Basic Facts • What are they? • Why should students memorize basic facts?
Why Basic Facts Instruction? • Frees up working memory to master algorithms and math applications. • Cognitive psychology research points to the value of automatic recall of facts. • Students who do not memorize facts flounder when more complex math is introduced—progress for these student can end at elementary school.
Math Wars(Strauss, Washington Post, Jan. 05) • Among the topics the NCTM and the mathematicians said they agreed on: • Heavy reliance on calculators in the early elementary grades is a bad idea. • Elementary school children must have automatic recall of number facts, meaning that, yes, they have to memorize multiplication tables. • Children must master basic algorithms. The meeting participants spent time defining the word "algorithm,"which means a set of rules for solving a problem in a finite number of steps.
Teaching Math Facts • Three types of instructional activities: • Understanding • Relationships • Memorization • What are these activities? Why?
Relationship Activities • Exercises based on a series (e.g., 3x1, 3x2, 3x2) • Exercises based on fact families and inverse operations (e.g., 5-2=3, 5-3=2, 2+3=5, 3+2=5) • Purpose of these exercises is to make memorization easier
Preskill for Relationships • Plus-one Facts—Format 6.1, page 90 • Teaches facts and relationships. • Rule: When you plus one you say the next number. • Model Part A and B.
Plus-one Facts Part C 8 + 1 4 + 1 7 + 1 5 + 1 9 + 1 (Your turn—format practice!)
Series Saying Prompts students to notice the counting relationship: 6 + 2 = 8 5 x 2 = 10 7 + 2 = 9 5x 3 = 15 8 + 2 = 10 5x 4 = 20 5 x 5 = 25
Series Saying • Format 6.2 Model and Practice • Part A: Reading the Statements • Part B: Reading the Statements with the Answer Erased • Part C: Saying the Statements (No visual prompts) • Part D: Random Fact Drill
Series Saying Format 6.2 Model and Practice Teaching behaviors: • 3-2 seconds per statement • Timing for each statement • Practice, practice, practice and make it fun! • Correction for slow pace—lead and work on increasing the pace • Correction for statement error?
Three Number Fact Families • Sets of three numbers from which students can create 4 statements. • Either—addition and subtraction or multiplication and division. • Teach commutative property of addition (a + b = c and b + a = c) and multiplication (a x b = c and b x a = c). • Why is the commutative property important?
Three Number Fact Families • Format 6.3 • Part A—how to construct pairs (big number introduced) • Part B—oral test on the “reverse” fact • Part C—worksheet, filling in the big number and generating the two facts
Three Number Fact Families • Format 6.4 • Family of Facts for Subtraction and Division. Teaches students how to generate 4 statements (2 addition and 2 subtractions or 2 multiplication and 2 division).: • 2 x 5 = 10, 5 x 2 = 10, 10/5 = 2, 10/2 = 5 • Students learn the rule: when you subtract (or divide), you always start with the big number.
Three Number Fact Families: Model and Practice ____ + ____ = _____ ____ + ____ = _____ ____ - ____ = _____ ____ - ____ = _____
Sequencing Introduction of Facts • Systematic, cumulatively introduction of facts—Figures 6.2, 6.3, 6.4, and 6.5. • Separate similar facts • Teach easier facts first • Teach related facts together • Reverse of specific series taught soon after initial series
Sequencing Introduction of Facts • Systematic, cumulatively introduction of facts—Figures 6.2, 6.3, 6.4, and 6.5. • Figures indicate the order in which to teach facts and the format to use for each set. • A set is presented for several days then included on memorization worksheets.
Sequencing Introduction of Facts • Addition facts are introduced first. • Little research is available to guide us regarding when to introduce subtraction facts—after addition is completed—after part of addition is mastered and extend to subtraction? • Recommendation—delay subtraction until students have learned about half of the addition facts (Sets a-m addition mastered). Then alternate—Subtraction A, Addition N, Subtraction B, Addition O, etc.
Sequencing Introduction of Facts • When to start multiplication? • Start in third grade even if students are still working on addition and subtraction. • Provide a “double dose” of facts instruction for students who need it.
Mastery Activities Programs for Fact Memorization include: • A specific performance criterion • Intensive practice on new facts • Systematic practice on previously introduced facts • Adequate allotted time • A record keeping system • A motivation system
Performance Criterion • Oral criterion: saying an entire fact every 2 seconds • Written criterion: 2/3 rate at which student is able to write digits • How do you determine students’ writing rate?
Performance Criterion • 40 facts per minute is the low end of fluent performance. • However, common expectation around the country is100 facts in five minutes. Otter Creek Institute (Don Crawford)
Intensive Practice • Instruction of new facts using relationship activity • Oral practice on new facts • Written practice on new facts and old facts
Adequate Allotted Time • Approximately 10-15 minutes per day • Preferably, time in addition to math instructional time • Before school, during lunch recess, after school etc.
Record Keeping System • Purpose: monitor student progress • Recommendation: keep paperwork at minimum—see Student Record Form on page 86
Motivation • Integrate with record keeping system • Motivation comes with SUCCESS • Teacher’s responsibility to set students up for success: Must teach facts
Two Fact Mastery Programs • Homogeneous Group Program • Teacher Led – Group Oral Practice • Materials • Pretesting • Timed test • Record keeping/Motivation
Materials • Written fact practice worksheet divided in half • Smaller number of facts to master at once • One minute timings (more than once?) • Top half: Practice on new facts – current set and two previously introduced sets • Bottom half: Current fact set presented twice; practice on previously introduced sets
Pretest • Develop a written pretest with all 100 facts of one operation. • Allow the students 2 minutes to work as many problems as they can. • 30 facts per minute—start at set G • 45 facts per minute—start at set M • 60 facts per minute—start at set R • >85 test on next operation
Oral Group Practice • Using the worksheets—oral drill of top part saying the problem and answer in unison • Repeat the first line until students can answer correctly at with about 2 seconds think time • Model and Practice
Timed Test • Bottom half of the worksheet • About 1 min. 15 seconds
Mastery Criteria • If ¾ or more of the students got 28 of 30 facts correct go on to the next worksheet. • If not, repeat the same worksheet.
Two Fact Mastery Programs • Heterogeneous Group Program • Partner Practice • Materials: folders for each student with their level of worksheet (one folder with answers and one without) • Pretesting: same • Timed Test • Record keeping/Motivation • Modifications
Two Fact Mastery Programs Heterogeneous Group Program • Daily Routine: • Pairs at same level, one with answers • Each student practices the worksheet twice, saying the problem and the answer • If student makes an error the partner with the answer sheet corrects • Teacher times for 1 ½ minutes on the top and one minute on the bottom • Written test—Bottom half of worksheet
Compare/Contrast Traditional Programs • Not teacher-directed • Length of time to mastery • Not cumulative Effective Programs • Teacher-directed (Oral practice) • Quicker mastery of smaller sets • Cumulative introduction and review
Resources • Math facts at: http://depts.washington.edu/facts/ (files are huge but free) • Otter Creek Institute • Mastering Math Fact Families • http://www.oci-sems.com/home.htm