had to read a literature book, but I could only spell.” Without the tools to grasp math, and without an accessibility tied to the ‘real world’, so many students feel lost.

Learning in a Vacuum

Whether math seems to come naturally or requires concerted effort, most students struggle with understanding why they are learning it in the first place. Most other subjects have obvious applications: the natural sciences outline how the world works; literature, art, and music convey deeper meanings; history aids us in not repeating it.

The two most obvious applications for math are (1) calculations in the natural sciences and (2) practical daily use (e.g., budgeting, taxes, etc.). Students often aren’t shown these connections, leaving them to believe that it is an arbitrary field taught for the sake of learning and perhaps technical careers. Emily said that her teachers “didn’t try to make it relevant to [their] lives;” and, although she particularly enjoyed biology, she never made connection between the two disciplines.

I don’t recall us ever understanding why we were doing what we were doing. (Ali, Ph.D. candidate, USA)

Ali did, however, recall liking algebra because it connected to chemistry, but didn’t appreciate this application of math until in high school when it felt too late to catch up. She had a natural curiosity for other subjects — which had “obvious utility” — but never felt curious about math. On one occasion in middle school, her class had a unit on using math to analyze bodily metrics (such a height, weight), which she remembers vividly as being the first engaging element they had been taught.

While Emily ended up be required to learn math for her nursing degree, she initially though it was boring and convinced herself that she didn’t need it. She would often skip class or sit there and tune out. Similarly, John — who now works with logical systems all day in his role as an operations manager — felt that higher level math was “pointless”. Math after algebra no longer seemed applicable and the teacher “didn’t try to bridge that gap,” leaving even naturally talented students without the desire to continue trying.

When in college John could finally choose his classes and had to do math as a part of his business curriculum, he had a deeper understanding of how it connected to the real world, which his professors reinforced. Finally, a student who despite doing well in his courses had felt a continual sense of anxiety started to understand why functions applied in the way they do. The way John described it, it previously felt like he was skating over the surface of the subject, hitting the right marks without feeling secure in its underpinnings. Once he understood those foundations, his anxiety finally melted away.

Arbitrary Constraints

Further exacerbating the issue for Ali was the existence of timed tests that measured precision rather than comprehension. After repeatedly flunking timed tests, she convinced her college statistics professor to allow her to write a rigorous statistical analysis rather than taking the final exam. The ‘A’ she received pulled up her entire grade; she claims this is because she’s “good at writing”, yet a statistical analysis is closer to what a real-life statistician would do than timed problem sets are, indicating that this grade was more reflective of her understanding of the material.

Repetitive testing is helpful to ensure that students can complete problems on their own. Yet, this is rarely employed for more qualitative subjects, which can absolutely be measured with timed, constrained essay writing. In my own college education — where I received a dual degree in philosophy and mathematical logic — I sat for only two timed philosophy tests, but on the converse, received only three take-home mathematics exams / analyses from a few very progressive professors.

Erin did exceptionally well on her math homework because she had the luxury of time, but tanked pressurized testing. Conversely, John was always a good test taker, but refused to do his homework because it felt arbitrary and meaningless, and he could coast along, passing tests on natural ability alone. In fact, he could not understand what was going on and still get a passing grade, resulting in a feeling of “resignation: I’m just going to do the best I can do.” Where Erin was wrongly penalized for her inability to perform on standardized tests, John’s abilities too were not accurately reflected, as he received good grades but did not understand the material.

Across three countries with different educational systems but a commonality of highly standardized and constrained testing, these experiences indicate a wider issues of how arbitrary metrics affect students’ ability to perform well and even think highly of their own competency.

Can’t Catch Up

Once students believe they’ve fallen behind in the subject, it can feel difficult to catch up to where they’re expected to be. This is further exacerbated by the tiering system most educational institutions use to each math, which arguably benefits those who place into higher sections (and most likely have a natural proclivity for math) but demoralizes those in lower sets.

Ali and Emily — both educated in the U.K. — knew that they weren’t in the top tier of math and believed this indicated that they could never make it past a certain level. In a way, they weren’t wrong: in the British educational system, students are capped at a highest possible grade based on the set they’re in, even though they all take the same test. Similarly, in the USA, weighted averages keep high school students from ever seeing beyond a 4.0 if they are in a lower-level course, making it difficult to compete with those who ace the upper-tier sets for admission into top universities.

“I felt resistant,” Ali recalled, as soon as her class divided into tiers. Emily elaborated, “if you’re not good, you almost think you shouldn’t try […] you think you must be bad.” This is an experience these individuals never had outside of math, which was unequivocally segmented as a subject. Emily and Ali both were in lower tier classes and suffered by being around apathetic teachers (and often students). Similarly, Erin struggled to stay out of these low tiered classes, preferring to not understand in upper tiered courses rather than “be put into a class with the ‘dumb’ math kids.”

John believed that he was placed in a reasonable math level, solidly in the middle of the tiering system, even though he was in the top tier for all other subjects. While the class felt like a comfortable fit, he was conscious that his placement was incongruent to the rest of his education. This wasn’t demoralizing, but did allow him to coast along, relying on his natural talent because he had no reason to try, believing that math was not a useful subject. Because he coasted, he struggled once he reached upper level classes in university, possessing only foundational knowledge based natural aptitude rather than concerted learning.

The one exception to tiering was the Greek system, described by Helen, who conversely empathized with teachers who “were in a bad position to teach different levels of students”, since they could not tailor the single class to particular levels. With Helen, she said, “It was a matter of preparation.” By the time she reached high school, she felt “insecure” as the material suddenly became more complicated. “If you’re not a strong student from an early age […] there isn’t enough of a foundation.” When she switched to a private school midway through high school she received more help but “my gaps were in the basic stuff”, and it was too late to fill them in.

Interestingly, this idea that once you’re doing poorly you ‘can’t catch up’ spanned all of these students, both in systems with and without tiering. In the school systems with tiering, the levels served to highlight when students were doing poorly, but Helen’s experience indicates that it may not be the root cause of their difficulty. And yet no matter what the cause, all of these students struggled to repair their foundations when they eventually wanted or needed to take upper level math for university — which they all eventually did.

