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Implementing an Effective Mathematics Fact Fluency Practice Activity

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Procedural Fluency in Mathematics

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Procedural Fluency in Mathematics - National Council of Teachers of Mathematics

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Procedural fluency is a critical component mathematical proficiency and is more than memorizing facts and procedures.

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Interesting perspective on what "fluency" means

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By NCTM President Linda M. Gojak
NCTM Summing Up, November 1, 2012

As mathematics educators at all levels consider effective implementation and instruction related to state or Common Core standards, a frequently asked question is, "What does it mean to be fluent in mathematics?" The answer, more often than not, is, "Fast and accurate." Building fluency should involve more than speed and accuracy. It must reach beyond procedures and computation.

Principles and Standards for School Mathematics states,"Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships" (p. 152). What a wonderful description of fluency! It reminds us that a student cannot be fluent without conceptual understanding and flexible thinking.

Focusing on efficiency rather than speed means valuing students' ability to use strategic thinking to carry out a computation without being hindered by many unnecessary or confusing steps in the solution process. Accuracy extends beyond just getting the correct answer. It involves considering the meaning of an operation, recording work carefully, and asking oneself whether the solution is reasonable.

Fluency encompasses more than memorizing facts and procedures. In fact, I believe memorization is one of the least effective ways to reach fluency. Anyone who has spent time teaching in the elementary grades realizes how many students are unsuccessful at rote memorization and how often they revert to counting on their fingers. We would agree that third or fourth graders who are counting on their fingers certainly have not reached a level of fluency, even though they may do it pretty quickly and accurately!

How do we help students progress from the early stages of counting to mathematical fluency? Let me give you a personal example. At the beginning of the school year, I gave a class of third-grade students a sheet with 10 addition facts. Under each fact was the word "explain," followed by a line. I asked one of the students the sum of the first fact, 8 + 9, and she immediately began to count on her fingers-certainly not the action of a student who is fluent with addition facts. Before she reached the sum I asked her, "What do you know that would help you find the sum of 8 and 9?" She thought for a brief time and replied, "Oh, it's 17." When I asked her how she had gotten that without counting, she looked at me and said, "I just took 1 off the 8 and gave it to the 9. That made it 7 + 10. That's easy-it's 17."

One might argue that child was not fluent. I believe, however, that she demonstrated fluency and more. She was able to use her understanding of place value, addition, and the associative property to arrive at a correct response. She was efficient, accurate, and flexible in her thinking-all in a matter of seconds. What made the difference between her fumbling first attempt and her successful second one? It was being provided with the chance to stop and think about what she already knew and apply that understanding to 8 + 9.

Do we give students the opportunity to think about what they know and understand and use it in ways that make sense to them? Do we model questions that students should be asking themselves as they strive to reach fluency in mathematics? As the student completed that assignment, she didn't need much more prompting. She continued to work on the rest of the facts efficiently and flexibly. She no longer needed to count on her fingers to complete the assignment.

It is interesting to note that fluency isn't mentioned in the high school Common Core Standards. The standards for grades K-8 refer to fluency in relation to mastery of basic facts and computational skills. As we think about fluency, we should realize that it is more than procedural. Are there mathematical topics in which we want students' thinking to be flexible, efficient, and accurate beyond computation and procedures? Can a student reach fluency in areas of geometric thinking, algebraic thinking, statistical reasoning, or measurement? What does geometric fluency look like? What are the characteristics of a student who is fluent in algebra? What areas of fluency in the K-12 curriculum reach beyond procedures and calculations but are not mentioned in the standards?

Our students enter school with the misconception that the goal in math is to do it fast and get it right. Do we promote that thinking in our teaching without realizing it? Do we praise students who get the right answer quickly? Do we become impatient with students who need a little more time to think? As we strive for a balance between conceptual understanding and procedural skill with mathematical practices, we must remember that there is a very strong link between the two. Our planning, our instruction, and our assessments must build on and value that connection. Fluency entails so much more than being fast and accurate!

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Fluency: Simply Fast and Accurate? I Think Not! - National Council of Teachers of Mathematics

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Linda M. Gojak As mathematics educators at all levels consider effective implementation and instruction related to state or Common Core standards, a frequently asked question is, "What does it mean to be fluent in mathematics?" The answer, more often than not, is, "Fast and accurate." Building fluency should involve more than speed and accuracy.

