Transcript Math Fact Instruction: Deciphering Fact from Fiction April Summey Introduction • Every teacher who teaches math has complained about students not knowing their math facts.

Math Fact Instruction:
Deciphering Fact from
Fiction
April Summey
Introduction
• Every teacher who teaches math has complained
about students not knowing their math facts at
some point in time. Throughout the hallways of
Upward Elementary School, you can hear teachers
saying the same phrase over and over again each
year, “They don’t know their math facts!” The
lower grade teachers always respond, “Well, they
should! We made them practice!” The main issue
is that from year to year, students aren’t
mastering their basic math facts. This
presentation will focus mainly on multiplication
and division facts.
Why is this important?
• Mastery of math facts go hand and hand with
many computation skills that are taught during
the school year such as adding and subtracting
whole numbers, multiplying larger numbers, long
division, and adding fractions. Joanne Legg, a
fourth grade teacher at Upward asserts, “I tell
my parents every year that if their child knows
their math facts, then I can teach them
everything they need to know in math very easily”
(2009). Fixing this issue is important because
math facts are embedded everywhere.
More reasons why it is
important..
• Caron (2007) points out that without the mastery
of math facts “students are virtually denied
anything but minimal growth in any serious use of
mathematics or related subjects for the
remainder of their school years.” (p. 279).
• Woodward concludes that “automaticity in math
facts is fundamental to success in many areas of
higher mathematics” (2006, p. 269).
• “Rapid math-fact retrieval has been shown to be a
strong predictor of performance on mathematics
achievement tests” (Scholastic, 2008, p. 1).
What students should know..
• According to the North Carolina Standard Course
of Study (2008) students should “develop fluency
with multiplication from 1x1 to 12x12” by the end
of the third grade.
• The National Council of Teachers of Mathematics
(NCTM) assert that Pre-K-2 students should
develop fluency with addition and subtraction
facts and that 3-5 students should be fluent with
multiplication and division facts as well (2000).
What is really happening…
• Unfortunately, research shows that many
students have trouble learning their math facts
(Woodward, 2006).
• According to the National Assessment of
Educational Progress (NAEP) basic math fact
performance declined in the 1990’s (Scholastic,
2008).
The Great Debate Among
Educators
rote
memorization
(drill and
practice)
VERSUS
explicit
strategy
instruction
The Debate…
• There is a debate about whether math facts should be
taught through rote memorization (drill and practice) or
through explicit strategy instruction (Woodward, 2006).
• Most educators (70%) believe that drill and practice or rote
memorization help students successfully learn their math
facts (Caron, 2007). However, research shows that rote
rehearsal alone does not produce automaticity of math
facts (Caron, 2007).
• Wakefield (1997) points out that requiring students to learn
math facts through rote memorization is counterproductive.
Students need to be actively thinking about what they are
learning in order to apply it to more complex math tasks.
• Therefore, the next part of the presentation is dedicated
to solutions and interventions that are proven to work.
Research Based Solutions and
Interventions
• Dr. Steve Tipps, a retired mathematics professor from NC
State University facilitated a math facts workshop at
Upward this year. He stated that “In order for student’s
to successfully master their math facts, they must be
exposed to activities with math embedded within them.”
• According to Kennedy, Tipps, and Johnson (2008), math
fact instruction should go through four phases: Conceptual,
Strategic, Mastery, and Maintenance. Let’s go through the
phases and the specific teaching strategies in each phase. I
implemented these strategies with my students during the
project.
Conceptual Stage of Math
Fact Instruction
• This stage involves representing problems in
story, physically, and with pictures and symbols
• One of the best ways to help students
conceptually understand number systems is
through chip trading. Chip trading scaffolds
math concepts for children for all ages. (Tipps,
2009).
• The next slide shows a sample game board for
chip trading and the instructions
Conceptual Stage: Chip
Trading Activity
To start, you will need some red, green, blue, and yellow
colored paper clips or chips. First, students roll a number
cube and place the number of chips, color tiles, links, or
paper clips in the first column. Students start on yellow
and move all the way to red throughout the activity. A
trading rule is established for each game. It is easiest to
start with 4. For example, when a student gets 4 yellows
they can trade them in for one blue and when they get 4
blues they can trade them in for one green and when they
get 4 greens they trade them in for one red. The object is
to get all the way to red. You can change the trading rule
as students progress and eventually get to the trading rule
of 10, which goes along with our number system. Higher
level students can use two dice and practice tax rounds
when they have to trade backwards. As students practice
more, they start to instantly put one blue down and one
yellow when they roll a five without even having to trade.
