Mental math

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See also Fact of the day and Practicing skills video

Contents

There is, of course, more to math than mental computation--geometry, graphing, problem solving, for just a few examples--but the ability to be flexible, fast, and accurate at mental computation makes all of mathematics easier.
This article focuses only on mental math and techniques of effective practice to become competent at mental arithmetic.

Why practice?

Whether one is learning mathematics or baseball or violin, practice is essential. And to develop real skill, one needs a lot of practice. But there are good and not-so-good ways to practice.

Good practice strips away some of the complexities, subtleties, and distractions of real problem solving -- we play tee-ball, for example, to establish batting before adding the extra challenge of following a pitched ball -- but if practice becomes too mechanical and divorced from the "game" one is playing, minds turn off; people “sleepwalk” through the practice, and not only enjoy it less, but gain less from it. And different kinds of learning may need less, or more, or different kinds of practice.

Drill and thrill: ideal practice

Ideal practice (see also Fact of the day)

  • generally focuses on a single skill (not random "mixed drill"),
  • includes enough repetition of that skill to strengthen it,
  • includes some age-appropriate element that keeps the practicer mentally alert, and[1]
  • proves its worth by making the progress noticeable, thus leaving one feeling competent and successful.[2]

Tee-ball removes the challenge of the pitch, but preserves the nature (and fun) of the game. Each new Suzuki violin piece introduces a single new skill and uses it a lot, but preserves the spirit (and fun) of music. It is not just an exercise; it is a piece of music that one can play and enjoy as music. Ideal practice in elementary mathematics must also preserve the nature of mathematics, even though it must necessarily simplify in some way.

For two examples of ideal practice designed according to these principles, see addition fact practice and multiplication fact practice.

Notes:

  1. This is one reason why addition practice that had been designed for first graders may not be equally suitable as remedial work for fifth graders.
  2. This is one reason why practice should be focused enough so that success is possible within the amount of time one has for the practice.

Mental arithmetic skills to be mastered in the early grades

All traditional "arithmetic facts" are essential mental skills, but they're not the only mental math skills that are needed for fluent computation. When we need to subtract 1 from 5000, we don't picture crossing out zeros and borrowing 1s and writing 9s. We just think 4999.

The way to master mental arithmetic early is to be strategic about it. Certain facts are both easier to learn and more useful in building strong mental arithmetic than other facts. Learning just those facts first, and waiting with others, lets students become and feel competent very quickly.

The following lists describe the most strategic order we've found, one that is easy to accomplish with almost all students. Mastery of these mental arithmetic skills, will assure a solid foundation for success in mathematics.

First grade

Every first grader needs to know six addition/subtraction skills with total fluency. Most children will be able to do much more, but at the minimum, they should all master these skills:

  1. Count backwards from 40 to 0;
  2. Double numbers up through 12;
  3. Find half of even numbers that are 20 or less;
  4. Make “pairs to 10” (e.g., if you say 6, they respond with 4);
  5. Add 10 (find 10 more) to any number from 0 through 90, and subtract 10 from any number from 10 through 100; and
  6. Recognize how to use their knowledge of addition for any related subtraction.

These are all mental arithmetic skills. All other facts build off of these, and so even if these are the only ones that are mastered, as long as they are mastered, children are in excellent shape for grade 2. Children will, of course, also record numbers on paper, learn to recognize problems in written form, and learn other mathematical content, but the essential arithmetic skills are all mental ones, as no formal algorithms are used in first grade.

Second grade

Second graders may repeat first-grade skills in brief, lively fact-of-the-day-type drills and then, early on, start building on them. Again, only mental arithmetic skills are listed here. All skills listed here are to be mastered by the end of grade 2.

  1. Addition/subtraction and place value: Over the course of the year, pairs to 10 (e.g., 3, 7) prepare them for pairs to 100 (e.g., 30, 70) and 1000 (e.g., 300, 700).
  2. Addition/subtraction: They are fluent with pairs to 20 early in the year.
  3. Addition/subtraction: By the end of second grade, they can add or subtract 8 or 9 (or 11 or 12) by being so facile at adding and subtracting 10 that they can use that, and a minor adjustment, for the other operations. (E.g., to subtract 8 from 35, they think “35 - 10 = 25, but then I've subtracted too much, so,” knowing that 8 and 2 are 10, “I must put back 2, and the answer is 27.”)
  4. Addition/subtraction: They can invent pairs to 11, or to 9, by thinking about pairs to 10 and adjusting.
  5. Addition/subtraction: They deconstruct numbers under 10 as 5 + more, and regroup to “explain” sums like 8 + 7. E.g., 8 + 7 = (5 + 3) + (5 + 2) = (5 + 5) + (3 + 2). (See Four-hand addition )
  6. Subtraction and place value: By mid-year, they can solve 23 - 20 by applying a linguistic idea -- "Laura G—— minus Laura = G——" and "Laura G—— minus G—— = Laura" (see explanation at article on "language and mathematics") -- to a mathematical context, to make it easy to do (twenty three – twenty = three) and (twenty three – three = twenty).
  7. Multiplication/division: By the end of the year, they double numbers mentally through 100 and halve “easy” even numbers (all digits even) through 800. (Half of 860 is half of 800 and half of 60.)
  8. Multiplication: By the end of the year, they know that 0 × anything is 0 and that 1 × any number is that number. Using the intersection model, they also understand why these are true. They know that "2 × a number" refers to the doubling that they already do well; they know that a number × 10 is that many tens, and they know what value that is; and they recognize, by sight, these arrays 3×3, 3×4, 3×5, 3×6, 4×4, 4×5, 4×6, 5×5, and consequently know those number facts.

