The meaning of numbers and implications for how we teach mathematics.
knowledge, learning, memory
Reported by LAK (on behalf of the New Bretten mind-brain consortium)
You can learn a good deal about someone’s knowledge of fractions by asking “about how many times do you think 1/8 goes into 2? or approximately how much is 15 times 25?” The question to be answered is not whether someone gets exactly the right answer but what are their guess like.
The New York Times Obit (October 8, 2009) featured Israel Gelfand who died at age 96. He was one of the 20th century’s leading mathematicians. He influenced the best and brightest in his field, loved math and felt it should be part of our everyday lives. He was also a dedicated teacher of young students including adolescents. He was quoted as saying “It is important not to separate mathematics from life. You can explain fractions even to heavy drinkers. If you ask them, ‘Which is larger, 2/3 or 3/5 it is likely that they will not know. But if you ask, ‘Which is better, 2 bottles of vodka for 3 people, or 3 bottles of vodka for 5 people?’ they will answer immediately. They will say 2 for 3 of course.”
Glen Whitney, a mathematician who recently left the quantitative hedge fund Renaissance Technologies has now turned to trying to develop a museum devoted to math. He is also someone interested as are many others on our understanding of large numbers such as the national debt as tracked on the National Debt Clock, on Forty-fourth Street New York City, ($11,518,960,404,062) and the average family share of the debt $96,700. So how big are those numbers? Whitney uses comparisons of the size of astronomical bodies in space as indices of the size of these numbers. In Whitney’s view, progression in how math is taught, algebra, geometry, trig, pre-calculus, calculus is not based on evidence, or increasing difficulty. We do know that the outcome of math education is often mathematical ignorance with lots of examples such as the popularity of the lottery. “The lottery is a tax on the mathematically illiterate.”
How we ask questions about the nature of the math knowledge someone knows matters. Our drinkers knew more than they thought they knew. Likewise, just because someone can tell us the size of our national debt tells does not mean they understand the meaning of the size of that number. Even if someone can point out the probability of winning the lottery does not mean that they understand how unlikely it is that they will win or the cash cow lotteries are for the states that run them.
We have been privy to lots of reports and studies that document the sorry state of math knowledge in kids (and adults). Another report just came out today (October 15, 2009) in most of our nation’s newspapers demonstrating generally continuing declines in math abilities in virtually all states (part of the testing associated with No Child Left Behind). The testing associated with Trends in International Mathematics and Science Study which provides reliable data on the mathematics and science achievement of U.S. 4th- and 8th-grade students compared to that of students in other countries has published their findings in 1995, 1999, 2003, and 2007.That study shows that we continue to lose ground in math achievement (compared to other countries) and what is worse the longer students are in school the larger is the gap between our students and those of many other countries. How are we going to remain competitive in a global economy with a workforce that has limited marketable skills? Perhaps ideas such as teaching math to elementary students should require teachers that really know math.
There is more to this story which has been updated on October 4, 2010.
A feature article by Winnie Hu entitled ‘Making math as easy as 1, pause, 2, pause describes the value of importing The National Math System of Singapore into the suburban New York City elementary schools. That system for teaching math to kids has been hugely successful for at least 2 decades. It is designed to provide elementary school children with a deeper understanding of the meaning of numbers. Without that kind of understanding kids will continue to struggle throughout the school experience leaving them math impaired.
A recent article (Dev Psychol. 2010, 545-51) by S. Dehaene and colleagues describes numerical estimation skills in preschoolers. He is one of many who have provided loads of evidence that children must learn in depth rather than be provided with a superficial treatment of many math topics. That has to begin starting in the earliest grades in elementary school when kids are supposed to learn the meaning of number. So much important and useful work gets lost in translation. How can we speed up making use of what we know?