Chip Trading Board Using Base
Ten Blocks
This board can be used to help
students see how to trade ones for
tens and tens for hundreds. You
can also make boards which
progress from .01 to 10 and many
more.
•
•
•
More conceptual stage
activities
Before moving on to the strategic phase of math fact instruction
students must understand that multiplication can be represented
in 3 different situations and division in 2. It is a good idea to have
students write a story, make a model, or draw a picture of the
different situations. Experiences with these situations “extend
children’s experiences with counting” (Kennedy, Tipps, & Johnson,
2008, p. 212).
Multiplication situations:
- Equal sets, repeated addition
- Arrays, geometric interpretation
- Cartesian product
Division situations:
- Repeated Subtraction or repeated measurement
- Partitioning or sharing
Sample Sheet for Multiplication and Division Situations
Multiplication - equal sets, repeated addition
Write a story, make a model, or draw a picture
Kobe scored 6 points, Juanita scored 6 points, Jeremy
scored 6 points for the Raiders. How many points did
they score?
Multiplication – arrays, geometric interpretation
Write a story, make a model, or draw a picture
Mr. Moore is setting up the cafeteria for a meeting.
He wants 10 rows with 9 chairs in each row. How many
chairs does he need for the meeting?
Multiplication – combinations
Write a story, make a model, or draw a picture
At the carnival, they had strawberry, vanilla, and
chocolate ice cream and three containers: cups, cones,
or waffle cones. How many combinations of one scoop
of ice cream in one container were possible?
Division – Repeated subtraction
Write a story, make a model, or draw a picture
Scott was packing apples for in each box. How many
boxes did he need for 21 apples? Did he have any
extra apples?
Division – Sharing
Scott had 19 apples to pack into 6 boxes. How many
apples were in each box? Did he have any extra
apples?
Write a story, make a model, or draw a picture
Picture Books and Music
•
•
•
Picture books and songs are great tools to use during the conceptual
stage of math fact instruction (Tipps, 2009). Below are some
suggested books to use:
-The Doorbell Rang by Pat Hutchins models sharing
- The Sundae Scoop by Stuart Murphy models combinations
-Spaghetti and Meatballs for All by Marilyn Burns models arrays and
geometric interpretation
- Anno's Mysterious Multiplying Jar by Masaichiro Anno
-A Remainder of One by Elinor Pinczes
-One Hundred Hungry Ants by Elinor Pinczes
Click here to visit a website with more great picture books organized
by mathematical concepts.
Clicking here will take you to a site that has a variety of
mathematical songs.
Using Manipulatives
• During the conceptual stage, students should be using many
manipulatives such as base ten blocks and linking cubes.
• The website below uses virtual manipulatives so students
can explore operations. See the following activities: Base
Blocks, Number Line Bounce, Number Line Bars, Abacus, and
Chip Abacus
http://nlvm.usu.edu/EN/NAV/topic_t_1.html
• This website also allows students to manipulative five and
ten frames:
http://illuminations.nctm.org/ActivitySearch.aspx
Strategic Phase
• Once students have a good conceptual
understanding of numbers, it is time to move on to
the strategic phase of instruction.
• This phase involves students understanding and
learning facts using rules, properties, and laws of
number operations.
• During this stage, students develop understanding
rather than rote memorization.
• The following slides include specific strategies
from this stage that I implemented in my
classroom.
Strategic Phase Activities
• Use skip counting as a foundation for this phase. Have
students skip count with a hundreds chart. A volunteer did
this with struggling students in my class. Students shaded
in the hundreds chart with dry erase markers as they
practiced. You will find an interactive hundreds chart by
clicking here. Students can easily learn their 2’s, 5’s, and
10’s with skip counting (Kennedy, Tipps, & Johnson, 2008).
Strategic Activities Continued
• Teach students multiplication rules:
– The Commutative Property of Multiplication – Model with
different colored linking cubes and arrays.
=
– Associative Property of Multiplication – Practice different
groupings of numbers
– Identity Property: 6 x 1= 6
– Multiplying by 0: 10 x 0 = 0
– Multiplying by 2 – Is related to double facts in addition. These
facts can be illustrated with linking cubes. The example below
shows that 3 + 3 is 6 and 2 groups of 3 is 6.
Strategic Activities Continued
– Squared facts – 4 x 4 Students can practice these by making
geometric arrays, which make squares.
– Near squares or square neighbors – Once students have
mastered the square facts, they can easily add or subtract one
to memorize near squares.
– Teach patterns such as doubles, doubles plus one, times five,
and halving (Woodward, 2006).
Strategic Activities Continued
•
•
•
Multiplying with 9 – Students can multiply by 10 and subtract 9. Also many
students learn by using their hands. See the following activity for teaching the
9’s using your hands.
