a math education discussion

A Parent’s Guide to The Bright: Smart is Their Superpower

The_Bright_Cover_for_Kindle (6)


11-year-old Gauss wakes up to find that his friend has been kidnapped. Little does he know that this event is just the beginning. Evil is coming to the small town of Franklin, and it’s coming for a very specific reason.

It’s up to Gauss and his friends to stop the threat.

In most ways, Gauss is an average boy. Actually, that’s too kind. Physically, he’s the opposite of imposing. One might guess that he weighs only slightly more than a feather and has the muscular strength of a mouse.

If there is anything extraordinary about Gauss, it is his math.

In a popularity contest, math ability is the equivalent of forfeiting. However, it’s appreciated in Gauss’s circle of friends that includes a young hacker and a budding engineer.

Gauss’s crew is talented, but they’ve never seen a challenge like what’s coming. What has brought this danger to their town? Can they piece together the clues in time? Can they outsmart evil?

What Parents Need to Know

The story contains instances of kidnapping and arson. There is one scene where an 11-year-old girl is struck in the leg with a blunt object. In another scene, a young character learns that her father has died in a car accident.

The Math!

In one scene, 4 friends must determine which of 16 dishes is tainted with a laughing potion. There isn’t enough time for each child to try a dish, wait to see if it’s tainted and then try another one. All eating has to be done immediately. Can the 4 children find a way to determine exactly which dish has the laughing potion?

In another scene, 4 of 9 scientists are impostors (and up to no good). Gauss must find a real scientist to thwart the evil plan. Gauss can talk to the scientists, but he knows that the impostors might lie to him. How will Gauss solve the puzzle?

The math in this book is challenging enough to spark interesting mathematical conversations. Parents are encouraged to read along with their children and to discuss why Gauss’s methods are successful.

It’s Not About the Math!

Gauss’s greatest challenge is an ethical one. Ultimately, he’ll confront the source of the evil in his town and hold that individual’s fate in his hands. After experiencing kindness repaid and learning from a role model who always turns the other cheek, will Gauss choose to punish or save?

Kindle ($2.99)

Paperback ($8.99)

I hope my kids fail

May 27, 2016


I want my kids to fail, and I don’t mean the type of “fails” found in the YouTube vines that resemble a blooper reel.

I hope my kids pour their hearts into an endeavor, such as making the school soccer team, getting the lead role in a musical, or passing an important exam.   I hope they practice and rehearse in a way that would make any parent proud. I want them to fret for weeks in anticipation of the tryout or test, and to be overwhelmed by nerves while waiting for the results.   I hope my kids face a challenge that completely consumes them, that feels like the most important thing in their young worlds, and when they’ve got what they think is all their chips on the table, when they’ve opened themselves up for an honest assessment that could crush their egos, I hope they fail.

I’m not immune to the pain that comes with defeat. I hate to lose, and I rarely admit my mistakes.

I don’t take pleasure in other people’s misery either. If there is anything that stings more than my own failures, it’s watching my children struggle.

Failing is difficult. However, in direct contrast to what society tells us, failing is not the opposite of success. Failing is the path to achievement.

Michael Jordan was once cut from his high school basketball team. It’s been claimed that Albert Einstein couldn’t read until age 9. “Walt Disney was fired from a local newspaper for not being creative enough.” These types of stories abound, because it usually takes time, practice and perseverance to be truly great. Along the way, there are going to be failures.

If my kids are good at something the first time they try it, they’ll be better the next time. If they are excellent on the first try, they should try something harder. If their first attempt was a complete disaster, then they may have found a challenge that’s truly worth pursuing.

To succeed, my kids probably must first fail … a lot.   But, failing itself is sooooo challenging. In other words, to fail successfully, they must first fail at failing.

Failing is painful, and it should be. We should be sad, frustrated, and curse-the-gods mad when we don’t succeed. That comes with ambition.

(Ambition usually leads to an honest attempt, and without an honest attempt, failing is…well…just failing, and that’s not so good.)

