Math Question and Answer

This Question and Answer blog relies on your questions! Please feel free to submit a question on math content, pedagogy, curricula, and the like here by emailing Allison at qanda @ (remove the spaces, please.) We will endeavor to answer in a timely fashion.

We accept standards in about in quality math instruction?

posted Jul 29, 2013, 11:37 AM by allison coates

I have heard many parents and teachers echo the sentiment that standards, and using standards to choose a curriculum, is too restrictive. I've yet to meet a single person who felt that having standards for levels in sports was "restrictive"  for coaches. I've never even heard of a coach who felt that standards for player skills was a negative, either. Sure, some may quibble about which skill when, but never that someone should not prescribe a list of skills students need to have.

Generally, people agree that youth sports should teach fundamentals well in ages 8 - 13. Whether or not that child will play sports competitively in high school or beyond, competency in mastering these skills is a requirement for even the opportunity to play at the high school level. They agree as well that certain skills come first as building blocks for other skills. Parents seek out and join youth sports organizations with quality instruction programs and quality coaches. Nearly every one of those has a set of standards for their levels. Those standards are what the coaches are expected to know, and expected to instruct students so they master those standards before moving on. Parents don't typically *know* the list of skills being taught, or in what order, but they know what the results are.

Let's take skating as an example. 

There are two major skating orgs in the US, and nearly every rink here has a learn to skate program that corresponds to one or the other. At every International Skating Institute (ISI) skate school in the world, they have a beginning alpha-delta system, and then more for competitive programs, like freestyle or pairs or speed skating. To progress to the beta, you must complete alpha.  It doesn't matter how many times you've been to class, or your age, that's the standard for moving up a level. The teachers know to teach that--how, up to them, but those skills before progressing. You can't earn a delta badge without shooting the duck. 

Why do they bother with such rigid standards? Because e.g. you will not be able to do the freestyle one half flip if you can't do a one-foot turn (delta level). 

Math is not different in this way. To suceed at the abstraction of algebra, you need to be able to handle fractions. To handle fractions, you need to have mastered whole number arithmetic.

US Figure Skating Assoc. has a different list, levels 1-10, and some skills come at different times, emphasized in different ways. But again, a set of standards--this is what they teach at level 4 to go to 5, etc.

Their coaches are expected to teach these levels, in this order. They don't get to call themselves certified coaches if they deviate. They still have the freedom to figure how to teach these skills, and they make changes in their methods depending on their students. But their students need to be able to do a regimented set of skills to progress.

A parent can get a kid up on skates and around a rink, but unless that parent knows every supporting skill, and how to break them down, they aren't likely to get their child proper enough technique to support that one half flip.

Likewise, a homeschooling parent or a generalist teacher needs to know the supporting skills in math, and how to break them down to ensure proper mathematical technique.

Let's look at youth hockey.

USA Hockey is the sponsor of nearly every youth hockey program in the country. Their programs have coaches that must follow a prescribed curriculum (now called the American Development Model, ADM) with handbooks, DVDs and trainings. Inside that curriculum, which describes how every practice should be run, down to how many minutes per week and weeks the program should be, they also have standard for skills at every level.

Here's just their list of standards for individual hockey skills mites, youth 8 and under (mite):

Individual Hockey Skills:
Players must learn and master:
1. Skating: • edge controlling • ready position • forward start • forward stride • control stop (two-foot snowplow, one-foot
snowplow) • backward skating • backward stop • control turn • forward crossover 
2. Puck Control: • lateral dribble • forward-to-backward dribble • diagonal dribble • attacking the triangle • forehand shift • accelerating with the puck
3. Passing and Receiving: • forehand • backhand • receiving (stick) • eye contact 
4. Shooting: • wrist • backhand
5. Checking: • poke check • hook check • lift the stick check
6. Goalkeeping: • basic stance • parallel shuffle • lateral t-glide • forward and backward moves • stick save • body save • glove save

They also have standards in Knowledge, Goal Setting, Team Play, Nutrition, Fitness and Training, Injury Prevention, Sports Psychology, Character Development and Life Skills. And yes, they push up by age, even if not every skill is met because some skills will come with different physical and mental development. But kids who don't develop and complete this mastery on this track won't make it to Bantams.  By then, it all had to come together.

