Drill and Thrill: Mindful Practice That Connects Skill with Understanding

Liz Uccello

LizzieBeth528@gmail.comhttp://thinkmath.edc.org

Find the Difference

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Example:           5−2=3

Got that one figured out? Try figuring it out with the numbers 1 – 9…

http://nrich.maths.org/public/

Solutions

The Importance of Fact Fluency…

When students are unable to retrieve facts quickly and accurately, they have a higher cognitive load.

This leads to inefficient processing strategies (i.e., counting on fingers), which can lead to computation errors.

Math skills build upon one another, so by having automatic retrieval of facts, students are able to quickly solve more complex problems, rather than be bogged down in computation.

Woodward, J. (2006). Developing automaticity in multiplication facts: Integrating strategyinstruction with timed practice drills. Learning Disability Quarterly, 29, 269-289.

Cumming, J. & Elkins, J. (1999). Lack of automaticity in the basic addition facts as a characteristic of arithmetic learning problems and instructional needs. Mathematical Cognition, 5 (2), 149-180.

Pellegrino, J. & Goldman, S. (1987). Information processing and elementary mathematics. Journal of Learning Disabilities, 20, 23-32, 57.

“Critics of current educational practice indict "drill and kill" methods for two crimes against mathematics: disinterest and anxiety. Yet despite the

earnest efforts to focus mathematics on reasoning, one out of every two students thinks that learning mathematics is mostly memorization (Kenney

and Silver 1997).

Research shows rather convincingly that real competence comes only with extensive practice (Bjork and Druckman 1994). Nevertheless, practice is certainly not sufficient to ensure understanding. Both the evidence of research and the wisdom of experience suggest that students who can draw on both recalled and deduced mathematical facts make more progress than those who rely on one without the other (Askey and William 1995).”

- Lynn Arthur Steen, 1999

“Can Drill Help Develop Mathematical Reasoning?”

Teaching for Fact Fluency

Carve a small amount of focused time out of your day (7 – 10 minutes).

Find time EVERY DAY. Teach facts in a sequential order that builds on itself

(see handout). Use manipulatives and visual representations when

possible. Show students the relationship between numbers (if I

know 5+5=10, then I know 50+50=100).

Mental ArithmeticKindergarten: Match “finger numbers” to visual numbers Hop forward and backward on a number line Say how many fingers are “down” as well as “up”First Grade: Count backwards from 40 Double numbers through 12 Make “pairs of 10” Add and subtract 10 to any number from 10 -- 100Second Grade: Expanding “pairs of 10” Use 10 to add and subtract other numbers (adding 9) Use number knowledge to solve trickier problems (half of 860) Visualize multiplication and division using arrays and intersectionsThird Grade: Doubling and Halving Two-Digit numbers Multiply numbers by 4 and 8 (using number knowledge) Square numbers Multiplication fact fluency

Match To

Students work to name off the number pairs that add to a particular number.

http://thinkmath.edc.org/index.php/Practicing_skills_video Part 1

(also at http://www.youtube.com/watch?v=p7w0A2T6Eyk)

Half and Double Bingo

1. Roll a pair of dice and call out the sum. (Record sums on the board for reference).

2. Students can place a counter on either half of the sum or double the sum. (You may want to record the half and double on the board initially).

3. Continue tossing cubes and calling out sums until someone gets Bingo!

Think Math!, grade 1, Ch. 8 http://thinkmath.edc.org

Using the Hundreds Chart

Circle multiples in different colors (i.e., 2s in yellow, 3s in green)

Leave out some numbers and see if students can figure them out.

Mix up the 100s chart and see if students can work together to put the numbers back in order.

Cut the 100 chart into pieces and have students reassemble.

An alternative 100s chart.

Numbers get larger towards the top rather than towards the bottom.

Correlates with graphing.

http://thinkmath.edc.org/index.php/Hundreds_chart

Dice GamesPig: To be the first to score 100 points or more.

1. On their turn, students roll a pair of dice as many times as they like, keeping a running total mentally. When they stop, they add this sum to their previous score.

2. However, if the student rolls a 1, their sum for that turn is 0 and it is the other player’s turn. If a student rolls both 1s, then their entire score returns to 0.

Two-Dice Sums: To remove all the counters in the fewest rolls possible.

Setup: Each player needs 11 counters, a game strip that lists the numbers from 2 to 12 far enough apart so the counters can fit on top of each number, and a recording sheet.

Play:

1.Each player arranges 11 counters on the game strip and records the arrangement.

2.Once the counters are arranged, players take turns rolling the dice.

3.For each roll, all players can remove one counter if it is on the sum rolled. Players keep track of the number of rolls of the dice it takes to clear their game board.

Card Games

Hi-Lo: for two studentsSetup: Remove all face cards from a regular deck of cards. (These will not be needed for the

game.) Shuffle the deck and deal out all cards evenly, in two face down piles, to both players.Play: At the same time, each student draws two cards from their own pile. Depending on the

game version (see below), they must have a high or low card combination to win. The person who creates the highest/lowest number wins the cards (4 cards each round) The person with the most cards at the end of the game is the "winner.”Three game versions: Place Value Game: Example: Player #1 draws 3 and 6, and so can create 36 or 63.

Player #2 draws, 4 and 8. Because player 2 can create 84, and 84 is greater than 63, Player #2 keeps all four cards. (Or play it so lowest combination wins!)

Addition Game: Example: Player #1 draws 3 and 6. The sum is 9. Player #2 draws 4 and 8and can make the greater sum (12), so Player #2 keeps all four cards.

Subtraction Game: Example: Player #1 draws 3 and 6. The difference is 3. 6 – 3 = 3 Player #2 draws 4 and 8. The difference of these cards is 4. The smaller difference wins, so Player #1 keeps all four cards.

Seventeen:Prepare your materials: From a regular deck of cards, remove all nines, tens, jacks,

queens, and kings. Aces are valued as 1, and all numbered cards between 2 and 8 have their own value.

While this game can be played by up to four players, you’ll probably want to start with just two. Shuffle the number cards and put them face down. Each player takes five cards. Take turns putting cards down, one at a time, and counting the total made when you add the pile together.

“Winning” and “Losing”: The goal is to get as close to 17 as possible. Let’s say, for example, that Player 1 puts down a “7” card, and then Player 2 puts down a “5” card. If Player 1 can add another “5,” she wins the round and gets a score of 17! That’s the clean way to win a round. But she can also win if she goes slightly over—say, to 19—but she must subtract the extra “2” from her score, so she only gets 15 points. The goal of the game (aside from complete Math Facts Mastery, of course!), is to have the largest number of points when the game is done.

Card Games (continued)

Websites

http://www.Fun4thebrain.com

http://www.learningplanet.com

http://nlvm.usu.edu/en/nav/vlibrary.html

http://thinkmath.edc.org

Thank you!!

Liz UccelloLizzieBeth528@gmail.comhttp://thinkmath.edc.org