Math is Uplifting

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Last week as teachers returned to school for faculty in-service, the school where I teach (Regents School of Austin) offered several talks/presentations that were broadly labeled as “Classical Christian Development.” There was a talk on western civilization, a talk on the importance of story, a talk on the centrality of theology, and a talk on math. I was asked to give the talk on math and this post is acting as a summary/recap. You can click the image above to download the slides I used in the presentation.

The title of my talk was “Math is _________ .” In introducing the talk I let that title just linger there for a while, asking the audience to consider what words or phrases come to their mind for filling in the blank. As the speaker I also enjoy soaking in the facial reactions of each member of the audience when it is announced that this next 45 minute talk will be about math. I contend that 100% of people (and as a statistics teacher, it means something when I say 100%) have a memorable, visceral experience from a math class. There are no neutral expressions on the faces of audience members. The sad thing from my perspective as a math teacher is that the majority of those memorable experiences are negative. My hope in giving this talk was to encourage people to consider some new words for the blank that they maybe had not thought of before.

I start by offering some familiar suggestions for the blank (familiar at least to our Classical Christian context where we teach). Here is the mission statement of Regents School of Austin:

The mission of Regents School is to provide a classical and Christian education, founded upon and informed by a Christian worldview, that equips students to know, love and practice that which is true, good and beautiful, and challenges them to strive for excellence as they live purposefully and intelligently in the service of God and man.

The bold emphasis is mine to point out a few words that might fit in the blank.

Math is TRUE.  This isn’t something I need to sell people on. I mean, 2 + 2 = 4 every time, amirite? To take it a step further though, I encouraged people to consider some ideas I put forward in another post: God, Math, and Order.

“To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology.”

~ Hilda Phoebe Hudson

When discussing mathematics from a Christian perspective, one statement that always seem to bubble to the top of the conversation is that mathematics reveals God as a God of order. This is true. This is also way underselling the connection between God and math.

Does God use mathematics because He is a God of order or does math have order because God uses it? I would argue that order is not a characteristic God displays but a quality that He defines by His nature and math gives us a glimpse into that nature. “Our God is a God of order” – By this claim we shouldn’t merely mean that God acts in an orderly fashion. We should mean God defines what an orderly fashion is. Order is not a quality God decided to portray, rather order flows from His nature.

If this can become our perspective, then when we speak of mathematics portraying God as a God of order, that description will carry so much more meaning. Instead of just correlating our mathematical results with some quality that God displays, we can realize those results are better understood as a manifestation of God’s nature. In a way we are communing with Him in our work as mathematicians, gaining deeper insight into His character.

Math is BEAUTIFUL. This is another category that I don’t have to do much convincing on. So many people have put together so many amazing presentations on the beauty of mathematics that any rational person could be convinced of math being beautiful after a quick Google search. Here is one my favorite videos in this regard and a few quotes.

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

~ G.H. Hardy, A Mathematician’s Apology

“The mathematical sciences particularly exhibit order, symmetry, and limitation; these are the greatest forms of the beautiful.”

~ Aristotle

Math is GOOD

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Here is where the sell gets a little harder. As I mentioned above, a lot of people associate very negative things with math class. When asked to complete the phrase “Math is _________” they may think of words like “stressful,” “confusing,” “too abstract,” “not applicable to me,” or “the exact opposite of all that is good and holy.” Here is where I focused the remainder of my talk in an attempt to get anyone who fell into this boat to start seeing math in a different way.

Whenever I am presenting at conferences I like to do the following exercise: I ask people what the number one question asked in math class is, and without fail I always hear back “when am I ever going to use this?” The reality is that this is not a question, it is a statement. It is a statement of confusion and frustration. In other words the answer to “when am I ever going to use this?” has already formed in the student’s mind as “I am never going to use this” and then they withdraw from the mental activity at hand.

I would argue that what a student is really asking is “why should I value this?” It is not a question of finding application but of finding meaning. Maybe another rephrasing would be “why is this worth learning?” As Christian educators this deep longing should be familiar. If we believe Augustine in the Confessions that “Thou hast made us for thyself, O Lord, and our heart is restless until it finds its rest in thee,” then that doesn’t stop when students walk into math class. The most fundamental thing that is happening in math class is that students are seeking value (something we as teachers need to address in our curriculum) and are seeking to be valued (something we as teachers need to address in our pedagogy). In other words, the foundational issue of math class is an affective one as opposed to a cognitive one.

