Making Mathematics Count: Exploring Mathematical Education Reform in America

Dread filled my body—my turn was only two classmates away. I began shifting around in my seat. Trying to get comfortable and prepare myself. It was Around The World Day, the same thing we did every Friday before class let out. It was an exercise that some of my peers enjoyed, but I despised. The rules of the game were this: someone begins by standing behind a classmate, they are given a simple math calculation (multiplication, division, addition, or subtraction) and whoever answers first correctly moves on to stand behind the next person in the line. What a fun and interactive game for fourth graders! Maybe for the three or four students who stood up during the class and could make it around the classroom, but for me and probably many other students, it was frustrating, painful, and embarrassing.

Despite the fact that 300 years has passed since public education was instated in the U.S., little has changed in our approach to the teaching of mathematics. Math education needs to be reformed in a way that reflects the true nature of mathematics as an art form and allows for students to explore problem solving and critical thinking. Like any art, mathematics strives to explain the world surrounding in an elegant way, but instead of paint, numbers, logic, and reason support the mathematician’s thoughts. Currently, a focus on memorization of algorithms and rules holds back mathematics education. Students certainly need to learn the basics of arithmetic, but they are being robbed of authentic mathematics. Instead of discovering reasons behind things, like true mathematics, students are lectured at and then sent home with a sheet of examples to practice the things they were taught in class the way the teacher taught them. Reasoning, logic, and critical thinking tie in closely with mathematics; exposure to these aspects of life would help students to become free, critical thinkers who are better prepared for the world.

The purpose of public education in the United States has always centered around the idea of training people for citizenship in a democracy (Bereday and Volpicelli 18) through teaching “scholarship, self development, and social responsibility” (Bereday and Volpicelli 1). Responsibility for the education of the people lies with the public (Bereday and Volpicelli 19), almost all funding for schools and power to organize curriculum lies locally within states (US Dept. Education). This allows for states to be more responsive to the needs of their children when determining the best way to educate students. This also means that within any community, education should be a means of bettering students, not only for the benefit of the child, but also for the benefit of the community, the country, and the world. To assess current public opinions about the purpose of education I conducted an informal survey of friends and family, asking about the current and intended purpose of public education in the US. The answers for what the intended purpose of education in the US is answers focused on teaching important life skills, guiding students to their ambitions, and provides a basic knowledge of all things. The answers for what the purpose of education should be were not significantly different; answers tended to focus more on challenging students and engaging them in critical thinking and problem solving, and better preparing students to enter “the real world.” (To see all survey results, look at the survey results page!)

Randall Bass, assistant professor of educational leadership, claims that education needs to serve a dual purpose in preserving values and beliefs of society and providing change (130), and while it is undoubtedly important to teach the past and give baseline knowledge about topics, society cannot function with only “yesterday’s solution”. “All students should have opportunities to go beyond what they have been taught to stretch their minds and invent new things.” (Bass 131) A large focus of primary and secondary educations should be on problem solving, this provides students with more tools to function in society, teaches students how to critically think, and gives opportunities to learn how to create unique solutions to problems. The most elegant and creative way to solve problems and think critically is mathematics. And because learning occurs best during adolescence (Zuckerman and Purcell), there is no better time to begin teaching mathematics.

The history of math education is long and filled with conflict, a seemingly dualistic battle between rote memorization of procedures versus freedom for students to learn what they want. The progressive movement in the 1920’s strived towards open classrooms that educated students to think critically. That was replaced with The Activity Movement, which sought to integrate subjects and “teach children, not subject matter.” Then the Life Adjustment movement came along in the 40’s, this hoped to prepare more students for work, and not for college, and math education beyond arithmetic was considered unnecessary. In the 1950’s lots of technological developments in fields that are heavy on science, technology, and math like radars, cryptography, navigation, atomic energy, along with the era of the space race, made discussion about mathematics education come back to the forefront. The age of “New Math” began, where educators placed importance onto logical explanations for mathematics, trying to combine skills and understanding for more comprehensive math education. The National Council of Teachers of Mathematics (NCTM) published multiple agendas advising important changes in the way mathematics was taught through the 80’s (Klein). These agendas focused on teaching students how to understand concepts about mathematics, instead of just regurgitating algorithms, all potentially great, but not implemented because of the system they are placed in.