Martial arts, anyone? The belt system is a standards system. I'm sure there are proverbs explaining how the restrictions lead to freedom in each martial art.

I could find you lists of standards for youth skill development in tennis, golf, soccer, swimming, and more. Keeping the doors open to the possibility of playing these sports after 13 requires a specific set of fundamentals must have been mastered, year over year.

Being able to succeed in high school math requires training just as sports in 8th grade and beyond do. If you don't have the fundamentals down by 8th grade, you will never catch up. But math is harder as a subject, since it is so linear. Skip something, and you will not understand, and at some point, the struggle to stay afloat will be too much.

The coach or teacher is the most important element of the student's learning in these programs, bar none. Solid standards don't magically make great teachers. But good coaches and teachers know the skills they are trying to develop, know how they fit together year after year, and work on improving what they don't know. They teach 
the standards, and they don't make excuses that the standards are restrictive.

US Figure Skating standards:
ISI testing requirements:
Hockey skills handbook:

Common core state standards in math:

Math Puzzles for your Kids

posted Jun 29, 2013, 8:54 PM by allison coates

Just about everyone has now seen Sudoku, but there are another dozen similar and fun puzzles that teach even more math skills.  You can find many free versions of these puzzles on the web at or other sites on the web.

Here are some to get you started (From:

Ken-Ken is terrific! There are even books of KenKen puzzles in the US that you can purchase.


Rules For Playing KenKen®

The numbers you use in a KenKen puzzle depend on the size of the grid you choose. A 3 x 3 grid (3 squares across, 3 squares down) means you use the numbers 1, 2, and 3. In a 4 x 4 grid, use numbers 1 to 4. A 5x5 grid requires you use the numbers 1 to 5, and so on.

The numbers in each heavily outlined set of squares, called cages, must combine (in any order) to produce the target number in the top corner using the mathematic operation indicated (+, -, ×, ÷).

Here's how you play:

1.   Use each number only once per row, once per column.

2.   Cages with just one square should be filled in with the target number in the top corner.

3.   A number can be repeated within a cage as long as it is not in the same row or column.

That's it! Can you solve the sample above?

Kakuro                       From:

 Each grid has horizontal and vertical lines containing either clues or empty cells. The clue number is the sum of the empty cells it represents. These clues are printed either above a diagonal line or below a diagonal line.


If the clue is printed above the line, the clues is the sum of the empty cells to the right. When the clue is printed below the line, it is meant for the row of empty cells below the clue number. Each number between 1 and 9 can only be filled in once! So when the clue says 4, and there are 2 empty cells, you cannot fill in 2 and 2. The answer must automatically be 1 and 3.





          What is Shikaku or Rectangles ?


Sikaku is all about the areas of squares and rectangles. The player has to fill a grid with rectangles or squares with areas of already printed numbers. When an 'eight' is printed in the puzzle the player has to figure out whether to draw a '8 by 1' rectangle or a '2 by 4' rectangle. When a 'nine' is given, a '3 by 3' or '9 by 1' rectangle/square must be filled in.


The areas of prime numbers are easier! A 'thirteen' printed in the grid can only mean a rectangle of '13 by 1'! The more clues in the puzzle, the easier it is to solve. The difficult Rectangle puzzles contain no clues at all! Use your logic skills to do the math!





Fubuki  (like magic squares)

What is Fubuki?


Fubuki is and great math game, or number puzzle, in which your addition and multiplication skills are needed. Each puzzle consists of a 3 by 3 grid which has to be filled with the numbers 1 to 9. Each number can and must be used only once and, in case of the addition Fubuki, add up to the totals of each horizontal or vertical row or column of 3 numbers. In the case of the multiplication Fubuki, the 3 numbers are the multiples of the given products. The rules of this maths game are pretty simple, but solving these puzzles is another story. Brain training to the max!













Math Games to Play with Your Kids

posted Jun 29, 2013, 8:32 PM by allison coates   [ updated Jun 29, 2013, 8:33 PM ]

Summer is a time for fun, not necessarily a bunch of math worksheets. But you can increase mental math skills and practice number sense by playing games.