Affective issues are just present for students but for teachers as well. Another exercise I do at conference presentations: I ask people to close their eyes an imagine their best/ideal teaching moments (the O Captain My Captain moments). I then ask volunteers to share a word or phrase that describes that moment. Not once in all my years of doing this has a teacher mentioned anything about content. The language that is used is always affective – “engaging,” “curious,” “joyful.” Don’t get me wrong – I know the content was still there in the lesson and probably operating at a high level to produce those affective moments. The point of the exercise is simply to illuminate how central issues of affect are to the math classroom.

This is not just an anecdotal observation, but it is also affirmed in educational research:

When teachers talk about their mathematics classes, they seem just as likely to mention their students’ enthusiasm or hostility toward mathematics as to report their cognitive achievements.

Similarly, inquiries of students are just as likely to produce affective as cognitive responses, comments about liking (or hating) mathematics are as common as reports of instructional activities.

Affective issues play a central role in mathematics learning and instruction.

~ Douglas McLeod in Handbook of research on mathematics teaching and learning (1992)

It is also affirmed in national policy documents on math education (even though those documents never really develop how to go about obtaining these stated results – hence the motivation for my dissertation).

“Being mathematically literate includes having an appreciation of the value and beauty of mathematics as well as being able and inclined to appraise and use quantitative information.”

~ NCTM Standards for Teaching Mathematics

“Mathematical proficiency has five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Productive disposition is the habitual inclination to see mathematics as sensible, useful, and worthwhile.”

~ Adding it Up: Helping Children Learn Mathematics (National Research Council)

OK – so students and teachers both would admit that affect plays an important role in math education, this is supported by research, and it is affirmed in national policy documents and recommendations. With all of this motivation how are we (math teachers) doing?

As it stands our current methods of teaching mathematics are producing untold numbers of students who see mathematics more about natural ability rather than effort, who are willing to accept poor performance in mathematics, who often openly proclaim their ignorance of math without embarrassment, and who treat their lack of accomplishment in mathematics as permanent state over which they have little control.

~ McLeod (1992)

This quote may seem a little dated as far as research goes but I think it perfectly sums up the situation. No matter how dated the quote is, I know this is still true today because… well, I’m a math teacher. Plus I introduce myself to people in social situations. Other math teachers will quickly confirm this: whenever you meet someone and they ask “what do you do?” and you respond “I teach math” the next response will typically be something like “I was never any good at math.”

Math teachers are probably second only to priests in terms of the number of confessions we take from people.

Also, through these conversations it gets revealed that what these people really didn’t like about math were more factors of the math classroom schooling environment than the discipline of math itself. To me though, these actions are very foreign to the actual discipline of mathematics. For instance, people might say “I hated memorizing all of those formulas.” No mathematician would describe math as memorizing formulas. In essence what these people are doing is gossiping about math.

It as if they are saying “My friend’s cousin’s roommate’s teacher said that math is a jerk. He saw math behind the bleachers making out with history behind science’s back. No thank you – I want no part of math.”

To which I’d have to say “First, math is everywhere so math is probably making out with all of the subjects. Second, have you actually met math? Maybe you should talk to math face to face to sort this out.”

I think people have this false perception of what mathematics is because their experience of the math classroom was through forced, awkward, artificial “relevance” of math topics. For example:

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The previous day another teacher had shared a story of how he actually jumped a cow in his car and miraculously survived – so I turned it into a word problem. Like most word problems it successfully takes and interesting event/story and kills it dead by now making it a chore for students to slog through. The artificial relevance of this problem makes it seem as if Mr. Williams was in the car doing this:

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(NOTE: prior to this example coming up, I had planned to share an example I saw in a textbook of calculating parabolic motion on Steph Curry’s jump shot – this is clearly mathematical but we have to be careful not to oversell relevance as if Steph Curry makes his shots because he knows the math).

So how do we avoid this artificial relevance? How do we teach math differently at Regents? Because we teach at a Christian school, does that mean 2 + 2 = Jesus now? Here I had to share some thoughts from another previous post: 2+2=Jesus? Ultimately this type of question fails to see math as anything more than calculations. Math has calculations, but it is more than that. The question also sees Christianity as simply a new way of thinking (Jesus is now the answer to everything). As a pastor of mine would always say:

Christianity is always more than thinking, but never less.

~ Neil Tomba, Senior Pastor, Northwest Bible Church, Dallas, TX

A better understanding of how Regents approaches math differently can be summed up in the following quote:

“If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.”

~ Antoine de Saint-Exupery

I pulled this from the first page of A Mathematician’s Lament, by Paul Lockhart, and it hits on the deeply affective aspect of what we do as teachers. I also proceed to read the opening imagery of Lockhart’s initial paper to communicate to everyone in the audience how we as math teachers feel about what often is described as math teaching.

