Ethan Brown, Ekam Rai, Cadyn Clark
May 21, 2013
Capstone Research Paper
May 21, 2013
Capstone Research Paper
Mending Mathematics Education
Picture this: a sixth grade mathematics classroom where a student asks the teacher a more advanced question. The teacher responds with “That will be taught next year,” or “That’s not going to be on the test.” Or, maybe a student got creative with the way he/she solved a problem, and lost points for not using the method that the teacher specifically taught him/her. This would make mathematics seem like a difficult, boring, and disciplined subject. Sound familiar? Believe it or not, these types of occurrences have happened, and are still happening, to children in schools across America. In fact, problems like this have dropped America’s ranking in mathematics in relation to other countries down to 32nd. This is a huge decrease from its 2003 ranking of 16th. On the other hand, the Organization for Economic Co-operation and Development found that the best countries include Canada in fifth, Japan in fourth, Switzerland in third, Finland in second, and South Korea in first (Shepherd). Though China and Singapore were not included in this study, they are also among the top mathematics nations in the world. Though many would argue that this is just a result of culture and attitude, in reality, it’s just a matter of America’s teaching methods, emphasis on standardized testing, and curriculum.
CHAPTER 1: TEACHERS
In order to fix math education in the USA, a focus needs to be put on improving the people teaching math, or the teachers. In the presidential election, both Barack Obama and Mitt Romney addressed America’s status on knowledge and technology, saying that more math and science teachers need to be hired. This is indeed essential, but the focus needs to be more specific.
TEACHERS AS EXPERTS
The teachers that are hired need to have a mastery of the subject. Unlike other high-achieving countries, U.S. teaching candidates often learn little mathematical content. Because of this, 100% of teachers in Canada, Japan, South Korea, and Finland had SAT scores in the top third of their class, while America had only 20% of teachers in the top third of their class, and 47% in the bottom third (Benjamin). This proves that other countries choose their teachers a lot more carefully than America does. The teachers who are chosen need to have proved they know the content and mastered it, like the high-achieving countries.
Money Matters. In order to attract top talent, the salary of math teachers needs to be increased. Math-based majors get an average pay of $93,000 a year (19% more than the average profession). On the other hand, teachers get an average pay of $51,299. This difference in annual net income really discourages smart math minds to pursue teaching. Also, studies have found that 40% of students planning math-related majors either pursue another major or fail to get a degree, and over the past sixty years, students majoring in math-intensive subjects decreased by half (America’s). Not only do teachers need to get paid more for their mastery, but the decrease of math majors can be caused by this low salary. America needs a smarter and brighter nation with mathematics, so why not pay more for teachers with mastery of the subject? It will definitely increase the number of math majors in America, so America can keep up with high-achieving nations.
Respect = Results. American teachers also need to receive more admiration and respect. Teachers in Canada, Japan, South Korea, and Finland get treated like lawyers or doctors, while American teachers aren’t receiving this recognition (Benjamin). Doctors and lawyers are valued a lot in society, mostly because what they do is difficult. That means that people in other countries believe that the teacher’s job is difficult, because of all the requirements and dedication they need in order to become one. That makes the teachers over there feel like they’re actually doing something worth their time and effort. On the other hand, though American teachers also work very hard to get the job and continue their profession, most Americans don’t think of the teaching profession as a top-notch occupation, which makes teachers feel that they aren’t having as much of an impact on society. Teaching, in fact, has an important effect on society; teachers educate students, and students apply that education to innovating the world around them. Teachers deserve more respect, and everybody feels better and more confident when their peers respect what they do. Overall, Americans should consider the difficulty of teaching, and realize the type of effect it has on modern society.
LIKE WHAT YOU TEACH
Not only should there be a focus on the teachers’ mastery, but also the teachers’ interest in mathematics. Interest level determines how skilled people are at something, and if they’re willing to continue pursuing it. If a teacher teaches all academic subjects, they might not have much interest specifically in mathematics; an elementary education has a huge focus on education methods and child psychology (Bureau), but a mere 15 semester hour requirement in mathematics (Elementary). If a teacher-candidate did not like mathematics, they could still easily get through these 15 semester hours.
No More Hugs and Kisses. Elementary school teachers tend to teach because of their love for children, not because of their love for mathematics (Benjamin). This is a big problem because if teachers aren’t interested in math, they’re more likely to teach the subject incorrectly. Not only that, teachers are also more likely to get into conversation with the students about their lives, not about mathematics. Teachers can be part of the reason of kids being distracted from math at a young age. In fact, kids in first grade who don’t have proper number sense are likely to test poorly by seventh grade. For these teachers, having a classroom of young children is like having their son or daughter with them all the time. Current elementary school teacher candidates with this mindset should be in search of a daycare-like job, not a teaching job. Liking children is important too; they love affection and care paid towards them, but more success will come in their future if they are taught with a balance; not having too much emphasis on liking children or too much emphasis on specific subjects, like math.
Elementary Enthusiasm. Most people probably would associate interested and knowledgeable math teachers with higher-level math content, such as teachers teaching in high schools or universities. Although this may be the case right now, the focus could rather be on emphasizing it earlier on in the students’ educations. If America decides to have interested math teachers at higher levels, the students may not be interested because they weren’t used to being taught mathematics like that in elementary school. This may have to do with the maturity-level of the students as they get older, but America needs to put more emphasis on this part of the issue. This is part of the reason why there are less people majoring in math in America than sixty years ago. The teachers do teach all subjects, and putting too much emphasis on one of them can ruin students’ interests, so a balance should be maintained to see better results in America’s math progression.
Interest is Mastery. Students cannot be expected to like something more than their teacher does (Benjamin). The students will begin to think that the specific subject is hard to do, if their teacher doesn’t enjoy a specific area of the subject, because teachers are role models for students to succeed. Teachers who tend to dislike a certain area of a subject may make math a bit more difficult than it has to be for the students. More students will be gifted in math if they’re interested in math. Many kids in America say they hate math because it’s hard or they aren’t interested. When people aren’t good at math, they hate it even more because people who hate math actually experience activity in the brain’s pain centers when solving math problems (America’s). There’s so much out there to love about mathematics like number theory, game theory, graph theory, combinatorics, or just the simple satisfaction of solving a difficult problem. This can’t be stressed enough; teachers need to express interest in the subject and pass that interest on to their students.
KNOW HOW TO TEACH
Teachers not only have to master the subject and have interest in it, they also need to have pedagogical content knowledge, or know how to teach. In Japan and Singapore, teacher training focuses on content as well as pedagogy, and both are highly valued in those countries. This way, students in high-achieving countries both learn the content while being intrigued in the lesson.
