Math Fact Instruction: Deciphering Fact from Fiction

1
Math Fact Instruction Deciphering Fact from
Fiction

• April Summey

2
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 dont 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 arent
  mastering their basic math facts. This
  presentation will focus mainly on multiplication
  and division facts.

3
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.

4
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).

5
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).
  

6
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 1990s (Scholastic,
  2008).

7
The Great Debate Among Educators
VERSUS
8
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.

9
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 students 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. Lets go through the phases and the
  specific teaching strategies in each phase. I
  implemented these strategies with my students
  during the project.

10
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

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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.
12
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.
13
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
  childrens 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

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Sample Sheet for Multiplication and Division
Situations
Multiplication - equal sets, repeated addition Kobe scored 6 points, Juanita scored 6 points, Jeremy scored 6 points for the Raiders. How many points did they score? Write a story, make a model, or draw a picture
Multiplication arrays, geometric interpretation 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? Write a story, make a model, or draw a picture
Multiplication combinations 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? Write a story, make a model, or draw a picture
Division Repeated subtraction Scott was packing apples for in each box. How many boxes did he need for 21 apples? Did he have any extra apples? Write a story, make a model, or draw a picture
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
15
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.

16
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

17
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.

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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
  2s, 5s, and 10s with skip counting (Kennedy,
  Tipps, Johnson, 2008).

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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.

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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).

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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 9s using your hands.
• Through exploration students can also see the
  patterns in the 9s 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
• 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.

0 9
1 8
2 7
3 6
4 5
5 4
6 3
7 2
8 1
9 0
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Strategic Phase

• Once students explore and practice the
  multiplication rules, they will find that there
  arent 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 didnt have that
  many to work on and they didnt feel overwhelmed.
  Below is a student example

23
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.

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Fact Family Practice with Triangle Flash Cards
Click here for the template.
25
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.

26
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.

27
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 2 3 5 0 1
0
4
2
1
5
x 6 2
4 28 8 8
16
3 18
25
9 63 9
28
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
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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.

30
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.ht
  m (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)

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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.

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Self-Assessment Implementation

• I used Carons 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

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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.

34
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.
• Im sure all teachers wish their students were
  here when they arrived, but unfortunately it
  takes a lot of work to get here.

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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.

36
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.

37
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 cant practice what you dont 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!

38
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!

39
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 childrens 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
•  

40
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/mathemat
  ics/.
•  
• 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/d
  ocument/index.htm.
• Scholastic (2008). Math fluency. Retrieved
  November 28, 2008 from http//www2.scholastic.com/
  browse/article.jsp?id324
• Tipps, S. (2009). Personal communication,
  February 16, 2009.
•  

41
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
•