Through exploration students can also see the patterns in the 9’s times table.
My students were fascinated that the multiples go from 0 to 9 in the tens place
and 9 to 0 in the ones place. See example below:
0
9
1
8
2
7
3
6
4
5
5
4
6
3
7
2
8
1
9
0
Gravemeijer & van Galen (as cited in Van de Walle, 2007) encourage using a
guide intervention approach where math facts are connected to the prior
knowledge students have about number relationships. For example, students
make up their own rules about certain facts, which make sense to them.
Strategic Phase
•
Once students explore and practice the multiplication rules, they will find
that there aren’t that many math facts left to memorize. My students
were given a multiplication chart and they shaded in which facts they
already knew. Students were amazed that they didn’t have that many to
work on and they didn’t feel overwhelmed. Below is a student example:
Strategically Moving to
Division
• Once students have explored the multiplication
facts, division facts should be practiced as the
inverse. Students should do and undo
multiplication facts to explore. They can also use
a division chart to do this.
• A focus on fact families is a great way to explore
as well (Woodward, 2006). My students explored
fact families by making triangle flash cards during
the strategic phase. The next slide has an
example of the triangle card. Students also made
handheld versions to practice with.
Fact Family Practice with
Triangle Flash Cards
Click here for the
template.
Moving on to Mastery
• The next phase in math fact instruction is the
mastery phase.
• During this phase students work on building
accuracy with reasonable speed. They are ready
for this stage with they know enough facts to
feel successful.
• This stage uses flash cards, puzzles, and games
(electronic and non-electronic)
• Students do a lot of self-assessments and keep
records of their accuracy and speed.
Flash Cards
•
•
Kennedy, Tipps, and Johnson (2008) recommend using triangle
flash cards because they help reinforce fact families.
Students can work individually or in groups to build speed.
Burns (2005) advocates using incremental research, which
means that students practice with flash cards orally with
known and unknown facts. The goal is to make the known facts
greater over time. Students can code the flash cards with
different colors as they become automatic. The greens ones
they know instantly, the yellow they hesitate slightly with, and
the reds require more time. As students begin to master the
facts they can change the colors on the flash cards. Students
can use stickers to keep track. My students really enjoyed this
and it made them feel successful to be able to go from red to
yellow and then to green.
Puzzles
• Below is an example of a puzzle that helps
students practice their facts. Students have to
complete the table. The right one is more
challenging. You can also encourage students to
make them for their friends.
x
0
2
3
5
0
1
x
4
6
28
8
4
2
8
16
3
18
1
5
2
25
9
63
9
Another Puzzle
•
Tipps (2009) provided this puzzle to teachers during our math
facts workshop. It is called a multiplication/division hunt. Students
have to search for 3 numbers in a row forward, back, down or
diagonally that make multiplication or division sentences. They
have to circle the three numbers and write the number sentences
they find.
56
42
6
18
48
16
12
12
20
4
7
3
21
3
4
3
5
60
42
6
8
2
16
4
4
5
20
2
12
24
35
12
1
12
25
3
19
3
8
24
4
6
24
2
12
23
4
6
14
3
6
18
6
3
2
12
11
18
12
36
3
12
2
3
1
30
7
6
42
9
4
6
9
5
2
10
1
13
27
4
12
Games
•
•
•
•
Using games and secret codes can help students learn math facts
(Mastering the math facts, 2001).
Use dice, cards, and board games because these include active
thinking along with math fact practice (Wakefield, 1997).
Card games and dominoes can be used to practice math facts.
Click here for directions on how to play War.
Tipps (2009) also recommends battle – you can play addition,
subtraction, or multiplication versions.
Interactive Games
•
Every week in the computer lab, my students practiced their math
facts. They had a lot of fun with the games and they seemed to
make a difference. Below are a list of some of the websites my
students visited:
http://www.primarygames.com/flashcards/twomin.htm (They enjoyed
tracking their progress and competing against me)
http://www.funbrain.com/math/index.html (Math facts and baseball
game)
http://www.multiplication.com/interactive_games.htm (All types of
interactive games)
http://www.aplusmath.com/Games/index.html (BINGO and hidden
puzzles)
http://www.gamequarium.com/mixedpractice.html (All types of
interactive games)
Self-Assessment
•
Caron (2007) developed the following assessment to help his students
develop automaticity with their math facts. The test was not a
competition and students had no excuse for leaving answers blank because
they were given to them.
Self-Assessment Implementation
•
I used Caron’s assessment from the last slide to help students practice
their facts. Students kept track of their progress using stop watches.
