It occurred to me recently that there are several perspectives on K-12 mathematics education, that it may be helpful for educators, parents, and students to consider and compare:

**The Utilitarian View:**This view argues that for most people that are not pursuing a career in science, technology, engineering or mathematics (STEM), all the mathematics being taught in schools is overkill - all that really need to be taught to all students are the basics of arithmetic operations and percentages, learning how to operate a calculator, and some basic geometry such as calculating areas of rectangles. These are the only essential life skills that would be needed to handle personal financial decisions including making purchases, budgeting, savings and investments. The focus in this view is placed somewhat narrowly on essential life skills, so more advanced topics are not seen as necessary. In an article (that goes beyond merely stating this view) titled "Why Teach Math", Paul Ernest writes that "my claim is that higher mathematical knowledge and competence, i.e., beyond the level of numeracy achieved at primary or elementary school, is not needed by the majority of the populace to ensure the economic success of modern industrialised society."**The Traditional View:**This view, which is the de facto norm encoded in most school textbooks and curricula worldwide today, argues for a deep and detailed study of mathematics, going beyond arithmetic to algebra and trigonometry. It entails a progression of concepts presented with an ever-increasing level of difficulty, and does emphasize learning how to execute many detailed calculation procedures such as long-division, simplifying fractions, etc. These procedures are often motivated and justified through applications in the form of word problems. There continues to be some debate about the most effective ways to teach this approach to math, with some emphasizing the need to proceed very systematically and build on successes (as in Mighton's JUMP math approach that he describes in "The End of Ignorance", or the Khan Academy math program), others like Jo Boaler emphasizing encouraging a growth mindset in kids, showing them that there many ways of solving problems and advocating the end of timed testing, and yet others like the proponents of the Common Core standards emphasizing conceptual understanding over rote memorization of procedures. But they still share much in common.**The Aesthetic View:**This view, exemplified by Paul Lockhart in his famous article "A Mathematician's Lament", and dating back really to the ancient Greeks, emphasizes that the essence of mathematics is the discovery of patterns, posing of relevant conjectures, and proving them rigorously through deductive reasoning. Mathematics, as a form of aesthetic creation, is understood to be a meaningful activity even if it has no applied end. In other words, this view advocates exposing students to pure mathematics, beyond applications, to get them to appreciate its beauty and value as an end in itself. Going through some of the original theorems and proofs in Euclid's Elements, or studying some of the basic properties of Prime Numbers are things that middle to high school students can do to gain an appreciation of mathematics from this pure perspective. Lockhart himself has written an excellent book titled "Measurement" in an almost conversational style that is intended to show how a mathematician would think about abstract ideas. Aligned with this perspective is the notion that mathematics isn't merely a set of known facts to be "taught", but a lens with which to view and discover (in a self-driven, self-motivated way) interesting new abstract facts, ideas, patterns. I myself benefited greatly from the excellent NCERT mathematics textbooks used in many Indian schools, which in my view do a great job of introducing topics in pure mathematics, and encourage some amount of open-ended creative exploration. Another way for students to appreciate the aesthetic beauty of mathematics is to read first-person accounts by professional pure mathematicians, a good example of this genre is Edward Frenkel's "Love and Math."**The Modeling View:**Dan Meyer's TED talk titled "Math class needs a makeover" presents the view that focusing on teaching students various algorithmic recipes for arithmetic calculations is not useful. He argues instead that what is worth teaching and emphasizing in the classroom is how to formulate problems in the real world (i.e., model the real world) using the language of mathematics, that this is the essence of mathematical reasoning. An excellent resource for this view is the SIAM guidebook on Math Modeling: Getting Started and Getting Solutions, by Bliss, Fowler and Galluzzo.**The Computational View:**Something that goes hand in hand with the modeling view is the view espoused by Conrad Wolfram in his TED talk on "Teaching kids real math with computers", that students should learn how to get computers to do the required computations, by programming them. I have had some positive experiences myself introducing middle-school to high-school students to some mathematical concepts via matlab programming, focusing on modeling various physical phenomena from gravity to the propagation of waves. Although this perspective is bread and butter at the college level in engineering, science and mathematical departments, with the growing trend of introducing programming into middle and high school programs, and events such as the Hour of Code, and beginner-friendly programming environments such as Scratch and Greenfoot Java, this view may grow in prominence for K-12 education as well.**The Recreational View:**While this cannot be the entire basis of a curriculum, an additional perspective on mathematics education is that it is helpful, even important, to integrate various recreational mathematical and logical "puzzles" and games. These include classic puzzles like the ones about measuring out fluid and boat trips with constraints and Sudoku's. Games of chance involving coin-tosses, dice, or cards naturally offer many opportunities to engage in probabilistic reasoning to figure out various odds and acceptable bets. More recently, there has been a lot of work on developing mathematically oriented video games, two particularly outstanding ones that I recommend heartily are Dragonbox, and Dragonbox Elements. Besides improving learner engagement with and interest in mathematics, these puzzles and games also provide practice in more creative and open-ended mathematical thinking. Indeed, many mathematically-oriented researchers I come across in academia have had a great life-long love for recreational puzzles. On a related note, there are now many fascinating youtube channels such as numberphile and Vihart that provide entertaining treatments of mathematical subjects.**The Historical View:**Another valuable perspective on mathematics and why it is worth appreciating is that it is very much an essential part of the history of mankind. From Babylon to Egypt, Greece, India, Arabia, to Europe, we have many interesting stories about the development of mathematics and the people behind them. In a class I taught to undergrads about probability, I felt it would enhance their interest in the subject to learn about the very origins of probability theory in the games of chance played in 17th century France: the problems posed by de Méré, and the correspondence between Pascal and Fermat. Excellent resources including several mathematical history and biography books such as "Men of Mathematics" and others. The BBC video series "The Story of Maths" and the podcast "A Brief History of Mathematics" are outstanding resources in this regard as well.

In "Why Teach Mathematics?" Paul Ernest presents some interesting arguments on different aims that five different "interest groups" in society have with respect to mathematics education. His categorization aligns partially with the above perspectives, but is not quite the same, as he applies it primarily to argue that mathematics curricula are determined ultimately by a political process arising from the frictions and compromises between these groups. My intention here has rather been to try and tease out different ways in which mathematics can be taught and experienced in a classroom and beyond, with pointers to different arguments in favor of these different ways, and present some relevant resources.

My own thought is that while some of these perspectives appear distinct, they are in fact complementary and can be harmonized to a great extent. Perhaps even someone with a hardline view that most math is not essential to daily living may not be averse to learning more about its interesting historical development, or trying out a puzzle or two as recreation, and may possibly be persuaded to appreciate how it is being used to model and engineer the world around us, even if they themselves don't feel inclined to engage in that kind of modeling themselves. School mathematics educators would benefit from keeping in mind all of these perspectives as they teach to give their students a more holistic and engaging experience of the subject.

Have I left out a perspective? Do you have some thoughts of your own on this matter? Please write.