David Coffey, a professor of mathematics education, points out that despite ongoing debate over math education, we have made no real progress or changes in the way mathematics is taught; partially due to the fact that when these discussions only focus on specific parts of a system: teaching, curriculum, and culture. If you get a better teacher but they are being placed in the same system with the same curriculum and the same culture surrounding education there will be no change. If they curriculum is changed to include meaningful mathematics, it is still taught in the same old way, and parents push back because they cannot help their children with the “new” mathematics. Change needs to occur within the culture, the teaching, and the curriculum, simultaneously to make a positive shift towards true mathematical learning. In an interesting study, Mathematician and popular writer, Keith Devlin showed that a child working as a street vendor with relatively no schooling would get a math question correct 98% of the time when asked about pricing of something they sold, given the same mathematical problem with no context they would get it correct 37% of the time. Clearly, the human brain can handle basic arithmetic without formal training, but only when in realistic situations; most people cannot comprehend numerical abstractions. Everyone has a head for figures, but not necessarily abstractions.

“The idea of utilizing arrays of dots makes sense in the hands of a skilled teacher, who can use them to help a student understand how multiplication actually works…But if a teacher doesn’t use the dots to illustrate bigger ideas, they become just another meaningless exercise. Instead of memorizing familiar steps, students now practice even stranger rituals, like drawing dots only to count them or breaking simple addition problems into complicated forms…without understanding why.” (Green) Teaching tools like this often turn into abstract practices when a teacher does not fully understand the basic concepts enough to explain them, or when they are not explained properly. These kinds of complaints from parents are common with parents who’s children come home confused about math class, and are not able to help them with their homework. (Lahey)

Misconceptions about mathematics are ubiquitous in society; perverse insistence keeps educators in the dark about the true, beautiful, elegant form of mathematics, and preserves “pseudo-mathematics” (Lockhart 6) taught in public schools. Like any art, mathematics holds itself to high standards of beauty; this means it is open to criticism like any painting or poem, “Is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter?” (Lockhart 6). But the mathematics done in schools today does not make arguments, or hold beauty, or strive to solve problems. A boring, useless, “vapid, hollow shell” of mathematics remains in school systems (Lockhart 7).

You may ask how can mathematics be so beautiful? All I remember doing in school is mindlessly pushing numbers around so I could answer a question and get a good grade! This is where the culture, the understanding of mathematics and its potential uses, come into play. Time for a quick mathematical history lesson (I bet you never got one of those in school): In 1202 Leonardo Pisano Bigollo, later known as Fibonacci, wrote the first high school algebra text. In it, he posed this question: “If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth?” This small and relatively simply question has wide reaching application from computer science, biology, probability, and architecture (Reich). Fibonacci sequences are fun, exploring the patterns within Fibonacci numbers is endlessly exciting, and as Arthur Benjamin shows in an exciting Ted Talk, it is fun to explore the reasons behind the patterns with geometry. To quote him “Mathematics is not just solving for X, it’s also figuring out why” (or Y, if you want to be cheesy), and currently, neither curriculum nor teaching styles allow for students to ask why.

(Check out the Math Art page for more examples of the beauty of mathematics!)

Most math classes use the familiar teaching format of “I, We, You”, first I am going to show you multiplication, then we will do an example together, then you will do a worksheet about multiplication so you can practice the skill. Even with great ideas from the NCTM on how to change curriculum and teaching, the teachers who are asked to make the changes are unable to do so because as with the common core standards, they are on their own in learning a new teaching approach and applying new curriculum. Most teachers were taught in the same system they now teach in. 13,000 hours of education, most likely with the “I, We, You” pattern, and so they instinctively step into that role (Green) and with no support system in place to change it, how can teachers possibly get more effective at teaching. “In Japan, teachers had always depended on *jugyokenkyu*, which translates literally as “lesson study,” a set of practices that Japanese teachers use to hone their craft*. *A teacher first plans lessons, then teaches in front of an audience of students and other teachers along with at least one university observer. Then the observers talk with the teacher about what has just taken place. Each public lesson poses a hypothesis, a new idea about how to help children learn. And each discussion offers a chance to determine whether it worked. Without *jugyokenkyu*, it was no wonder the American teachers’ work fell short of the model set by their best thinkers.” (Green) Giving teachers more of an opportunity to keep learning about their trade and more resources to improve their teaching is absolutely necessary if improvements to teaching are to be made.