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The standards, whether the CCSSM or the TEKS in Texas, definitively state that students should develop fluency with calculations, and that means automaticity with basic facts. Knowing math facts is similar to knowing sight words-it frees up the mind to solve real math problems. If a child has to struggle to solve 8 + 3, they have no mental energy (or desire) left to grapple with the types of problems that will increase their capacity as a mathematician. The difference is the approach we now take to teaching basic facts. Traditionally, the emphasis has been on memorization and speed, much to the detriment of countless students. For a great read on the problems with memorization, check out Jo Boaler's Fluency Without Fear.
So the shift has been away from rote memorization of math facts and toward a strategy-based approach for learning math facts. There's a big difference between memorizing and understanding. Sure, we want kiddos to have automaticity with their facts, but we want that fluency to be rooted in number sense-an understanding of how numbers are related. A great resource for strategy-based fact instruction is Mastering the Basic Math Facts in Addition and Subtraction: Strategies, Activities, and Interventions to Move Students Beyond Memorization. There's one for multiplication and division facts, too.

To develop automaticity, students also need to engage in meaningful practice. Here are some suggestions for practice games and activities:

1. Provide concrete or pictorial support. The ability to create mental images of numbers and facts helps students make sense of numbers. A student who "guesses" that 7 x 9 might be a number in the 30s clearly does not have a mental image of 7 x 9. Tasks that either use pictures to represent facts or have students draw representations for facts help them develop that ability to form mental images.

2. Focus on strategies, not facts. When learning addition facts, strategies like Make a 10 and Using Doubles are very powerful. With multiplication, students learn that facts are related, for example by doubling. A game focusing on the 2s, 4s, and 8s highlights that doubling relationship.

3. Spotlight a specific number. In Kinder and 1st, students need lots of practice composing and decomposing the numbers to ten. Because of it's importance in our number system, special emphasis should be given to making ten. Use a fun skip-counting game to practice all the multiples of a given factor.

4. Add a twist of strategy. Let's face it, who doesn't like a game of tic-tac-toe? The strategy that's involved makes it almost addictive. Anytime you can incorporate a little strategy into a fact practice game, you're golden. And speaking of tic-tac-toe, I've created a little freebie for Making 10 (addition) and Making 24 (multiplication). Click here to grab yours!

I'd love for you to share your comments about how you work with your kiddos to develop automaticity with facts!

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Developing Automaticity with Basic Math Facts - Math Coach's Corner

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The standards, whether the CCSSM or the TEKS in Texas, definitively state that students should develop fluency with calculations, and that means automaticity with basic facts.

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Should I focus my research on building fact fluency while working with a smaller group of students instead of the entire class?

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How to Use Flashcards with Constant Time Delay: Letters, Numbers, Sight Words, Math Facts, Vocabulary

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Description. Flashcard review with constant time delay is a motivating academic intervention. It helps students to memorize foundation academic content such as letter and number names, sight words, spelling words, and vocabulary definitions-while largely eliminating errors (Hua et al., 2013; Schuster, Stevens, & Doak, 1990; Stevens & Schuster, 1988).

Here is the general sequence for flashcards with constant time delay:

The teacher prompts the student to identify an academic item on a flashcard.

If the student is unable to respond correctly within a fixed span of time, the teacher always provides the correct answer.

The strategy combines regular performance feedback with consistent praise for correct responses.

While constant time-delay procedures across academic areas are highly similar, there are notable differences in how the cards are presented and feedback given. Instructions are given below for using constant time delay to teach letter and number identification, sight words, math facts, spelling words, and vocabulary terms and definitions.
Constant Time Delay: Letters/Numbers/Sight Words
Preparation. The set of upper- or lower-case letters, numbers, or sight words to be used in the session are written on the front of flashcards, one item per card.

Directions. The teacher opens each session with these instructions (adapted from Schuster et al., 1990): I am going to show you cards with letters/numbers/words. If you know the letter/number/word, say it. If you don't know the letter/number/word, say nothing or say 'I don't know'.

Phase 1: Teaching Sequence: No Delay. When first introducing the flashcard items, the teacher presents each item, provides the answer with a 0-second delay (i.e., no delay), and immediately prompts the student to provide the answer.

The teacher shows each card and says, This letter/number/word is [insert letter/number/word]. What is this letter/number/word?

If the student answers correctly, the teacher states, Right. The letter/number/word is [insert letter/number/word]. Good job! or similar feedback and praise. The teacher then advances to the next card.

If the student fails to respond or responds incorrectly, the teacher repeats the initial prompt: This letter/number/word is [insert letter/number/word]. What is this letter/number/word?