Board Games: Board games can teach number sense, addition, and virtues such as patience and stamina.

For children in preschool through primary grades, playing games where they practice moving the correct number of steps is good practice for the number line. Emphasize that a move of 3 means counting 3 hops, and that adding means the same thing. Chutes and Ladders, Uncle Wiggly, Sorry, Parcheesi, etc. help addition skills. For children in lower primary and above, Monopoly and Scrabble both practice addition and multiplication, optimization of sums and mental math.

Card Games: These games can be played using a standard deck of cards. Some are traditional card games, others are more mathy, but all can be fun. Having your child be the score keeper is an excellent way to practice mental math or standard algorithms.

    1. Salute! (For any child learning addition or beyond)

This is a game for three players. The goal is to be the first person to determine their own card’s value. This game is played with an ordinary deck with face cards removed. Aces have value 1; all other cards match their number. Play is as follows: one of three players is chosen as Dealer; Dealer hands out the cards evenly to two players who sit facing each other; each holds the stack of cards face down. Dealer says “Salute,” and the two players simultaneously take the top cards off their respective piles and hold them on their foreheads with the card facing OUT, so each player can only see opponent’s card. Dealer announces the sum of the two cards. First player to correctly announce his own card wins the hand, and keeps both cards. 

For short games, play continues until all cards have been played once; winner is player with most cards. Players then rotate who becomes Dealer and repeat until everyone has once played Dealer. For longer games, play continues in the style of War, repeating the game with cards won on previous rounds, until one player’s cards are exhausted. Players then rotate who becomes Dealer and repeat until everyone has once played Dealer. More advanced variations have the Dealer finding the product of the cards, or 3 times the sum, or ….

    2. 24 (For children who have learned all four basic arithmetic operations. Middle school or beyond)

24 is a game where the object is to use arithmetic operations on a set of 4 numbers to produce an expression whose result is 24. Suitable for any number of players. This game is played with an ordinary deck of playing cards with all the face cards removed. The aces have value 1; all other cards match their number. The basic game proceeds as follows: dealer places 4 cards face up, and each player tries to form a mathematical expression whose result is 24 using only addition, subtraction, multiplication, division, and parentheses. The first player to do so wins the hand. (Some groups of players allow exponentiation, or even further operations such as roots or logarithms.)
For example, if the four cards are 3 of hearts, 8 of diamonds, 4 of clubs, 3 of clubs, then one possible answer is (3 + 3)  × (8 - 4) = 24. Another is 3 × 8 × (4 – 3). Yet another is ∛8  ×4 ×3.

Players may either write or verbally express their answer. For short games of 24, once a hand is won, the cards go to the player that won. If everyone gives up, the cards are shuffled back into the deck. The game ends when the deck is exhausted and the player with the most cards wins. Longer games of 24 proceed by first dealing the cards out to the players, each of whom contributes to each set of cards exposed. A player who solves a set takes its cards and replenishes their pile, after the fashion of War; players are eliminated when they no longer have any cards. (Rules from Wikipedia.)

3. War! (For any child learning numbers)

A standard game of War is perfect for teaching young children to count and compare. The object is to win all of the cards. Suitable for 2 or more players. Dealer deals out the cards evenly between game participants. Number cards are worth their number; face cards are worth ten; aces 11. (Alternative rules: Aces represent one and face cards are ten.) Play proceeds as follows: each player turns over the top card from their own pile. Player with the largest value card wins all the cards. In the event of ties between highest valued cards, players who tied then shout “War!”, and count out three cards face down from their pile, and then display an additional fourth card. Winner takes all played cards. 

4. 21 (For any child learning to add)

This is a game of Blackjack with all of the complicated rules removed. The object of the game is to get a hand of cards whose sum is as close to 21 as possible without going over. Suitable for any number of players. Number cards are worth their number; face cards are worth ten; aces are worth either 1 or 11, as determined by player’s choice for each ace they hold. Dealer deals two cards to each player. First player then chooses based on his hand whether to take another card (“a hit”) or to receive no more cards (“stand”). If he chooses another card, dealer lays it down face up for all players to see. Player then repeats this choice as often as he likes or until he goes over 21 (“bust”.)  Dealer then moves on to next player and repeats the choice. (First player may not take more cards later in this round.) When all players have chosen to stand or gone bust, all players then lay down their cads so all players can see. Player whose cards are closest to 21 without going over is winner. 

