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Mathematics: What's the Big Idea?

Workshop #5

Whole Numbers:Castles and Shadows

Content Guide - Timothy Erickson

Supplies Needed for Workshop #5:
six-sided dice (5 dice for every 2 participants)
copies of Workshop #5 Worksheetfor every participant

pencils, paper, scissors, rulers, calculators, tape, a variety of colored markers


About the Workshop

What is the theme of the workshop?
Nothing says "math" to the public louder than computation with whole numbers. But what role should whole numbers play in today's mathematics classrooms? What should students memorize? When should they use calculators? There are no easy answers to these questions; we'll work together to address them as best we can.

Whom do we see? What happens in the videoclips?
We'll see elementary school students doing a variety of activities either focusing on or requiring basic computation. These range from doing number puzzles with multiple solutions to elaborate estimation activities.

What issues does this workshop address?
This program addresses a range of issues. For example, what are "the basics"? How might we distinguish basic number skills such as the times tables from basic paper-and-pencil algorithms such as borrowing and carrying? What should the role of mental math and estimation be for today's students?

What teaching strategy does this workshop offer?
We'll see a number of effective ways to introduce and reinforce number skills in the classroom. We'll see ways to practice basic number relationships (e.g., 3x3=9) in rich, engaging contexts, and also "pure math" lessons that have no immediate real-world counterparts. We'll discuss games as a medium to reinforce facts. And we'll see how some teachers have used artifacts from real life and activities from other curriculum areas to give their students opportunities to do a lot of estimation and mental math.

To which NCTM Standards does this workshop relate?
This workshop focuses on Standards 5-8: Estimation; Number Sense and Numeration; Concepts of Whole Number Operations; and Whole Number Computation (middle grades Standards 5 and 7). All four "process" standards are represented, but we will especially see Standard 3: Mathematics as Reasoning.


Suggested Strategies and Classroom Activities

Seven-Minute Boogie
Playing basic-fact games can be both fun and effective, as long as they are either non-competitive or "self-competitive." In Seven-Minute Boogie, students roll two dice, make a multiplication problem out of the numbers, write it down (with the answer), and repeat the process as many times as they can in seven minutes. Their score for the day is the number of multiplication sentences they got right in the seven minutes. The next day, they try to improve their scores. With regular dice, they practice low numbers, which are the most useful facts anyway. But you can alter dice to cover more facts: try "2-3-4-6-7-8" and "3-5-6-7-8-9," which miss only a few of the non-trivial facts.

Comparing Procedures
Have students explain to the group how they do various computational procedures. Have students discuss what they like or would change about each procedure. This helps students see, first, that which procedure is best may depend on context, and second, that there is room for disagreement and flexibility in which procedure they use. Some examples: in single-digit addition (for example, as part of a larger algorithm), do you find "friendly and complementary numbers" (e.g., 6 and 4), or just add one after another or count on with your pencil? When you multiply 25 x 36, do you first convert it to 25 x 4 x 9 (even in your head), or do you start with the 5 x 6 is 30, put down zero, and carry three? Be sure to ask what the dangers are of using a "trick."

Peer Assessment
Any of these small projects are perfect fodder for peer assessment. If students make posters they put up around the room, they can also be asked to look at two or three other posters and comment on them. Consider applying "writers' workshop" commenting guidelines. For example, students write answers to "What did you like about this poster/presentation?" and "What additional questions do you have?" to help make the situation safe for criticism. This way, each student gets feedback and has the experience of giving it, which supports the goal of fostering communication skills in mathematics.

Finding estimation opportunities
Chances to do the kind of estimation that requires mental math and calculation - especially "back-of-the-envelope" problems - arise frequently and from all sides. How many cars pass the school in a day? How many people do you think are in a newspaper picture of St. Peter's Square in Rome? How long will it be until the school runs out of copier paper? How many little dots are there on the ceiling tiles of this room? Remember: the point is not to guess but to calculate!


Post-Workshop Questions

  1. Some teachers let students use calculators at any time - but only after they've gotten their "calculator license." A student gets this license by proving that he/she can calculate well enough without the calculator. Do you like this idea? What do you think would be a reasonable requirement for getting and maintaining one's calculator license?

  2. Think about your own computational strategies in your daily life. For example, when do you use long division? When do you use a calculator? Do you estimate by saying, for example, "29 is almost 30, so …"? Do you look for friendly or complementary numbers (e.g., 7 and 3) when adding? When you're faced with subtraction, do you line up the columns and borrow? You might ask your colleagues the same questions: Is there much variation in the strategies you use?

  3. Ignoring the requirements of standardized tests for a moment, at what grade do you think students really need to know the times tables up to 10 times 10? When do these multiplication facts become genuinely useful?

  4. In this workshop, we saw teachers using a variety of contexts, ranging from rich contexts to no context at all. Think about your mix of contexts and where you want it to be. Do you usually struggle to fit everything into a context? Or do you always reach for "naked numbers?" And what do you think is best?

  5. Debate this argument pro and con: "There's not a lot you can do in mathematics (outside of geometry) without number skills. Number permeates and supports mathematics. So it doesn't make sense to have students do open-ended problems in rich contexts until they master the basics. It's a waste of their time and only leads to frustration and misconceptions."

  6. Being a math teacher, you probably learned the traditional algorithms and memorized your times tables pretty early. You also probably have pretty good number sense. Do you think that you developed number sense from your computational ability, or did you learn to compute well because you had number sense? Or were you just the kid who "got it" no matter what? What helped you learn math well, and how does that apply to your instruction and your students?


Pre-Workshop Assignment for Workshop #6

Look at the large triange on Worksheet #6-2.

If the large triangle is equal to one whole until, what is the value of:

  • a small triangle?
  • a parallelogram
  • a trapezoid
  • a hexagon (made up on 2 striped trapezoids)





Mathematics: What's the Big Idea?

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