So how do we go about cultivating mathematical affections? Well, I’ve written a lot about that here, but to quickly summarize:

Education is not primarily a heady project concerned with providing information; rather, education is most fundamentally a matter of formation, a task of shaping and creating a certain kind of people…. What makes them a distinctive kind of people is what they love or desire or value….. An education, then, is a constellation of practices, rituals, and routines that inculcates a particular vision of the good life by inscribing or infusing that vision into the heart (the gut) by means of material, embodied practices…. There is no neutral, nonformative education.

~ James K.A. Smith, Desiring the Kingdom (2009)

And also:

Mathematics educators who set out to modify existing, strongly-held belief structures of their students are not likely to be successful addressing only the content of their students’ beliefs…it will be important to provide experiences that are sufficiently rich, varied, and powerful in their emotional content.

~ G.A. Goldin in Beliefs: A hidden variable in mathematics education? (2002)

In other words, it is the practices of the math classrooms that shape mathematical affections. So I challenged the teachers in the audience to consider:

  • Students want to know your story…
  • What are the touchstone moments you can recall from a math classroom?
  • What would you say are the “thick” practices/routines/liturgies of a math classroom?
  • How has your experience of those practices shaped your perspective of mathematics?
  • In light of our own experience of mathematics how do we work to shape our students’ experience of mathematics? How do we cultivate their mathematical affections?

To help answer these questions, I closed by offering three new words to fill in the blank of “Math is ____________ .”

Math is INVITING

Here I got to share about my role as an ambassador for the Global Math Project. First, an introductory video:

I challenged people to tell me how this math problem was different than the cow-jumping math problem above. A couple of different responses: this one makes you curious – you want to solve it. This problem has no words only images. This problem makes you ask questions rather than asking them for you.

Often our invitation into mathematics is already excluding some students. The words or terminology we use to introduce the problem may already shut people down. I’m not saying we shouldn’t use proper terminology, I am just asking to consider if it is always necessary. For instance, this example is teaching binary numbers but that term is never used. We don’t start by telling students “Ok, let’s learn the properties of binary numbers.” We have this interesting video instead. I would rather students understand the underlying concepts and connections (which this video very clearly portrays) than parrot terminology without understanding.

Math is ANALOGY.

I have written elsewhere about Flatland: A project of many dimensions. I love Flatland, and occasionally you’ll even see it’s title referenced as a parable. I love taking students through the though process of what would it look like for a sphere to enter into the 2D world of Flatland. In Flatland the inhabitants would only see the sphere as a circular cross section, completely unaware of the concept of a 3D sphere outside their literal plane of existence.

In this way the concept of dimension in mathematics offers a great analogy for issues of faith. How can Jesus be both fully God and fully man? Well, maybe it is kinda like the sphere still being a sphere but also a circle. I also love the illustration put forward in the chapter on dimension in Mathematics Through the Eyes of Faith about what if a hand entered Flatland. As your fingers went into Flatland the inhabitants would see cross sectional circles, none of which are connected. But if they could zoom out to 3D space then they could see fingers all connected to one hand. Maybe this can help us understand how the church is composed of many separate members but is still referred to as one body.

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Finally there is Salvador Dali’s Crucifixion (Corpus Hypercubus). Just as a 3D cube can unfold into a 2D net that appears like a cross, a 4D hypercube can unfold into the 3D net seen above. Dali is speaking of the mystery of the crucifixion – of something that originated in a higher plane of existence unfolding itself into our world. These moments of insight for our students are made richer by the use of math as an analogy for faith.

Math is SERVING.

I have written a lot here about service-learning in mathematics. I won’t expand here, I’ll just summarize why I think teacher’s should consider service-learning:

  • Affective learning objective is primary
  • Cognitive learning objective is still present and operating at a high level
  • Opportunity to communicate the value of affective learning outcomes through assessment
    • “It is through our assessment that we communicate most clearly to students which activities and learning outcomes we value.”
    • David J. Clarke, NCTM Assessment Standards for School Mathematics
  • Reflection is key
  • Moves toward inculcating a servant’s heart

Here is a quote from a student who would adamantly describe themselves as “not a math person” at the end of the year after going through a service learning project. When asked if they think their attitude towards math has become more positive:

“Yeah, definitely, much more positive. It was hard, don’t get me wrong and I’m not saying the ‘I’m no good at math thing’ didn’t change, but I do think … I am sure that I can learn it, because I am sure I can learn it. It just will take longer and when you don’t feel so completely discouraged about it … When you do feel that you do have shot to understand it and learn it, for me at least it really raises my attitude towards it. It doesn’t feel like it’s this hopeless thing that I just have to suffer through. It is kind-of just a hill you climb, right?”