Practice Pedagogy. The improvement of American teachers’ PCK (pedagogical content knowledge) is crucial. Studies have shown that American teachers’ PCK is significantly lower than the teachers in Japan and Singapore (Daily). Currently in America, the content is being taught, but teachers are unable to take students beyond the minimal material. In America, people believe that a good teacher has a natural ability to teach; they figure out how to engage students into their lessons all by themselves. However, in high-achieving nations like Japan and Singapore, good teachers are carefully developed over time; they aren’t expected to just be born with a natural talent of teaching. Though a teacher can teach using many different methods, most teachers tend to teach the same way they were taught, so countries with top-notch mathematics programs can easily maintain them (Daily). This is part of the reason why many foreign countries are consistently ranked as the best in the world, since their teachers are always able to teach correctly. In foreign countries like Japan and Singapore, teachers are developed and assessed, they aren’t just hired because they are smart. So, if their math programs started out good, then the students becoming the future teachers will know how to teach the future students properly, and those students will know how to teach their future students properly, and so on. The math programs over in those high-achieving countries will actually improve because teachers are developed, not hired. So, if this trend maintains, those countries’ math schooling (math majors, test scores, teacher evaluations, etc.) will skyrocket. But, if teachers in America continue to teach the way they’ve been teaching (which is definitely not at the level of Japanese and Singaporean teaching), the future for America (in regards to mathematics) doesn’t look pretty. As of now, teachers haven’t been fulfilling their expectations to educate students the right way, but if they start changing that now, this part of the issue will be forever fixed. Math teachers need to use the methods discussed earlier so America can catch up to other high-achieving countries.
Another problem here in America is that after teacher trainees complete their training process, they are immediately in the classroom, while teachers in Japan and Singapore continue to train and develop their skills for the rest of their teaching career (Daily). Yes, American teachers are required to continue professional development, but require only 15-24 hours of it annually (Ferreira). This needs to change; American teachers need to go through more training and develop their skills like other countries do. If this continues, rookie teachers will know the content and teaching methods, but the training system in America isn't as advanced as it is in other countries.
Lesson Study. In Japan and Singapore, “Lesson Study,” a recently implanted system, is where students get a half-day while teachers gather and watch one of their fellow teachers teach a lesson to a group of student test subjects, and then collaborate on the teacher’s performance with the guidance of an accomplished university professor (Daily). This really helps to develop teacher skills, and learn from their mistakes. Evaluation is key to make someone better at something, especially with teaching. Having other teachers critique one teacher really benefits that teacher, because not only are they developing their skills, but they’re getting expert analysis from other teachers, and even guidance from an accomplished university professor. Most teachers in America are evaluated by the administrators, which is a very helpful evaluation, but teachers shouldn’t only rely on the administrator's evaluations of their teaching skills. Although teachers want to improve and make the best classroom environment possible, America's teachers should rely on other teachers' evaluations of them because all teachers have similar, but unique experience in the field. That’s why lesson study is a big boost for teacher development. Tips and suggestions from other teachers are reliable and logical because teachers understand the need to improve in order to better promote understanding in the students.
Additional Alternatives. In addition to Lesson Study, Japanese teachers must also conduct public lessons and open houses to be evaluated more often. This gives teachers there even more of an opportunity to get evaluated and improve upon their skills. Not only that, but teachers in Japan and Singapore rotate through grades, teaching multiple levels and content. For example, a teacher in Japan could teach fractions to fourth graders one day and algebra to eighth graders the next day. They even rotate through schools to gain experience and spread their knowledge to various school districts (Daily). This proves that Japan and Singapore really take education seriously; they’ll do whatever it takes to be able to provide top-class teachers. This also gives the teachers an opportunity to go through other schools’ and grades’ curricula, and brush up on their knowledge in other levels of math. This system really benefits everyone there because not only are the teachers receiving teaching skills and being influenced by other teachers, but the students are learning new things from new teachers, and these things being learned go beyond their curriculum, which provides students with a more diverse education. Here in America, the teachers don’t rotate through grades or schools and don’t conduct Lesson Study programs, which is a big part of why teachers aren’t learning pedagogical content knowledge as much as they should be.
Teachers just cannot be efficient teachers without a good understanding of pedagogy. If American schools start to undergo these practices and start to collaborate with other schools and other teachers, it would provide the same benefits for the teachers and students in America as in Japan and Singapore.
Teachers need experience to be an efficient teacher. Currently, teaching is considered a minority compared to other occupations. “Despite its complexity, from a cross-occupational perspective, teaching has long been characterized as an easy-entry occupation. Compared with other work and occupations and, in particular compared to high-status traditional professionals, such as physicians, professors, attorneys, engineers, architects, accountants, and dentists, teaching has a relatively low entry ‘bar,’ and a relatively wide entry ‘gate’” (CPRE). This quote proves that people don’t consider teaching as a real occupation, and how it isn’t really thought of an admirable occupation.
Bring in the Experience. Despite these stereotypes, it take a lot to become a good teacher, and it’s important that people understand that and don’t underestimate the teaching position. The requirements for becoming a math teacher in Connecticut are: receive a Bachelor’s Degree, pass the Praxis 1 test (which is waived if SAT scores are high enough), pass the Praxis 2 test (a mandatory test assessing the specific content in the teacher’s future curriculum), student teach, and get certified. A Master’s Degree is required to get the teacher’s “Professional Certification,” which is needed to keep their job by their eighth teaching year. The Praxis 2 test focuses specifically on math for math teacher candidates for grades 7-12, while the K-6 teachers are only required to know the math content in that range. That means that a fifth or sixth grade teacher only needs to know up to fractions and decimals to pass the Praxis 2 and get a job (Bowman). This definitely limits their understanding of the subject.
Keep Up With What You Teach. The above requirements can lead to many problems. For example, if a student asks the teacher a higher-level question, the teacher might not be able to answer it because the teacher wasn’t tested on it when preparing for their job. Not only does this say that teachers should become more experienced like teachers in other countries, but it also shows how narrow the curriculum can be. Since none of the teachers can teach beyond the curriculum, students get a very limited mathematics education. Also, there isn’t much emphasis on maintaining the expertise that they did have. In order to maintain that knowledge, a test similar to the Praxis 2 should be given to a random selection of teachers in every school annually.
Teaching and teachers are an area where America needs to put more emphasis on, to become a smarter and brighter nation, and to catch up with high-achieving countries.
CHAPTER 2: STANDARDIZED TESTING
On January 8, 2002, the No Child Left Behind Act was signed into law (No). This was supposed to be a way to ensure that all students across America were getting a top-notch education by testing every student in every school. If a school was slacking on these assessments, the government could then approach the school and fix the problem (Benjamin). Though it sounds like a great proposition, this plan ended up backfiring; schools were strategically teaching to increase test results, which ends up hindering the students’ education. In order to improve America’s mathematics education, the standardized testing methods need to be altered.
