Nancy Anderson was great. She asked us to estimate the quotient for a problem such as 3 1/6 divided by 1/3. We very naturally decomposed the mixed number into a whole number and a fraction. None of us made it into an improper fraction. It made more sense to think how many 1/3's would be in 1: 3. So how many would be in 3: 9. How many 1/3's would be in 1/6? Well it takes 2 1/6's to make 1/3, so it must be half. So the answer would be 9 1/2. We divided mixed numbers by fractions without pen or paper! We made sense of it by using the distributive property.
I was reminded of that night recently when reading an article in Mathematics Teaching in Middle School, Oct. 2009 edition. Susan Taber writes, in Capitalizing on the Unexpected, "If ...the teacher invites various students to describe and explain the differing ways they combined the blocks to represent adding the two numbers, the students will...develop a more complete understanding of the meaning of what is represented by each of the digits and the relationship among units, tens, and hundreds. An Additional benefit is that ...(it) allows the class to discuss the associative property of addition. ...They ...learn that numbers can be decomposed and composed in various ways."
The article goes on to describe work with fraction circles. It states that, "Using a set of fraction pieces with denominators of 2, 4, 6, 8, and 12 provides many more opportunities for students to explore equivalent fractions than does using pattern blocks, because the pattern blocks can represent only halves, thirds, and sixths."
This is no longer true. You can now purchase an additional set of pattern blocks which includes right trapezoids (half the size of the regular trapezoid) and right triangles (half the size of the equilateral triangle). Thus you can now work with fourths and twelves as well. So, in actuality, you are only missing 1/5's, 1/7's, 1/9's, and 1/10's.
The best manipulative I know for 1/10's is money. Beyond that cuisenaire rods work well, as there are ten differing sizes, ranging from one to ten.
We did some great work with cuisenaire rods at that seminar. We not only explored proportions and percents using an activity from Math Matters, but Nancy encouraged us to write our own lessons, with guiding questions to take back to our students.
This is the power of good teaching. This is the power of encouraging students to use manipulatives, mental math, and various strategies to solve problems, rather than the traditional algorithm. Then, be ready to take advantage of the unexpected. Explore the properties of numbers and operations the way a scientist would: does it work now? Now? Does it still work? Why does it work?
The article asks, "How do you decide how many students' solutions should be presented during the class discussion?" This is very telling. You don't want to overload the students with alternative solutions, but you do want to explore similarities and differences in strategies, and why each works, or doesn't. Keep in mind the big idea: what do you most wish the students to understand at this point? Which examples of student work might best highlight and illuminate this concept?
In summary, students learn best when allowed to play around, explore, and test ideas: both mentally and through hands-on manipulatives. Our job is to encourage and guide this exploration. Please write to me and let me know what your favorite manipulatives are and how you use them. --Lauren
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