I like this quote because it is honest. The point of cultivating mathematical affection is not to have every student now love math and have it be their favorite subject. The hope is that students who once saw math as this hopeless thing to be endured now see value in working hard at it. They start to see why they should value math.

Finally, returning to the Global Math Project as the inspiration for my talk in the first place:

Math is UPLIFTING.

I love that the motivation behind the Global Math Project is to change people’s experience of mathematics. I would love to see the students who would say that math is “confusing” and “stressful” now start to use words like “uplifting” to describe mathematics.

Returning to the Regents mission statement:

The mission of Regents School is to provide a classical and Christian education, founded upon and informed by a Christian worldview, that equips students to know, love and practice that which is true, good and beautiful, and challenges them to strive for excellence (inviting) as they live purposefully and intelligently in the service of God and man.

Notice the new points of emphasis. Sometimes focusing on math as true, good, and beautiful can still be an abstract exercise. Let’s start looking for math after the comma. Let’s look at the experience students are having of mathematics. Let’s care about the practices and liturgies of the math classroom so we can impact the mathematical affections of students.

If you are interested in starting the conversation with math (and leaving gossiping about math behind) I ended but sharing two great talks given by Francis Su, former president of the Mathematical Association of America:

The Lesson of Grace in Teaching

Math for Human Flourishing

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Making it count: the importance of emphasizing mathematical concepts and positive math mindsets in elementary school

By Ariana Forsythe

Every senior at my school has to deliver and defend a senior thesis at the end of the year. I have previously shared the resulting work of one of my students. This year one of my students did a marvelous job in exploring mathematical mindsets. I am sharing (with permission) Ariana’s work below because 1) readers of this site may find it insightful, and 2) I’m just so darn proud of her.

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“Dear Maths. Here are 10 things I hate about you…With maths, you’re either right or you’re wrong. You can’t argue your way out of an incorrect answer….Maths teachers seemed angrier than all others…There seems to be no middle ground with maths. You’re either a maths person or you’re not. It’s unwelcoming. Exclusive. It doesn’t reach out” (Hunter). These are just a few of the reasons why blogger Kate Hunter, a self-proclaimed math hater who has allegedly never been good at math, has so much disdain for the subject. Her voice reflects those of countless Americans, as the phrase “I’m just not a math person” becomes a more and more prevalent excuse for poor math performance. In one tragic example, Joseph Cabral, now a successful freelance writer, was a straight-A student until he had to take algebra in middle school. Since then, he has taken and failed the course seven times throughout high school and college and has explained, “I started to question my character, my brain, my capabilities, and even my values” (Cabral). Countless American students like Joseph Cabral are struggling so much with math that they feel like failures, have limited career options, and hate the subject of mathematics.

Background

Americans’ reputation for poor math abilities is not a recent development. Over a century ago, education experts were already noting Americans’ “meager results” in mathematics, and they have been disputing how to correct these results since then (Phillips). Throughout the 20th century, most traditional public education was dominated by the progressive ideals of John Dewey and William Heard Kilpatrick who believed worthwhile information was that which was useful in everyday life or for a student’s specific career interests. This more pragmatic but less academic strategy backfired during World War II when army recruits had such poor understanding of basic arithmetic for bookkeeping and gunnery that math education had to be added to military training (Klein). By the Cold War, math education reform was popularized, and a model called “new math” was pushed throughout American classrooms in attempts to improve students’ abilities to grasp concepts through abstract thinking rather than by memorization of facts (Phillips). In 1958, the American Mathematical Society founded the School Mathematics Study Group which led to math as advanced as calculus being incorporated into high school curriculums (Klein). In the seventies, however, education professionals noticed declining math test scores as a result of the new math model, so curriculum returned to the more traditional, simplified, memorization-centered approach (Philips). By the nineties, a so-called “math war” arose between the U.S. Department of Education and university mathematicians. Since then, professionals have continued their efforts to reform standardized tests, skills requirements, curriculum, and teacher training in efforts to raise Americans’ scores (Klein).