Currently, when America take a standardized test, teachers want as many students as possible to get above a certain score. This optimal score is referred to as the Proficient Bar. Since this bar is the main statistic that teachers and schools get judged on, it is natural for them to teach in a way that focuses on pushing the average kids above the bar and putting minimal energy into the gifted or struggling students.
Jump Math. There’s a program mentioned in the book The Myth of Ability that suggests a program called “Jump Math” that promotes students to get over the proficient bar. Currently, Jump Math is utilized in Canada and UK. Pushing students too hard will stress them; it takes time to develop the students. In any classroom, there are a small percentage of below average kids, a large percentage of average kids, and a small percentage of gifted kids. If one were to graph the popularity of the different ability levels of students, a bell-shaped curve would be created. This representation is commonly referred to as a bell curve. Jump Math takes this curve into consideration, and pushes the curve toward mastery at a constant rate rather than just skewing it in the mid-section. This strategy actually ended up moving almost every student above the proficient bar (Mighton).
Above the Bar. Every teacher’s ultimate goal is to get as many students above the bar as possible. Currently, the students that are average in this subject get the most attention because the teacher has hope in them being able to pass the bar and make it to the next level. Some of the students that are really gifted in this subject are already above this bar, so they get minimal enrichment and little encouragement because they are already ahead and are not in desperate need of assistance (Benjamin). Because of this, on the TIMSS test used for the mathematics rankings around the world, 7% of students scored in the advanced range in America while in Singapore, 48% of their students scored in the advanced range and in South Korea, 47% of their students scored in the advanced range (America’s). Since gifted students are the ones who go on to innovate society, this lack of American students in the advanced category is a huge problem. But, the teacher already has these gifted students above this bar. So, the teacher’s focus is entirely shifted to the average students (Benjamin). In fact, American teachers worked so hard on these students that America’s average students scored almost as high as the gifted ones (Rich). These students are saved from being under the bar. Those students take up all of the teacher’s time to get there though. Once all of the dust settles, the gifted and average students have made it above the bar.
Below the Bar. Though this set of students made it above the proficient bar, the below average students were forgotten. The program was called No Child Left Behind, but many students were left behind as a result (Benjamin). Because of this, they don’t really understand the subject and most of the time, they hate the class because they don’t get a very good grade in it. People assume that they just don’t have the skill to understand it. Lots of the time though, this is not true. The student does not get any assistance because their teacher believes that the student doesn’t have any hope and thus, doesn’t care because they just need to get enough people above that bar. Therefore, these students are neglected and don’t know what to do. This can cause them to fall so far behind and they just simply can’t do anything about it.
Would teachers really do this? Well, they only do it because they want to keep their job and be recognized as a skilled teacher by that proficient bar. The simplest solution is to take away or rely less on the proficient bar so that the teacher pays attention to all of the students that need help in the subject, so that the below average students will become closer to an average student, instead of getting farther and farther behind. America needs to focus on all students equally regardless of what the proficient bar says (Benjamin). If America did this, it would lead to more intelligent students not only in math, but in all subjects that have tests. This would all lead to a smarter USA.
JUDGEMENT OF SCHOOLS
In America, society judges schools based on test scores and program offerings. Because of this, schools try as hard as they can to push their students into accelerated and Advanced Placement courses.
Advanced Placement. When high school graduates have AP courses on their resume, this is supposed to show colleges that they are above average students. However, this opportunity is starting to be abused by schools. Most people would argue that AP courses are a good thing; students can take a college level course and get college credit for it. Although it seems promising, schools end up offering more AP courses than the top-notch students can fill. As a result, kids that are not as gifted are pushed into this class (Packer). Statistics show that in 1956 only 1,229 students enrolled in an AP course, but in 2009 this number went up 1,691,905 which is a 127,665% increase (Annual). That is a lot of people! Some might argue that our population has increased a fair bit since 1956, making it natural for more AP students. However, it does not come close to meeting the mere 82% population increase from 1956 to 2009 (America’s population in 1956 was 168,903,031 (US) and was 307,006,550 in 2009 (Stterr2009)). When schools put students into these higher math classes, it’s because they want to look statistically better than everybody else without realizing that they are hurting their students and making it too hard for them. This will end up hindering their education.
Slow Down! Whether it is AP courses for high school students or accelerated math for younger grades, students that should be learning at a slower pace need their curriculum slowed down. At about 8th grade, studies show that students who are accelerated into Algebra I do not perform as well as their non-accelerated peers in higher math classes (America’s). Students should only get moved up if they are really highly accelerated. They need to be the top of the top in their school, meaning that they need to get extremely good grades on all of their class work, and in reality, be able to teach the subject to other students (Benjamin). If students don’t understand what they are learning in school, they aren’t going to do very well on anything that is coming up. Some teachers recommend their students just because they finish first in class, and get most them right. Well, that could just mean that they understand this part and work at a fast pace. That doesn’t mean that they are ready for the next level. They still have learning to do; compare them to the students that are really accelerated and see if they can get the grades that they were getting. If not, they are not ready for the accelerated course. That doesn’t mean that they won’t be later on in their career. That just means that they need time to get there and they can’t be stuck into the higher level at the moment. America needs students to be as smart as they can be in mathematics. With these changes, all students would be in the correct class to receive the best understanding possible.
NARROWS THE CURRICULUM
A standardized test assesses a specific set of skills. Because of the importance of these tests, the curriculum only covers what is asked on the test. So, students do not have the opportunity to get a well-rounded mathematics education.
Connecticut Mastery Test vs. Common Core. In Connecticut, the CMT (Connecticut Mastery Test) is the most prominent standardized test for elementary and middle school students. As a result, all teachers do is prepare for this test from around December to the time it happens in March. However, mathematics is a lot more than the things that are going to be on the test. Since it doesn’t cover everything there is to learn, teachers don’t worry about the content that the test does not cover (Benjamin). In fact, the teacher isn’t even expected to worry about it. This isn’t always the school’s fault; the CMT standards are written directly into the curriculum. Almost all public schools in the USA abide by the Common Core Standards, which give an outline of what each grade’s curriculum should contain. So, the people writing these standards can easily just reword the standards of the tests, and ask schools to teach that. The CMT standards strictly cover numerical and proportional reasoning, geometry and measurement, working with data: probability and statistics, algebraic reasoning: patterns and functions, and integrated understandings (Connecticut). At the same time, the Common Core Standards expect students to know number and operations in base ten and fractions, geometry, measurement and data, and operations and algebraic thinking (Common). This is basically just rewording the CMT standards! Students are just expected to know what will be on their test, but they should learn everything that could be relevant to the unit. Currently, if students don’t know something, they almost always ask the teacher if it is going to be on the test and the teachers responds with either yes or no. If the teacher says yes then the student goes home and studies it until they know it for the test. If the teacher says no then the student doesn’t bother putting any effort into learning it. However, the student should know that part because they might need it for a future occupation or scenario. All in all, by eliminating these tests, the curriculum will become much more flexible.