Closing Invisible Doors

At a certain point, it becomes difficult to catch up, but not impossible. In high school, Ali discovered a love of geography and chemistry, pushing herself to succeed in subjects that she hadn’t naturally taken to but really enjoyed. Yet in her schooling system, she was required to select 3-4 ‘A-levels’ for her final two years of school, leaving no room for error. In order to get into a good college, she believed that she had to take classes in which she could clearly excel.

I wanted to do the things I was passionate about but was forced to choose what I was actually good at. (Ali, Ph.D. candidate, USA)

Once in university, it felt too difficult to catch up. There were no tutors available and even rudimentary math classes required preexisting knowledge that she didn’t have. She tried to supplement her academics with online sources, such as Khan Academy and mathematically-inclined friends, but ultimately believed that she couldn’t catch up in time to shift her career trajectory.

Now, as a university lecturer and Ph.D. candidate, she says that when she has to do basic arithmetic in front of her students (e.g., dividing them into equal groups) they “know” she can’t do math. This situation still causes her anxiety, as do other arithmetical functions of everyday life, such as banking, loans, budgeting, and even adding tax to the end of a bill.

In Helen’s case, she was held back from actually entering university to begin with. During the last few years of high school she faced problems because of math, and ultimately was not able to pass the admittance exams to the public Greek university because of them. Her workaround was to enter the American university in Greece, which didn’t have the same requirements, although there are systemic barriers to others doing the same. Equally fortunately, her brother ended up going to school in the USA, as he also did not pass his exams. Remarkably: in a way that is highly indicative of the arbitrary nature of these exams, her brother ended up flourishing in mathematics in college and majored in the subject at a top-tier school.

Ending up in a relatively quantitative business program, John was consistently confronted with math at the university level. He was able to recoup his understanding of the ‘why’ behind math to succeed in these courses, although this required a true reimagining of what math meant. It was only when he was surrounded by professors who finally took the time to explain the ‘why’ behind mathematical problems that he found the enjoyment in math. Now an operations manager, he has to use Boolean systems regularly and is confident that he has the fundamental skills and understanding to figure it out. He’s currently taking extra programming courses on the side to diversify his skillset, and says he gains a “sense of accomplishment” from solving quantitative problems. “I love this kind of stuff,” John says, and luckily, he’s been able to catch up and seize it.

Emily, who realized in high school that she wanted to be a nurse, forced herself to learn math outside of school. In the educational system at the time, she wasn’t required to take quantitative courses to be admitted into a top diploma program but was verbally confronted with her GCSE scores in the interview. Despite the gap in math courses that she had between 16 (when she could drop mathematics all together) and 18 (when she began her nursing diploma), she was required to do complex calculations in her practical training.

To fulfill this passion, Emily sought help from a tutor outside of her main program. She met with the tutor throughout the entirety of nursing school and found success because the teacher “explained math in an entirely new way.” There were no math classes in her college, which freed her from arbitrary constraints — like timed testing — to focus on what worked for her cognitive disposition.

For Erin, the realization that math was a necessary tool occurred the earliest. Erin described how her elementary math education bordered on gimmicky, learning times tables through rhymes: “Eight squared is two snowmen together (8). They have sticks (arms) for the fire, so it’s sixty-four.” When she entered middle school math suddenly became far more challenging, but she wasn’t set up for success. While she disliked math in elementary school, her anxiety skyrocketed from “I see this anxiety” to “this anxiety controls me”. After seventh grade, Erin confided that she never took a math test without crying.

The effects of this lag magnified as she moved through school. In high school, she enjoyed other subjects more and started to avoid subjects like math. She could “just get by” without them, so when she didn’t initially pass the test to receive college credit for her final math class, she refused to retake it since she could graduate without it. This came back to bite her in college, however, when she could have used the credit to opt out of the distribution requirement for math. “My entire college career was spent identifying what degree I could get without math,” recalled Erin. She all-but graduated from college, participating in the commencement ceremony but not receiving her degree because of the unmet math requirement.

Having successfully navigated her way into a fulfilling career, Erin doesn’t feel held back by her math education and no longer has anxiety when she has to perform quantitative functions in her role as a marketer, such as budgeting and reporting. Erin isn’t the only one who feels calm around math now: Emily, in an effort to remove the “blemish on her record”, is planning to voluntarily retake her GCSEs and feels much more confident now. Taking care of patients is a massive responsibility and her employer encourages all nurses to take their time with calculations, which has provided her with the platform to take pride in her newfound math skills. Math no longer feels scary, especially when done collaboratively with the other nurses.

An Ideal Pedagogy

For one golden year, Ali had a teacher who focused on explaining the underlying structures of mathematics and conducted the course as an exercise in exploration, which was the class she most enjoyed. It’s critical to “associate to people why it matters in the real world,” said John. Children and adults are searching for the ‘why’ behind the problem; since math is not as clearly connected to the real world as other subjects, being utterly foundational as a subject, this needs to be elucidated by professors.

Math is also often taught by removed teachers who don’t appreciate the emotional impact that the subject has on their students. Helen believed that unlike teachers in other subjects, “most of the math teachers were not very emotionally intelligent.” Similarly, Emily also didn’t feel that teachers were “approachable” or warm, although she didn’t have this experience outside of math. When in university Helen had more intuitive professors, she had a much easier time accessing help and understanding the material, because they provided individualized help. Because math is such a fraught subject for so many people, teachers must be empathic to the impact each class has on growing minds. Students cannot universally succeed at a subject when we take it for granted that they will intuit the material.

Furthermore, teachers are unable to accurately gauge how students are doing — and consequently, how much help they need — if we continue to rely on standardized testing. As early as six years old, Ali’s parents were told she was incapable of math and had a learning disability, after which an independent tutor contested that she had perfect natural ability but “hadn’t been taught properly”. We must accommodate for different learning styles and veer away from rote memorization, as that neither teaches true mathematical exploration nor does it work for many students. As Emily said, the “brain is a muscle with no one-size-fits-all approach.” She suggests that teachers outline multiple ways to solve the problem, rather than assuming that one explanation will work for the full class.