5. Uno!

Uno! is terrific for preschoolers through lower primary grades, teaching number recognition and matching. Most easily played by removing the face cards from a traditional deck, but local variations are numerous. Increase the difficulty by varying rules so the next playable card matches the same number or is the same color but only either 1 greater or 1 less than the current played card. Encourage score keeping by keeping track of the sum of losing players’ hands across many rounds; lowest sum after n rounds wins.

6. Go Fish!

Play Go Fish with a standard deck to help a preschooler or kindergartener practice counting and matching. To increase difficulty, keep score, counting the face value of all cards each player wins. Look for patterns in how to count things up more quickly.

7. Gin Rummy 

Variations of this game are numerous. Score keeping versions encourage the most practice with mental math and standard algorithms.

8. Sum Fun (For primary grades and above)

Working with a deck of cards with face cards removed, player whose 2 2-digit sum is as close to 100 without going over wins the round. To begin, play an easier version until the concept is mastered: player whose sum is highest wins the round. 

Dealer deals out 5 cards face down to each player. On a turn, each player lays down 4 of 5 cards, face up, in 2 pairs, showing cards a 2 2-digit numbers. 
Example: Player receives 8 of hearts, 2 of spades, 3 hearts, 6 of clubs, 5 of spades. She plays 8h5s and 6c3h, because 85 + 63 = 148. She doesn't play the 2s, because using the 2 anywhere would have lowered her score.

Player whose 2 2-digit numbers sum highest wins the round. Winner receives all cards. Dealer deals out 4 more cards to each player, repeating until deck is dealt. Player with most cards wins. Once this version is mastered, play this real version: player whose 2 2-digit sum is as close to 100 without going over wins the round.

What are "standards"? Are standards curricula?

posted Mar 28, 2013, 7:28 PM by allison coates   [ updated Mar 28, 2013, 7:32 PM ]

Q: Why do schools need standards? What are standards for? What do they have to do with curriculum?

A: A set of standards is a *minimum set of requirements* that a product must meet. Let's imagine a house; the building codes are standards. An individual standard says something like "a wall must withstand straight line winds of up to 75 mph" or "studs must be places at least every 24 inches." Standards tell *what* you need to do, but don't tell you how to do it. They are used so that in the event of a serious risk, the whole structure stays standing. The building code for homes applies whether building a tiny townhouse or a 5 bedroom multi-story rambler. They are meant to force people not to cut too many corners. Standards here in math education are similar: they tell you what must be taught to a student, when, year over year. They are still *minimum set of requirements*, the ones that need to apply to every classroom, every school, every student.

A set of building codes aren't plans for a building. They are not a complete design.

A design tells you what *this building* going to look like. It answers the question of what purpose this house serves. It shows you how the floors and rooms fit together, how space is utilized. Architectural plans are a design for a home. They explain what the floor plan is, what the foundation is shaped like, where the plumbing goes, where the electricity goes, how the building will maintain even temperatures, etc. It's a map for everything that needs to be built.  A solid design takes into account the what the end result should be like and plans the structure accordingly. A good design is also relatively easy to build, because it anticipates what elements of the design depend on the other elements.

In math education, this design is the curriculum. The curriculum is the plan for each year of math education, starting with a foundation, then the walls, the plumbing, the electrical, the floors, the roof, the interior. Like a good design for a building, a good curriculum is easy to implement, anticipating not just the end product, but how to ease the building of the elements as they are put in place.

A design is necessary for a home, but we don't live in an architectural drawing. The home must still be built So there are builders. Clearly the builders need to know the standards, and build according to them, so the structure is safe. But they build according to the standards while the build up the plan, starting with the foundation, and then moving on through the structure. They complete the plan.

