By: Cindy Nebel

The use of concrete examples is one of the six strategies for effective learning that we discuss throughout this website. Our goal is to present strategies that are evidence-based and we have blogs devoted to the effectiveness of concrete examples (here, here, and here) but we have discussed before that these strategies do not work in every situation. We have blogs about times when retrieval practice might not work (here and here) as well as other learning strategies, such as writing or the use of manipulatives that do not always work the way that you might think.

In 2008, researchers Kaminsky, Sloutsky, and Heckler, demonstrated that concrete examples might also not always be beneficial for learning (1). The research was picked up by popular news articles with titles such as, “Concrete Examples Don’t Help Students Learn Math, Study Finds” (2). Well, perhaps we should delete concrete examples from our downloadable materials! Or perhaps not…

Let’s look more closely at the study by Kaminsky et al. They were particularly interested in transfer, or the idea that students should be able to use the knowledge they acquire in novel situations. Megan talked about why concrete examples are good for transfer in this blog post. Essentially, the benefits of using lots of examples are that students can see the commonalities between the examples, demonstrating the underlying concept. But, concrete examples can hinder learning if the surface details are too distracting for students to realize the underlying concept, which is discussed in more detail here. Kaminsky et al. taught students a new mathematical concept using either abstract symbols or concrete examples of real-life situations. Here is a figure showing their materials:

On the left is a generic, abstract symbol language which shows the rules associated with this mathematical concept. On the right, you see an example of one of the real-life situations in which measuring cups are combined and a remaining amount is leftover. There were two other real-life situations – slices of pizza or tennis balls in a container – which followed the same rules as the measuring cup example. Participants were then divided into groups who received either the generic condition or either one, two, or three of the concrete examples. After training on these tasks, participants watched a game in which a child was shown two objects and had to point to the third “correct” object based on the rules of the task they had just completed. After watching this game played, they then took a multiple-choice test to see if they had learned the rules of the game.

Participants in the generic group performed very well, whereas individuals in the other three conditions performed poorly and not statistically better than if they’d been randomly guessing. Uh oh. Should we, as the news article suggests, stop using concrete examples in all math classes?

I believe that we have here another example of trying to use a learning strategy the wrong way. As Megan described in her blog post about concrete examples, an important feature of using examples is that the surface details need to vary or students will pay attention to these details and ignore the underlying meaning. In the Kaminsky et al. study, all of the concrete examples involved a similar surface feature. You add the two items together and produce a remainder. The underlying rule of the abstract concept did not contain this surface detail. Therefore students learn that the surface detail IS the concept and are unable to transfer to situations where that surface detail no longer exists.

The other issue that I believe may matter in this study is one of practical importance. In a later experiment, they either gave students just the generic symbols or they gave them a concrete example followed by the generic symbols. Learning was higher without the example. However, there’s an important condition missing. Students were never given the generic symbols followed by a concrete example. At least in my own classes, I teach the abstract concept and then provide examples of it instead of the other way around. It is possible that students are less likely to focus on the surface details if they already have a rudimentary understanding that there is an underlying concept.

Take Away Points

1. Don’t believe catchy news titles. Always go and read the original research yourself with a critical eye. In this case, people might erroneously believe that concrete examples should never be used in math, but the implications from this study are not nearly so profound.

2. Use varied examples. We know that students tend to pay attention to surface details, so if all of your examples are similar, students will probably believe that those similarities are a part of the concept. Try to use examples that are very different.

3. The order (might) matter. There have been lots of studies done looking at how to bridge concrete to abstract understanding, but the jury is still out on whether higher education math classes might benefit from explicitly teaching abstract concepts before demonstrating their utility in the real world.

References:

(1) Kaminsky, J.A., Sloutsky, V.M., & Heckler, A.F. (2008). The advantage of abstract examples in learning math. Science, 320, 454-455.

(2) Ohio State University. (2008, April 25). Concrete examples don't help students learn math, study finds. ScienceDaily. Retrieved from www.sciencedaily.com/releases/2008/04/080424140410.htm