SAVE THE TEST FOR LAST
If America is going to resort to a test, it should be placed right before high school graduation. The use of any test before that will lead to all of the problems mentioned earlier. However, by twelfth grade, students’ ability will not change drastically into a bell curve hovering around the proficient bar. If students take just that one test, teachers would assist all of the students equally because they won’t know where their class falls in relation to the bar. The teacher will help everybody and nobody would be left behind in the dust.
Don’t Judge a Book by its Cover. Having just this high school test would decrease most of the judgement within schools. This is because the teachers would have no test to look at coming into high school they would all have a clean slate to work off of. The teacher would make time to help the students that are less advanced in the system. America wouldn’t have everybody going into accelerated math unless they can prove that they can handle the heat and do well at that high of a level. So if America were to do this, it would eliminate the proficient bar and all of the teachers would just teach normally. It also keeps students from moving up into accelerated math when they are not ready for it. If a student isn’t capable of the accelerated pace, there would not be a test that they happened to score exceptionally on that resulted in them in the advanced class. This end of high school test would be late enough where the results would not affect the student’s class placement.
Benefits of the SAT. This test would not rule the school’s curriculum. Currently, the SAT (which is the most popular end of high school test) has no influence on teachers. In fact, lots of schools don’t even mention it in class. Rather than the school living up to the test, the test would live up to the school; it would test students’ understanding of mathematics as a whole rather than specific concepts of algebra, geometry, and calculus that are listed in the standards. Rather than being a typical test that throws in trick questions, it would be straightforward and fair to all types of students. By having this type of test, colleges are then able to assess the applying students in comparison to each other, which is essential for their admissions (About).
Aside from this high school test, all standardized testing needs to be eliminated for the benefit of America’s students and schools.
CHAPTER 3: CURRICULUM
Not only are there problems with the execution of the mathematics curriculum, but there are also problems with the curriculum itself. The Common Core Standards call themselves “The mathematics that all students should study in order to be college and career ready.” If America is 32nd in mathematics, this claim is clearly not being fulfilled. There are many changes to these standards that must be done in order to achieve this goal of “college and career ready mathematics.”
NEED FOR NUMERACY
Throughout the whole curriculum, more focus on developing number sense must be integrated. Studies show that one in five American adults are lacking number sense, or are “innumerate.” That means that 20% of our society cannot perform simple arithmetic, compute measurements, or comprehend fractional quantities (America’s). This is far worse than our literacy rate, and it needs to be fixed.
Linguistic Leverage. America does have a bit of an unfair disadvantage in earlier development. Chinese, Japanese, and Korean numbers are easier to learn and compute because of the structure of their language. While English speakers refer to 11 as “eleven,” Chinese speakers refer to it as the equivalent of “ten-one.” English speakers refer to 0.50 as “one-half” while Chinese speakers call it “fifty-out-of-one-hundred.” By having this sort of language, Chinese, Japanese, and Korean students are able to learn the number system much quicker and earlier (Chinese).
Enforce the Building Blocks. Cultural differences are not the only problem; these Asian students are taught in a way that develops some number sense by first grade, while Americans take far longer to develop a proper first grade number sense, if they ever develop it. A numerate first grader should know what numbers are, phrases like “greater than,” “more than,” “less than,” and “equal to,” as well as know basic arithmetic without relying on counting. In fact, first graders without this level of numeracy will most likely have trouble keeping up in seventh grade and beyond (Baker). In America, first grade standards require students to be able to count to 120, compare two-digit numbers using <, >, and =, understand that subtraction is the opposite of addition, be able to perform 2x1 addition problems, know two-digit numbers to be an amount of tens combined with an amount of ones (19 = 10 + 9), and utilize the communative (4 + 5 = 5 + 4) and associative (1 + 2 + 3 = 2 + 3 + 1) properties (Common). This curriculum definitely provides first graders with an understanding of what numbers are and how to compare them, but the basic arithmetic without need for counting part is lagging. Having students perform 2x1 addition problems teaches them to add by counting their fingers, which too many Americans are doing as a method of simple addition. Drills like breaking two-digit numbers into tens and ones lends itself directly to turning the problem into a countable quantity. Many schools will use small blocks, where they have single blocks (units), a stuck together row of ten blocks (rods), a chessboard like formation of one hundred blocks (flats), and a 10x10x10 cube of blocks, which is one thousand blocks (cubes), to learn to solve 2x1 problems. This might reinforce what adding is, but it ends up with students counting a bunch of rods and flats. The Common Core Standards state, “Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value.” In other words, the curriculum actually demands that students learn to add by counting, whether it be blocks or dots in a drawing. Since first grade is such an important year in number sense development, the methods of teaching foundational operations must be fine-tuned to fit future demands.
A Mathemagical Solution. As students move forward in school, just like their literacy should develop, their numeracy should develop as well. Not only their ability to perform computational mathematics, but their comfort with numbers as well, which is something Americans often neglect. A perfect, and actually fun, way to continue developing this numeracy is to teach mental mathematics. Not only your times tables through 10, but how to compute two and three-digit numbers in your head. This would not be memorization, but actual methods that students learn to apply in bigger problems. By adding an alternative to pencil and paper arithmetic, students learn to think of numbers in a different way (Benjamin).
For example, the problem 24 x 7 would be approached from right to left on paper. Students would multiply 7 by 4 to get 28, put down the eight and carry the two, multiply 7 by 2 to get 14, and add the carried two to get 16, which yields a final answer of 168. However, mental mathematics shows students that these numbers can be looked at in a different way. This problem can also be looked at as the sum of 7 x 20 and 7 x 4. 7 x 20 = 140, and 7 x 4 = 28, and 140 + 28 = 168. Rather than approaching it from right-to-left, it can be approached from left-to-right. Even the final addition can and should be done from left-to-right. Students will learn that numbers and problems can be analyzed in a new light, which will increase numeracy.
