The people at Gen Yes blog did a very in depth analysis of Khan Academy from a constructivist point of view, which is a good read if you are interested in the debate about the value of the Khan Academy. This is where I found a link to Conrad Wolfram’s TED talk on computing and math education. Wolfram stated that there are really four parts of math: posing the question, translating a real world problem into a math problem, computation, and then turning the answer back into a real world answer and verifying. He said that we spend most of our time in math education teaching the third step, computation, which is the least important step to learn today. Throughout history doing math meant doing loads of computation. Not today. He asserted that people doing math in real life are not working out computations themselves –they have computers to do that. The important thing is to know how to problem solve, which the other 3 steps relate to – and which we spend the least amount of time on in education. I am going to spend much more time going into the problem solving aspects of math education in a later post, where I will talk about my experience with Cognitively Guided Instruction.
Today, however, I want to continue to look at this third step, computation, in relation to Wolfram and the Khan Academy. The Khan Academy provides instruction solely in computation. Both their video tutorials and their practice activities are geared towards learning computation. They teach primarily traditional methods of computation, which were very efficient strategies to use in the past, but are less useful now.
Wolfram asserted that there is one time when it makes sense for people to do calculations themselves and that is when they are doing mental math. What does it mean to do mental math? When people are doing mental math are they doing the same thing they would do on paper - traditional algorithms, like carrying and borrowing, only doing it in their heads? The answer generally is no. People are not simply doing traditional algorithms in their head, when they are doing mental math. Mental math is also not simply proficient fact recall, though people with good mental math skills usually do know their math facts. Mental math is an internal representation of core number relationships. This internal representation makes it easy for someone to solve math problems in their head, and also to estimate. It is great that we have calculators and computers to do the hard work of computation for us now, but we can’t simply type in a problem, and trust whatever comes out. People need to have enough knowledge of numbers and number relationships to know if an answer seems right or way off, so that they can easily identify mistakes, before these mistakes cause big problems.
John Van de Walle (author of Teaching Student-Centered Mathematics, which I highly recommend to anyone involved in math education) asserted that the basis for mental math is flexible thinking about numbers, where students can easily think of one number in a variety of ways. For example, with the number 77 a student might think of it as 70 and 7, or 3 away from 80, or 75 and two more, or one less than 78, or three quarters and 2 pennies, or 23 away from 100. When a student is able to quickly think of a number in a variety of ways than it makes it easier to solve a variety of different problem types quickly and easily – quicker and easier even than solving it with the traditional algorithm. Here is a screen shot from the Khan Academy with a subtraction problem involving the number 77.
I solved it the traditional way, the way that Khan showed in his video tutorials, and the way you were probably taught in school - borrowing. First I borrowed one from the ten's place, since there weren't enough ones to take away 7. Then I computed 14 - 7 = 7, and moved over to the ten's place. There were 0 tens and I needed to take away 7, so I had to borrow from the hundred's place. Then I found that 10 - 7 = 3 and moved to the hundred's place. There were no hundreds to take away, so I brought down the 5 and finally had the answer of 537. A student who had learned to flexibly think about these numbers might solve it in an assortment of different ways, many of which would be more efficient than the traditional algorithm, and could probably be done in the head.
For example, a student might think: I know that 77 is 23 away from 100, and 614 is 514 more than 100, so the difference between 77 and 614 is 23 + 514, which is 537. This thought process is illustrated on the above number line.
Or a student may think: If I take 80 away from 614 than that is 534. 80 is three more than 77 I took away 3 extra, so 614 minus 77 must be three more than 534, so the answer is 537. This thought process is illustrated on the above number line.
Or a student may think: I want to go back 77 from 614. If I just go 14 back I get to 600, and then I still need to go 63 more back. I know that if I took 63 off of a whole hundred than there would be 37 left, so the answer must be 537. This thought process is illustrated on the above number line.
All three of these mental math strategies would be more efficient than the traditional algorithm taught on the Khan Academy. I believe that there are types of math practice activities that could be done on the computer in a format similar to that of the Khan Academy which would be more effective at developing these skills in students than the drills of traditional algorithms that are currently on the site. A student would still get the benefit of immediate feedback, and teachers would still get useful data on student progress. The activities would not simply be the rote drill of traditional algorithms, but activities that would build knowledge of number relationships that would increase flexible mathematical thinking. I hope to follow this post shortly with mock-ups of ideas for such practice activities.
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