Approaches to teaching styles which offer flipped classroom settings like Khan Academy as a learning tool for students got a lot of attention the past few years as being a potentially new tool for learning. The learning and lecturing occurs outside of class and then more meaningful experiences can take place in the classroom with more personalized teaching (Parslow). David Coffey took issue with how the media was toting Khan’s ‘flipped classroom ideas as innovative and revolutionary because flipped classrooms have been around forever, for example, any take home reading assignment. Coffey does believe that the flipped classroom is a useful tool, and that learning things outside of class then having discussions can lead to greater understanding of material, but Khan academy may not be the best way to go about it. Salman Khan has no formal teaching skills, some of the mathematical concepts and content in his lessons are shaky.

Utilizing that idea, a teacher can make a video for students to watch outside of class or an interactive book, these would assure that the students get the correct information taught in a good way. A fantastic concept like this, which similarly utilizes technology, is the 3-Acts series by Dan Meyer, a former high school math teacher and advocate for mathematics education reform. A short video representation of a problem allows students to visualize the problem and start to come up with solutions, possibly outside of class. Then in class talking to peers about the solutions and finally a class discussion with the teacher about the solution. This also moves classes into the “You, Y’all, Us” classroom narrative which Green discusses. First you try the problem, then get in groups and discuss, then the class can come up with a final solution together with the teacher. It also allows for an introduction into proofing.

An examples of some student proofs from a 3-act activity.

Proofs are the heart of mathematics; in the words of the famous mathematician Vladimir Arnold “Proofs are to mathematics what spelling (or even calligraphy) is to poetry. Mathematical works do consist of proofs, just as poems do consist of characters.” So if true mathematics is to be taught in school, then teaching proof making should be integrated into education. Proofs are not magical, they are logical statements that show why something is, or is not (Introduction to Proofs). Lockhart gives a great example of proofs that young students can do in a class, a problem that he gave to seventh graders and asked them to come up with a solution for:

This could be proven by listing lots of rules:

But one student came up with a very elegant, simple proof for this question (Lockhart 21).

Many small things like this can be proven easily for instance, why a negative multiplied by a negative equals a positive. Students given a chance to try and prove this on their own are far more likely to remember the rule, and to start thinking creatively about all sorts of problems and things that are presently taken as rules to be memorized. This kind of exploratory teaching and learning has been an expanding part of Japanese mathematics education for a few decades, “Instead of having students memorize and then practice endless lists of equations… Matsuyama taught his college students to encourage passionate discussions among children so they would come to uncover math’s procedures, properties and proofs for themselves. One day, for example, the young students would derive the formula for finding the area of a rectangle; the next, they would use what they learned to do the same for parallelograms. Taught this new way, math itself seemed transformed. It was not dull misery but challenging, stimulating and even fun.” (Green)

Bret Victor, a user interface dreamer and creator, claims that people don’t use any more math than arithmetic in their lives, which is mostly true. However, when the American public thinks that 1/4 is bigger than 1/3 (Green), there must be a problem in the way in which even basic arithmetic is being presented. Even if arithmetic was being taught in a way that everyone understood, teaching more mathematics in classes allows for more critical thinking. The relationship between mathematics and problem solving is substantial, if teaching begins to consist of true mathematics, seeking beautiful solutions to problems, then students have more skills to function in society. David Coffey mentioned that with so much data available to everyone with technology, students need some critical thinking skills to interpret numbers that happen all around you. Most Americans are innumerate. Even if we only need arithmetic on a regular basis, about 30% of people can only answer a one-step arithmetic question. (Green)

Coffey spoke about the change in the way we teach mathematics needing to be a slow one. Subtle changes have to be made, otherwise people revolt against them teachers, parents, and students get lost. Because math is an art, and should be treated as such, a first step towards changing math education could be making a math class styled after an art class. I made a Facebook post , asking friends to tell me how often they did art class in elementary school and if it was in a different room.