While I never like to see my children suffer, I can handle when they cry after losing a Saturday soccer game or get visibly frustrated when they struggle to read. (On the other hand, when Dad flips the Candyland board because he’s about to lose, it’s not a praiseworthy demonstration of his competitive fire. He’s just being a jerk.)

There is a part of me that’s happy when my kids are upset with a mistake or a loss. Their feelings show me that they truly want to succeed.

I don’t want my children to be comfortable with failing. However, I do want them to recognize that it’s a byproduct of a constant quest for self-improvement. I want my children to learn that failure is agonizing and good!

Unfortunately, my kids are growing up at a time when we don’t keep score at children’s sporting events, and we’re afraid to mark homeworks with red ink.

I’m aware of the science that suggests that kids have trouble losing or feel bad when their exam is graffitied with red marker. I’m not sure that bad feelings are a problem (and those studies aren’t telling parents anything they don’t already know.)

I’m also aware that competitiveness can interfere with development, and a lack of confidence can be an obstacle to learning. Those are real problems that need to be addressed, but is the solution to avoid the obstacle?

When we stop keeping score and don’t tell students they are wrong, we’re avoiding the problem, and we aren’t providing the students the opportunity to experience and learn from failing.

Instead, could we figure out how to use these situations as teaching moments? Can we change children’s mindsets so that they don’t view their mistake or misfortune as an indication of an irreversible condemnation of their academic (or other) career? Can we teach them that it’s ok to be frustrated, but that they should be proud of their honest attempts, and through their efforts, they’ve improved themselves?

Most children aren’t very good at math. If we treated math like we do losing, we’d wait until children were 18 to teach multiplication. We can probably guess how that would turn out. Our children would grow up to be as likely to attempt a math problem as current college students are to raise their hand when they aren’t sure of the answer.

At my college, we will soon be allowing students to take some courses pass/fail. This change was largely motivated by students requesting the opportunity to explore new topics without fear of it impacting their GPA. Ponder that for a moment. We’ve created a system where students go off to college and then avoid exploring new areas of study that interest them.

I completely understand why students are afraid of poor grades, why B has become the new F. Future employers and graduate schools review students’ transcripts like their academic record is a pair of mint condition original Air Jordans from 1985, which have been autographed by Jordan himself. Any blemish on the student’s record greatly decreases the student’s value.

Instead, a student’s academic history should more closely resemble a marathon runner’s training shoes. The holes and worn-out soles suggest a runner that has trained hard and is prepared for the big race.

Yes, consistently low marks can indicate the type of poor performance or general apathy that employers should avoid. However, poor marks in challenging courses early in a student’s career can indicate a willingness to challenge oneself, and improved grades over the four years suggest a student can persevere.

As a mathematician, I have an unhealthy obsession with quantifying things, but I’m also aware that a student’s GPA cannot capture all of her academic qualities. It can’t even capture the most important ones.

Students’ obsessions to preserve their GPA aren’t a reflection of their fear of failing. Their obsessions are a rational response to society’s inability to properly interpret failure.

In regards to the students, I’m more concerned with their unwillingness to participate in class unless they are absolutely certain they are correct.   I’m troubled over their extreme reactions to a homework problem they view as too challenging.

If students immediately know the answer to every question, then they aren’t in an appropriate class. If they are going to learn, they need to challenge themselves. They must encounter problems that they don’t know how to solve. Inevitably, they will get some wrong. That’s more than acceptable; that’s learning.

So, the next time a teacher asks for a volunteer to attempt a difficult problem on the board or the next time the school choir holds auditions for solos, I hope my children are first in line. Then, with the force of a heavyweight champion’s right hook, I hope defeat knocks them on their asses.

When they come to their senses and regain feeling in their legs, I hope they are proud of their efforts, they learn from the experience, and they try again.

At that point, I’ll know that my children aren’t afraid to take chances and that they are up for a challenge. That’s what they will need to succeed.