Also, mental math brings lots of creativity to the table. For instance, the problem 24 x 18 can be approached mentally, and in probably over a half-dozen ways. One can use the addition method, where they multiply 18 by 20, and then 18 by 4. This would result in 360 + 72 = 432. One can also use the subtraction method, where instead of breaking a number into a sum, they break it into a difference. In this case, one would probably turn 18 into 20 - 2. 24 x 20 = 480, and 24 x 2 = 48. 480 - 48 gives the same 432. They can also use the factoring method, where they split one of the numbers into two of its factors, and multiply all three numbers together. For instance, 24 x 18 = 24 x 9 x 2, which can be approached with the left-to-right analysis. 24 x 9 x 2 = 216 x 2 = 432. It can also be done by splitting 18 into 6 x 3, or splitting 24 into 12 x 2, 8 x 3, or 6 x 4. It can also be done with the squaring method, which utilizes one of the special products taught in most Algebra II classes. This product states that (a + b)(a - b) = a^2 - b^2. So, 24 x 18 can also be seen as (21 + 3)(21 - 3), which is equivalent to 21^2 - 3^2. Since many schools already demand that students memorize square numbers through 25, students would know this to be 441 - 9, which again, yields the answer of 432. If schools do not mandate this square number memorization, there are also mental shortcuts for squaring and cubing numbers that should be taught in the new curriculum. This problem can also be approached with the close-together method, which utilizes a Vedic mathematics identity that is commonly taught in classrooms in India (where mental mathematics is a big part of the curriculum). This identity states that (a + x)(a - y) = a(a +x - y) - xy. So, students should find an easy number to work with as their a, which would probably be 20. They would then multiply twenty by twenty-two (basically, since they went up two from 18, they will come down two from 24), which gives them 440. Then, they subtract the product of two and four (18 is two away from twenty and 24 is four away from twenty) to give them the 432 (Shermer).
Pencil and paper math only allows for one way to complete this problem, which leads to people’s misconception of math as being a very disciplined subject, with only one right answer and one right process. However, the same problem of 24 x 18 could be approached ten different ways with mental math. There is a lone correct answer, but students can choose what strategies they are best at, and this will incorporate creativity into mathematics. Once students need to figure out how to solve problems rather than be told how to do it, their number sense will skyrocket (Benjamin).
Problem Solved. Moving forward to middle school and high school courses, this numeracy must still be maintained. A big part of why number sense has been unnecessary in these higher level classes is the way that their textbooks are structured. In math class, textbooks are used a lot for sample problems that students have to solve. However, the way that textbooks in America are structured, students are watching the textbook solve the problem for them. Many problems are given where the textbook gives a bunch of information, and then takes the student through step-by-step to complete the problem where the student writes down each answer found along the way. Problem nine might want students to find the graph of a parabola when given an equation, but it might have beneath it problem 9a asking for the vertex, 9b asking for the x-intercepts, 9c asking for the y-intercept, 9d asking for the mirror of the y-intercept, and 9e asking for the graph. By having the problem like that, the student doesn’t actually learn how to graph a parabola. They are being told what to do each step of the way! If the student is supposed to graph the parabola, the question should simply be graph the following equation. The students can then go through and find each piece of information however they want and in whatever order they want, which increases creativity, problem solving skills, and requires number sense (Dan).
In other countries, textbooks are not only lighter and cheaper, but strategically write problems that incorporate previous concepts, summarize each grade level into its most important points, and do not baby students along, but have them figure out what to do (Daily). In fact, Singapore’s math curriculum is largely based on problem solving. Each Singaporean class focuses on word problems, and very carefully chosen textbook problems, while American classes randomly choose the examples, and often skip over the sections with the word problems. By having students learn to solve these more practical and difficult problems, they will develop number sense.
JAZZ UP CLASS
Not only is this numeracy important, but but it has to be fostered with an enriching program. The current curriculum teaches the same way as it did twenty years ago, despite the drastic changes in our society.
Chronological Cognition. The problem solving skills discussed earlier need to be expanded on to create procedural thinking. Students should be able to figure out certain concepts without it being directly taught to them. For example, when an American Algebra II, Geometry, Trigonometry, or Precalculus class teaches the Law of Sines, they probably just tell the students what it is and how to use it. But, they never go over why it works, or how it was developed. That is because America’s standards do not demand this. They restrict our curriculum to eight requirements: make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning. Nowhere in there is “understand the reasoning behind processes and strategies” (Common).
This reasoning is something that students not only should be taught, but should figure out on their own. It may sound like something only an ingenious mathematician could discover, but with some guidance, any numerate student can personally rediscover it. The teacher would tell students how to start and what their end goal is (not the exact formula or strategy, but what the formula or strategy should be able to do), and allow the students to break into partnerships or small groups and figure out the rest. After the students do this, the teacher would then show how to use the formula or strategy and continue through their curriculum. By letting the students figure out the material, each unit would become more personal to them (they would remember their discoveries), they would be more sure of how to apply it (they would know why they tried to figure it out, so they could use it for that same purpose), and they might even gain more of an interest in the material (when people find something easy and have a personal connection to it, they tend to enjoy it as well).
In Japan, a math class begins by the teacher putting a carefully chosen problem - usually a word problem - on the board, and the students do not instinctively know how to solve it. It requires knowledge in the next unit. Students then try to solve this problem, using their prior knowledge to come up with new methods and approaches. After some time, the teacher has students discuss what approaches they used, and praises the students who used the most efficient approach. The teacher then goes on to teach the lesson material (Jackson). By having students construct all of their concepts, they will definitely be more mathematically adept.
Real Data. The curriculum also needs to be enriched by having students use real data. If students are learning to create scatter plots, they should actually go survey friends and family to get data for their graph (Reshaping). With a textbook problem asking for a scatter plot, the data used has already been analyzed by the person who gathered the data, the textbook authors, and the teacher who chose that problem for their class. By the time it gets to the students, it doesn’t require any more analyzing. Students just stick it onto a graph and say they’re done. In contrast, students who collect the data will need to analyze it to see how reliable it is. If it looks unreliable, they will have to survey more people. This will teach students how to apply critical thinking to math, which would pay off a lot in society. Lots of newspapers use very deceptive statistics, that Americans do not know how to properly analyze. This math-critical thinking combo will give this ability to analyze things properly. Also, collecting and using real data shows how mathematics is applied to science. In fact, the math and science teachers and curricula can coordinate on data analysis requirements to create more of a STEM (Science Technology Engineering Mathematics) atmosphere.
Let’s Go Digital! This change will provide the science part of STEM, but what about the technology part? Mathematics doesn’t seem like a technological subject, but it has a lot to do with technology. Mathematics is the language of computers, after all. In the past decade, this world has gone from analog to digital. It went from reading news in newspapers to watching it on television, from hearing music from a stereo to listening to an iPod, from mailing letters to sending text messages. America has to make the same switchover in mathematics (Reshaping).
