Monday, March 02, 2015

Fraction Framework: Outstanding Presentation Snapshot

Mahesh Sharma provided a meaningful framework for fraction instruction today at a one-day conference. For years I've heard about Sharma's expertise with regard to math and math education, and today I was finally able to experience this gifted and knowledgeable educator's presentation. I recommend that every math teacher and program leader seek out Sharma's expertise if possible.

Now, after a great day of learning, the question is how will I apply all that I learned. I'll begin by stepping back a bit with my students and reviewing many fraction concepts I've covered, but this time I'll use the depth that Sharma modeled today.

General Teaching Tips
I'll apply the following teaching tips:
  • Be mindful of the message of the nonverbal communication you use during math class.
  • Encourage students to write it down for better retention.
  • Don't let the class be "hijacked" by the four top performers.
  • Let students know that math is hard work and it does take perseverance to learn it well.
Review Multiplication, Division, and Decimal Computation
We'll review the fact that multiplication can be thought of as arrays, equal groups, area and repeated addition. We'll also review that division is repeated subtraction, partitioning, arrays, and related to the dimensions of the area of a rectangle. Also show the meaning behind the decimal point with regard to aligning the decimal points while adding and subtracting and counting the places for decimal multiplication.

Review Prime Factorization, Lowest Common Multiple and Greatest Common Factor
  • Lead students to the fact that all terminating decimals have prime factors of 2 and 5.
  • Use the double method of finding prime factors.

Review Symmetry

Naming Fractions
With a focus on using his language, concept, procedure roll-out of content, we'll first revisit benchmark fractions by drawing bar and number line models of those fractions and naming the fractions in multiple ways. For example 3/4 can be named:
  • 3/4
  • three fourths (1/4 + 1/4 + 1/4 or 3(1/4)
  • three fourth
  • 75%
  • .75 or 75/100
  • 3 divided by 4 or 3/4
  • 3 X 1/4 = 1/4 X 3
  • 3 groups of 1/4
  • 3:4 the ratio of 3 to 4
  • three fourths of the whole
Make Fraction Bar Models By Folding Paper
Make a large number of one inch strips of paper. Pass out eight pieces to each student. Begin with a focus on one whole, then move onto halves, thirds, fourths, sixths, eighths, and tenths. Each time you focus on the paper, ask the following questions:
  • What am I putting through the fraction "machine?" (one whole)
  • What machine am I using? (halves, thirds, fourths. . . .)
  • How many parts did I make?
  • Are the parts equal?
  • What is the name of each part?
  • How many equal parts make a whole?
  • What is the new name?
Develop Fraction Concepts
  • Name fractions
  • Locate fractions
  • Relate fractions to multiplication and division, use variables.
  • Compare the relationships within and amongst fractions.
  • Prompt students to observe patterns, conjecture, and summarize.
Develop Concept and Procedural Knowledge of Fractions
  • Apply congruency as you develop a concept with the concrete, pictorial and abstract. Use a multi-sensory approach.
  • Manifest fractions:
    • parts to whole
    • comparing quantities: ration, scale
    • comparison of quantities with a standard such as decimals or percent.
    • comparison of comparison: proportion
    • rational numbers.
  • Procedure
    • intuitive: mix, combine with conceptual schema
    • concrete: exact number, efficient, elegant, generalize
    • pictorial representation with numbers and models
    • application: intra, interdisciplinary, extra curricular
    • communication/teaching: mastery
Fraction Computation
  • To add or subtract fractions they must have a "common experience" or common denominator.
  • Study fraction size, denominators, relationships, different "experiences" or behaviors.
  • Determine:
    • What is the whole under discussion.
    • How many parts is the whole divided into?
    • Are the parts equal?
    • Name each part?
    • What is the new name?
  • Use area models with cuisenaire rods, base ten blocks, folding paper, and pictoral models.
I highly recommend Mahesh Sharma's math workshops for your math teaching development. He focuses on deep understanding and mathematical thinking. The framework and pedagogy he presents helps one to meet the standards with depth and meaning. It was a great day of learning, one I look forward to sharing with my students.