Should colleges stop giving grades?

Educators at any level can question the value of grades. My experience is at the college level, and I will approach the question for that domain specifically.

What if colleges stopped giving grades?

THAT’S ABSURD! Grades are an essential component of our educational system. Without them we’d have a building with no scaffolding. Sure, in an ideal world where every student was self-motivated and driven to succeed, grades could be seen as an unnecessary complication. But we don’t live in an ideal world! The slightest disruption, such as students that haven’t yet awakened to the value of education, will send the entire institution crashing to the ground, like if a hurricane hit a skyscraper with no support beams. Students will stop trying.

Was that your reaction to the question? That’s how I first reacted.

But would that really happen? If it would, then why? Why would removing grades undermine efforts to educate? I’m admittedly afraid of the answer to that question because I believe it reveals a major problem in many educational systems.

“Is this going to be on the exam?”

That question irks me like no other. It’s not that I dislike students questioning the value in learning a particular topic. Students should absolutely ask, “Why should I learn this?” The answer is never that it’s on an exam.

I know that if I indicate any topic that we are discussing is not on the exam, half the students (or more) will check out. Why?

I informally surveyed students, “What would happen if we stopped giving grades?” The most common response was that students would stop doing the work. That answer troubles me, but what troubles me more is that it didn’t trouble many of the students. They fully acknowledged that students learn to get a good grade, and they were perfectly content with that reality.

Grades are a motivator for students to learn. That alone may not be a major problem. Unfortunately, the students’ response suggests that grades are often the only motivator to learn. That is a BIG PROBLEM!

Some motivators are better than others

At some point in my math graduate studies, my advisor suggested I read a book on an advanced topic. At the time, I wasn’t sure how the topic related to my research. However, I trusted and respected my advisor, and so I began a several-week study into the book. I worked hard. I took very detailed notes. I read and reread, I worked out examples on my own, and I tried to understand how this new material could be helpful in my future math work.

When I was done, I brought the book back to the library and set my notes aside. Several months later, I was tidying up things and I came across a curious binder. Inside were very detailed notes, but the subject was foreign to me. At first I thought someone had mistakenly left his work on my desk.

They were my notes from the book my advisor had recommended. I had worked so hard, and almost none of it stuck.

I was several years into a graduate program in a subject that I was passionate about. I ate, breathed and dreamed mathematics in those years. I was shocked that I had retained so little.

I compare that experience to the times I raced to the library to pull a book that I thought might hold the key to unlocking the problem I had been working on for months. I would devour those readings, often working late into the evening (by choice). The lessons learned from those activities are chiseled in my brain never to be forgotten.

Grades as a motivator

Education is a lifelong process. In a sense, it’s a marathon, but I don’t mean that to imply it needs to be painful.

When students are motivated solely by grades, it turns education into a series of sprints. At the college level, students can sprint so hard (via caffeine and cramming) that they vomit back most of what they learned immediately after crossing the finish line (after the final).

The shortsighted goal of getting a grade invites strategies of memorization and tricks in place of true understanding. In the worst cases, it leads to cheating, which is the ultimate demonstration that the pressure for grades is outweighing any intrinsic desire to learn.

If the primary objective of students in algebra class is to get a good grade, what reason do they have to explore how algebra might be of value to them in the future? What reason do they have to explore the connections between subfields of mathematics? What reason do they have to learn how math might be connected to music, religion, or politics? After the course is complete, what reason do they have to reflect on their experience or to use the foundation laid through classwork to continue an unassigned investigation into the subject?

Grades have become a permissible placeholder to answer, “Why should we learn this?” Students accept it, and so teachers need not delve into a complicated discussion of a subject’s true value. Even if the teacher tried to explain, a student might ask, “Is this going to be on the exam?”

We should strive to instill in our students a sense of the value of the subjects we teach so that they can have the mindset of a lifelong learner. Grades are an obstacle to impressing proper motivation.