Phase 2: Practice Sequence: 5-Second Delay. Once the student has been introduced to all flashcards, the teacher institutes the 5-second-delay phase, which continues for the duration of the session.

The teacher shows each card and says, What is this letter/number/word?

If the student initiates a response within 5 seconds and answers correctly, the teacher states, Right. The letter/number/word is [insert letter/number/word]. Good job! or similar feedback and praise. The teacher then advances to the next card.

If the student hesitates longer than 5 seconds before responding or answers incorrectly, the teacher says, This letter/number/word is [insert letter/number/word]. What is this letter/number/word?
Constant Time Delay: Math Facts
Preparation. A set of math facts (without answers) to be used in the session are written on the front of flashcards, one fact per card. The answer to the math fact appears on the back of each card.

Directions. The teacher opens each session with these instructions (adapted from Schuster et al., 1990): I am going to show you cards with math facts. If you know the answer to the math fact, say it. If you don't know the answer to the math fact, say nothing or say 'I don't know'.

Phase 1: Teaching Sequence: No Delay. When first introducing the flashcard items, the teacher shows each card, reads the math-fact appearing on the front, then reads the math fact plus the answer appearing on the back of the card, and finally has the student read the math fact and give the answer: This fact is [insert math fact]. [Math fact] is [insert answer]. What is [insert math fact]?; e.g., This fact is 3 times 8. Three times 8 equals 24. What is 3 times 8?

If the student answers correctly, the teacher states, Right. [Insert math fact] equals [insert answer]. Good job! The teacher then advances to the next card.

If the student fails to respond or responds incorrectly, the teacher repeats the initial prompt: This fact is [insert math fact]. [Insert math fact] is [insert answer]. What is [insert math fact]?

Phase 2: Practice Sequence: 5-Second Delay. Once the student has been introduced to all flashcards, the teacher institutes the 5-second-delay phase, which continues for the duration of the session.

The teacher shows each math-fact card and says, What is [insert math fact]?

If the student initiates a response within 5 seconds and answers correctly, the teacher states, Right. [Insert math fact] equals [insert answer]. Good job! The teacher then advances to the next card.

If the student hesitates longer than 5 seconds before responding or answers incorrectly, the teacher says, [Math fact] is [insert answer]. What is [insert math fact]?
Constant Time Delay: Spelling Words
Preparation. The set of spelling words to be used in the session are written on the front of flashcards, one item per card. The student also has a lined worksheet with room sufficient to write all spelling words reviewed in the session.

Directions. The teacher opens each session with these instructions (adapted from Schuster et al., 1990): I am going to give you words to spell. If you know how to spell the word, write it on your sheet. If you don't know how to spell the word, just say 'I don't know' or wait for me to show you the word.

Phase 1: Teaching Sequence: No Delay. When first introducing the flashcard items, the teacher prompts the student to spell the word, shows the correctly spelled word on the flashcard with a 0-second delay (i.e., no delay), and immediately prompts the student to spell the word.

The teacher says, Spell [insert spelling word] and immediately shows the student the flashcard with the spelling word, which the student copies onto the worksheet.

If the student spells the word correctly, the teacher states, Right. You spelled [insert spelling word] correctly. Good job! or similar feedback and praise. The teacher then advances to the next card.

If the student fails to respond or spells the word incorrectly, the teacher repeats the initial prompt: Spell [insert spelling word] and immediately shows the student the flashcard with the spelling word, which the student copies onto the worksheet.

Phase 2: Practice Sequence: 5-Second Delay. Once the student has been introduced to all flashcards, the teacher institutes the 5-second-delay phase, which continues for the duration of the session.

The teacher looks at the spelling word on each card but keeps that word hidden from the student. The teacher says, Spell [insert spelling word].

If the student initiates a response within 5 seconds and spells the word correctly, the teacher states, Right. You spelled [insert spelling word] correctly. Good job! or similar feedback and praise. The teacher then advances to the next card.

If the student fails to start writing the spelling word within 5 seconds or spells the word incorrectly, the teacher says: Spell [insert spelling word], and immediately shows the student the flashcard with the spelling word, which the student copies onto the worksheet.
Constant Time Delay: Vocabulary Terms & Definitions
Preparation. A set of vocabulary terms to be used in the session are written on flashcards, one term per card. For each card, the vocabulary term appears on the front; e.g., Chlorophyll. Both term and definition appear on the back in this or similar format: [Term] is [definition]. E.g., "Chlorophyll is the green pigment in plants that converts sunlight to energy for photosynthesis."

