Math Fact Instruction: Math Fact Instruction: Deciphering Fact from Deciphering Fact from

FictionFiction

Math Fact Instruction: Math Fact Instruction: Deciphering Fact from Deciphering Fact from

FictionFictionApril SummeyApril 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

VERSUS

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 SituationsMultiplication - 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

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 Abacushttp://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:

• 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

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 2 3 5 0 1

0

4

2

1

5

x 6 2

4 28 8 8

16

3 18

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!

ReferencesBurns, 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

 

ReferencesNorth 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.

 

ReferencesVan 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