Nevertheless, American adults and children continue to demonstrate meagre math abilities on the international level to this day. Adults between the ages of 16 and 65 from 23 different economically advanced countries were surveyed in 2012 by the Organization for Economic Cooperation and Development in a test called the Program for the International Assessment of Adult Competencies, or the PIAAC. On the math portion of this survey, American adults scored 16th out of the 23 participating countries, with the average American score 26 points below the average of the highest scoring country, Japan, and only 20 points above the average of the lowest scoring country, Italy (Literacy, Numeracy, and Problem Solving in Technology). Similarly, when tested amongst 15-year-old international high school students, Americans scored 35th out of 70 participating countries and 13 points below the international math average on the Programme for International Student Assessment, or PISA (PISA 2012 Results in Focus). While the United States is a leader in world economics and politics, it continues to lag behind educationally and perform below average in mathematics, as seen in both children and adults.

A common psychological phenomenon relevant to the discussion of education reform concerns the debate about whether intelligence is genetically fixed or acquired through effort. Is it nature or nurture? For a long time, the prevailing conception was that genetics determined brain power, as seen in geniuses like Einstein and Beethoven, but scientists such as Bruce Wexler, a neuroscientist and professor of psychiatry at Yale University School of Medicine in New Haven, recently confirmed that any genetic predispositions from birth are surpassed by superior education and effort (Boaler 5). While many Americans see their high schoolers underperforming in mathematics on a global scale and conclude that they are genetically predisposed to struggle with math, studies have revealed that proper encouragement can greatly increase students capabilities (Kimball). This is evident in PhD Carol Dweck’s idea of “the mindset” as developed in her 2006 book Mindset: The New Psychology of Success. Through her research, Dweck discovered that people’s mindsets, or perceptions of themselves and the world around them, dictate their success. Those with fixed mindsets believe that they have fixed potential; to them, challenge means inadequacy, and failure reflects directly on their person. Those with growth mindsets know they have untapped potential; they know they can always keep growing, so they know that challenges will eventually lead to success (Dweck). For over a century, America has lagged behind in mathematics achievement and continues to demonstrate poor math abilities on international tests, despite being a great and prosperous international power. There is now substantial scientific proof, however, that genetics have little to do with students’ intellectual abilities, but learning experiences do. Past educational reforms have dealt primarily with curriculum reform, but one important component in math education that has not been fully addressed is the mindset students have about math.

The argument for math

Americans’ poor performance in mathematics needs to be addressed in elementary education before it can be improved at the secondary level. This math crisis should be addressed earlier because the foundations of basic conceptualization necessary for higher-level, abstract, mathematical reasoning are set at this time. Furthermore, children between the ages of 4 and 10 have the highest brain activity of their lives and are mentally equipped to absorb not only procedural but also conceptual teachings of elementary school. Unfortunately, students’ mindsets about math are also set at this time, and they are often discouraged by teachers not specialized in mathematical instruction. Ultimately, elementary schools need to employ math specialists to teach elementary math classes or, at a minimum, to instruct generalist teachers on the latest and most effective strategies of presenting math curriculum in constructive and positive ways.

Students are currently lacking adequate foundations in basic math concepts that they need to understand algebra and higher quantitative reasoning. As mentioned earlier, on the most common international test taken by 15-year-old high school students, the PISA, the U.S. scored 35th out of the 70 participating countries and 13 points below the average (PISA 2012 Results in Focus). On a different type of test, the Trends in International Mathematics and Science Study, or TIMSS, American fourth graders ranked 14th out of the 49 participating countries and 39 points above the average on the 2015 math section (“Mathematics for Grades 4 and 8: Averages”). American students are performing competitively on international tests at younger ages but are scoring below average by the time they reach high school, a trend directly opposite to higher scoring countries. One possible explanation is that American schools are teaching students for immediate success through math facts, memorization, and procedural approaches but not amply preparing them for the abstract reasoning required in subsequent study of higher level math that begins in middle school, because elementary math education does not emphasize mathematical concepts as a stepping stone to abstraction. In Singapore, for example, elementary students are required to use problem solving and visual models to learn mathematical concepts while American students are primarily taught procedures for obtaining similar results. The problem solving approach is initially more confusing and challenging, but the concepts learned through problem-solving are the building blocks for abstraction, which ultimately prepares students better for higher level mathematics than a procedural approach.

This defense for strong conceptual foundations is discussed in the book Mind, Brain, and Education Science by Tracey Tokuhama-Espinosa, PhD and Director of the Institute for Teaching and Learning at San Francisco University of Quito. Her research shows how children need points of reference to connect to new information and how this means that poor instruction early on will continue to cause students to fail in math, a cumulative subject. She says, “Without a firm foundation in basic mathematical conceptualization… a student will have a lot of trouble moving on to build more complex conceptual understandings” (Tokuhama-Espinosa 35). All students must make this jump from concrete to abstract—from arithmetic to algebra and beyond—but American students have not been given the tools in elementary school to make this leap smoothly.