Students loved competing against themselves. Before timing themselves
students ranked the math facts by degree of difficultly using the colors
green, yellow, and red. Below is a student example:
Keeping Records
•
Students kept track of their progress using the following chart.
All of my students were successful with this method except two
and those students went back and did more strategic phase
activities before moving on to the mastery phase ones.
Maintaining
• The last phase of math fact instruction is the
maintenance phase. During this phase, students
use facts in real life and games.
• They identify their strengths and weaknesses and
continue to work on them.
• I’m sure all teachers wish their students were
here when they arrived, but unfortunately it
takes a lot of work to get here.
Implementation
• I implemented all of the strategies in my classroom starting
with the conceptual stage ones. My students absolutely loved
to play the chip trading games. They gained a lot of
knowledge about base number systems from the experiences
as well. I could instantly see who had trouble with math
concepts during the activities.
• Having students illustrate different multiplication and
division situations seemed to build a deeper understanding
for many students.
• Learning all of the rules and properties made the facts seem
not so overwhelming to students because they began to see
that they really knew more facts than they thought.
Implementation Continued
• The use of triangle flash cards helped my class
become more familiar with fact families.
• My students really enjoyed the multiplication and
division puzzles. They liked making them for their
peers to try.
• The self-assessments were effective, but they
did require a lot of time. A volunteer helped keep
track of times. After students got quicker, I
took the answers off and you could tell that they
really knew their facts because their times
remained the same without the answers.
Reflection
•
•
Overall, I feel like the strategies implemented with my class were
very effective. They required a lot of time and dedication though.
It is hard to make students magically know their facts. It is not a
quick fix! In the long run, if teachers help their students gain a
conceptual understanding, I think the math facts will stick from year
to year. Instead of giving 50 timed tests, teachers need to start
doing other activities that encourage understanding and not just
memorization.
I really enjoyed this quote from Dr. Tipps (2009): “You can’t
practice what you don’t know.” This is so true yet many teachers
make students take multiple timed tests even though they fail them
over and over again. Before this project I did that, but not anymore!
Conclusion
•
•
Mastery of math facts is an important skill that affects many concepts in
math. Caron (2007) reminds us that “computation and problem solving
virtually demands that students know multiplication” (p. 278). Even though
there is a lot of debate about whether drill and practice should be used
when teaching math facts, there is a general consensus that it must be
done in combination with strategic teaching of math understanding in order
for it to have a positive effect (Caron, 2008). Teachers would greatly
benefit if students successfully mastered their math facts. They could
focus on math concepts without worrying about students lacking math fact
knowledge. According to research (Scholastic, 2008), end-of-grade test
scores would possibly increase as well.
Teachers should keep the following conclusions in mind:
-Build student confidence
-Beware of group tests, which create pressure and stress. Instead focus
on individual improvement and progress.
-Use a variety of strategies
-Focus on strategic instruction instead of rote memorization and drills!
References
Burns, M. (2005, August). Using incremental rehearsal to increase fluency of
single-digit multiplication facts with children identified as learning disabled
in mathematics computation. Education and Treatment of Children, 28,
237-249.
Caron, Thomas A. (2007, July). Learning multiplication the easy way. Clearing
House, 80(6), 278-282.
Kennedy, L., Tipps, S., & Johnson, Art (2008). Guiding children’s learning of
mathematics (11th ed.). Belmont, CA: Thomson Wadsworth.
Legg, J. (2009). Personal communication, February 13, 2009.
Mastering the math facts. (2001, April). Instructor, Retrieved January 30,
2009 from Academic Search Premier database
References
North Carolina Department of Public Instruction (2008). Fifth grade North
Carolina Mathematics Standard Course of Study. Raleigh, NC: Author.
Retrieved January 28, 2009 from
http://www.ncpublicschools.org/curriculum/mathematics/.
National Council of Teachers of Mathematics. (2000). Principles and standards
for school mathematics. Reston, VA: Author. Retrieved February 27,
2009 from http://standards.nctm.org/document/index.htm.
Scholastic (2008). Math fluency. Retrieved November 28, 2008 from
http://www2.scholastic.com/browse/article.jsp?id=324
Tipps, S. (2009). Personal communication, February 16, 2009.
References
Van de Walle, J. (2007). Elementary and middle school mathematics: Teaching
developmentally (6th ed.). Boston, MA: Allyn and Bacon.
Wakefield, A. (1997, November). Supporting math thinking. Phi Delta Kappan,
79(3), 233. Retrieved February 1, 2009 from Academic Search Premier
database.
Woodward, J (2006, Fall). Developing automaticity in multiplication facts:
Integrating strategy instruction with timed practice drills. Learning
Disability Quarterly, 29, 269-289