Sometimes clever and capable students stopped because they didn’t have the courage to carry on. (Helen, Opera Singer, Greece)

Helen felt an intense loneliness and “had something like a psychological trauma” as a result of her math schooling. Once she “realized it wasn’t [her] fault; it was the system,” she gained clarity around the failings of her education. Prior to her singing career, she studied university-level psychology to understand how the Greek educational system is set up and ultimately fails many students. It is imperative for all of us to have compassion and empathy for those students who are not doing well, who seem checked out, or who don’t appear to be trying. In the current world order, sometimes carrying on is the biggest step.

Words from the Wise

To help those already in a broken system, the people I interviewed had different takes on how students and families should approach their math education. “There’s not many things you can do because the problem has already been created,” lamented Helen. If we cannot fix the system, she advised, parents should take a concerted interest in helping their young children understand the fundamentals and help them “have patience and courage.” Yet, she did not encourage these students to ask for help from the educational systems that have failed them in the first place.

Perhaps in part influenced by her role as an educator, Ali had a different approach: “I’d tell [students] to ask for help and keep asking for help, and not be embarrassed. [You] just have to keep asking.” John echoed this sentiment, advising students to “make sure you’re speaking up and asking questions in class […] ask teachers why it matters — put that on them.”

As someone that didn’t complete homework throughout school, John continued that “homework is not a time-waster”, and that you need to keep on practicing. Finally, he advised: “if you do like math, go to college for it.” Since math is something that so many people struggle with, we need to fight the fear and encourage people to pursue their passions. Notice that he didn’t say that those who feel they are ‘good’ at math should pursue it, only those who enjoy it. This enjoyment is difficult to take when you’re anxious, but if you can break through that membrane into the feeling of learning and growing in the subject, that’s the goal.

The very fact that we’re having this conversation and that there’s a term called ‘math anxiety’ [represents] 99% of the fight. (Erin, Marketer, USA)

Through these individual accounts of the educational system, we can discern a number of tactical areas in which math educators can improve. Furthermore, for students currently struggling through a system that is not set up for their success, it seems like the most promising solution is to relentlessly ask for help — whether from parents, educators, peers, or online — and just keep asking.

This is a lot to expect of young children and teenagers, who have other subjects with which to be concerned, social development, home lives, and more. We are all failing these kids. To start, as Erin said, it is critical that we as a society admit that there is a problem with our pedagogy and that we have a long way to go.

* All names have been changed for privacy.
** GCSE stands for “General Certificate of Secondary Education.” These exams are taken in the U.K. at the end of mandatory schooling (within the older iteration of the system, which has since changed). They encompass all core subjects and are the closest equivalent to passing a GED (‘General Equivalency Degree’) in the USA, although they are part of everyone’s schooling and cannot be taken in lieu of high school.

The Voices of Math Anxiety was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

The second sentence of the title was inserted not to belie the great work of math educators everywhere, but rather remind us all that maybe the shading of arithmetic/algebra into obscurity is blinding us to the magnitude of our failures…

________________________________________________________________

Peter Harrison, my mentor who I taught with for 4 years from 1998 to 2002, and who was a huge influence in the landscape of math education during his career, started losing hope for its future a few years after his retirement.

Once over dinner, I distinctly remember him saying “…they should just turn math into Latin”. Implying that mathematics was losing its lustre as a language, and maybe just offer it as an “option” for anyone who is interested in honoring its central theme — the connection between arithmetic to algebra.

I don’t know anyone past or present who thought as deeply about building an organic and accessible bridge between these two powerful ideas more than Peter Harrison.

Unfortunately, the disjointedness between arithmetic and algebra has only gotten worse. Andrew Hacker’s New York Times article, Is Algebra Necessary, only widened the divide. Ironically, I agreed with much of Hacker’s article.

But, Hacker, like so many others, are lamenting about algebra in its present state of delivery — the way that it is taught. And, because of the way it taught, I would beat him to the front of the line of “Get Rid of Algebra”. Peter Harrison would have walked through that door 20 years ago…

However, the erosion of algebra in our education system has been going on for 50 years. What kids were able to do algebraically in the 60’s/70’s by the beginning of high school is what we hope our kids exiting high school can aspire to do. Somewhere along the way, a whole generation of students has become several years behind in algebraic thinking. If we trace the breadcrumbs, we will see that the “land” on the other side of the bridge — that was never been built for everyone to journey across safely and enthusiastically — is just as parched as algebra.

That would be arithmetic.

The treatment of Arithmetic in school is an abomination. What should be a joyful excursion of numbers, teeming with play, historical development, and deep connections, gets reduced to algorithms which are prefaced by the word “standard”. Its only headlining gig are those — as in the words of my 12 year-old son — boring PEDMAS/BEDMAS questions.

Algebra is the generalization of arithmetic. Separating the two is criminal.

Arithmetic and Algebra, once highly prized and valued — and connected — now have their discussions in math education’s discount bins. If these things could be thought of as wine, then their current status should put them on par with a jug of Carlo Rossi.

If we are truncating the experience of arithmetic, then we are definitely going to be shortchanging algebra. So yes Andrew Hacker, with the current state of the mathematical union, let’s ditch algebra.

But, I am not prepared to ditch algebra or arithmetic. I am prepared to fight valiantly for its resurrection through a prism of historical delight and imagination. The current lens of practicality and perfunctory practice is a complete distortion. It’s broken. It’s murky. It’s waste of time.

Students and teachers deserve better. Hopefully, the current road of play that math education seems to be embarked on, will guarantee some sightings of authentic arithmetic and algebra. And just maybe, enough will begin to understand and care about the idea that any restoration of mathematics must go beyond fixing and maintaining the bridge between these two giants of mathematics.

It must be celebrating. You can learn a new language out of practicality or for pleasure. Mathematics should be learned for both — and, for me, that includes a rich exploration of arithmetic and algebra.

The Bridge Between Arithmetic and Algebra is Broken. Which Means Math Education is Broken. was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

It took 358 years to crack Fermat’s Last Theorem. And, prior to its proof in 1994 by Andrew Wiles, it held The Guinness World Book of Records for having the most number of unsuccessful proofs. It is kind of deliriously maddening to think that something like the Pythagorean Theorem, that is almost universally known by every kid by their teens, had an “innocent” idea about exponent solutions other than two become entrenched in the lore of mathematics for 4 centuries.