Another reason to focus stronger math instruction toward elementary students is their comparatively greater brain activity at this age. In the 1980s, Harold Chugani was the director of the PET Center at the Children’s Hospital of Michigan and used developing technology to record children’s brain activity at different ages. He found that 4-year-old brains consume copious amounts of glucose and are twice as active as adult brains. This extremely high level of activity continues until age 10 and gradually decreases until about age 16, when brain activity levels to that of adults. Children really are sponges for information at this early age, and as Michael Phelps, a biophysicist at UCLA, not the Olympic swimmer, who develops and researches with neuroscience technologies, explains, “If we teach our children early enough, it will affect the organization, or ‘wiring,’ of their brains” (Nadia). This easily applies to the instruction and development of math concepts for elementary students and demonstrates how math education is critical in the early years for setting up success later.

Similarly to how elementary math education sets the conceptual foundation for higher level mathematics, elementary math education also shapes the mindsets students have about the subject. Unfortunately, the current education system fosters negative mindsets about mathematics, especially when students at young ages are indirectly taught that mistakes equal failure. Dr. Jo Boaler, author of Mathematical Mindsets, recounts a lecture with Stanford professor Li Ka Shing who explained, “Every time a student makes a mistake in math, they grow a synapse” (Boaler 11). These synapses occur both when a student first makes a mistake—when the answer is not reconcilable with prior experiences—and when an outside source, or teacher, corrects the mistake. Accumulations of synapses lead to brain growth, meaning that mistakes in math also lead to brain growth. Because of this, teachers should treat their students’ mistakes as opportunities for instruction and growth (Boaler 11). In reality, mathematics is a diverse, creative subject as there are countless, different, equally correct ways to approach each problem, yet math continues to be taught from the perspective that there is only one correct answer. Unfortunately, field researchers like Dr. Jo Boaler believe American students are too often discouraged from making mistakes, from freely exploring different solutions, and from joyfully embracing math (Boaler).

One solution to this issue of negative mindsets is to employ math specialists, who are more knowledgable on how best to teach math and foster their students’ love of the subject. Francis Fennell, former president of the National Council of Teachers of Mathematics, explained that generalists usually take “two or three courses in mathematics content and one course in the teaching of mathematics” and that “their teaching load generally consists of a full range of subjects, with particular attention to reading or language arts in a self-contained classroom” (Fennell). On the other hand, math specialists are required to take more math courses and have more math teaching experience in addition to the requirements of generalist teachers. In Arizona, for example, endorsed math specialists must have an “elementary or special education teaching certificate, 3 years in full time K-8 teaching experience…18 semester hours of courses in mathematics content, 3…in mathematics classroom assessment, and 3…in research-based practices, pedagogy, and instructional leadership in mathematics” (“Mathematics Specialist Certifications (by State) with Descriptions”).What is needed, then, are math specialists who can keep up with the constantly evolving and perfected mathematical teaching approaches, who know ways to teach to different kinds of learners, and whose love of math can inspire their students. Math specialists can be actively involved in the on-going discussions of how best to teach math with time that generalists simply do not have, as they must be well rounded in several areas of study. If Americans want to remedy poor performance in math, they need to call for math specialists to properly lay foundational, conceptual knowledge and foster positive mathematical mindsets at an early age for their students.

The case against math

Some skeptics argue that America, on the whole, is not actually performing poorly but that the scores of low scoring students are dragging down those of high scoring students, affecting America’s average math test score. The results from the 2012 PISA international math test revealed that 15% of American’s score variances, or the difference between high and low scoring students, was due to socio-economic status, a percentage twice as large as those of high scoring countries such as Finland, Japan, and Hong Kong (Ryan). America is an extremely diverse nation, especially compared to countries such as Finland or Japan, so some experts believe that whatever attempts are made to close this achievement gap or raise overall scores will always be futile or will always leave some students behind.

While it is true that part of Americans’ poor results on the PISA math test is due to diversity that many other participating countries do not have, it is important to address another perspective that the data shows: resiliency. This is when economically disadvantaged students from the bottom 25% income bracket perform better than predicted, or overcome their negative odds. The U.S. has a lower than average percentage of resilient students (Ryan), which can be attributed to the overall negative mindsets about mathematics that leads to complacency. Ethnic minority and socio-economically disadvantaged students especially often have more difficulties excelling academically. This is largely due to a phenomenon experts call “stereotype threat,” or when teachers’ and administrators’ intentional or unintentional racist or sexist stereotypes negatively impact students. America as a whole needs a mathematical mindset shift, and math specialists are studying how to address social stigmas in reforming cultural mindsets to help close the achievement gap.

