As soon as I heard him say that his friend had sixteen twelve-string guitars, I put the 0 on the end of 16 for 160 and added two more 16s, aka 32, and had the sum of 192 strings.
I could have given that answer in less time than it takes to multiply the 6 by 2 and carry the one, etcetera, all on paper. And in far less time than I’d have needed to do it the familiar way in my head, carrying numbers and trying to keep the columns straight below the drawn line.
Had he said the 16 instruments were worth an average of $743 apiece, I’d have found a pen and paper and done it the usual way.
Now and then we hear complaints about what is called “new math,” or “common core,” new ways of teaching math, simple arithmetic really, in elementary schools. Naturally, we have a tendency to ask and perhaps object: What’s wrong with the “old” way?
Often, the issue devolves into a debate to decide which is better, and as seems unavoidable these days, it becomes a standoff between two sides unwilling to budge. For the most part, parents who want education left as they themselves experienced it, and young teachers convinced that “new” methods represent the future.
This is all for naught because the mistake is thinking that it is a debate at all.
Done on paper, the new way appears to have more steps and take more time, while t he old way is obviously quicker and seemingly simpler, even with the carry-overs. This is why opponents of the “new” math always never use numbers with more than two-digits in the pictures, memes, and videos they post on social media. The small numbers allow them to exaggerate the amount of work involved–and add several curved arrows making it appear hopelessly convoluted and confusing.
Use larger numbers, and the difference becomes slight while the confusion disappears. Here’s an example using just one three-digit number:
Old: Underneath 543 x 4, you write 2, carry 1, write 7, carry 1, write 21. Result: 2,172
New: Underneath 543 x 4, you write 2000, then 160, then 12, after which you add the three numbers. Result: 2,172
Common Core: Change to 550 x 4 for 2200, then subtract 7 x 4 (28) for 2,172.
In what is called “new,” you can see the extra step: “Old” is straight multiplication of a three-digit number; “new” is three multiplications turned into one addition. No question but that “old” is more efficient if you have paper and something with which to write.
Now, try doing that in your head the first way, and the carry-overs blur your process, forcing you to stop and restart. The second will give you the answer with relative ease, at least for numbers that aren’t too long. Depends on one’s ability to remember. Multiplications and divisions of three, four, or even five digit numbers by any single digit should not be difficult. Beyond that, I’m sure that I, too, would rely on the old method–even if it meant waiting until I parked my car, finished the dishes, or got out of the shower.
Problem in this debate, which at times waxes indignant and sarcastic, is how misleading the labels “new” and “old” are. Are we unthinkingly turning this into yet another front of what we call “the culture wars”? If we are to be honest, it is the same math, as the identical results prove. Explain both, and I bet more school children–and adults–would find both easier to grasp.
Numerous other tricks anyone can play in the numbers game are loosely called “common core.” This is best understood as rounding out the numbers for the process, and then restoring the rough edge at the end. Or you can break a number into single-digit denominators for two or three simple calculations easily combined, instead of one that is difficult.
Can you divide 6,795 by 45 in your head? No, neither than I, not until I break 45 into 9 and 5. Now it becomes easy to divide by 9, making it 755, then by 5 for 151. And I can tell you that in half the time it took to type that sentence.
Can you multiply 131 by 18? Maybe in time with considerable effort and concentration. Why not 131 by 20? Now you just put a 0 after 131 and double it for 2,620. Then you simply subtract two 131s, aka 262, from 2,620. Shouldn’t take long to figure 2,358 out of that.
Many numbers–those high prime numbers, will defy the tricks. If the division above was 6,795 by 47, or the multiplication was 131 by 181, I’d find a pen and paper. I recall using a ladder to pick the apples at the top of MacIntosh trees, but I stayed on the ground to pick the branches within my reach. No one in the harvest thought we should either always or never use a ladder.
We all agree that mathematics, unlike other educational subjects including most sciences, is always exact. The expression which holds that “the whole is greater than the sum of its parts” is true for many other things. But the math in that adage stops at the word “sum,” which cannot be altered, improvised, or changed in any way.
Ways of reaching a sum, however, are many. What is happening in elementary school math classes should not be a debate or a one-way-only showdown. Each way offers a useful option to fit an occasion.
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