Directions. The teacher opens each session with these instructions (adapted from Hua et al., 2013): I am going to show you cards with vocabulary terms and definitions. If you know the meaning of the term, tell me it If you don't know the meaning of the term, say nothing or say 'I don't know'.

Phase 1: Teaching Sequence: No Delay. When introducing the flashcard items, the teacher initiates a practice sequence in which the student first pronounces each term, then reads that term's definition, then states both term and definition.

1. Term Pronunciation: Practice. For each card, the teacher shows the vocabulary term on the front of the card and says, This vocabulary term is [insert term]. What term?
If the student reads and pronounces the term correctly, the teacher states, The term is [insert term]. Good job! or similar feedback and praise. If the student fails to respond or responds incorrectly, the teacher repeats the prompt: This vocabulary term is [insert vocabulary term]. What term?

2. Definition: Practice. The teacher turns the card over and reads aloud the full term and definition written on the back. The teacher next hands the student the card and directs the student to read the term-and-definition statement aloud. The teacher then takes the flashcard back.

3. Term and Definition: Practice. Before moving to the next card, the teacher again holds the current card up so that the student can see only the vocabulary term on the front. The teacher says, What is this term. What does it mean?
If the student begins to answer within 5 seconds and correctly names the word and provides an acceptable definition, the teacher says, The term [insert term] means [insert definition]. Good job! The teacher then advances to the next card.

If the student hesitates in answering for longer than 5 seconds or answers incorrectly, the teacher repeats the teaching sequence above for this vocabulary term: (1) Term Pronunciation: Practice; (2) Definition: Practice; and (3) Term and Definition: Practice.

Phase 2: Practice Sequence: 5-Second Delay. Once the student has been introduced to terms and definitions on all flashcards, the teacher institutes the 5-second-delay phase, which continues for the duration of the session.

The teacher shows the term on the front of each vocabulary card and says, What is this term. What does it mean?

If the student initiates a response within 5 seconds and answers correctly, the teacher states, Right. The term [insert term] means [insert definition]. Good job! The teacher then advances to the next card.

If the student hesitates longer than 5 seconds before responding or answers incorrectly, the teacher turns the card over and reads aloud the full term and definition written on the back. The teacher then shows only the term on the front of that vocabulary card and says, What is this term. What does it mean?

Tips for Use. Here are a couple of suggestions to enhance your use of this intervention:

Alter the length of the time delay if needed. The procedures presented here for using flashcards with constant time delay include a 5-second delay during the teaching sequence. However, teachers are free to exercise their own judgment to shorten (e.g., 3 seconds) or lengthen (e.g., 10 seconds) the time delay to match the abilities and age level of the student. While that time delay can be changed, however, it is important that, once set, it is applied consistently. Also, when documenting this academic strategy in a student intervention plan, the teacher should record the length in seconds of the time-delay value selected.

Monitor the student's responses. There are four possible types of significant student responses using constant time delay. The student can answer (1) correctly or (2) incorrectly during the several-second span BEFORE the teacher gives the cue (correct answer). Or the student can answer (3) correctly or (4) incorrectly AFTER being given the cue (correct answer).All of these responses are meaningful and all should be recorded. For example, it is noteworthy when a student who initially answers correctlyonly after being given the cue begins to respond correctly immediately and without prompting. See the attached form, Flashcards with Constant Time Delay: Recording Sheet, as an example of a convenient instrument for tracking student responses for any time-delay intervention.

References

Hua, Y., Woods-Groves, S., Kaldenberg, E. R., & Scheidecker, B. J. (2013). Effects of vocabulary instruction using constant time delay on expository reading of young adults with intellectual disability. Focus on Autism and Other Disabilities, 28(2), 89-100.

Schuster, J. W., Stevens, K. B., & Doak, P. K. (1990). Using constant time delay to teach word definitions. Journal of Special Education, 24, 306-318.

Stevens, K. B., & Schuster, J. W. (1988). Time delay: Systematic instruction for academic tasks. Remedial and Special Education, 9(5), 16-21.

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How to Use Flashcards with Constant Time Delay: Letters, Numbers, Sight Words, Math Facts, Vocabulary

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Description. Flashcard review with constant time delay is a motivating academic intervention. It helps students to memorize foundation academic content such as letter and number names, sight words, spelling words, and vocabulary definitions-while largely eliminating errors (Hua et al., 2013; Schuster, Stevens, & Doak, 1990; Stevens & Schuster, 1988).