Overwhelmingly, the typical elementary art class model was once a week, in a separate room, with artists teaching or guiding the class. Once a week, students could go to a different room, set up with circular tables and no podium, where they take an hour or two out of the day to work on a problem and come up with creative solutions on their own, with a mathematician present to guide them in their creative learning. This is only a beginning to the solution of America’s math problems. Teachers also need to get more support. Feedback on their lessons and supplementary math classes at a university so they have a better grasp on the mathematics they teach. Students may need more support at home as well, a monthly or weekly newsletter about what parents should expect and explanations of the mathematics they are learning, like Takahashi began to do in his math classes (Green), could reduce parents anxiety about not being capable of helping their kids with math past 6^{th} grade.

Around The World may have been good practice for the three students who hardly ever sat down, but for the rest of the class it was an exercise that showed our inadequacy with memorizing times tables. I still struggle with times tables, because once I began to think that math was something I could never be good at, I stopped trying to learn. As mathematics curriculum is reexamined in the 21^{st} century, there will be a movement away from being able to do quick arithmetic in your head, and towards focusing on learning how to problem solve, because we have calculators more time can be spent on thinking about questions instead of calculating. Computers do all of our data analysis; on Microsoft Excel you can quickly do calculations on lots of different data points. With less time spent on mindless averaging, more time can be spent on figuring out what the data means.

To cure American innumeracy and improve critical thinking and problem solving skills changes must be made to improve the curriculum and ways mathematics is taught to engage students in exploratory mathematical learning.

For a list of sources and other interesting links to check out if you are interested in learning more, look at the sources and links page!

Those repeating digit multiplication patterns are interesting. Try using a fixed number of identical digits for the first factor, and append digits to the second factor. I would bet that any consistent digit-pattern in the factors creates some kind of digit-pattern in the products, though often difficult to see. (There is probably a proof for that somewhere, and maybe even a generalized formula for the patterns.)

Most of my own grade school and high school math teachers were pretty good. We were always given a rationale for the algorithms. But a flaw of my own was placing most of my attention on the steps required to do the homework, while somewhat neglecting the meaning of the equations. We weren’t tested on the meanings.

“You, Y’all, We” is a striking idea that I will not forget. My best teachers did lean toward that. But the question was only posed at the beginning of the lecture. A hand or two would go up, taking us straight to the “Y’all”, and quickly on to “We”. We weren’t given more time to reflect on the question, or to consider the possibilities in each other’s ideas.

“Lesson study” (continual peer review and teacher training) was another striking idea in Green’s article. We should never lose sight of the fact that there is no single best way to teach something, because every student is different. We can probably identify the methods that work best on average. And we can probably analyze each student to make good guesses about which of several methods will be easiest for them.

Students should receive frequent overviews of the entire K-12 curriculum. K-12 math and English grammar are not vast subjects. But the way we walk kids through them tends to imply that they consist of a neverending series of new topics. Let’s start showing students the whole path, and point out that many of them can master it in a few years.

I have my own criticisms of Khan Academy. But one should try to do better than Salman Khan before nitpicking his flaws. (Anybody. Go on. I will wait very patiently.)

With each year of K-12, it becomes increasingly absurd to have everyone cover the same material at the same rate. Even books and audio recordings make this unnecessary. The web just makes it far more obvious. Kids who are having trouble with arithmetic should not be shoved into algebra. Kids who have mastered a topic should be encouraged to move on.

Homework should be eliminated: 8 hours per day is enough. No. It’s more than enough. That goes for the teachers, too. The only time a kid should be working on a school project at home is when they have great enthusiasm for it.

Kids beyond 5th grade should have freedom more similar to that of college students; even younger kids, if they demonstrate the ability to behave. K-12 is far too much like jail. It isn’t normal. It incites abnormal behavior. Think about it for a while. Kids do things in school that they would be quite embarrassed to be seen doing in most public situations. So let’s make it more like typical public situations, and less like a prison.

Concise, accurate lessons are priceless, and harder to find than I would like. I hope to gather an online community of lifelong learners eager to share what they have learned. To begin with, we will trade links, curricula, and personal notes. I’m not interested in a wiki; we already have those. I want the individual thoughts of thousands of bright people, so I can pick and choose from their ideas in order to refine my own. As a goal, I will publish an initial site by September 1st, and announce it on Twitter with the #thoughtvectors hashtag. I hope you’ll be interested.

Best wishes,

David A. Hale