The New York Times reported in 2008 on a study from Ohio State University suggesting that using concrete, real world examples and manipulatives to teach math do not enable students to transfer their knowledge to new problems. They found that students who learn through examples do not do as well as those who learn only the abstract symbols.

There undoubtedly are some important “real world” lessons for us to reflect on in this study. Mere activity with manipulatives does not guarantee that learning will take place. Manipulative lessons are not an exception to the need for well designed and delivered instruction. Do we give proper attention to making the connection between manipulation and computation? Are students guided in making generalizations based on specific actions and observations involved in real world lessons? It is not automatic. It is not enough to simply present students with the abstract symbols after working with manipulatives as this article suggests was done. How many teachers are comfortable with teaching abstract formulas, but less skilled in designing and presenting lessons that draw effective learning from tangible objects?

While there is much to think about, there are also questions about the study itself. What about developmental readiness for abstract thinking? This study focused on college not elementary school students. What about multiple learning styles? Were the students guided in making generalizations based on their specific observations? Were they guided in connecting manipulation and computation? Again, the article suggests that the students were simply presented with the abstract symbols after working with real world examples. In other words, were the teachers skilled in designing and implementing effective manipulative lessons? Pre-existing bias and relative skill with different styles of teaching make a difference.

Not recognized in this study is that students who are taught a number of abstract formulas can remember them poorly, confuse them with each other, unwittingly mix parts of different formulas together, apply them incorrectly, and most importantly, fail to understand the actual meaning behind the formulas. For example, students who are taught several formulas for determining perimeters, circumferences, areas, volumes, and surface areas for different kinds of shapes can still find it confusing to determine the perimeter of a simple rectangle. They are so focused on trying to recall the abstract formula, that they are unable to think about what it actually means, or to check the reasonableness of their calculations.

And what do the researchers think is the purpose of math education? Is the goal just to pass a test? Or is the goal to be able to harness math concepts and skills for use in the real world of objects and actions? If we expect children to learn to apply math in the real world, then would it not be useful to make that connection all along the way?