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.

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

  1. Dr. R. Scott Eberle June 1, 2017 / 8:59 AM

    What an excellent thesis! This was well researched, well written, and very thoughtful. Congratulations!

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