Hoping this post finds you having enjoyed your holiday weekend! Today we continue our book study of Teaching Student-Centered Mathematics Pre K-2 by Van de Walle, Lovin, Karp, and Bay-Williams with a discussion of chapter 10. It’s all about helping children master basic facts. To read past posts, simply visit our book study archive.
Chapter 10 – Helping Children Master the Basic Facts
At the beginning of chapters 8 through 17, the authors present the big ideas of the chapter to readers. I chose to share one of the big ideas presented at the beginning of this chapter that really says it all.
“Mastery of basic facts is a developmental process. Children move through stages starting with counting, then moving to more efficient reasoning strategies and eventually to quick recall. Instruction must help students move through these phases without rushing them to memorization.”
In the second paragraph, the authors define mastery of basic facts. Mastery means that a child can give a quick response (in about 3 seconds) without having to revert back to much less efficient ways of figuring out an answer such as counting. We all know how important it is for our students to have mastered basic addition and subtraction facts, but how we help them do this is crucial.
I appreciate how the authors stress that mastery of facts does not simply lie in the hands of second grade teachers (as knowing facts from memory is part of the Common Core Standards for Mathematics at second grade). Teachers PreK-1 play an important part in helping students get ready to master facts in second grade. An emphasis on helping students develop number sense is essential. You can visit our discussion of chapter 8 (Developing Early Number Concepts and Number Sense) to read/reread the relationships between numbers that students MUST develop before moving on to basic facts. Just as important, students MUST develop an understanding of operations before moving on to basic facts, as discussed in Chapter 9 (Developing Meanings for the Operations).
The authors overview three approaches to fact mastery that are commonly used and research to back up their effectiveness or ineffectiveness. Memorization is too often, BUT SHOULD NOT BE, the focus. When students are asked to memorize a vast number of facts by simply practicing them a lot, an emphasis is not placed on using strategies or known facts. The result? As the authors state, “Too many fourth and fifth graders have not mastered addition and subtraction facts and continue to count on their fingers.”
So what do we do?
The authors suggest using a combination of explicit strategy instruction and guided invention.
When we explicitly teach students efficient strategies that they can use to figure out a group of facts, we help them understand how strategies can be effective. Students will then ultimately be able to choose what works best for them. There are many great resources out there helping teachers do just that. I feel one of the best to be Mastering the Basic Math Facts in Addition and Subtraction: Strategies, Activities, and Interventions to Move Students Beyond Memorization by Susan O’Connel and John SanGiovanni. You can read more about this helpful resource in my past post, What are the other kids doing? Wednesday Linky – Basic Facts.
Guided invention focuses on supporting students as they create relationships between numbers and what they already know. Ultimately, when we help students explore the relationships between numbers, they naturally begin to use these relationships to help them reason with more difficult problems. This will lead to flexibility when working with basic facts and beyond. This is something my second graders are able to do, yet it needs to be instilled in them through much exploration. An example of flexibility would be a student who cannot yet recall the sum of 8 + 9 but can use his/her current knowledge of number to create further relationships. He/she might choose to:
- use doubles, “I know 8 + 8. That’s 16. 9 is just one more than 8, so 8 + 9 is 17.”
- use tens, “Well, one more than 9 is ten, so I can move one from 8 to make a ten. Then 10 + 7 is 17.”
- create landmark/benchmark or “friendly” addends, “I can think 8 + 10, and that’s 18. 18 – 1 = 17.”
We must give students the opportunity to explore relationships in a guided environment, as the authors say, this kind of thinking will not “magically happen”. The type of reasoning described above must come before mastery. Consider this example. A student has an understanding of what is needed to make ten with any given number. When presented with situations that requires the addition of 9, she consistently chooses to make a landmark 10 using the other added (reasoning stage). Eventually she is able to quickly add 9 because she knows what the result is (mastery within 3 seconds). BUT this would not have come without the reasoning stage in which the student explored the use of the strategy and proved its effectiveness with experience. The student has developed mastery–knows her add 9 facts from memory. This DOES not happen with memorization.
As I shared in a previous post, here is a great resources for helping students develop flexibility with numbers, Fluency Through Flexibility: How to Build Number Sense 0-20 by Christina Tondevold.
One of the best features of this chapter is the overview of reasoning strategies for addition and subtraction as presented by the authors. The authors also do well to remind us that we need to have an understanding of the different ways students might think about a problem because there is not one right way. Our understanding is vital in supporting our students’ development.
As expected, chapter 10 is also chocked full of activities and games to engage students in developing important reasoning strategies for addition and subtraction (and a couple for building a foundation for multiplication facts). If you do not already have a copy of the book, I say again–you need to get a copy so you can have access to all of the activities and games and the blackline master materials that accompany them.
Speaking of games, here is a MUST-READ article—Enriching Addition and Subtraction Fact Mastery through Games by Jennifer M. Bay-Williams and Gina Kling from November 2014’s Teaching Children Mathematics. This article also outlines the important phases of progression to mastery.
I would like to end with some DON’Ts for teaching basic facts as shared by the authors. It is always helpful to know what not to do as well. DON’T…
- use public comparisons of mastery
- work through fact in order from 0 to 9
- work on all facts at once
- move to quick-recall activities too soon
- use facts as a barrier to good mathematics—it’s not all about computation
- use fact mastery as a prerequisite for calculator use–there are times when a student does not have mastery of all facts but he/she will benefit from the use of a calculator to attain the learning goal of a lesson
As always, please feel free to leave your thoughts in a comment, and come back and join us on Sunday when Courtney discusses chapter 11 – Developing Whole-Number Place-Value Concepts. Until then, enjoy the rest of your week!
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