Assessments need not be graded

10-year old Ava comes home from school. She drops her backpack at the door, but never steps inside her house. Instead, she picks up a basketball and heads to the driveway. She spends some time imagining she’s Maya Moore or Britney Griner of the WNBA going up for the game-winning shot. She then runs through some drills she picked up in practice to improve her post game.

Ava’s basketball game has been and will continue to be assessed. Her coach pays attention to her strengths and weakness. Right now, that means he spends extra time coaching Ava in her weakness, and it has some influence on how Ava is used in games. For example, her coach tends to have her play on the post because she’s a good rebounder and finisher there. Ball handling isn’t her strength. So, she doesn’t usually bring the ball up the court.

The game itself is an assessment. Ava is aware of when her team wins and loses, and by how much. She understands that losing can indicate that her team didn’t perform well, and she’s part of that performance.

In future years, Ava’s performance will impact her playing time. It could also determine whether she makes the team or gets cut. It could be the difference between a college scholarship or not.

Athletes are assessed all the time. However, they never receive a report card in the same sense as we see in school.

Much like Ava’s coach, teachers are constantly assessing students. Students are assessed as they respond to and pose questions, work in groups, and present to the class. They are assessed through their papers and assignments.

As I observe a student presenting a solution at the board, I am trying to gain insight into their understanding of the subject. I am asking myself, “Do they understand?” I never put those questions in the context of a grading system. I never observe and think, “That student is a C.”

A school can stop giving grades and still assess effectively.

No participation trophies

You want to drop grades? Why? Are you afraid to hurt the students’ feelings? Are you scared of crushing their fragile egos?


I’m not suggesting that academic assessment should mimic sport assessments, but it’s important to notice that athletic assessment can be much harsher than academic assessments.

Compare what a student might experience in math class to what they would see when trying out for the basketball team.  Right now, a student can get through an algebra course and “earn” a C with perhaps minimal engagement. Imagine if instead, the teacher walked into the algebra class of 30 students on the first day and said, “There are 15 seats in this class; you’ve got one week to show me you deserve one of them.”

Removing grades from an educational system does not imply that a student’s experience will be less challenging. It does not imply that there won’t be consequences for a lack of performance.

Can school be fun?

Ava comes home and picks up a basketball, not because she’s been assigned basketball homework, but because she wants to. To Ava, basketball is fun.

Most agree that children have a natural desire to learn. There is an innate curiosity. Educators try to encourage that curiosity, not stifle it. But that’s hard.

A young child’s interests do not always align with what might be valuable for him to learn. For example, a child might not show an interest in reading until age 9 or 10. There have been instances of individuals that went on to be incredibly successful after not learning to read until such an age. However, it’s not likely we’d recommend that path for many. If for no other reason, it would pose practical challenges if teacher’s had to cater their lessons to fourth grade students with vastly different reading abilities.

Any college that forces a core curriculum is stating that there are certain topics that students should learn and is recognizing that students wouldn’t necessarily choose these topics on their own.  A core curriculum can put a student in course A when he’d rather be in course B.  This poses a problem for motivation.

College instructors can hope to motivate students by convincing them of the long-term value of the material they are learning.  (Any course that is required should be required for a good reason.)  They should also try to make the experience fun.

Grades as feedback

Grades can serve as feedback for students. When a typically good student gets a low grade on an exam, it elicits an emotional response. I also know that this response is highly dependent on the number grade on the exam. I could give back the exact same exam with exactly the same mistakes marked, but if I put a 95 on the cover instead of a 65, I will get a much different response.

But this is just another indication of the flaws behind grades as a major motivator. Students should be more interested in what they got wrong than their total grade.

When a student gets a low grade on an exam, he often shows up in my office with a request. He asks, “What can I do to bring up my grade?” Note that he doesn’t ask, “What can I do to better understand the material?” If I offered to give him an extra 10 points for doing 100 jumping jacks, he’d happily take that deal, even though it wouldn’t mean he had improved his understanding of the material.

