The Argument -- Technique vs. Innovation in Mathematics
One of Shenk's main points regarding the distinction between "child prodigy" and "adult creator" (92). He claims they have different skill sets; a child prodigy is characterized by technical skill, while an adult creator is characterized by innovation. Mature success seems to have a very different basis than the success of a child. However, regarding mathematics education, most math classes focus on technical skill -- how to solve the same kind of problem over and over again -- rather than creativity and innovation in problem solving. Furthermore, students are rewarded for technical skill, and often have trouble adjusting to the innovation and creativity required for adult success. Obviously, Shenk would argue that this focus on technical skill as opposed to innovation and creativity is a problem in mathematics education. How serious of a problem is this? How must mathematics education be altered in order to foster successful innovation and creativity? Include information regarding child prodigies vs. adult achievers. It may be helpful to read Lockhart's Lament (www.maa.org/devlin/LockhartsLament.pdf) in order to develop your answer.
Brandon Axe
brandona0701@aol.com
Currently, mathematics education involves teaching students technical skills on how to solve math problems that heavily rely on formulas to solve; however, a serious problem arises from teaching children only technical skills. Like Pavlov’s classical conditioning experiment mentioned in Campbell page 1127, children are being taught to heavily rely on formulas then follow a standard procedure and expect to get an answer; therefore, children are being conditioned to not have to creatively think to get an answer as the problems are always straight-forward. As a result, like Shenk suggests, children who become extremely good at technical skills “develop a natural aversion to stepping outside their comfort zone” (92). In other words, because at a young age children associate following a formula and mundane procedure to receiving praise, they become conditioned to believe that continuing to use standard formulas as they mature will allow them to continue to receive praise, which is often not true. With regards to child prodigies and adult achievers, child prodigies are praised for technical skill because knowing more technical skill is the only thing that helps make them unique from other children as they are not exposed to the challenges of the real-world whereas adults are praised for their creative thinking because they have been exposed to the real-world, where not everything occurs perfectly. In relation to the biology themes, the current mathematics education system relates to the theme of evolution. With society today being highly favorable for creative thinkers that can create solutions to problems that occur, teaching children solely technical skills in mathematics creates a selective disadvantage which puts those children at a disadvantage to survive and reproduce. Ultimately, teaching children only technical skills is a serious problem because it doesn’t allow them to think creatively which is required as an adult.
ReplyDeleteMathematics education should be altered to help children understand how the procedures they are taught work. According to Lament, students should be “given time to make discoveries and formulate conjectures” (www.maa.org/devlin/LockhartsLament.pdf). Instead of anticipating a result from following the same procedure over and over, like Pavlov’s dogs, students should be using the cognitive activity of problem solving (Campbell 1128) to create their own result.
-Edward Wu (edwardwu0@gmail.com)
Yes, an education strictly in mathematics would be problematic, but that is why education has become more balanced; it is required that students take English and PE and electives. Having this balanced education ensures that children are well rounded and not only technical or creative. I believe that if an education was only a technical mathematical education the child would not understand any of the practical applications of math. A purely technical education would be good for creating people with bland personalities. This type of person works best alone, and in a community of innovative people, a purely technical person sticks out like a sore thumb.
ReplyDeleteIn order to create a mathematical education that is both innovative and technical, it must be alerted to resemble more of an engineering education. An engineering education includes the technical aspect of calculating forces and stress capabilities while also including the innovative designing of bridges and houses. Another type of education that would be both innovative and technical would be a strong physics education.
This idea relates to mutualistic symbiosis interaction between people as studied in chapter 54. Mutualistic relationships are “(+/+)” (Campbell 1203), meaning that both people benefit from interacting with each other. People who have gotten a strictly mathematical education lack the ability to communicate with other people, and therefore lack the ability to engage in successful mutualistic exchanges. Paul Lockhart says “it would be ludicrous to expect a child to sing a song or play an instrument without having thorough grounding in music notation and theory.” (Lockhart n.p.) Wouldn’t it also then be ludicrous to expect a person who received a purely mathematical education to know how to lead a group of researchers, a family, or a math team? Education needs be a healthy balance of technical and innovative aspects.
This idea relates to the biological theme of interdependence in nature. A person that has a technical education and cannot communicate efficiently will not excel in the community. However, a person with a balanced education that can interact efficiently with the community, exchanging knowledge, will excel far beyond the purely technical education.