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Assessing Basic Facts Fluency

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Math Computation: Promote Mastery of Math Facts Through Incremental Rehearsal

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Incremental rehearsal builds student fluency in basic math facts ('arithmetic combinations') by pairing unknown computation items with a steadily increasing collection of known items. This intervention makes use of concentrated practice to promote fluency and guarantees that the student will experience a high rate of success.

Materials

Steps to Implementing This Intervention

In preparation for this intervention:

1. The tutor first writes down on an index card in ink each math fact that a student is expected to master-but without the answer. NOTE: Educators can use the A-Plus Math Flashcard Creator, a free on-line application, to make and print flashcards in addition, subtraction, multiplication, and division. The web address for the flashcard creator is: www.aplusmath.com/Flashcards/Flashcard_Creator.html

2. The tutor reviews the collection of math-fact cards with the student. Any of the math facts that the student can orally answer correctly within two seconds are considered to be known problems and are separated into one pile. Math facts that the student cannot yet answer correctly within two seconds are considered 'unknown' and collected in a second pile -- the 'unknown facts' deck.

3. The tutor next randomly selects 9 cards from the pile of known math facts and sets this subset of cards aside as the 'known facts' deck. The rest of the pile of cards containing known math facts is put away ('discard deck'), not to be used further in this intervention.

During each day of the intervention:

The tutor follows an incremental-rehearsal sequence each day when working with the student:

First, the tutor takes a single card from the 'unknown facts' deck. The tutor reads the math fact on the card aloud, provides the answer, and prompts the student to read off and answer the same unknown problem.

Next the tutor takes one math fact from the 'known facts' deck and pairs it with the unknown problem. When shown the two problems in sequence, the student is asked during the presentation of each math fact to read off the problem and answer it. The student is judged to be successful on a problem if he or she orally provides the correct answer to that problem within 2 seconds. If the student commits an error on any card or hesitates for longer than two seconds, the tutor reads the math fact on the card aloud, gives the answer, then prompts the student to read off the same unknown problem and provide the answer. This review sequence continues until the student answers all cards within two seconds without errors.

The tutor then repeats the sequence--taking yet another problem from the 'known facts' deck to add to the expanding collection of math facts being reviewed ('review deck'). Each time, the tutor prompts the student to read off and answer the whole series of math facts in the review deck, beginning with the unknown fact and then moving through the growing series of known facts that follow it.

When the review deck has expanded to include one 'unknown' math fact followed by nine 'known' math facts (a ratio of 90 percent 'known' material to 10 percent 'unknown' material), the last 'known' math fact that was added to the student's review deck is discarded (put away with the 'discard deck'). The previously 'unknown' math fact that the student has just successfully practiced in multiple trials is now treated as a 'known' math fact and is included as the first item in the nine-card 'known facts' deck for future drills.

The student is then presented with a new math fact to answer, taken from the 'unknown facts' deck. With each new 'unknown' math fact, the review sequence is again repeated as described above until the 'unknown' math fact is grouped incrementally with nine math facts from the 'known facts' deck-and on and on.

Daily review sessions are discontinued either when time runs out or when the student answers an 'unknown' math fact incorrectly three times.

References

Burns, M. K. (2005). 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.

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Math Computation: Promote Mastery of Math Facts Through Incremental Rehearsal

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Incremental rehearsal builds student fluency in basic math facts ('arithmetic combinations') by pairing unknown computation items with a steadily increasing collection of known items. This intervention makes use of concentrated practice to promote fluency and guarantees that the student will experience a high rate of success. In preparation for this intervention: 1.

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How do I get consent from all students/families in my classroom when I conduct my research? Obtaining parental consent/trying to contact families is often a struggle.

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How can I approve automaticity in my students?

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"Math fact fluency is the ability to accurately and quickly recall basic addition, subtraction, multiplication, and division facts (Burns, 2005; McCallum, Skinner, & Turner, 2006; Poncy, Skinner, & Jaspers, 2006). Math fluency is often calculated by determining a student’s digits correct per minute for a specific set of facts (e.g., addition, division). Students who possess fluency can recall facts with automaticity, which means they typically think no longer than two seconds before responding with the correct answer. "
(C. Frawley, 2012)

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Are timed math tests/drills the most effective way to build fluency? I've often witnessed this be the "go to" strategy for building fluency. From my perspective, many students seem to develop anxiety over these types of drills.

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Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts

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Download PDF Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts By Jo Boaler, Professor of Mathematics Education, co-founder youcubed With the help of Cathy Williams, […]

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Students seem to struggle more with subtraction skills compared to addition skills.

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How can I improve addition and subtraction math fact fluency skills (#s 0-20) in 2nd graders?