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February 24, 2014

Mental Math Tricks

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New Math was a teaching method introduced in the 1960s to give the United States a competitive edge in innovation in response to contemporary advances in technology by the U.S.S.R. The term has become far more generic that what it once was. Fifty years after its failed implementation in American schools, people who never even learnt the New Math jest about it when they encounter methods for teaching basic math skills that don’t quite coincide with their own experience in grade school. So too was my reaction when I was presented with a list of mental math tricks presently being taught to first graders.

Mental Math Hits Home

The following is a summary of mental math tricks that I recently obtained from a first-grade teacher in the form of a classroom handout. I know that the curriculum in question includes Saxon math, but web searches have failed to confirm that these tricks are a part of that specific program, at least by the names of the tricks I was given. The tricks rely heavily on the ability to recognize patterns.

Doubles. The method for memorizing doubles (5+5=10, 8+8=16, etc.) is the first trick on the page and is the basis for several other tricks. Unfortunately, it employs the use of a mnemonic device, a song to which the lyrics were not included on the page. I did hear the song recited once and I have to question the efficacy of this particular device solely because the connection between the numbers and the rhyming words was non-existent. I don’t recall them now, but they were something akin to Four and four is eight. Isn’t that great. What’s so great about it? Wouldn’t it be more practical to learn lyrics with meaning? Four and four is eight, like when I roller skate, referring to the number of wheels on a pair of skates, contains a visual cue to which kids can relate. In either case, there’s only so many words that rhyme with -teen. I suppose it works in the short term, and hopefully the children outgrow the need for this trick quickly.

Counting Numbers. The “Plus 1” and “Minus 1” tricks require that a student know their counting numbers backward and forward. Anytime a simple addition or subtraction problem is encountered that involved a “1”, the student just remembers to count in one direction or the other. Not a lot of magic here.

Counting By Twos. The “Plus 2” and “Minus 2” tricks require knowing how to count by twos, which also introduces the concept of even and odd numbers.

Counting By N. Actually, there are no tricks that involve counting by threes, fours, or even fives and tens! I only mention it because it seems like an odd omission to me.

Partner Numbers. Partner numbers are any two consecutive counting numbers. To add two partner numbers the “Doubles Plus 1” trick is used. As implied in the name, this is a combination of tricks listed above: the double of the lower number is found first and then the next number counting by one is identified. A subtraction problem containing partner numbers will always result in “1”. Similarly, subtracting “Odd or Even Partners” — that is to say, consecutive counting numbers on number lines containing only odd or even numbers — will always result in an answer of “2”.

Tens. The “Making 10” trick is just memorization of number combinations whose sums equal ten (1+9, 2+8, 3+7, etc.). The “Minus 10” is the corollary that informally crosses over into algebra (7+x=10 solve for x). The “Add/Subtract 10” trick is an abbreviated form of “long” addition: the student knows that to add ten to a number one need only increment the tens-digit column.

Nines. The “Plus 9” trick could just as easily been called the “Adding 10 Minus 1” trick, for it is a combination of the two tricks above.

More Doubles. Two more tricks extend the concept of doubles. Similar to the rules for finding the difference between partner numbers (where the answer is always “1”), the “Doubles Minus” trick states that the subtraction of any number from itself will always be “0”. Also, the “Half of Double” trick is similar to the “Minus 10” trick in that it approaches algebra (14-x=7, 16-x=8, etc.). Students are encouraged to recognize combinations of numbers memorized in the Doubles song.

Zero. Any number plus or minus zero is the same number.

Everything Else. The catch-all is to memorize a small subset of one-digit addition problems. Specifically, these are the number combinations that do not trigger one of the rules above (3+5, 3+6, 5+7, etc.).

I have mixed emotions about these rules. On one hand, I learned “long” math (carry the one, and no jokes about walking to school uphill both directions through three feet of snow in August) and after enough rote practice patterns would start to emerge. Eventually, the math happens happens almost automatically. With that experience as a baseline for comparison, the tricks described above seem to be a real grab bag and the inconsistency in naming doesn’t help make any of them particularly memorable in my opinion. On the other hand, I recognize that the essence of a few of them has been programmed into my own head over time. For example, I use an expanded variant of the “Making 10” trick all the time (e.g. to add 87 to another number in my head I would usually add 90 and subtract 3). I’ve taken many tests at the university level for which using a calculator (much less long math) meant not having time to finish the exam. Mental calculation proved to be an essential skill.

Mental Math Hits Facebook

UPDATE! 3/26/2014. A post on the Federalist Papers blog is making the rounds on Facebook, highlighting a Common Core math problem that completely baffled a frustrated parent (who happened to be a degreed engineer). The parent’s reaction is typical. Why must one draw a number line to subtract 316 from 427? I think the parent’s response would be completely apropos if the worksheet simply required finding the numeric difference. But this is not the question at hand. The instructions are to identify the flaw in Jack’s approach to solving the problem. One of the comments to this post provides a lot of insight: this is the type of question used to train math teachers on how to teach math. Substitute “your student” for “Jack” and “write on his paper” instead of “write a letter” and you will see what I mean.

The worksheet illustrates how to approach the problem mentally, in bite-sized chunks. If I needed to do this calculation on the spot and without any tools (pencil, calculator), I’d probably start by taking away the biggest chunks first, that is to say the hundreds digit, and work my way down from there. Patheos blogger Hemant Metha addresses a similar Facebook favorite that shows how the student counts up in stepwise fashion, basically working a subtraction in reverse. If you expect the student to solve this problem mathematically, then yes, this is so very backward; however, it makes perfect sense from a mental math perspective. Consider his example of how you would figure out (on the spot) how much change you should receive when paying for $4.70 worth of merchandise with a $20 bill (let’s see…30¢ brings us up to $5, then another $15 to make $20 — there we go! $15.30).

Unfortunately, what Metha brings out in his post, and what is mentioned in the comments of the FP post, is the political agenda behind these viral Facebook posts. Conservatives apparently view new methods for teaching math as a threat to traditional values and an effort to degenerate the ability of our citizens to think for themselves so that they can more easily be swayed by the government. As compelling as this argument may sound (in all of its various manifestations), it is irrational. After all, conservatives exercise mental math skills for more often than liberals do. Being successful business people, they are more likely to have $20 bills in their pockets, and thus they find themselves calculating the change they should receive from liberal neo-hippie baristas on the spot more often. Also, liberals don’t have to get any number right as long as they sound convincing when they deliver it in a campaign speech. See how ugly this can get? Of course, I’m being extremely cheeky to prove a point. Math is a highly objective discipline and different people learn it in different ways. For either side to use this as a mark of political partisanship makes them look petty and ignorant.

By the way, Jack, you forgot to subtract 10.


1 Comment

  1. In my opinion there is too much time spent learning the “tricks”. Of course, I learned “long” math and, granted, “carrying the one” was a learning adventure but we learned how the the numbers worked together. Over time we all learned “tricks”, like the example you cited but we learned it on our own, it wasn’t taught to us. How do I feel about it? I think things should be taught the “long” way and then the shortcuts — if not, the shortcuts mean nothing.

    Comment by melissaboyettbrinkley — February 25, 2014 @ 7:14 am

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