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As we talk about the reform in mathematics education, it is important to make it clear that mastery of basic facts remains as essential today as ever. I've included a quotation from Principles and Standards to that effect (T-64). See also the Standards quotations on pages 168 and 183. A major issue for me is to help teachers at all levels see the necessity and the power of students developing number relationships (Chapter 9) and the connections between addition and subtraction (Chapter 10). It is critical, especially for teachers of grades 4 and above to realize that these same number concepts and operations meanings are still the keys to fact mastery for their students - not drill.

Three Potential Approaches to Fact Mastery

Teachers essentially have three options for helping their students with basic facts: drill, directed teaching of fact strategies, and what Gravemeijer and van Galen refer to as "guided invention." As discussed on pp. 165 - 166, the first two options have drawbacks, especially a reliance on drill before efficient strategies have been developed. The intended outcome of the chapter is for teachers to believe in the third option and understand how it can be carried out in the classroom.

Efficient Strategies

Drill can be both important and effective. HOWEVER, and this is a major caveat to the preceding statement, the drill must be drill of an efficient strategy with the emphasis on efficient. Efficient strategies do not develop as a result of drill, but as Brownell once said, they develop in spite of the drill we provide. Do not let students get the wrong message from this chapter by simply observing all of the flash cards and other drill activities found in the illustrations. Practice in the sense defined in Chapter 5 - repeated problem-based experiences with the same topic - is absolutely necessary for the development of efficient strategies.

Your students may not know what is meant by an efficient strategy, one that can be used quickly and mentally, without recourse to counting. Efficient strategies should be based on meaning and should not be seen as cute or clever tricks. Once your students see an efficient strategy such as a "doubles plus one" approach to 7+6, they also need to believe that the strategies will fade into the background with sufficient practice.

Addition Facts

The text focuses on the most popular or well-known strategies currently found in many resources and textbooks: zero facts, doubles, near doubles, and make-ten. There is one significant exception. Many authors promote counting on as a strategy for facts with an addend of 1, 2, or 3. I have argued against any form of counting and instead have encouraged teachers to base addition facts on number concepts. To that end, you will find a "strategy" called One- and two-more-than facts, or facts with an addend of either 1 or 2. These main strategies then leave 12 unaccounted for facts or 6 facts and their commutative partners. Several additional strategies (including counting on) are suggested.

When these additional options are discussed, I want to help students see strategies as flexible options and not mandated prescriptions. Adults who happen to use the strategy 10+7 is 17 so 9+7 is 16, feel uncomfortable when that approach is not included in the text. When there are choices to be made, it is also comforting to know that the choice is the child's. The emphasis is on anything that is efficient and understood - not on doing it the one "best" way.

Subtraction Facts

There is evidence to suggest that think addition is the principal method used by most of us for mastering subtraction facts. To this end, the mastery of addition facts should come first and the conceptual development of subtraction as naming the missing part should be the critical link. I focus on the 36 facts that have a sum greater than 10. In addition to think addition or think missing part, it is also quite reasonable to use some strategy that utilizes 10 as a bridge. First, you can add on or up through ten. For other facts it may be useful to first take away an amount that gets you down to ten and then take away a few more, or work down through ten. As noted earlier, the emphasis is on students' flexible use of strategies, not forcing one of ours on their way of thinking. This is a real value of the Missing Part worksheets described in Figures 11.9 and 11.10.

Multiplication Facts

Most authors will use the strategies of zeros, ones, doubles, fives, and nines facts. The two popular nines strategies create an interesting discussion - one may be easier to use while the other is easier to justify. Is the Nifty Nines approach just a meaningless trick? These strategies leave 25 facts not accounted for, including those most people consider the hardest to learn. The book suggests exclusively the approach of deriving the unknown fact from a known fact. The justification is either by appealing to the repeated addition concept or the use of partitioning an array. In either case, these methods occasionally require mental addition capability. For example, if you use the idea of double and double again for 4 x 8 then there is a need to know 16 + 16. This is a great time to briefly discuss mental computation.

In keeping with the spirit of this book, I strongly discourage the use of rhymes, mnemonic devices or other tricks for fact mastery. This may be contrary to the beliefs of some teachers who have a favorite collection of such tricks. These tend to be more prevalent for multiplication than other facts.

Division Facts

Although the Principles and Standards quotation mentions division facts, there is actually little need to spend a great deal of time on these. A fact such as 42 ÷ 6 appears with no greater frequency than a non-fact such as 43 ÷ 6 or 41 ÷ 6. A related argument is that virtually no one memorizes 42 ÷ 6. Instead, they know this because they know 6 x 7 = 42.

Fact Remediation in the Upper Grades

For teachers in grades 4 to 8, the approaches in this chapter often seem less than useful because they do not have the class time in their curriculum to help students develop strategies. Furthermore, a large number of their students do not need fact remediation. Thee message in the section near the end of the chapter is that these same methods can be used with older students but they may well require individual diagnosis and students working on their facts out of class.

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