For instance, mathematics classes have taught for years how to read an analog clock. One of the second grade standards is “Tell and write time from analog and digital clocks to the nearest five minutes, using AM and PM” (Common). Is it necessary to take time out of math class to teach how to read a type of clock that is slowly, but surely going extinct? Many argue that everyone should know how to read an analog clock. However, taking it out of the mathematics curriculum won’t make the skill disappear. It should just be valued like writing a letter or reading a newspaper. Schools do not take time out of their schedule to do these things, but people still end up able to do it. Reading analog clocks should be the same thing.
But technology in mathematics isn’t limited to clocks; there are devices out there that aid students with math problems. The most common one is probably the calculator, whether it be a handheld calculator or a smartphone application. Currently, our curriculum frowns upon calculators; it wants students using pencil and paper mathematics to solve problems. One of the fifth grade standards is “Fluently multiply multi-digit whole numbers using the standard algorithm.” Similar standards exist for the other three operations (Common). These “standard algorithms” refer to adding, subtracting, and multiplying in columns, and using long division, which is probably how most Americans would do it now. However, there is no standard way of performing problems, as there are five pencil and paper ways to solve a multiplication problem, which does not include the ten mental ways and three ways to type it into a calculator. Yes, the pencil and paper methods should be taught, as should the mental methods. But once they are mastered, why inconvenience students by requiring them to show all of their work? Students have this technology at their disposal, and they should be able to use it. Many argue that students fall into the trap of typing something into the calculator wrong, which loses them credit since their work is not shown. However, students will learn from their mistake, and might use a different method next time. If they prefer using pencil and paper math, great. If they prefer using mental math, that’s fine too. If they prefer using their calculator, that can also be fostered. By having technology, time doesn’t have to be wasted on doing the computations. This also puts lots of emphasis on procedural thinking, which is essential. If students don’t have to do the computations, their grade will be based solely on their ability to structure and solve the problem. This will be a big step in the right direction.
Technology can also be used in the form of computers. Not only for instructional videos and math forums, but students can actually use the computer to enrich their education. In fact, there are whole branches of mathematics that have had no relevance until the invention of computers. Computer science is a whole area of math that is specifically exploring the mathematics behind computers. Computers can also be used to analyze statistical problems through simulations and animations. By turning on that neglected laptop on the back table, math class has just become a new world. Technology has already entered the humanities, and it is waiting for mathematics to be ready for it as well (Reshaping).
MAKE MATH REALISTIC
Also, mathematics needs to be taught like a practical subject. It is currently treated like a practical subject; all standardized tests have a mathematics section on them, but rarely do they contain science, history, or art. Though these areas can be practical too, this attitude about mathematics should be supported with its curriculum. Unfortunately, the mathematics that America teaches in school is not nearly as practical as it could and should be.
Statistics vs. Calculus. The high school mathematics curriculum has six components: number and quantity, algebra, functions, modeling, geometry, and probability and statistics. This sounds somewhat practical on the face, but look a little deeper. “Number and Quantity” requires students to compute with rational exponents, irrational and complex numbers, vectors and matrices, and various units of measure. “Algebra” requires students to write equations and expressions, solve equations (one and two variable) and inequalities, and compute polynomial and rational expressions. “Functions” requires students to understand what a function is, and use that understanding to construct and interpret linear, quadratic, exponential, and trigonometric functions. “Modeling” requires students to apply their knowledge in algebra, geometry, functions, and probability to create models that can relate to science or public policy. “Geometry” requires students to understand proofs and transformations of congruence and similarity, define trigonometric ratios needed to measure triangles, find measurements of circles and three-dimensional solids, and apply algebra and modeling to geometry. “Probability and Statistics” requires students to analyze, represent, and synthesize data, understand randomness that underlies many experiments, calculate expected value, and compute elementary and compound probabilities (Common). Aside from parts of modeling and probability and statistics, these specific skills are not at all needed in people’s day-to-day lives.
Out of these standards, statistics is definitely the most practical; it is the area of mathematics that teaches risk, randomness, upside, downside, variability, and critical thinking, all things that we encounter every day. But, how much does statistics get taught? Most American high schools take these standards, and rather than collectively developing each one, they split them up into five classes: Algebra I, Geometry, Algebra II, Trigonometry/Precalculus, and Calculus. There is no specific class to teach statistics; it is just sprinkled around the math and science curricula (yes, math teachers will ask the science teacher to teach data analysis for them since it can be used in both fields). Schools may choose to teach Algebra I in eighth grade or leave out Calculus, but the curriculum pretty much follows the same path. Everything is preparing students to take Calculus, when understanding Statistics would be far more beneficial. If this whole mathematics pyramid built students up to Statistics, people would have a much more practical mathematics education. This requires not only a change in curriculum, but a change in college admissions. Many colleges require students to have taken Calculus, when they should require Calculus or Statistics, just Statistics, or even any senior level math course. That would make sure that schools taught the right classes (Benjamin).
If Statistics were to replace Calculus at the top, then what would the foundational classes be like? The algebra standards still should exist, since most of mathematics requires a firm grasp of variables and how to manipulate them. However, an algebra class shouldn’t resemble a modern Algebra I or II class. Current algebra classes spend lots of time with functions: writing them, graphing them, interpreting them, etc. Functions need to be understood for statistics, but not as in depth as they are taught now. The current two years of algebra can be shrunk down to one year, where the most important concepts are taught, with more of an emphasis on the actual algebra and less on the functions.
Citizen Statistics. Another year in high school should really focus on mathematics in society, which is referred to as Citizen Statistics. This would cover things like personal finance, quantitative urbanism, and mathematical logic. These are branches of math that people not only could use, but need to use in their careers (Hacker). Since school is supposed to educate students for their adulthood, why should these skills be ignored? Many Americans complain that they have trouble with their taxes or budgeting their money. Citizen Statistics will help teach these skills. If this is how high school starts, students will already know how practical and important mathematics really is.
After foundational algebra and citizen statistics, what should students learn? The correct answer is that there is nothing that students “should” learn. At this point in students’ high school career, they are choosing electives in english, science, history, art, and technology. Why can’t it be the same in mathematics? Since the new curriculum builds students up to an understanding of statistics, this is what a good percentage of students will probably choose to pursue. However, students should decide if they would rather go on to learn geometry, trigonometry, calculus, game theory, number theory, combinatorics, graph theory, real analysis, computer science, fractals, linear algebra, etc. Of course, not every school can offer every branch of mathematics, but each teacher can choose a diverse area that they have some background in and teach a class on it. Since the teacher will have extra knowledge on the specific branch, students therefore can continue with their pursuits.