Teachers are the builders. They are building the house of mathematics that students will live in. Some teachers are like builders who just doing the efficient thing, putting the pieces up in order. Others are like exquisite craftsmen. The most experienced teachers are like the most experienced builders who are able to look at the plans and figure out what won't quite work in this context, and where changes should be made. They know how long the job will take, and where the hard parts are before they start. They make very few mistakes, and correct them before they cause more problems. A great math teacher is building above and beyond the minimum standards where and when she can; she is on the lookout for the pitfalls in the curriculum, ready to point them out and rework them; she is improving her own skills, employing new innovations, and producing beautiful results.

Bad standards mean the structure is unsafe. Unfortunately, this isn't always visible to the eye. The owner may not find this out until a minor tremor comes and the whole house collapses. Likewise, a bad plan means the house is not a comfortable or useful place to live, or worse, could be dangerous. If the house was designed incorrectly, then the second story may not be able to hold up the third story, and it could collapse. A truly terrible design usually can't be rebuilt, but needs to be torn down. And a bad builder means things go wrong and break in your home, and again, if bad enough, the errors could mean that the house collapses. 

Likewise, in math education, bad standards means the whole structure of math that a student is building up is faulty, and sooner or later, it is going to collapse. But good standards aren't the same as a good design. Students themselves don't need to learn the standards, any more than a homeowner needs to know the building code. Schools need to have a design, a curriculum, for their teachers, the builders to work from. A bad design won't support another story being built on top, year after year after year. And a bad builder on a floor means nothing can be built on top of it until a great builder comes in and rips out the bad work and rebuilds.

Every part of the process matters. Good standards, good curriculum, good teachers.

What's wrong with a spiral curriculum?

posted Feb 24, 2013, 10:07 PM by allison coates   [ updated Mar 28, 2013, 7:30 PM ]

Q: What's wrong with a spiral curriculum? Isn't distributed practice a good idea?

A:  Nearly all US mathematics textbooks are incoherent. That is, they don't teach that math builds upon in itself, block by block. For math to make sense, what you learn today should be completely derivable from what you did yesterday. You would never use a history book that taught history by teaching why the Puritans left Britain for two days, then spent two days talking about the Incas, then a unit on the geography of Antarctica followed by a bunch of other topics about the Ming dynasty, Reconstruction after the Civil War, and Medieval architecture, and the only several dozen or hundred pages later taught about what happened when the Puritans arrived at Plymouth. But that is what spiral curricula textbooks do in math. 

By jumbling up all of those concepts, kids can't keep track of why what they are doing is true, or how it's related to what they did before, anymore than they would keep track of why the society the Puritans built in America was about religious freedom when they hadn't seen nor heard about religious persecution for several weeks.

Any well phrased problem that built on prior math would allow for distributed practice. But skills are built in order to build up other skills, and should do so in a sequential manner, not haphazardly.  When we teach students that math is incoherent, they cannot learn how to reason mathematically. But algebra is all about reasoning. We must teach math coherently.

What does an incoherent curriculum look like? Here's a list of lessons in Saxon Intermediate 5: (All lessons from Section 3):
 21 Word Problems About Equal Groups
22 Division with and without remainder
23 Recognizing Halves
24 Parentheses and the Associative Property
25 Listing the Factors of Whole Numbers
26 Division Algorithm
27 Reading Scales
28 Measuring Time
29 Multiplying by Multiples of 10 and 100
30 Interpreting Pictures of Fractions, Decimals, Percents

By comparison, here's a list of lessons from Singapore Math 5A, Standards Ed:
Section 3: Fractions
1. Comparing Fractions
2. Fractions and Division
3. Addition and Subtraction of Unlike Fractions
4. Addition and Subtraction of Mixed Numbers
5. Multiplying a Fraction and a Whole Number
6. Fraction of a Set
7. Word Problems
Section 4: Multiply and Divide Fractions
1. Product of Fractions
2. Word Problems
3. Dividing a Fraction by a Whole Number
4. Dividing by a Fraction
5. More Word Problems

Until American students are taught in a sequential, coherent manner, they will continue to fare poorly at algebra and beyond.

1-5 of 5