Third grade

Third graders continue to build on second grade mental computation abilities -- pairs to 10; adding and subtracting 10s; equal comfort with 70 + 80 as with 7 + 8; ability to do 1000 – 7 mentally, and to solve 1000 – 27 by thinking 1000 – 7 and then subtracting 20 from that (or subtracting 20 and then 7); deconstructing/regrouping numbers to be able to add and subtract two digit numbers with ease; total comfort with all second grade multiplication facts and ideas (understanding and knowing that 0 × anything is 0, that 1 × any number is that number, 2 × a number is its double…) -- and extend these mental computation abilities to the following:

  1. Double two-digit numbers mentally and take ½ of any (even) two-digit number;
  2. Multiply any number by 4 or 8 by doubling the appropriate number of times (to multiply 4, double twice; to multiply by 8 three times);
  3. Multiply any number by 5 by using (and understanding) the fact that the result is ½ of multiplying that number by 10; use this skill both to help with their 5 × facts and to multiply 5 × easy (both-digits-even) two-digit numbers mentally (e.g., 5 × 84 is half of 10 × 84);
  4. Know square numbers (3×3, 4×4, 5×5, 6×6, ... through 12×12) like their best friends' names;
  5. Learn the remaining multiplication facts fluently (only six facts are left to memorize: 6×7, 6×8, 6×9, 7×8, 7×9, 8×9).

You want third grade students to acquire arithmetic skills beyond this, including a solid foundation for multi-digit addition and subtraction algorithms (and, of course, lots of mathematics that is not arithmetic). As in all of these lists, the above are only the essential mental math skills.

Very fast one-a-day method for multiplication facts

A child is asked to be the "class specialist" for one math fact for that day. Any time anyone needs help with that fact -- or any time anyone (including the teacher) asks for that fact -- that child is the expert-of-the-day in charge.

Method

At the beginning of math class or, even better, at the beginning of the day, assign one math fact to each of three or, at most, four kids (not the whole class and not more than one fact per child). These few kids are that day's “experts,” each “specializing” on just one fact. So, for example, one child might be in charge of 7 × 8 = 56.

A good way to make the assignments is to ask a child to name one fact that the child often has trouble with and then assign just that fact to that child. Periodically during the day, ask about the facts, varying how you ask.

  • Sometimes, "who's in charge of 56 today?" or "who's in charge of 7 × 8" or even, "who are our experts today?" followed by "and what are you in charge of?"
  • Feel free to revisit previous days’ facts in a kind of backwards way, like “who was in charge of 49 yesterday?” and then, to that child, “what was the fact that gave us 49?” (7 × 7).

How it works

Purpose: At the rate of one fact a day, the task of memorizing multiplication facts is much less daunting than “learn all your facts,” and it proceeds to mastery very rapidly. The child who is in charge feels "in charge," which helps, but many other children pick up that one fact at a time, too.

Asking a child which fact he or she is in charge of -- that is, not asking “what is 7 × 8?” but literally “what fact are you today’s expert on?”) puts the child in charge of remembering the whole package, and not just the “answer part.” Sometimes they get 56 and have to remember 7 × 8; sometimes they get 7 × 8 and have to remember 56; and sometimes they get nothing at all, and need to remember what they're responsible for.

And it's not just the expert who learns one fact. Most of the time, most children in the class will learn all three or four of the day's experts' facts, and will associate those facts with the experts, too!

Fact of the day exercises help children master doubles, multiplication by 5, and squares (multiplication of numbers times themselves, like 7 × 7). Principles (and the intersections imagery) help children with multiplication by 1 and 0 (anything taken zero times is, well, not there, so 0; anything taken exactly one time is whatever it is). The multiplication pattern exercise helps them memorize still more facts, like 6 × 8 and 7 × 9. "Being a specialist" makes quick work of the very few facts that remain.

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