Making Meaning in Maths with Manipulatives

Ever since I was an undergraduate student learning how to be an elementary school teacher I have always believed passionately in the use of manipulatives to introduce children to new concepts, ideas and skills in mathematics.  It has always seemed the most logical way of starting children on their journey of mathematization, as Bob Wright of Math Recovery would say.

Well for the first time in my professional life someone asked me about the research basis for using manipulatives for teaching math. In the math ed. texts i use in my courses the use of manipulatives is advocated and illustrated with the introduction of every new idea. It's as if it's a no-brainer, something that is as natural as the sun rising each morning or snow falling at some point in a Vermont winter. So I took to Google this morning and tracked down an array of really interesting research-based articles in support of the use of manipulatives for teaching math at the elementary school level. It felt a bit like conducting research to see if more people used raincoats and umbrellas when it was raining than when the sun was shining but the results were startlingly interesting. Here they are;

Here's an NCSM article. NCSM is the national math leadership council and reminds us that we must also include the use of virtual manipulatives. This one is from the Journal of Instructional Pedagogies and gives an overview of the history of using manipulatives as well as the current research.

There are many publishing companies that also produce manipulative materials so it is probably only natural that they should also produce research to support the use of their products. This is a particularly good article from the folks at ETA.

And here's a student research paper on the topic from Marygrove College. Finally, here's a neat article from Sage Publishing that mentions Montessori education which is really where the use of manipulatives in teaching math all began.


 

Mathematize and the CCSSM

Some time ago when I was learning about the Math Recovery program and reading the wonderful books co-authored by Bob Wright I came across the verb 'mathematize' and its noun counterpart "mathematization". I always thought it was a wonderful way of describing what math education at the elementary school is all about.

In his book, Developing Number Knowledge,  Wright defines the term (p15) this way;
                  Mathematization means bringing a more mathematical approach
                  to some activity. For example, when a student pushes some
                  counters aside and solves an addition  task without them,
                  we say they are mathematizing, since it is mathematically 
                  important to reason about relations independent of concrete
                  materials.

Others define it as "reduction to mathematical form" (Merriam Webster), "to treat or regard mathematically" (The Free Dictionary) and "explaining mathematically" the Collins dictionary.

The really, really interesting thing about all these definitions is the idea of reduction or movement from real life, concrete situations such as that described by Wright, to the symbolic form of symbols and algorithms typically used in math. This is completely opposite to the way math has traditionally been taught and  how it is sadly still taught in poorly taught math classes.

A classic example occurred in my math class yesterday when I asked students what 1/2 ÷ 1/4 meant. No one knew. My hunch is that if you randomly asked 100 people on the street only a handful would be able to tell you that this meant how many quarters are in a half. What really makes this intriguing is that most people would tell you to change the sign, flip the second fraction, multiply and get the answer 2. 

In other words people have not gone through the process of  mathematization when they have learned this procedure. A real world, concrete idea has not been "reduced to a mathematical form". They learned the mathematical procedure without any sense of what it meant or connection to any concept or relationship. There was no derivation, if you like, from a concrete experience or idea  to a symbolic,  mathematical relationship. This happens all the time in math.

Students are taught a square number is the result of "a number times itself" instead of a number that makes a square.

They are taught a prime number is "a number divisible only by 1 and itself" instead of a number that can only make one rectangle (e.g 1 x 7 or 1 x 13).

Students are taught the symbolic mathematics first and not the idea so they cannot be mathematized. Instead, we should be mathematizing them from concrete experience  to symbolic representation. This is what the Common Core State Standards for Mathematics is trying to achieve.

 

No Cursive; no hand calculations?

I recently read this neat Huffington Post piece about the demise of cursive handwriting. Well, demise maybe premature as the conclusion in the article was that cursive writing now belongs as part of the elementary school art curriculum.

So, if we can cast aside the century-old idea of creative handwriting as a mainstay of the elementary school curriculum then it's time we cast out the four algorithms, hand calculations, used to perform addition, subtraction, multiplication and division. I hasten to add that I am referring to the vertical form of algorithm where you put one number above the other with a line below and follow a procedure. We must, of course, still teach the concepts of addition, subtraction, multiplication and division in all their forms but it's time we used technology to do the tedious part, so to speak. Learning the algorithmic procedure does nothing to enhance students' mathematical skills. But, there it is in the Common Core Math Standards (CCMS) due to be implemented in 2014. 
 
Conrad Wolfram suggests we should be using computers to take the drudgery out of hand calculating in maths by allowing students to use calculators and computers to perform these tedious hand calculations. 

I was beginning to go along with this idea until I read this piece by Karen Fuson and Sybilla Beckman about the inclusion  of the "standard algorithm" in the CCMSs. Their argument is persuasive and cites the practice students gain in developing their understanding of place value and base ten when they are involved in the completion of hand calculations. It is part of the process of mathematization that students go through when they are learning elementary school maths. The use of manipulative materials and technology are important in developing ideas but it is what happens in the student's mind that is the most important aspect of learning math; the ability to abstract and manipulate ideas mentally.