Grades can serve as feedback for students, but so can an assessment that clearly articulates their strengths and deficiencies. I argue that when grades are the source of feedback, it only furthers the students’ misguided motivation for their participation in class and drives a mindset of grade improvement over comprehension.

A loose college plan

If a school were to simply stop giving grades while leaving all other activities the same, they might slip into a temporary state of chaos as students over-indulged in their new freedom. Many students see grades as the primary motivation for performing in class. If this were removed, the students might stop trying.

Removing grades is freeing the students. It’s not freeing them of their responsibility to engage in their work as they might initially suspect. It’s freeing them of the primary obstacle to seeing the bigger picture of education as a long-term pursuit and their many activities across a variety of disciplines as collaborating towards a common goal of creating a lifelong learner.

I doubt that simply removing grades will lead immediately to this enlightened state. Educators will have to guide their students.

Furthermore, any school or school system that chooses to stop grading should take full advantage of this new situation by tailoring their curriculum and pedagogy to students that can begin to view education in a holistic manner.

I work at a small liberal arts college, and so I will present a general structure under which a curriculum with no grades could be implemented in that environment.

Years 1-2: Foundations and Exploration

In the first year, and possibly the second for some, students will engage in courses for two primary purposes. The first is to develop a foundation of knowledge, understanding and skills that is appropriate for all students of the school regardless of eventual choice of program. This might include, for example, very practical skills such as the ability to communicate effectively both orally and through writing. It might also include the modes of thinking that we see as essential for a liberally educated individual.

It is important to note that if these courses are not graded, then the burden is on the institution and the instructors to sell the value of these courses to the students. They can no longer be boxes to check for the students. They cannot be courses the students simply endure. The goal cannot be survival and a decent grade. The college must demonstrate the purpose of each course, and that’s a good thing.

Students will also explore. They will explore new subjects of which they’ve had no prior acquaintance. They’ll explore subjects that they enjoyed or didn’t enjoy in high school and prior. They might find that, for example, college mathematics is much more creative and enjoyable than the “math” they had encountered in the past. Students will explore without fear of failure.

Consider the following scenario. Jane is a first semester freshman in college. She’s not quite sure what philosophy is, but what she’s heard intrigues her. She knows that her college’s philosophy program has an excellent reputation for preparing students for life and careers after college. She’s also heard that it is an incredibly challenging program. Even though Jane has no prior experience in the subject, and she’s not sure of her ability in the field, she boldly enrolls in a philosophy course.

It turns out that Jane was not as interested in philosophy as she had hoped. She also struggled to grasp the material in her first semester. As a result, she earned a C.

Jane ventured out of her comfort zone in a manner that most educators would greatly encourage, but now when she’s applying for summer internships, or when she’s applying for graduate programs or jobs after graduation, that C weighs down her GPA.

Jane could have taken the safe route and stuck to what her prior experience had shown were her strengths. There were courses available that would have been an easy A for her. Our overemphasis on grades is causing Jane to regret her brave excursion into philosophy.

That’s a problem.

Certain colleges already implement procedures that free freshman of the burden of grades. (The reputable and demanding) MIT doesn’t grade freshman in their first semester beyond a pass/fail. This is done to help students with the transition from high school to the increased demands of college, but it can also free students to explore new areas without fear of GPA repercussions.

At the very least, MIT’s program demonstrates that a school won’t implode if they stop giving grades for all courses.

Year 2: Reflections and Strategies

Throughout the educational process, students should be asked to reflect on what they have done in the past, their current engagement, and their plans for the future. What have I done, what am I doing, and where am I going? How do all the pieces of the puzzle fit together? Do they fit? Have my plans and expectations for the future changed? These are questions that make sense when a student sees her education as a lifelong and connected process.

In my loose plan for this “no grades curriculum,” there would be a formal process of reflection. In year 2, a big part of this process would be to prepare the students for important decisions regarding their curriculum over the next couple years.

