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Practice

Practice in Think Math! comes in many forms. This article focuses on the practice that leads to full and fluent mastery of mental mathematics. Such mastery requires what cognitive science calls "distributed practice," practice given in small parts spaced out over time rather than "massed" into a short unit and then abandoned (see, e.g., How People Learn, Bransford, Brown, and Cocking, 2000). The Mental Math ("Skill Practice and Review") feature of Think Math! carefully sequences a highly focused and particularly strategic selection of exercises, and presents students with the distributed practice they need -- a few minutes at a time, spaced out over a few days and recurring weeks later -- to master the skill.

There is more to math than facts, and there's more to learning than practice.
This article focuses only on the role and techniques of effective practice of mental math skills, and which skills require that practice. The particular skills chosen, and the focused way in which they are developed, also contribute to conceptual growth, particularly around properties of arithmetic, which is needed for algebraic thinking.Anchor

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

Think Math! treats all traditional arithmetic facts as essential mental skills, and develops additional mental skills that are needed for fluent computation. Like any responsible mathematics program, Think Math! teaches other content as well. The following lists include only mental arithmetic using whole numbers. Mastery of these mental arithmetic skills, along with the plentiful opportunities for discovery and algebraic reasoning provided in Think Math!, will assure a solid foundation for success in mathematics.

First grade

Every first grader needs to know six addition/subtraction skills with total fluency. Much more is built into Think Math!, and most children will be able to do much more, but at the minimum, they will all:

  1. Count backwards from 30 to 0 and, later, from numbers as high as 100, one number per second for about 30 seconds;
  2. Double numbers up through doubling 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 gradeAnchor

Second grade

Think Math! second graders repeat many of the first-grade skills in their brief, lively Skill practice and review drills and then build on them. Second graders do begin building understandings that lead to the arithmetic algorithms, but 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 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).

Third grade Think Math! students acquire arithmetic skills beyond this, including a solid foundation for multi-digit addition and subtraction algorithms. (Of course, they also learn 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 a handful of 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.

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.

Opportunities for fact practice

Addition/subtraction

AnchorInitial Models

  • Number line jumps backward and forward
  • Number line distance
  • Combining and partitioning collections (including base 10 blocks)
  • Cuisenaire rods -- a "number-free" model

Building the foundation skills

  • fingers: mirroring (see Show how many), Make this number, pairs to 5, Making 10, using fingers as dimes, pairs to 100 (see also gr1, ch9, les2)
  • pairs to 10, uses and extensions
    • pairs to 10 (mental/verbal) (paper: recognizing pairs-to-10 sheet)
    • pairs to 11 (one more than a pair to 10) or 9 (one less than a pair to 10)
    • pairs to 20, arbitrary pairs to 100 (mental/verbal) (paper for 20, only; optional writing of the pair to 100)
    • circling dominos
  • Other complements after fractions/decimals: pairs to 1, pairs to 10 with fractions/decimals
  • adding/subtracting 10 or 20, 100 or 200 to anything; Number Line Hotel; linguistics "poetry" of adding 10 (connecting this process to skip counting)
    • adding/subtracting 8, 12 to anything
    • adding/subtracting 19
  • Four-hand addition
    • with four hands (uses excess over 5, and adding 10 to anything)
    • with two hands (invisible fingers) (also a case of building memory)

Practicing the larger facts

  • games
    • Gr2 Ch8 Lesson 6 Race to 100 adding two-digit numbers
    • Gr2 Ch10 Lesson 11 Race back to zero subtracting two- and three-digit numbers
    • Gr3 Ch3 Lesson 8 Least to greatest two-digit version; Ch14 Les6 three-digit version: estimation and addition
    • Gr3 Ch14 Lesson 4 Addition scramble Adding multi-digit numbers
  • cross number puzzles
  • magic squares
  • practice embedded in multiplication-building activities (skip counting, "counting" elements in arrays and incrementing a row or column at a time, etc.)
  • skip counting by 49, SPR in gr4 ch3 les1, (TG pg 147)

Multiplication/division

Building the foundation images and facts

Together, these build facts through 5x5. Building early by capitalizing on what students already know or find easy.

