By Francis (Skip) Fennell, Posted January 5, 2015 –   

To me a—perhaps the—“signature expectation” of any Pre-K–grade 6 mathematics experience is the ongoing nurturing and development of a sense of number. Yes, number sense. Although elusive, establishing and maintaining flexibility with number (and this is certainly not the last chapter or topic in a textbook or curriculum guide) is simultaneously ongoing and foundational to work with operations with whole numbers and fractions.

The focus of the next four Teaching Children Mathematics blog entries will be place value and fraction equivalence. The ongoing instructional development and understanding of these pillars of a sense of number are must-haves for all mathematics learners. So, let’s get started.

Place value, which for many young learners begins with “tens and ones” activities that typically involve base-ten manipulative materials and pictorial and symbolic representations of two-digit whole numbers, is much more—so much more. Consider that place value is centrally connected within a continuum that begins with counting and essentially extends counting to understanding numbers greater than 9, which then presents the need for a direct focus on place value related to two-digit whole numbers, and extends to comparing and ordering such numbers as well as early work with addition and subtraction. When developed with understanding, these concepts, from counting to work with operations, are all connected. See my admittedly crude rendering below, which attempts (successfully, I hope) to convey this point.

Also note that place value understandings extend to multidigit whole numbers and to those other numbers on the other side of the decimal point—yes, decimals. How can we continually work to engage students in activities that truly develop understandings related to place value but also serve as a bridge into activities and problem-based tasks that extend place value to comparing and ordering and operations? This two-part blog entry will suggest some basic, popular activities and tasks that connect place value to mathematics topics within the continuum proposed above. I hope that what you read in this post and the next will generate discussion and a literal swap shop of comments and ideas of what works for you as well as questions to help all of us in our efforts to deepen place value understandings. Try these suggestions:

1.   Multiple representations. Here’s a favorite: Pick your own representations, but do this often. Pick a number and have students represent it four different ways; for example, 57 as base-ten blocks, expanded notation, a written number (fifty-seven), and its location on a number line from 0–100. And this could be expanded to consider three-digit numbers or larger as well as decimals.

2.   Number of the day. Pick a number (e.g., 78). Post it and have students (throughout the day) think about and write different ways they could represent the number. Share and discuss the number. Because some numbers are “friendlier” than others, compare numbers from day to day (e.g., 24 hours in a day; two dozen versus 79, which is prime and frankly boring).

• 70 + 8
• 40 + 30 + 8
• 80 – 2
• 100 – 22
• Using base ten blocks

3.   Hundred Charts. I haven’t forgotten them, but I wanted to get your thinking about them. How do you use hundred charts for counting, for comparing, and for mental mathematics? How do you see their use as important in our quest for developing place-value understandings and helping to build this number-sense foundation? Post your responses!

Let’s think about decimals for a bit. Consider the following:

4.   Different Takes: Have students consider and then discuss where they might place the following on a number line: 23; 23.84; 23.845; 0.23; 0.2384; 0.23845.

5.   What happens here: Start with 0.7 (or any decimal amount). Have students respond to each of the following:

• What happens to the value of 0.7 if a 0 is inserted after the 7?
• What happens to the value of 0.7 if a 0 is inserted before the 7?

  A sampling of place-value related opportunities would not be complete without some tasks for students to consider. Try these:

6.   Task: Chase tried to make a decimal number as close to 50 as he could (using the numbers 1, 4, 5, 9, and a decimal point). He arranged the numbers in this order: 51.49. Paige thought she could arrange the same digits to get a number that is even closer to 50. Do you agree or disagree? Explain your thinking.

7.   Task: For an online place-value–based lesson, try Next Lesson’s “That Video Got How Many Views?”  and see what you think. 

Your Turn

Now it’s your turn. A rationale for the foundational importance of place value and a half dozen + 1 set of activities and tasks has been provided for you to consider. Try them. What do you think? What works? How do you develop place-value understandings? How does what you do connect to developing the level of flexibility with number so necessary for developing and establishing a sense of number? 

We want to hear from you! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk, or get to me personally on Twitter @SkipFennell or at ffennell@mcdaniel.edu. Feel free to visit the following websites for information, resources, or just for fun:

• My personal site: www.ffennell.com
• Elementary Mathematics Specialists and Teacher Leaders Project site:  www.mathspecialists.org

  Francis (Skip) Fennell, ffennell@mcdaniel.edu, is the L. Stanley Bowlsbey Professor of Education and Graduate and Professional Studies at McDaniel College in Westminster, Maryland, where he directs the Brookhill Foundation–supported Elementary Mathematics Specialists and Teacher Leaders Project (http://www.mathspecialists.org). He is a past president of NCTM and a recipient of NCTM’s Lifetime Achievement Award. He is interested in the work of mathematics specialists, implementation of CCSSM, teacher education, number and fraction sense, and educational policy.  

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