How often do students pick many of their courses on a semester-to-semester basis? They choose courses based on what will be an easy grade or when the course meets. How often does the student reflect on the role a particular course would play in their total education? How will it set them up for the career they want to pursue or the graduate program they hope to get into? Do students even have a clear sense of their plans and objectives?

Under the direction of appropriate advisors, and after proper reflection on their experiences, students will form a strategy for their education over their final 2 to 3 years in the college.

They will decide on a preliminary objective to pursue, a motivation for their studies. This might be that they want to pursue a particular graduate program, such as med school. Perhaps the students want to enter a certain career. The student might want to start a business, launch a charity, master an art, write a book, or teach high school mathematics in a Spanish speaking community.

The students will find their motivation to learn. In the same way students can change majors, students can alter their objective down the road. Of course, the further down the road and the more dramatic the change, the harder it will be to switch (and finish in four years). It’s ok to change plans, but the absence of any plan can imply a lack of motivation to learn.

Once the student has chosen an objective, they will collaborate on a plan of courses, internships, activities, and undergraduate research that will align with their objective. The goal is for the student to have a reason for every course in their schedule. It’s fine if one of those reasons is that they have a free course and they want to pursue a topic because it sounds interesting even if it doesn’t appear directly related to their objective. However, the collection of their activities should properly prepare them to achieve their dreams.

From that point forward, the student should understand why each course is an important step towards her eventual goal.

I understand one possible objection to this setup is that we would be asking students for a possibly specific objective. College curriculums are often broad in their scope so that students have a lot of options after graduation. We can’t expect 19-year-olds to know precisely what they want to do with their lives.

Objectives can be specific while the training remains appropriately broad. For example, I do sports analytics. You can find some of my basketball work here. Students often approach me because they want to pursue the same topic. The reality is that there aren’t tons of job in sports analytics. However, training for sports analytics can be training in big data organization and analysis. It can be statistics. It can be economics, finance and business. It can be data communication and story telling.

Students can learn big data analytics for sports and then apply it elsewhere after graduation (and there are A LOT of jobs in big data.) More than that, if the students’ passion is sports, they will work harder and learn more if they are pursuing answers to sports questions. If their interests shift to politics, business or some other topic later, they will be better prepared to apply analytics in those fields than if they had drifted through a general analytics program that didn’t ask the type of questions that excited them at the time.

Years 3 and 4: Failures and Accomplishments

Students have to be challenged to learn. Challenges lead to mistakes and failures. That’s a good thing. This program will be challenging.

If an undergraduate tries to start a business, it will probably fail. If they write a book, it probably won’t be very good. Occasionally, a student’s work will be amazing. There will be charities that catch on. There will be businesses that succeed. However, we should not be surprised by failure.

We should not be surprised, and we should not be disappointed. Most successful people have a long list of failures in their past. Often what distinguishes successful people from those that are unsuccessful is the ability to learn from those failures and to persevere.

Students will be challenged not because we expect the end product to be amazing. They will be challenged for the value of the experience. Even if a student’s project fails, they can become quite accomplished in the process.

Years 3 and 4 will be about executing the strategy. Students will be asked to continuously reflect and may alter their plans. Students might fail. Failing in their challenge is not failing in the program. Students will pass the program if they’ve properly dedicated themselves to their strategy and can demonstrate they’ve learned from their experience and are prepared to continue on a productive path in the future.

Standards will be high. Throughout higher education there is a push for high “retention” and “perseverance” numbers. This means that we want a high percentage of the students who enter the program to finish. These are noble objectives, but we must be sure that our efforts to improve retention involve providing more support and motivation and not lowering the bar.

How would this look to outsiders?

We haven’t yet addressed the one potentially redeeming quality of grades. They quantify performance so that an outsider, someone who was not in the classroom to observe the learning and contributions of the student, can draw conclusions about that activity. A potential employer can look at a student’s transcript and glean insight into the candidate’s ability and work ethic.

Or can they?