Currently, the curriculum takes students directly through a path that focuses them in on calculus. It’s almost as if it is saying that all students should take a language in high school, but that language must be french. What if students would rather take Spanish, or German, or Italian, or Chinese? If someone who never heard music in their whole lifetime listened to one classical music piece and didn’t like it, they might come away from that thinking that they don’t like music. If they tried to play the piece, and struggled, they might think that they are bad at music. However, they didn’t know that there is rock, jazz, country, oprah, and even more music genres. This is exactly what is happening with mathematics students; they are only getting exposure to algebra and calculus, and failing to enjoy or master any mathematics because they give up within that single field (Benjamin). In fact, after recent high school exit exams, 33% of Oklahoma students and 35% of West Virginia students failed to pass the algebra portion (Hacker). A whole third of our high school population failed to grasp this subject, and probably even more kids have lost interest in it. A third of that generation of our country could have been saved by simply having more mathematics options. Though that generation has lost this exposure, generations to come will gain this mathematical diversity, and go on to have a much better mathematical mind. With all of these curricular changes, our country will start to thrive once more.
Although America is one of the most developed nations in the world, this is not represented with our mathematics ranking. However, this would be fixed by improving teachers, minimizing standardized tests, and revising the curriculum.
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Can you describe your background in mathematics?
I went to a pretty good public high school, middle school, and elementary school. I graduated high school in 1979. I was one in about 25 students who took AP Calculus back then. I went on and did a bachelor’s degree in Applied Mathematics concentrating in Statistics at Carnegie Melon University. I went on to get my masters and PhD in what was then called Mathematical Sciences at Johns Hopkins University. Ever since then, I have been a professor of mathematics at Harvey Mudd College, which is a small private school focusing on math, science, and engineering in Claremont, California. I have written close to 75 mathematics research papers. I have given a couple hundred research talks to groups. I have written one advanced mathematics book on my area in mathematics called combinatorics. I’ve also done a lot of work for the general public, such as books and DVD courses.
Can you show us an example of your Mathemagics?
That is something I developed as a kid, which is a way of multiplying numbers and other feats of mind that is hopefully entertaining. (Does a few problems.) That’s the sort of thing I like to do for audiences and I teach people how to do this, so this isn’t a case of numbers mysteriously appearing in my head; there is a method to this that anybody can learn.
What’s your favorite area of mathematics?
I like what is known as discrete mathematics, and the best way to describe that is to first say what it’s not. A lot of mathematics is continuous; things like moving around in a circle is a continuous movement whereas counting or hopping 1-2-3-4 is discrete. If you ask questions like how many different ways can the four of us sit among these four chairs is a discrete mathematics question. I think that because I’ve always enjoyed playing with numbers, I like mathematics that’s very numberful. There is a lot of mathematics that has less to do with numbers, like mathematics that describes the way heat is transferred or the way planets move, those sorts of things. Those are more of a continuous process.
Why don’t you think these are taught at lower levels?
At the lowest levels, we need to first concentrate on building people’s number sense through addition, subtraction, etc., and understanding these operations and why they work. I think that stuff is taught. How well it’s taught is another discussion, but the question is if at the highest level, are discrete mathematics, probability and statistics taught in the high school. Some schools offer these things and many don’t. I think the more and more we focus on only teaching the things that are going to appear on tests, the less we as a society are going to know collectively.
How would you like to change the curriculum?
I would change the curriculum by giving students more choice in the mathematics that they take. I think that after you take algebra, and maybe geometry, you should be able to decide what mathematics appeals to you more. Maybe you like trigonometry, or you want to take calculus, or maybe you want to learn about the mathematics of games and game theory, maybe you want to learn about statistics, maybe you want to learn about discrete mathematics, maybe you want to learn about fractals or chaos. It’s kind of like history; not everyone has to learn the same history. You might be interested in chinese history, you might be interested in russian history, you might be interested in ancient civilizations, you might be interested in twentieth century developments. We give students electivity in other subjects, but we tend to be very much one curriculum when it comes to mathematics; everything is taking students up to calculus even if they’re never going to need calculus in their life and a student that is never ever going to need calculus is taking all this mathematics that is preparing them for something that they’re never going to need. It’s as if we’re telling all high school students that they should all take a language in high school, but that language has to be french. What if you’d rather take spanish, or german, or russian, or chinese? That’s kind of what we’re doing in mathematics, saying that everyone has to take a language, and that language has to be french, or everyone has to take music, and it’s got to be just classical music. Maybe you don’t like classical music. You might come away thinking you don’t like music, but you didn’t get a chance to get exposed to opera, or jazz, or rock. We’re limiting our students that way.
How would these changes boost America up in math compared to other places?
I think that if we’re going to do that, we also need to improve who’s doing the teaching. If you want to have teachers who can do this kind of curriculum, then you need people who actually know and love and understand mathematics. I would say that a major step that’s needed is to improve the quality of incoming mathematics teachers. I’m willing to bet that if they did an international comparison one year, not just comparing the students from all these countries but they actually gave those tests to the teachers as well, that our teachers would also rank similarly to where our students do. I think that would shine a rather bright light as to where part of the problem is. If you look at the teachers that are teaching in Japan, South Korea, Singapore, Canada, you don’t have to go too far, you’ll see that they really understand their subject and they were math lovers from an early age. We don’t do that. The person who is teaching you mathematics in elementary school might very well be someone who never liked mathematics. They like kids, but do they like mathematics? Maybe not at all. I think that if I could make one change to the early curriculum is that I would make mathematics a specialty subject. We don’t let just anyone teach music, or anybody go in and teach physical education; you need some kind of credential for that. I think that we should do that in mathematics as well. It’s too important of a subject to be taught by people who are math phobic.
Are other countries using this method?
There was a great report that came out a few years ago that investigated the countries that were consistently scoring at the top of the mathematics comparison group, and the thing they discovered most that they had in common is that they were highly selective in who became teachers. If you measured the quality of teachers by what their incoming high school grades or SAT scores were, then you would find that 100% of the teachers in Canada, Japan, South Korea, Finland were in the top third of their academic cohort. In the United States, 47% were in the bottom third, and only 20% were in the top third. In Canada, you get taught by teachers who were A students in mathematics. In the United States, you might get taught mathematics by people who never had a great aptitude for the subject. How do you get top talent? You have to make it a more valued profession. You need to not just pay them more money, which you do have to do, but pay them more respect so that if somebody says I’m a teacher, that would be like saying I’m a doctor, I’m a professor, I’m a lawyer, some other kind of professional where people will say oh, that must mean you are a highly educated and intelligent person. I mean, when I say I’m a math professor, people somehow immediately assume that I must be an intelligent person. But I don’t know if somebody says I’m a high school teacher or elementary school teacher that they say oh, you must be really smart whereas I think in those other countries, I think it is like that. If you want to teach high school in Canada, they have a school of education, but if you want to get into it, that’s like getting into a good law school here, highly selective. And also, the teaching environment has to improve too. Highly talented people who have many professional options aren’t going to be willing to teach in very unpleasant conditions, which I think teachers do a lot.
