I’ve spent the past month visiting classrooms, reading blogs, attending workshops, sitting in grade level meetings and I’ve noticed a common theme which is concerning. Before I continue I want to acknowledge that the use, and proposed use, of manipulatives is rampant and that is awesome.
Here’s the trap…Manipulatives often have one use or they’re assigned a single value and that is how/when learning becomes static.
As teachers, we need to ensure that manipulatives are not dependent on values. Manipulatives are a tool that make mathematics accessible and the “pigeonholed” use of them can give students AND teachers a false sense of understanding, if not used flexibly.
Example: CCSS.MATH.CONTENT.3.NF.A.2: Understand a fraction as a number on the number line; represent fractions on a number line diagram.
What is the value of the “?” on the number line below?
Pretty straight forward right?!? I would say that most 3rd and 4th grade students could identify the “?” on the number line above. But here’s the catch, if you think they have mastery show them About How Much below:
Changing the value at the beginning of the number line allows for the learning to get messy (this is my happy place). Almost every time I’ve shown this to students they completely ignore the one-half which unveils misconceptions and a lack of understanding as to how number lines are used.
TRAP #1: Number lines do not have to start at 0 or 1.
Example: CCSS.MATH.CONTENT.4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
I love Tangrams!!! After students have identified the value of each piece, I show the picture of the candle and ask “What is the value of the candle AND base if the value of the flame is three-eighths?”
Most students immediately say “five-eighths” but I’m ready for it.
When we use pattern blocks, the majority of us assign a value of 1 to the yellow hexagon and go through the sequence of questions: If the yellow is a whole, what is the value of a blue rhombus? What is the value of 2 trapezoids? 4 triangles?
What if we assigned a value of two-thirds to the hexagon? Whoooa!!! The rigid use of manipulatives can mislead us to believe that our students have mastery of a concept which couldn’t be further from the truth.
TRAP #2: The value of manipulative remains constant.
My personal rule: Once I think a student understands a concept, I change what they know and make them reapply their understanding. Although this throws students into an uncomfortable state of disequilibrium, I believe this is what CCSS expects when they mention “going deeper”. But that’s just me.