At many institutions, grade inflation has pushed the average scores up and condensed the GPA window that could have differentiated quality students. A recent survey at Harvard found that more than half of their graduates had a GPA above 3.67. Are employers choosing employees based on whether they had a 3.8 or 3.7?

For several years, I’ve told students to find a way to distinguish themselves among their competition. A math major with good grades is not enough to get any job that you desire. I ask them, “How will you stand out?”

The answer could be internship work, undergraduate research, minors or a second major, or a blog that got local attention.

If a student can intelligently reflect on her experience trying to start her own business, I expect that will be far more impressive to many employers than an A in a macroeconomics course.


I used to be of the mindset that grades are a necessary evil in our educational system.

Grades are a means to quantify and record student performance. They’ve never been perfect, but are usually good enough.

Once a grading system is implemented, it inevitably becomes a target. The graded will be judged on their grades, and so it’s understandable that they shift their priorities towards optimizing their score.

This creates undesired consequences. Students develop a shortsighted mindset about education. They see their courses as a disjoint collection of hoops to jump through, and this inhibits their motivation to learn.

I now wonder if our society has gotten so grade-focused that the negatives outweigh the positives in the system. I think its close enough to seriously investigate the question.





Meet Vi Hart

Meet Vi Hart, and then watch ALL of her videos.  Doodling has never been so cool and informative. There’s also a bit of commentary on mathematics education.  Here is one of Vi Hart’s videos to get you hooked.  You can find the rest here.

Common Core Math is a Challenge for Teachers Too

A couple years ago, I attended a meeting about a new math program that a local town was planning to implement. The switch to a new program was largely motivated by new common core standards, and the meeting was called to prep parents for the “new” mathematics that their children would encounter.

A representative from the company behind the new math curriculum ran the meeting. The representative, who I’ll call Linda, was an experienced teacher and one of the first teachers to implement this new math program.

While at the meeting, Linda gave the parents a math problem. It went something like this. A basket half full of apples weighs 20 pounds. An empty basket weighs 5 pounds. How much does a basket full of apples weigh?

After allowing the parents enough time to work out solutions in small groups, Linda asked for volunteers to share their work.

The first parent shared how she arrived at a correct answer of 35 pounds. This parent took the 20 pounds that a half-full basket weighed and subtracted the weight of the basket to learn that the apples (without the basket) weighed 15 pounds. The parent then doubled 15 to see that the full load of apples (minus the basket) weighed 30 pounds. She then added the basket’s weight of 5 pounds to arrive at 35 pounds for a full basket of apples.

Linda congratulated the parent for getting the right answer and having a correct approach to the problem.

The common core holds dear the principle that the process is as important, if not more important, than the answer. Common core standards fully acknowledge that there are multiple correct approaches to the same problem. In fact, they encourage students to explore multiple paths to each solution.

So, I was very pleased when Linda asked the audience if anyone had arrived at 35 pounds in a different way.

A parent volunteered the following approach. He doubled the weight of a half-full basket to see that a full basket plus an extra basket weighs 40 pounds. He then subtracted the weight of the extra empty basket to arrive at the correct answer of 35 pounds.

BRAVO! It was a valid approach.

Linda did not agree. She suggested that although the answer was correct, the process was flawed. Her reasoning was that there weren’t two half-full baskets in the problem, and you couldn’t just create one for the sake of generating the correct answer.


For decades, math education has misled students to believe mathematics is all about calculations and symbolic manipulations. For example, students think they learn multiplication so that they can multiply six times nine. If that was the end goal of teaching multiplication, then it might as well be taught with a trick like the following.

Even better, we could just give everyone a calculator and use valuable school time teaching other concepts.

Mathematics is not calculations and symbolic manipulations. It’s a structured approach to problem solving. This is what students should take away from any mathematics education, and students aren’t going to develop this ability if they replace true understanding with tricks and memorization.

Parents are struggling with the common core for two reasons. One, it teaches mathematics in a new way. This means new terminology and notation. It means a totally new way of thinking about the same math that these parents thought they mastered years ago.