It has even made several guest cameos on The Simpsons.

The entire history of the problem, while woven with many advances and mathematicians, is really a story about one of the most important and overlooked ideas of mathematics — especially in the world of speed and competitiveness found in K to 12 math education — time and inertia.

Some of the most brilliant minds in mathematics were stuck on this problem,and were outlived by its final resolution. But, there is something hidden here which speaks to what binds strategic thinking and a solution — that is…nothing.

In my IGNITE Talk at the Annual NCTM Meeting in San Diego, my opening two slides were blank, and I didn’t speak for the first 5 seconds or so.

I was trying to pay symbolic homage to the most beautiful and unheralded gift of mathematics — stillness. We have all been given that gift. You know, when you have that feeling that this problem has gotten the best of you. No more thinking. No more strategies left. Pondering and mulling coming to a stop. The only think left in the wake of all this problem solving is some blank staring that is a mix of mild fatigue, daydreaming, and quiet reconciliation. That the problem in front of you is outside of your capabilities — or at least, for now.

And, that is more than okay. That is the world of mathematical thinking.

I had hardly any of those moments prior to high school. The best courses and the best teachers I had offered plenty of moments to hit that mathematical wall, and yield to stillness. And, of course, university was filled with those moments — more than I should be admitting!

Those days — sadly — are almost gone. I don’t have as much time to be beaten by time. Most of the mathematics I encounter and promote is in the wheelhouse of my understanding. I can only talk about mathematical struggle from a nostalgic lens. So many days I wish I could sit with a problem and struggle with it, yield to it, and share that story.

We have been asked to slow down by Carl Honore for almost this entire century.

Mathematics goes one better. It asks us to stop. Well, “asks” is being polite. It demands that we stop. That stopping is not a sign of weakness, ineptitude, or failure. That stopping is a pause button is absent from our lives. And, while we should aim for rich understanding and illumination of mathematical ideas for our students and ourselves, we are doing a great disservice to the learning of mathematics and to the general wellness of how we think if we are obsessed with just showing the colors of mathematics — without the needed white space.

Just remember, most of the mathematical canvas is blank. It is not to be filled in. It is to valued for exactly what it is — nothing.

Promoting productive struggle is wonderful. But, we should also be promoting “letting go”, stopping, and maybe doing something else — like go for a walk(a James Tanton strategy). If a solution comes back to you. Great. If it does not. Great. Try again, or try another problem. Appreciate those blotches of vivid color you find, but be respectful, mindful, and equally appreciative of all that is not.

It is just as important.

The Most Underrated Gift of Mathematics is Stillness was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

A little over a year ago I wrote a blog post on what I like to call “The Era of Resource Abundance.” You can find that post here. To continue the conversation, I presented at the National Council for Teachers of Mathematics Annual 2019 Conference on April 5, 2019 on the topic. This blog post serves as a reflection of the talk, especially for those who could not attend.

I think we have a serious problem in math education today. In simplest terms, there is just too much crap out there. As a former elementary educator, I know firsthand how challenging it is to teach multiple subject areas and do them all justice, especially mathematics if one does not have a strong pedagogical content knowledge of the subject. If a district provided curriculum (if one is even provided) doesn’t suffice, teachers are forced to supplement with alternative resources. How do they find those resources? How do they determine if those resources are good resources?

As I thought more about this issue, I came across a set of research studies conducted by Iyengar and Lepper (2000) in which these researchers sought to better understand how the presence of choice impacts our decision making. Using field experiments, they conducted three different studies, but the one I focused on for my talk was Study 1: The Jam Study. The study took place at Draeger’s Supermarket in Menlo Park, California. The site was chosen for two reasons: (1) the supermarket is an upscale grocery store that is known for its extraordinary collection of items (e.g., they sell 250 different types of mustards, 75 different types of olive oils, and 300 different types of jams); (2) there was a regular presence of tasting booths at this store, so the study would not seem out of the ordinary. On two consecutive Saturdays, researchers set up two different stands that were rotated every other hour so both tables were not present to the consumers at the same time.

Stand A had six different types of jams, while Stand B had 24 different types of jams. So, which stand do you think attracted more people? Well, what if I told you that 40% of the 260 people who passed by Stand A stopped to try the jams? What percentage of those who passed Stand B then would you guess stopped to try the jams? It turns out that 60% of the 242 people who passed Stand B stopped to try the jams. This doesn’t seem unusual — there were many more options at Stand B, and as humans, we are curious!

Of the 260 people who passed by Stand A, 104 of them (or 40%) stopped to try the jam. Of the 104 people, what percent do you think actually bought the jam? Well, what if I told you that 30%, or 31 of the 104 people, bought the jam? What percentage of the 145 people who stopped at Stand B do you think bought the jam? Surprisingly, only 4 people (3%) bought the jam! People were at least six times more likely to buy the jam from Stand A than Stand B, even though there were ¼ of the choices available.

Professor and Choice Expert Sheena Iyengar (2000) said it best when she said, “The presence of choice might be appealing as a theory, but in reality, people might find more and more choice to actually be debilitating.” This is literally how I feel about mathematics education today.

When I apply the Jam Study to Math Education, I realize why so many teachers, especially our elementary teachers, decide to stick to their traditional teaching methods or their district provided resource or resort to Pinterest or Teachers Pay Teachers — because there are just too many places to choose resources from and while the presence of choice seems appealing, it is actually debilitating. Case in point: if I want to find a Three-Act math task to use (assuming I’ve heard and know what that is), at this present time (if I’ve found them all) there are 11 different websites I can browse! I don’t know about you, but when I was an elementary educator, time was my most valuable resource of which I just didn’t have enough.

This lead me to learning more about what is called the paradox of choice and its ultimate decision or choice paralysis, which I face daily. The paradox is this: less is more, and more is stressful. One would think that the more choices there are, the happier we would be, but after a certain point, too much choice becomes overwhelming. Famous psychologist George Miller would say the magic number is 7 (+/- 2). More than 7 choices or things to remember and we are just overwhelmed. I connect to this understanding every time I go to the Cheesecake Factory.

