Transcript Slide 1

Drill and Thrill:
Mindful Practice That
Connects Skill with
Understanding
Liz Uccello
[email protected]
http://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…
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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 strategy
instruction 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.
“Can Drill Help Develop Mathematical
Reasoning?”
“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
Teaching for Fact Fluency
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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 Arithmetic
Kindergarten:
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Match “finger numbers” to visual numbers
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Hop forward and backward on a number line
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Say how many fingers are “down” as well as “up”
First Grade:
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Count backwards from 40
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Double numbers through 12
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Make “pairs of 10”
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Add and subtract 10 to any number from 10 -- 100
Second Grade:
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Expanding “pairs of 10”
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Use 10 to add and subtract other numbers (adding 9)
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Use number knowledge to solve trickier problems (half of 860)
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Visualize multiplication and division using arrays and intersections
Third Grade:
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Doubling and Halving Two-Digit numbers
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Multiply numbers by 4 and 8 (using number knowledge)
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Square numbers
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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).
Think Math!, grade 1, Ch. 8
3. Continue tossing cubes and
calling out sums until
http://thinkmath.edc.org someone gets Bingo!
Using the Hundreds Chart
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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.
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http://thinkmath.edc.org/index.php/Hundreds_chart
An alternative
100s chart.
Numbers get
larger towards the
top rather than
towards the
bottom.
Correlates with
graphing.
Dice Games
Pig: 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 students
Setup:
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Remove all face cards from a regular deck of cards. (These will not be needed for the game.)
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Shuffle the deck and deal out all cards evenly, in two face down piles, to both players.
Play:
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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.
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The person who creates the highest/lowest number wins the cards (4 cards each round)
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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 8
and 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.
Card Games
(continued)
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.
Websites
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http://www.Fun4thebrain.com
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http://www.learningplanet.com
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http://nlvm.usu.edu/en/nav/vlibrary.html
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http://thinkmath.edc.org
Thank you!!
Liz Uccello
[email protected]
http://thinkmath.edc.org