  • intersections, arrays, from first grade
  • single-color trains of Cuisenaire rods; variable base (if r=6, what is g?)
  • use of intersections/arrays/area to show commutativity: 3 x 5 = 5 x 3 (cuts in half the number of facts to "memorize")
  • principle of multiplying by 0, 1
  • Square numbers
  • doubling and halving
  • small arrays (sight recognition, tabular description)
  • skip counting
  • multiplying and dividing by 4, taking advantage of doubling, a skill already well developed
    • multiplying by 4 by doubling twice -- associative property n x (2 x 2) = (n x 2) x 2
    • dividing by 4 (taking a fourth, multiplying by 1/4) by halving twice
  • fact families, again reducing number of discrete facts to memorize

Images and strategies for the larger facts

  • partitioned arrays (two-part) -- distributive property n x 9 = (n x 4) + (n x 5)
    • building block for larger facts
    • setting the stage for multiplication algorithm (n by m partitioning of the array)
  • multiplication by 10, 100 (counting 10s, 100s, with blocks, or linguistically)
    • Gr1 Ch5 Lesson 3 pennies and dimes
    • Gr2 penny ages (multiples of 100),
    • Gr2 Ch4 Lesson 6 pennies and dimes
    • Gr3 Ch12 Lesson 3 Base 10 blocks;
    • Gr3 Ch15 Lesson 1 dimes;
    • Gr4 Ch6 Lesson 2 Eraser store;
    • Gr5 Ch2 Lessons 4 and 5, multiplying by large powers of 10, grounded in building arrays
  • deriving one result from another
    • additive strategies
      • If we know 6x6, what is 7x6? If we know 17 x 100, what is 17 x 101?
      • chain changes: 5x7, 6x7, 6x8, 6x9, 7x9, 8x9, 9x9, 9x10, 9x11, 9x12, 10x12, 11x12, 11x13, 12x13, 13x13, 13x14 (see Fact builder)
      • near numbers and corrections: 99 x 7, 99 x 12, 99 x 38, 49 x 6 (see eraser store) (see memory)
    • multiplicative strategies
      • deriving 5n by halving 10n (see memory), multiplying by 50 (aiding multiplication by 49)
      • multiplying by 4 by doubling twice, multiplying by 12 by multiplying by 6 and doubling, multiplying by 20
  • "Stray" facts: multiplication by 12 (dozens), 15 (clocks), 25 (coins), 60 (seconds or minutes)

Practicing the larger facts

  • patterns
    • difference of squares and extensions
  • games
    • Gr3, Ch9, Lesson 3: Factor tic-tac-toe
    • Gr3, Ch12, Lesson 6: Factor tic-tac-toe
    • Gr3, Ch15, Lesson 3: Partial claim multiplying two-digit numbers (and adding)
    • Gr4, Ch1, Lesson 6: Number builder
    • Gr4, Ch6, Lesson 1: Find a factor
    • Gr4, Ch6, Lesson 7: Profitable products
    • Gr5, Ch1, Lesson 7: Fact builder
    • Gr5, Ch5, Lesson 5: Favorable factors multiplication of two-digit numbers
    • Gr5, Ch8, Lesson 5: Don't overestimate estimate quotient of 2-digit into 4-digit without going overAnchor

Magic Squares–options for extra practice in 4th and 5th grades

If you have students who need more addition/subtraction practice (or just like extra math puzzles), incomplete Magic squares are great puzzles, much more fun than worksheets!

I have found a great website that makes magic squares for you (all you have to do is press a button!) and also tells you how to decide what cells of the square you need to leave filled in and what you can leave blank for the kids to puzzle out.

The generator is easy. You just click on start and it does the rest. It works really fast for making squares that are 5×5 or smaller. (Frankly, you don’t want larger ones, anyway, as they begin to feel like “work” rather than “puzzle”!)

Have some blanks for copying the squares that are generated how to make a magic square (by dr. mike). Then follow the rules at magic square puzzles (by dr. mike) (again very straightforward and fast!) for what to leave in and what to leave blank for the kids to fill in.

I like both the activities and the approach to them on this site but am not endorsing the other resources that google advertises on the site. They are not all the same quality as those on the dr-mikes site.Anchor

No need to supplement

Used properly, Think Math! provides a lot of practice, all designed according to those principles. There is no need to supplement Think Math! with extra practice materials. The Skills Practice and Review (SPR), if done daily, builds mastery of essential mental computational skills that are the foundation even for paper-and-pencil skills. The games are designed to offer practice beyond the confines of the math lesson. Paper-and-pencil skills get considerable practice in the Practice book. And Think Math! embeds useful (and considerable!) practice in many explorations and Lesson Activity Book (LAB) pages as puzzles, curiosity builders, and exploratory activities that anticipate algebraic ideas, taking advantage of (and fostering) children's natural curiosity at the same time that it builds skills. The Practice Book provides yet more practice.

Embedding drill and practice in puzzles, activities, and games helps students focus on them, enjoy them, and acquire the skills much more effectively than they would with pages of raw practice. Every grade of Think Math! puts considerable emphasis on mathematical experimentation, understanding mathematics, and algebraic thinking, but also provides ample practice for children to master all essential facts and skills. The research bears this out: if the lessons are done faithfully, children become very good at their facts.

See Mental Math/Skills Practice and Review for more about this key mastery component of Think Math! -- essential every day!

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