Who the heck needs 36 cheesecake options? I felt the same way this week at the Annual Conference. Just look at the program — for the 8:00 session on Day 1 there were 49 different options, not including the Exhibition Hall with over 110 booths, the Networking Lounge, Infinity Bar, and other things from which to choose. Now, I get it; we all have different likes and needs, so choice is needed, but there are times when too much choice is paralyzing and the problem with too much choice is that it causes us to overthink. Overthinking leads to: (1) lower performance; (2) lower creativity levels; (3) decision fatigue; and (4) dissatisfaction. I don’t know about you, but I don’t want teachers entering their classrooms with the impacts of overthinking before a lesson begins.

In further researching the impacts of overthinking, I came across a National Academy of Sciences study called Extraneous Factors in Judicial Decisions (2011). This study looked at how decision fatigue impacted the rulings made by parole judges. Look at Figure 4 to see a chart from the study.

Looking closely at the data, you can see that the percentage of favorable rulings in each session starts around 65% and then gradually drops to nearly zero within each decision session — with an abrupt return to about 65% after a break. Cases that came before the judges at the end of long sessions were more likely to be rejected. This phenomenon held true for over 1,100 cases, regardless of the severity of the crime. I fear a similar phenomenon is happening once a teacher has seen over 7 pins on Pinterest or any other website — their ability to make good judgment calls decreases and they end up settling on a poorly chosen task or feeling dissatisfied with what they did choose.

One way I’ve begun to combat this decision fatigue is through using a protocol. I looked everywhere to find one, and instead found many different ones, but none that met my personal needs. I decided to create one that worked for me — and I hope it works for you, too. When I see a task online, I take out my protocol and go through it as a checklist. If each and every criteria can be checked off, then I consider it a “good” task. The caveat here is that regardless of a good task or not, the pedagogy and instructional implementation is truly what matters. A good teacher can take a bland and boring worksheet and make it come alive. But, for those of you who struggle to do that, perhaps using this protocol might save you some mental energy. The protocol shown in Figure 5 encompasses what research and my personal philosophy states about high-quality tasks in what I think is an easy-to-use format.

During the workshop, I showed a variety of tasks and worksheets commonly seen in PK-2 classrooms and we used the protocol to determine whether or not the task was deemed as good. The conversations that ensued were tremendous. Perhaps in another blog I can go deeper into the rest of the workshop, but for now, give the protocol a try and let me know what you think. Always looking to improve. Oh, and be choosy about choosing, as Professor Sheena Iyengar (2011) says in her TED Talk.

References:

Danziger, S., Levav, J., & Avnaim-Pessoa, L. (2011). Extraneous factors in judicial decisions. Proc Natl Acad Sci USA, 108, 6889–6892. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3084045/.

Iyengar, S. (Columbia) & Lepper, M. (Stanford). 2000. When choice is demotivating: Can one desire too much of a good thing?, Journal of Personality and Social Psychology, 79, 995–1006.

Iyengar, S. (2011). TED Talk. Obtained from: https://www.ted.com/talks/sheena_iyengar_choosing_what_to_choose/transcript?language=en#t-225497.

The Era of Resource Abundance Part II: How to Navigate Through the Crap to Find the Rich and… was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

Here’s a strange question.

How many degrees are in a Martian circle?

And I am serious. I want you to answer it. Go!

The Nature of Mathematics

Mathematics is a human endeavour, created (or discovered?) by humans for humans, and as such is a fundamentally human experience. It is a glorious subject chock full of passion and emotion. Why else does Sir Andrew Wiles cry on camera in the 1996 BBC Horizon’s documentary when describing his journey in solving Fermat’s Last Theorem?

Every challenge or problem we encounter in mathematics (or life!) elicits a human response. The dryness of textbooks and worksheets in the school world might suggest otherwise, but connecting with one’s emotions is fundamental and vital for success — and of course, joy — in doing mathematics.

So… Experience mathematics as a human! Help your students do so too!

Answering Strange Questions

There are two key first steps to solving any given challenge. These steps are so fundamental, so important, and really will help you and your students make serious progress. I’ve never seen them explicitly stated — perhaps another consequence of the disconnect between school curriculum culture and the actual practice of mathematics — so let’s remedy that now.

STEP 1 to PROBLEM SOLVING: Have an emotional reaction.

We can’t help it. We are each human and we react emotionally to challenges. So acknowledge your human self by pausing to acknowledge your human response to a problem. That is, explicitly take note of your internal state and give it voice. Doing so gives one’s emotions a place to sit and simply be, and not overwhelm.

If a challenge looks scary, say to yourself, “This looks scary.” If it looks fun or intriguing, say “Wow! Cool!” or “Wow. Weird. Could this be true?” If you are suspicious, say “Ooh. Could matters be this straightforward?” If you are flummoxed and don’t have a clue what to do, acknowledge that and say “I don’t know what to do!”

Next step is to take a deep breath and …

STEP 2 to PROBLEM SOLVING: Do something! ANYTHING!

The key is to work past any emotional block that might hold you back. Turn the problem page upside down, draw a diagram, draw a tree, circle some words, answer a different question that might or might not be related. Simply force yourself to actually do something with no expectation of it leading you down the right path — or down any path for that matter. Really, just do something!

I cannot underestimate the power of taking a piece of action of any kind. (In fact this, I think, is my greatest wish for mankind, for everyone to have the confidence and sense of agency to do something — anything! — in reaction to a problem or challenge in life and not to sit completely stymied and stuck.)

An Example: Putting these Steps into Action

I chose the Martian question here as it is so “out of this world” that you really might be flummoxed by it. (Did you say, “I don’t know what to do!”?) What could the question possibly be asking?

In order to DO SOMETHING, let’s not answer that question and answer an easier question instead. Why not?

How many degrees are in an Earthling’s circle?

Well, we Earthling’s say that there are 360 degrees in a circle.

This now begs the question

Why that number? Who chose the number 360 for the count of degrees in a circle?

And when you sit with this question for a while you might realize that this number is very close to a count in a regular, cyclic phenomenon we humans experience: the count of days in a year.