Do you know why students are losing interest in mathematics?
I think if you have teachers that don’t have passion for the subject, students are going to lose interest. You can’t expect a student to be more talented or more passionate about a subject than the teacher they have. Also, in the last ten years, there has been a real emphasis on standardized test taking. The goal for this was to see that if you want to improve education, let’s measure how well the teachers are doing and a school that doesn’t measure up or a teacher that doesn’t measure up, we’ll try to fix that. That’s a good idea in principle, but the reality of the situation is that these tests become very high stakes and a school or a teacher may be judged by what fraction of their students get above a certain bar. If that’s what you as the teacher or you as the school are being judged on, how many people perform satisfactorily in mathematics, you’re going to work really hard to get as many people as you can just above that line. You’re not going to put really any extra effort to get people way above that line. In fact, you’ll probably say that the brilliant kids can take care of themselves. You might not even pay much attention to the people who are way below the line because they’re never going to get there, or it would take too much work to get there. They say “No Child Left Behind” but a lot of children get left behind because of that. A school might say uh-oh, here’s somebody that’s really struggling in math. Is there a way we can get them out of our pool? Maybe we can get them to drop out or not take the test. You’d say would they really do that? If that’s what your school is going to get funding based on or your reputation is at stake, that’s going to happen. So as a result, there’s all this emphasis on mastering the basic material beyond the point that’s it’s interesting where the kids who’ve got it say, “We’ve got it, we’ve got it, show us something interesting, show us something new.” “Well, that’s not going to be on the test.” That’s what’s happening at the public schools, it’s not always happening at the private schools. You’d think that if this was such a great idea, then your most exclusive, most selective schools would be doing that, but they don’t. They’re giving a lot of freedom, and their students do really well and if they have the right teachers, a good teacher can make all the difference and a bad teacher can make a big difference too.
How should the teachers be teaching to promote interest?
I think that it’s hard to change people. If you intrinsically don’t have a real love for the subject or a real talent for teaching, you can take steps to improve people’s ability a little bit. But I think that the best thing you can do at least for the long term is to see who it is you’re bringing into the system in the first place. Someone who’s been teaching for twenty years, you can’t fire them and I’m not saying you should, but I am saying that you should try to raise the bar or maybe have a different kind of bar for who it is who we’re letting teach mathematics in the first place. I’m not eligible to teach mathematics in a public high school, even though I’ve written books on the subject, I’ve won teaching awards, I really have a good understanding of what makes mathematics interesting and I have enthusiasm for the subject, but I haven’t taken the proper education courses that are needed for somebody to teach in a public high school. Are those education courses important? I’m sure they can’t hurt, but I think it’s easier to get someone who’s passionate about math to learn what’s important about education than for someone who comes in with these education credentials to all of the sudden become super charged about mathematics. I think you really have to see who you’re bringing in.
Do you think the current method of standardized testing is affecting it?
It has to. The testing assessment is way more important now and in the last ten years than it’s ever been before. It has to have an impact. At some point, we should be assessing these assessment tools. Are they really accomplishing what we want them to accomplish? Are students now more enthusiastic about mathematics and science than they were ten years ago? If they are, maybe these methods are working. But what we’re seeing is more students taking AP classes, more advanced placement classes. I mentioned at the beginning that there were 25 of us who took AP Calculus the year I graduated high school over thirty years ago. Today, that same high school, even though it has just as many students, there are something like 75 students taking AP Calculus. There’s one advanced section called BC Calculus and two other sections called AB. There’s essentially been a certain amount of inflation, a certain amount of rush to get more people into calculus, but I don’t think students have gotten necessarily smarter. I just think that schools want to be able to say we offer this many AP courses, we have this many students taking them. That’s a whole other issue.
Should statistics replace calculus at the top of the mathematics curriculum?
I’ve said that given my choice of calculus or statistics, I would say that it’s far more important for our education citizenry to have a good understanding of probability, statistics, risk, reward, upside, downside, variability. Statistics teaches you to think critically in the way that lots of mathematics teaches you to push symbols around. Statistics, you come away saying huh, did this just happen by chance or is there maybe something going on. That’s a whole extra level of thinking. I would also say that I would like students to have more choice in the matter, and that it shouldn’t just be statistics, but if you’re going to require a course, I’d sooner make it statistics than calculus because there’s more likely a chance that they’re going to use it or need it in their life.
Can you talk about your experience teaching in Taiwan a little bit?
This September, I was brought into the Taipei American School in Taiwan basically as a Scholar in Residence. I was there for four weeks, I would visit three classrooms every day for four weeks, sometimes I would be teaching a subject like discrete mathematics, other times I was trying out new lecture material on games and puzzles to see what students would find interesting. I definitely got to experience the life of a high school teacher. I was in an giant office with 20 other high school math teachers and I got to be good friends with them, and boy, I got to see how hard they work. As a college professor, I might teach 5 hours or 6 hours a week. You could teach 5 or 6 hours a day as a high school teacher, so I definitely came a way with a greater appreciation.
How would you say it differed teaching in a different country versus teaching in America?
What’s interesting is that the Taipei American School was intended to give students from that part of the world who had an interest in going on to college in the United States, giving them an American school-like experience. 3/4 of the teachers were Americans. Many spoke no chinese at all. All of the classes were in english. I thought it was a well run math department; they had a curriculum, but the people they hired were people I could tell really knew and liked their subject. They were hired first and foremost for their ability to communicate mathematics. That’s not always true at other schools. Sometimes, they first and foremost hire coaches and they say oh, we have to give them something to teach, let’s give them that Pre-Algebra course to teach.
Are there any other changes that you’d like to see that you think would benefit America in mathematics?
I’ve mentioned give more flexibility on the curriculum, I’ve mentioned trying to recruit and attract top teachers who really love their subject. I guess another side thing that I would do in the early grades would be to put more emphasis on mental mathematics. We put a lot of emphasis on doing everything by paper, pencil and paper, but I think that by learning how to do problems mentally, it improves your estimation skills, it develops your number sense ability. I think that goes hand-in-hand with mathematical understanding.
What have you been doing to try to help solve this problem?
I guess the thing that is having the most impact is that I have put out DVD courses on subjects ranging from mental math to general middle school and high school level mathematics called the Joy of Mathematics, introducing people to discrete mathematics which is my area of specialty, as we speak, I am in the final stages of putting out a course of mathematical games and puzzles. Hundreds of thousands of people have watched these courses and I’m hoping have come away with a new and improved way of thinking of math. Different people like different types of teaching styles. I’m hoping that my teaching style has at least inspired many people to go on and enjoy mathematics more than they did before.