The other reason for parents’ struggle runs deeper. Parents are having trouble understanding why the math has changed. They are having trouble understanding why their children can’t use tricks or just memorize the answers. For example, parents want to know why their children can’t add fractions with the “butterfly” method. The answer is because that’s not math.

Parents are struggling, and that’s an issue. What about the teachers? These teachers went through the same math curriculum as the parents. Many of our teachers also lack an understanding of the true nature of mathematics and the purpose of a math education.

The common core puts an emphasis on evaluating how students arrive at their solutions. Certain approaches are discouraged. These include tricks and memorization that undermine the students’ development of mathematical reasoning.

However, the common core is not discouraging mathematical exploration. Students are encouraged to find multiple ways to solve the same problem. The core embraces mathematical creativity.

In this new system, teachers are tasked with judging the merits of a students’ approach to each problem. The teachers must determine whether a particular approach is in line with the purpose of mathematics education. Is the approach consistent with the objective of developing the students’ ability to reason mathematically?

We’re learning that this is difficult for teachers like Linda.

The following is another example of teachers’ struggles to properly assess methodology.


Problem 1 is an attempt to see if students’ recognize that multiplication is repeated addition. The student accurately shows that 5 x 3 is 5 + 5 + 5. Why was a point deducted? My guess is that the teacher has taught the students to write 5 x 3 as 3 + 3 + 3 + 3 + 3. In other words, the teacher has asked students to always sum the second term in the product.

The teacher’s instructions are not consistent with the intention of the common core. Writing 5 x 3 = 5 + 5 + 5 is not a trick that undermines the student’s ability to understand the concepts. By not allowing this response, the teacher is stifling the students’ mathematical creativity as opposed to embracing it. The teacher is teaching mathematics as if it is mechanical or formulaic. Why? My guess is that the teacher was taught that way.

Problem 2 is a demonstration of the same issue, which shows that the teacher’s incorrect assessment of the first problem was not a fluke.

I want to be clear that these teachers and parents that are struggling with the common core are smart. The problem is not a lack of intelligence. The problem is that our educational system has been misleading.

The damage cannot be undone overnight.

The common core asks teachers to assess the validity of each student’s mathematical approach, but how can they do this if they haven’t been taught what mathematics is?

What is Mathematics?

What is Mathematics?  It sounds like a simple enough question.  Wouldn’t you expect that a high school or college student that has been formally engaged in mathematics education for more than 10 years would have an immediate (and relatively complete) response?

They don’t.

The truth is that mathematics is difficult to define, and even worse, there are many misconceptions regarding the true nature of math.

Fordham math professor, Dr. Robert H. Lewis, did his best to address the topic in an excellent article that first appeared online in 1999 (and has been revised several times since).

Many believe that mathematics is calculations and symbolic manipulations. This is both incorrect and problematic. When a calculus student believes that the primary objective of differential calculus is to be able to take derivatives, he focuses on the “tricks” such as the chain rule, product rule and power rule. The ability to take derivatives algebraically is a good thing, but if it becomes the focus, this same student might not be able to answer, “What is a derivative?” He might also lose sight of the mathematical process that led to the creation of these differentiation “tricks” as means to be more efficient.  This student might be frustrated by a lecture on limits or a proof of the chain rule.

The role of limits in calculus or the notion of a proof in mathematics are far more important for developing one’s sense of mathematics and developing one’s ability to reason mathematically.  If a proper understanding of math and a capability to reason mathematically are of value, then they must be taught as well.

Dr. Lewis’s article, through analogies (or “parables”), captures the misconceptions about math and the consequences of these misconceptions.

Unfortunately, the article does not provide anything better than a definition by analogy for mathematics.

In my courses, I regularly point to examples that I believe demonstrate mathematical reasoning. However, I can’t formulate a complete definition of what this is.  Like Dr. Lewis’s article, I tend to resort to analogies.

Why is it so difficult to answer, “What is mathematics?”

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