Babylonian scholars of 4000 years ago were very much aware that the count of days in year is 365¼. Shouldn’t we be saying then that there are 365¼ degrees in a circle?

The answer to this question might be clear, and very human: Who wants to do mathematics with the number 365¼? It’s a very awkward number! The natural thing to do is to round it to a friendlier value.

If we round the number 365¼ to the nearest five or the nearest ten we get 365 and 370, not the number 360. Why did we humans decide to round down to 360?

Let’s continue to be very human.

Thousands of years ago there were no calculators and all arithmetic had to be conducted by hand or in one’s head. It is natural to work then with a number that is amenable to easy calculation.

Often in mathematics we want to divide numbers by two and we see already that choosing 365 as the count of degrees in a circle is unfriendly. Both 370 and 360 are even at the least.

We often want to divide things by three as well, and 360 is now looking good! In fact one realizes that 360 is a much friendlier number for arithmetic over 370: it is divisible by three, four, five, six, eight, nine, ten, twelve, fifteen, eighteen, twenty, and more! Whoa!

So for two very human reasons — what we experience on this planet and our desire to avoid awkward work — we settled on the number 360 for the count of degrees in a circle.

Can we now answer how many degrees are in a Martian circle? What do we need to know?

Martians might follow same reasoning we humans did, but in their context. So we need to know how many Martian days (we call them sols) are in a Martian year.

Each sol is 24 hours and 37 minutes long and Martians experience 667 sols in their year. So we might argue that Martians might initially say that there are 667 degrees in a Martian circle. But given that this an awkward number for basic arithmetic, they too will likely round that count to a much friendlier number.

So … What number do you think that might be?

The Moral of this Story

You might be wondering in this essay: Where’s the math? Why is there no analysis of an actual mathematics problem?

The point of this essay is to simply illustrate the power of being true to one’s human self. We have each encountered math problems in our work that have stymied us and have threatened to shut us down. That happens and that is okay.

And the way to get unstuck is to acknowledge that you are stuck — acknowledge your emotions — take a deep breath and just do something, anything! Astonishingly, that alone can often be enough to break through an impasse and get you going on some fun thinking. And thinking is always fun!

The gist of this essay is courtesy of the Global Math Project.

Two Key — but ignored— Steps to Solving Any Math Problem was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

Both of my kids love math. Both of my kids find math to be their least favorite subject in school. That isn’t so much as contradictory, as it is history. What is sadder is that I am now indifferent to their situation. I have stopped asking what they are doing in math, because hearing monotone responses of “something about trapezoids” has become disheartening. I don’t worry that school isn’t going to make them functionally literate when it comes to math — although the tax for that will be antiquated tests, worksheets, statistically and emotionally meaningless grades, and other assorted goodies from math education’s 20th century time capsule.

What I worry about is my kids not loving mathematics.

Which means I worry about kids in general not loving mathematics. It turns out a lot of other people are “worrying”. Those Twitter numbers below transpired in just over 24 hrs.

The problem for mathematics’ marriage with education is that it has generally suffered from an orthodox mentality that can be traced back to the purpose of school from the Industrial Revolution. Practicality and Efficiency in the acquisition of mathematical knowledge. Math education has not deviated too much from that in the last 100 years. But, to be fair, education has not deviated too far from its Dickensian roots either.

Until this past decade.

This decade has given birth to what we can almost officially refer to as The Age of Disruption, where traditional pillars of trust — institutions — have started to crumble. As such, the historical, vertical movement of trust through traditional hierarchies is losing currency. Trust, as I have mentioned in previous articles, is moving horizontally with a robust engine of socialization/deeper human connection.

What is in abundant flow — a tsunami really — are deep dives into the human essence of learning and sharing. And so many of these ideas, emboldened by a revolution of the human spirit in education, are being committed to game-changing books.

One of the latest offerings from the wildly popular books published through Dave Burgess Publishing is Elizabeth Bostwick’s, Take The Leap.

I would like you to not only just pause on those words, but just the cover. The cover, with its colors, variety of fonts, and images of connected gears, symbolizes the changes that are rapidly occurring in education.

Mathematics cannot be immune from these changes. It cannot continue to operate and function with directives and mandates that occupy the narrow domain of outdated procedures and purposes.

And even with the tremendous work of people like Jo Boaler(youcubed) and James Tanton(The Global Math Project), math communities all over need to invite more holistic and human ideas about education — to ensure that the work of these math revolutionaries not just survives, but thrives.

It cannot think for one second that all the changes happening in education will bypass it — that there will be no cross-pollination of innovative/disruptive ideas that have become omnipresent in schools everywhere.

Francis Su, who gave the closing keynote at NCTM’s Annual Meeting in 2018, has raised the bar for all of us to “leap” over…

Human Flourishing. That is the ocean that education is trying to find. Math education, while still generally running like a constrained and stressed river, is also beginning that search. The revolution that was promised in the past or lurked in closets, wholly intimidated by math education’s elitist structure and machinery, is finally afoot — fearless and formidable.

The math revolution is being driven by play, truth, beauty, justice(equity), and love. My own children will most likely miss it — but, their kids won’t…

The revolution is for the next generation. I do not need to see it. I just need to know it will happen.

No more shadows. A sun awaits us.

The Math Revolution: No Longer In The Shadows of Subversiveness was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

There is a psychological tactic employed by some classroom teachers following a strict curriculum: to bond with one’s students by casting the enforced textbook as a common enemy. “We’re in the system and this is the game we have to play,” is the sad and perhaps demoralizing approach to this. But if conducted with grace and care, elements of this psychological stance can be respectful, uplifting, and pedagogically good.

After all, any passage of text provides opportunities to question content, probe inconsistencies, explore missed opportunities, counter seemingly inflexible assertions and definitions with exceptions and alternative approaches, and so on. That is, all texts can serve as an invitation to examine the place and context of content. And in this content-rich, content-at-the-ready world, isn’t it all the more important that 21st-century education should help students learn to be the arbiters and assessors of their own knowledge?