Some people, especially parents of struggling students, argue that many students are incapable of comprehending higher level math and that increased efforts are unnecessary since people can still find career success without advanced math understanding. One very prominent, passionate voice, Andrew Hacker, describes how millions of high school and college students are struggling with math courses such as algebra and asks, “Why do we subject American students to this ordeal?” (Hacker). He argues that supplementary courses such as finance and economics should be offered for students who do not have the natural aptitudes needed to complete algebra. Hacker is not alone in his belief that math is holding students with poor math aptitudes back from their academic goals. He cites Barbara Bonham of Appalachian State University who has seen students take and retake algebra up to five times without passing (Hacker). Their proposal to solve the American math crisis is to lower math course requirements so that students with poor math abilities can graduate from high school and college.

However, the issue lies not with genetic abilities but instead with how the different students were prepared differently for a challenging but doable course. As PhD Tracey Tokuhama-Espinosa explains in her book, students who are struggling in algebra and high school level math courses are encountering issues due to the crumbling foundation of elementary math education because math is a cumulative subject (Tokuhama-Espinosa), for if their misconceptions are not properly address early on, they will never be corrected and will continue to fail. As seen during World War II, when soldiers did not even understand enough mathematics for bookkeeping and gunnery, a pragmatic approach to math education only perpetuates Americans’ poor math skills on the international stage (Klein). Though many students may not need algebra or calculus for their music or social-work careers, that is not a reason to teach math poorly and ineffectively. Furthermore, giving students strong mathematical foundations in elementary school can decrease the number of students who are deficient in math.

Conclusion

Americans need to demand reforms to math education, specifically in elementary school. Children’s brain activity during this time allows them to be informational sponges. They are well-equipped to absorb complex conceptual understandings, but this also means they are easily susceptible to contracting negative mindsets about mathematics. When elementary math teachers can foster positive mathematical mindsets in their students, we will see classrooms filled with children actively working together and problem solving in pursuit of a common goal. Dear Math. Here is why I love you. With math, there are countless correct ways to get to the right answer. You can collaborate to solve problems. And math teachers, they help me to love math. Anyone can be a “math person.”

Works Cited

Boaler, Jo. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages, and Innovative Teaching. San Francisco, CA, Jossey-Bass & Pfeiffer Imprints, 2016.

Cabral, Javier. “How Algebra Ruined My Chances of Getting a College Education.” KCRW, 2 May 2013, curious.kcrw.com/2013/05/how-algebra-ruined-my-chances-of-getting-a-         college-education. Accessed 26 Jan. 2017.

Dweck, Carol S. Mindset: The New Psychology of Success. Ballantine Books, 2008.

Fennell, Francis. “We Need Elementary School Mathematics Specialists NOW.” National Council of Teachers of Mathematics, 2006, http://www.nctm.org/News-and-Calendar/Messages-from-the-President/Archive/Skip-Fennell/We-Need-Elementary-School-Mathematics-Specialists-NOW/. Accessed 3 Apr. 2017.

Hacker, Andrew. “Is Algebra Necessary?” The New York Times, 28 July 2012,                         http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html. Accessed 10         Dec. 2016.

Hunter, Kate. “Dear Maths. Here Are 10 Things I Hate about You.” MamaM!a, 5 Sept. 2012, http://www.mamamia.com.au/10-things-i-hate-about-maths/. Accessed 26 Apr. 2017.

Kimball, Miles, and Noah Smith. “The Myth of ‘I’m Bad at Math’.” The Atlantic, Atlantic Media Company, 28 Oct. 2013, http://www.theatlantic.com/education/archive/2013/10/the-myth-of-im-bad-at-math/280914/. Accessed 28 Oct. 2016.

Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century.” California State University, Northridge. Ed. James Royer. Information Age Publishing,   2003. http://www.csun.edu/~vcmth00m/AHistory.html. Accessed 18 Oct. 2016.

“Mathematics for Grades 4 and 8: Averages.” Trends in International Mathematics and Science Study (TIMSS), National Center for Education Statistics, 2015, nces.ed.gov/timss/timss2015/timss2015_table02.asp. Accessed 3 Apr. 2017.

“Mathematics Specialist Certifications (by State) with Descriptions.” Elementary Mathematics Specialists & Teacher Leaders Project. N.p., 2009-2015. Web. 19 Apr. 2017.