Edward Wu was correct in saying that children of a mathematical education "associate following a formula... to receiving praise" (Mr. Wu)These children would not be receiving praise for creative work, but for merely working with in their comfort zone using equations. As shown by Dweck's study on page 98, kids that praised for their work are more likely to try harder and excel more than those praised for succeeding by using things they already had (such as formulas and mundane procedures).
Josh Gerber (grbr_jsh@yahoo.com)
In Lockhart’s Lament, Lockhart explains that the “nightmare situation” of a painter would be that students study the visual arts technically, without picking up a brush, until high school, where they begin with “Paint-by-Numbers” (http://www.maa.org/devlin/LockhartsLament.pdf). The statement that Lockhart is making is more profound than painting or mathematics: the concept of information being spewed at young students who are supposed to repeat the information back to the teacher by regurgitating it onto the paper. A simple visit to an elementary school classroom can be a window into the world of technical teaching and learning…
ReplyDelete* On the white board, a teacher will explain how to multiply 3 digit numbers by other three digit numbers. For the next half hour, the teacher will circulate, helping students who either do not understand this concept, seeing it as foreign, or having trouble with the actual multiplying and carrying out of the process; meanwhile, a gifted few will understand the concept after it has been introduced to them and proceed to solve the problems independently faster than their peers. *
As delineated by Shenk, prodigious children often are exceptionally gifted in the mechanics or techniques of their respective fields—like mathematics or music—but lack true innovative vision that comes later in life. In the Stevenson community, elementary school and middle school mathematics is very technical, so the child-prodigies shine here because they immediately master the skills that they are handed. As soon as they take Geometry in High School, they are immersed into a new world. At Stevenson, Geometry is more of a crash course in innovative problem solving than in the actual mechanics of finding measurements and applications of shapes—the first encounter a prodigious, technical child may have with mature, innovative material. Geometry is an effective introduction to innovation in mathematics, but most students do not take that course until their sophomore year of high school—their tenth year of being taught mathematics.
Technical instruction of mathematics requires students to think and learn with general “cognition—the process of knowing represented by awareness, reasoning, recollection, and judgment” (Campbell 1128). By observing what their teachers are doing, thinking back on previous experiences, and deciding what method to use, students are able to find the answer to a problem with a singular path. While cognition is very impressive and important—without technical skills, innovation would be incomplete—it is necessary to incorporate problem solving: “the cognitive activity of devising a method to proceed from one state to another in the face of real or apparent obstacles” (Campbell 1128). Innovative problem-solvers are the cartographers drawing the maps, while technical thinkers applying general cognition are the ones who can follow the maps directions flawlessly after reading it once or twice. The current educational issue is that no happy medium is reached until the high school level where they are taught hand in hand. However, as Shenk states, adult achievers are those skilled with innovation, and younger students struggle with that style of sophisticated thinking whereas technical genius is more prevalent among kids.
(third paragraph in the following post)
-Kyle Mueting (kylemueting@comcast.net)
My proposal to incorporate problem-solving at the lower levels is surprisingly simple: a group project involving finding the solution to a complex issue. Often, those who become adult achievers are not the technical prodigies, so technically inclined students can help innovatively inclined students and vice versa. Campbell explains that “social learning forms the roots of culture, which can be defined as the system of information transfer through social learning or teaching that influences the behavior of individuals in a population” (Campbell 1140). By forming a mutualistic learning relationship with other students, each student can learn skills in the field that he or she was less experienced with, forming a more balanced culture. Teaching is the most effective form of learning, so students teaching each other in a group are a valuable skill asset. Studies have shown how effective these methods can be, and distancing themselves from their teachers allows students more room for innovative exploration (http://www.westga.edu/~distance/ojdla/fall113/hovermill113.html). By striking an educational balance between innovation and technical skill at a lower level, other institutions can adopt the balance more effectively (i.e. if elementary schools institute collaborative learning initiatives, middle schools can teach innovation as well as technical skills rather than waiting to find the balance until high school level mathematics). In a way, there is interdependence in the “food chain” of education: the high school dictates the curriculum of the feeder schools, but the middle schools and elementary schools are the ones supplying high schools with the prepared students. To alter the entire system, each level must reevaluate the balance between the two teaching styles to restore balance to the students’ learning.
ReplyDelete-Kyle Mueting (kylemueting@comcast.net)