This applies to mathematics education and its availability of content too. Once everyday number facts and facilities are at hand by middle school, the remaining six years of mathematics education could attend to the thinking and meta-thinking of mathematics, and the self-reliance of thought this induces. We could, with intention, help students learn how to learn, to personally assess what they know and how they know it, and to differentiate between familiarity and understanding: the familiarity that comes from repetition and rote doing versus the empowerment of knowledge.

So how does one foster student metacognition, self-confidence, and nuanced understanding when presented with a rigid, upper-school curriculum that is content focussed and content laden? The “system” need not be an enemy, per se, but can we identify opportunities within the system that extend beyond itself perhaps despite itself?

For starters, we might argue that the volume and nature of the mathematics content we are expected to cover in high school mathematics in and of itself holds a message. Factor trinomials, use 2x2 matrices to represent certain geometric transformations, analyse the ambiguous case of the Law of Sines, and so on and so on and so on. The count of disparate topics makes us realise that they can’t all be sacred topics that need to be taught. No topic is important because students will “need to know it later on.” (Or, if I am wrong about that, then the topic could wait until “later on” when it is actually needed!) The notion of detailed “sacred high-school content” that must be taught is a myth. Accepting this and being honest with your students about it is freeing and liberating! It gives the community of your classroom permission recognize pieces of content as beautiful vehicles for learning new potent thinking, and to enjoy the power this can bring each time. Sure, answering loads of “what” questions about content will be on the standardised exam, but all will know that that is not what defines the joy of mathematics. How lovely!

Second, in the push to cover content, many curricula present material briskly and with authority. It gives the impression it is removed from human story and exempt from questioning. And this is gorgeous as this is the very stance that should invite skepticism and question! So then, question it! You yourself, and you and your students together.

This point has struck home for me very recently as I am currently reviewing an international mathematics curriculum with a predilection for authoritative commands to students like “simplify, ” “factorise” (U.S. educators say “factor” rather than “factorise”), “solve via the so-and-so method,” and so on. It’s all very intimidating. My inner-student cowers — I grew up in a mathematics education system of this nature and recognise its effect — but now I have adult confidence and am feisty with the confidence to say: “Hang on! Let’s think about this. And let’s have students think about this too!” I want to empower students with the intellectual might and cleverness to push back when they deem it appropriate themselves to do so.

I’ll present next in this essay three examples on this matter that just arose for me in reviewing this particular curriculum. They are the value of questioning the importance of a “sacred” topic, the value of probing deeply into nuanced understanding, and the flawed use of the word “know.” I have taught each of the specific topics mentioned to high-school students and I have practiced what I preach in each of the examples below.

But I understand it is a challenge to change the culture of what defines success in the high-school mathematics classroom. I have the advantage of a Princeton PhD in mathematics under my belt (and a British-esque accent to boot) to help students — and parents — believe in my vision of long term mathematics meaning and value. But I am also very cognizant that assessment, in particular, standardised assessments, are the first order definors of high-school mathematics value. That is not going to change anytime soon. So the key is to take all I offer here as casual advice and simple first baby steps towards inducing cultural change. But please don’t underestimate the power of such simple first steps!

Factoring

Every algebra curriculum has students spend hours — many hours — factoring mathematical expressions that have been carefully crafted to magically factor. (Most expressions don’t.) So why not explore a question of the following type?

In a class of twenty students, each student can pick three quadratics at random and attempt to factor them. From a sample size of 60, maybe one can garner a sense of what proportion of quadratics are actually factorable? (Or why not use Wolframalpha and examine a larger sample size? Or use basic coding software to run through all 900 quadratics? Why stop at single-digit positive integer coefficients?)

The international curriculum I have been examining asks students to identify the “largest common factor” of three terms such as 12x³ , 6x⁵, and 10x⁴. The desired answer, to be written in the box on the exam page, is 2x³. But pause. Think! This is a question from algebra class where x represents a yet-to-be-specified value. It might well be a whole number quantity, but it could just as well be a fractional quantity or an irrational quantity. We don’t know! This question is akin to asking for the greatest common factor of 4, √3, and ⅚ and so is technically silly since every real number is a multiple of any other non-zero real number. (For instance, 4 is a multiple of 17 since 4 = 17*(4/17).) The notion of a greatest common factor is meaningless in the real number system. What a great conversation to have!

But the meta-conversation to be had then is: Okay, so what are the curriculum authors meaning to ask?

One realises that the curriculum chooses to focus on polynomials with integer coefficients involving non-negative powers of x. (Umm. Why?) The authors probably thus seek as an answer an expression of the form axᵇ, with a and b non-negative integers as large as possible. In which case, when later asked to “factorise fully” 6x⁵ + 10x⁴+12x³, one is probably expected to write 2x³(3x² +5x+6) and not x³(6x² +10x+12) or 10x⁴(0.6x+1+1.2/x) or some other expression that might be more relevant for some later, yet-to-be specified, context.

Question: Simplify 6x⁵ + 10x⁴+12x³.

The Actual Correct Answer: It looks fine as it is. It all depends on what you want to do next with this expression. If there is no “next,” then what more is there to do now?

Solving Equations

Mathematics is a language … literally! Every mathematical statement one writes is a sentence. For example, the statement 3 + 4 = 7 has a noun (the quantity “3+4”), a verb (“equals”), and an object (the quantity “7”). The statement 5>8 is also a sentence.

The first sentence happens to a true sentence about numbers and the second a false sentence about numbers. As mathematics tends to focus on truth, it is interested in sentences that represent true statements about numbers.

The statement w²+w = 2 is a sentence about an unspecified value w and is neither true nor false as it stands: it all depends what specific value one might want to assign to w. If we set w to be 1, then we have a true number sentence. If we set w to be 2, then we have a false one.

Similarly, the sentence x = 3 is currently neither true nor false. If x is 3, then we have a true number sentence. If x is set to be 4, then we have a false one.

When presented with an equation it is natural to seek the value(s) of the unknowns that lead to true number statements. For example, the set of all numbers that make the equation w²+w = 2 a true number sentence — its solution set — is {1,-2}. The solution set of x = 3 is {3}.