Literacy, Numeracy, and Problem Solving in Technology- Rich Environments Among U.S. Adults: Results from the Program for the International Assessment of Adult Competencies 2012. National Center for Education Statistics, Oct. 2013. https://nces.ed.gov/pubs2014/2014008.pdf. Accessed 20 Apr. 2017.

Nadia, Steve. “Riggs News.” The Riggs Institute, http://www.riggsinst.org/brainpower.aspx. Accessed 3 Apr. 2017.

Phillips, Christopher J. “The Politics of Math Education.” The New York Times, 2 Dec. 2015,    http://www.nytimes.com/2015/12/03/opinion/the-politics-of-math-education.html?_r=0. Accessed 15 Oct. 2016.

PISA 2012 Results in Focus. Organisation for Economic Co-Operation and Development, 2014, http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf. Accessed 3 Apr. 2017.

Ryan, Julia. “American Schools vs. the World: Expensive, Unequal, Bad at Math.” The Atlantic.   Atlantic Media Company, 3 Dec. 2013. http://www.theatlantic.com/education/archive/2013/12/american-schools-vs-the-world-expensive-unequal-bad-at-math/281983/. Accessed 20 Oct. 2016.

Tokuhama-Espinosa, Tracey, and Judy Willis. Mind, Brain, and Education Science: A             Comprehensive Guide to the New Brain-Based Teaching. W.W. Norton, 2011.

Love should be the goal of education

by Jacob Mohler

The following post comes from Jacob Mohler, Math department co-chair at Westminster Christian Academy in St. Louis, Missouri. From Jacob’s bio on the Westminster website:

I became a math teacher because I wanted students to learn math in a better environment than I did. A teacher who was trained to teach English taught me math. There are wonderful notions in math of how numbers found in the natural world point to a Creator. For this reason, as I noticed how God has left clues for us to see His handiwork, I wanted to find ways to show students similar things. Thinking about math as the language and logic God used to create the world was too good a secret for me to keep. Now I want students to think about their involvement in the mathematical enterprise as a means to join God as co-creators of interesting things.

Mathematics allows humans to make the “invisible” part of the created world become “visible” in a way to tame it, describe it, use it, wonder about it, write about it, have control over aspects of it. These ways of making the “invisible world become more visible” suggest a knowable and dependable world that is worth knowing and caring for to enhance the human experience. Furthermore, from the perspective of a Christian in the stream of the Reformation tradition, one might say that humans’ ability to know the world in these ways is “thinking God’s thoughts after Him” and living out our Imago Dei as creators, nay, co creators, as we do not Create from nothing as God did.

Humans endeavoring to learn math from our forerunners is necessary to continue the timeless knowledge of our past, but we must not think that simple memory of facts and procedures will guarantee that we will learn what is needed to be faithful messengers to the next generation or that we can rightly use the past understanding of mathematics to “better the affairs of mankind” or as Sir Francis Bacon so aptly put it, “relief of man’s estate.” Learning mathematics should be seen as a work in progress as a training and maturing and not as mechanical or a behavioral modification that tests learning by simple reciting of known and recognized nomenclature, although memorizing of agreed upon facts might be the minimum we ask of students, we need to stress that knowing the world of math is not the same as duplicating what a teacher shows.

The important facet of thinking about mathematics as a training and maturing is that each student’s background of mathematics, abilities, ways of thinking and even difficulties with learning ALL influence how teachers decide to design class lessons and assessments. Routine and simple problems are set in front of students as well as non routine and challenging ones. Whether in a public, private, parochial or protestant christian school the teaching and learning of mathematics may look similar because knowledge of math is similar to knowing God’s world in general terms. This is often referred to as general revelation. All humans have access to this form of knowing about the world.

The depth of knowing about the world is not the same for each person, partly because of interest in knowing, time committed to learning and personal abilities differ. Here is where notions of giftedness bumps up against various ways we as humans interact with the various aspects of enjoying the world. Some people are more athletic than others, for example, and some will be better surfers than others. This reality should not affect the the idea that humans were meant in the Imago Dei and can interact and know the world in small, large, and enjoyable ways. Just because I will never be an ukulele player invited to in Carnegie Hall does not mean that I should not work to improve my skills to bring enjoyment to me and others in my spheres of influence. In a similar way, learning subjects beyond personal giftedness or interest should be encouraged because learning broadly betters each person’s ability to interact in the natural world by enjoying it and the more people know the more they have ability to love more broadly.

Love should be the goal for education.