Pages

Friday, August 28, 2015

Seven Perspectives on K-12 Mathematics Education


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:
  1. 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."
  2. 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.
  3.  
  4. 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."
  5. 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.
  6. 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.
  7.  
  8. 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.
  9.   
  10. 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.  


15 comments:

skrishnamachari said...

Lovely compilation of options:

My two cents:

K-12 : Should ideally have atleast 2 (if possible 3) levels of curriculum in almost any subject.

1. Utilitarian / Basic: Make this pleasant and push it as the desired average
2. Basic ++ : Optional extras for those motivated or with extra support
3. Advanced: Only for the identified fast learners, potential much higher than the class average

These should not imply or create peer pressure and that should be well monitored.

Math / Science specially demands the segregation as they can be naturally challenging for many students.


Davis Garcia said...

Today i am feeling awesome because after long time later today i receive a fantastic blog where i found more idea about math solution. Actually i was finding your blog. I think maximum student choose your blog for valuable info. Actually i want to know about essay services essaywritingservices-help.blogspot.com becasue i will write a college essay for our collage student. Now i am feeling very excited. Thank you so much for your great blog post.

packers movers bangalore said...

Excellent post! I must thank you for this informative read. I hope you will post again soon. Warehousing of merchandise....
Packers And Movers Bangalore charges

packers and movers jaipur said...

really a vry nice blog i really appreciate all your efforts ,thank you so mch for sharing this valuable information with all of us.
Packers And Movers Bangalore


Packers And Movers Hyderabad said...

We Provide Best Packers And Movers Hyderabad List for Get Free Best Quotes,Compare Charges, Save Money And Time, Household Shifting Services
Packers And Movers Hyderabad

Packers And Movers Hyderabad said...

Packers And Movers Turkayamjal Hyderabad
Packers And Movers Toroor Hyderabad
Packers And Movers Tellapur Hyderabad
Packers And Movers Singaipalle Hyderabad
Packers And Movers Rampally Hyderabad

Manish Packers and Movers Pvt Ltd said...

Hire Manish Packers and Movers Pvt Ltd in India for hassle-free Household Shifting, Office Relocation, Car Transportation, Loading Unloading, packing Unpacking at affordable Price Quotation. Top Rated Safe and Secure Service Providers who can help you with 24x7 and make sure a Untroubled Relocation Services at Cheapest/Lowest Rate. Visit More :-
Manish Packers and Movers Pvt Ltd
Packers and Movers Bangalore
Packers and Movers Gurgaon
Packers and Movers Indore
Packers and Movers Kolkata
Packers and Movers Mumbai
Packers and Movers Nagpur
Packers and Movers Ahmedabad
Manish Packers in Indore
Manish Packers and Movers Pvt Ltd Sitemap

Manish Packers and Movers Pvt Ltd said...

Best Movers and Packers in India for Moving and Shifting Door to Door Transportation Services to all over India. Visit More :-
Manish Packers and Movers Pvt Ltd Blog
Packers and Movers Bhopal
Packers and Movers Jabalpur
Packers and Movers Vadodara
Packers and Movers Udaipur
Packers and Movers Jaipur
Packers and Movers Chandigarh
Packers and Movers Chennai
Packers and Movers Hyderabad
Packers and Movers in Indore

Manish Packers and Movers Pvt Ltd said...

Packers and Movers Dewas Naka Indore
Packers and Movers Vijay Nagar Indore
Packers and Movers Scheme 54 Indore
Packers and Movers Delhi
Packers and Movers Pune
Packers and Movers Jodhpur
Packers and Movers Kanpur
Packers and Movers Ghaziabad
Packers and Movers Noida
Packers and Movers Navi Mumbai

Manish Packers and Movers Pvt Ltd said...

Manish Packers and Movers Pvt Ltd as a Services providing company can make all the difference to your Home Relocation experience. Indore based Company which offers versatile solutions, Right team that easily reduce the stress associated with a Household Shifting, Vehicle Transportation. we help things run smoothly and reduce breakages and offer you seamless, Affordable, Reliable Shifting Services, Compare Shifting Charges, Visit :
Manish Packers and Movers Ptv Ltd @ https://www.manishpackersmoversindore.in/
Packers and Movers Indore @ https://www.manishpackersmoversindore.in/packers-and-movers-indore.html
Manish Packers Indore @ https://www.manishpackersmoversindore.in/manish-packers-and-movers-indore.html
Packers and Movers Gurgaon @ https://www.manishpackersmoversindore.in/packers-movers-gurgaon.html
Packers and Movers Kolkata @ https://www.manishpackersmoversindore.in/packers-movers-kolkata.html
Packers and Movers Mumbai @ https://www.manishpackersmoversindore.in/packers-movers-mumbai.html
Packers and Movers Bhopal @ https://www.manishpackersmoversindore.in/packers-movers-bhopal.html
Packers and Movers Nagpur @ https://www.manishpackersmoversindore.in/packers-movers-nagpur.html
Packers and Movers Raipur @ https://www.manishpackersmoversindore.in/packers-movers-raipur.html
Packers and Movers Ahmedabad @ https://www.manishpackersmoversindore.in/packers-movers-ahmedabad.html


Manish Packers and Movers Pvt Ltd said...

Packers and Movers Andheri Mumbai
Packers and Movers Thane Mumbai
Packers and Movers Bandra Mumbai
Packers and Movers Navi Mumbai
Packers and Movers Dadar Mumbai
Packers and Movers Goregaon Mumbai
Packers and Movers Kurla Mumbai
Packers and Movers Mhow
Packers and Movers Pithampur
Packers and Movers Khandwa

Manish Packers and Movers Pvt Ltd said...

Packers and Movers DLF Gurgaon
Packers and Movers Iffco Chowk Gurgaon
Packers and Movers MG Road Gurgaon
Packers and Movers Palam Vihar Gurgaon
Packers and Movers Udyog Vihar Gurgaon
Packers and Movers Asansol
Packers and Movers Durgapur
Packers and Movers Haldia
Packers and Movers Kharagpur
Packers and Movers Siliguri

Manish Packers and Movers Pvt Ltd said...

Packers and Movers Jamnagar
Packers and Movers Gandhidham
Packers and Movers Nashik
Packers and Movers Gwalior
Packers and Movers Visakhapatnam
Packers and Movers Patna
Packers and Movers Ranchi
Packers and Movers Bilaspur
Packers and Movers Korba
Packers and Movers Raigarh

Manish Packers and Movers Pvt Ltd said...

Packers and Movers Sector 14, 17, 19 Gurgaon
Packers and Movers Sector 23 Gurgaon
Packers and Movers Sector 45, 46, 47 Gurgaon
Packers and Movers Sector 55, 56, 57 Gurgaon

Unknown said...

Packers and Movers in India
Packers and Movers Raipur
Packers and Movers Nagpur
Packers and Movers Jaipur
Packers and Movers Udaipur
Packers and Movers Surat
Packers and Movers Vadodara
Packers and Movers Chandigarh
Packers and Movers Jabalpur
Packers and Movers Hyderabad