The Exit Ticket shown above should give full credit to the student who clearly recognizes that “x=3” is an equation and that the solution set to the equation is the set of numbers {3}, and, perhaps, more subtly recognizes that writing solution sets as “w = 1 or -2” or “x = 3” is technically not correct: these are both still equations that each might or might not be true, but their solution sets are so blatantly clear that folk tend to regard these equations as statements of solution sets. (Indeed subtle! A lot to be discussed here.)

Irrational Numbers

Every curriculum I have seen expects students to “know” that √2 and π are each irrational.

Umm ... How?

There are some texts that do share proofs of the irrationality of the √2 . (Start by assuming that we can write √2 = a/b as a reduced fraction and see what goes wrong. Squaring and some algebra gives a²= 2b² showing that a², and hence a, is even. Writing a = 2k then gives 4k² = 2b² leading to b², and hence b, being even too. But this then contradicts a/b being a reduced fraction in the first place. Our beginning assumption that we can write √2 as a reduced fraction just must be wrong.)

But how is one meant to “know” that π is irrational? One is just told!

Okay then, so let’s actually tell the story, the whole story. Scholars across the entire globe struggled over the question of the rationality or irrationality of π for well over 2000 years. That’s worth sharing!

To human eye, wrapping a string seven times around the lid of a jar seems to match perfectly 22 copies of the width of the jar. (Try it with your students!). This suggests that π might be 22/7. But is it?

Next have your students Google the work of Archimedes of Syracuse from the third-century BCE who showed that π actually has value just shy of this fraction, and of how fifth-century A.D. Chinese scholar Tsu Chung-Chi found an approximation we recognise as correct to seven decimal places, and of the general race over time to the digits of pi, more and more of them, all the while scholars still were still unsure whether or not the number was a fraction. (Could it be a fraction with some extraordinarily large numerator and equally large denominator?) Students will read that it wasn’t until 1760, after some 2000 years of wondering, that Swiss mathematician Johann Heinrich Lambert finally settled the question as to the rationality or irrationality of π once and for all. Using advanced techniques he proved that π is simply not a fraction. (And why not look at his proof on the internet too and see the reason why his argument does not appear in school textbooks?)

This story is good. But, as educators, can we go further and ask about what mathematics we could offer students to hold on to and own for their themselves from this story?

My answer is this. Let’s have budding scholars construct their very own irrational numbers! Try this:

Have students use long division to compute 1/3 as a decimal and then think about why one is trapped in a repeating cycle. (Exploding Dots makes this fun and easy.)

Next have students use long division to compute 4/7 as a decimal (do it too, right now!) and see that one is trapped again in a repeating cycle. (Why? As you try this you will see that there are only seven possible remainders that could appear in the division process and so one must soon repeat some remainder. As soon as you do, you are in a cycle.)

Next imagine — but don’t conduct! — long division to write 13/32 as a decimal. What remainders could appear? Must one eventually repeat a remainder and thus fall into a cycle? Yes!

In a jiffy we learn that every fraction has a decimal expansion that falls into a repeating pattern. (Even ¼ = 0.2500000…. falls into a repeating pattern of zeros.) So any number that has a decimal representation that does not fall into a repeating pattern cannot be a fraction. WHOA! And such numbers are possible to write down. For example, here is James’ Irrational Number

0.101001000100001000001000000100000001000…

This number is a little over a tenth. Its decimal digits have a pattern (for instance, I could tell you what the billionth digit if you really wanted me to) but it is not a repeating pattern. Therefore this number cannot be expressed as a fraction. It is an irrational number.

Exercise: Your turn! Now make up an you own irrational number that you actually KNOW to be irrational.

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Each and every curriculum is chock-full of intended and unintended nuances that invite questions and exploration. As they occur to you, share your questions and wonderings with your students. They too will notice and share their own wonderings, and a lovely snowball effect of genuine intellectual curiosity will evolve.

This is joy in empowered learning.

Breaking the Mathematics Education System Despite Itself was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.

A new, and wonderfully charming new friend on Twitter posted this reply to it:

This reply got me thinking about the practice of “Extra Credit” in math classrooms. Up front, I have to state emphatically and truthfully that I have engaged in the tradition that I am about to excoriate. So insidious are the effects of prior experiences on our teaching practices.

So here is my puzzlement on the teaching of mathematics today…Why are the good, fun, interesting problems left for “Extra Credit”? Why can’t Simon’s picture above be the starting place for some lessons? What effect on student learning would this have?

As I am working on this little piece my friend, co-author, and kindred mathematical spirit Sunil Singh has just published this short piece with a similar theme, check it out here. While I was reading this piece, I am struck with the idea that perhaps it would be better to grow our students into flexible, creative thinkers who can tackle novel situations rather than regurgitate a formula and use it correctly in a narrow application?

Anyway back to the good stuff and it not being left for extra-credit…Look at that picture again. When I look, I see lessons involving Geometry: area, shape, transformation, shape naming and describing, proportionality, composition, and decomposition, perhaps you see others… I also see opportunities to develop: Number Sense, factors, composition and decomposition of magnitudes, links to primality, algebraic thinking. All that just on the face of the original question, so much richness here. If a teacher spends the time and massages out and teases the students with a few probing questions, you could learn a lot about how they are thinking, and what they really do understand. Further, you could ask focusing questions (not funneling questions) to help bring their perceptions into greater clarity.

Also, this picture with that simply stated, easily understood question does not need to be the end of this experience. Put students in groups and give each group a different pattern block but lots of copies of it, tell them “This is your 1” construct a puzzle for the other groups in the same fashion as the one in the picture, be prepared to ask your fellow students focusing questions to guide them. Then let them go, let them play, design, create and generally LEARN mathematical thinking.

Good problems are where good lessons should begin, not be tagged on and left to the few who choose to do extra-credit. Perhaps, if all our lessons began well, connected to many topics, created deep understanding, and provided opportunities for play and creativity; the need for extra-credit problems would disappear altogether.

I hope you see my problem isn’t with the practice of extra-credit, but really in general lesson design. Rich contexts, provide plentiful opportunities for learning. Let us strive to always place our students in places of abundance, not paucity.

Extra Credit Should Not be Where the Good Stuff Lives was originally published in Q.E.D. on Medium, where people are continuing the conversation by highlighting and responding to this story.