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.
Please share when you’re done! I’m definitely intrigued!
Whoa! I love your questions! I want to use tangrams in my Alg II class and you have just given me some great ideas for number talks. I’ll flesh them out and share in another comment.
Probably my favorite post of yours! We’ve talked about this a lot and we both agree that the best manipulatives are those that have no meaning attached other than what students give them. This is where even base ten blocks can be problematic as Joe mentioned because the meanings assigned to each of the pieces are often assigned by teachers rather than through the sense making of students. I’d like to add to the list of multi-use manipulatives (cuisenaire rods are definitely at the top along with number lines) …..wait for it…….linking cubes. Great for all number concepts from counting through base ten to fractions and decimals, measurement, algebraic thinking and on and on.
Absolutely Mike! I think my favorite manipulatives are the ones used in kindergarten for counting (red yellow counters & colored tiles).
Probably my favorite example of how you did this was with the stacking counters for prime and composite. It would be sweet if you wrote that up so I can re-live you explaining it whenever I want…And no I’m not joking!
Excellent post Graham. You raise important points regarding “rigid use of manipulatives” (I never thought to categorize a number line as a manipulative but now you’ve got me thinking). We’ve thought about this too with pattern blocks and fractions. Our program (Everyday Math) has activities where the values of the blocks change; the hexagon is not always the “1 whole”. Using Cuisenaire rods is a good remedy; you can change the values of the rods. Our district invested some $ in classroom sets for just this purpose.
It also crops up when we start using base-10 blocks for decimals, and all of a sudden the “flat”, which has always represented 100, now represents 1 whole, and the “long”, which has always represented 10, now represents 0.1, and the cube, which has always represented 1 whole, is now 0.01. Talk about blowing a 10 year-old’s mind!
I like to try to re-purpose manipulatives when I can…i.e. using fraction bars in several different ways: for “fraction of” problems:
or creating number lines with highway exits;
or multiplying mixed numbers:
Mostly because we have so many in our school I don’t know what to do with them all!
These are great examples of why I totally dig your blog Joe! You’re always looking for ways to change it up and the multiplying mixed numbers is a perfect example.
I’m a huge fan of Cuisenaire rods as well because of how you can assign any value of them. It’s for that reason that I shy away from fraction bars whenever possible. The blue ALWAYS has to be a half because it’s written on there. Sure you can flip it over to the blank side but I’ve found that students still struggle to let go of the half.
I wonder how the multiplying mixed numbers would work with Cuisenaire rods…hmmm?
Will do, Graham! And when you make sense of multiplying fractions with Cuisenaire rods, don’t stop there! Look at division! It’s so revealing for adults and children when they build understanding based on the meaning they assign to the tools they use. And, as always, context is key!
Yes, so true about fraction bars. Those values are locked in. But you can play around with what they represent. So the “1 whole” can one day be a candy bar, and the next day be a mile, and another day…well you get the idea. I just hate to see all those tiles go to waste. Maybe we could start a #fractionbars and collect some ideas!
Mike, your point about allowing children to assign meaning is powerful, and very Danielson, but scary because teacher is no longer in control.
Mike and I saw Greg Tang speak 3 years ago and he was talking about this very thing, where teachers should remove the base-ten blocks and let students assign values to clothing items.
Love the #fractionbars to collect some ideas Joe. My wheels are turning now.
Yes, and while I didn’t completely agree with Greg Tang, some of what he said was validating. It can be scary, for sure. At some point, I think, students should assign a value to a tool in order to reason their way through a problem. If we can help students develop this early on, just think of how much more ready they’ll be to assign variables to represent situations/possible values. Stepping out of our comfort zones is difficult, bit it helps us grow, so let’s grow slowly and steadily!
Great ideas here. Great idea about using fraction bars to represent contextual wholes and fractions. A good use of a “too helpful” tool.