Blogging Note - it's somewhat frowned upon in the blogosphere to erase your mistakes once an item has been posted - frequently the mistake will be caught by readers and commented upon. If you erase the mistake, the comments become confusing or meaningless. Hence the common practice of using strike through text and adding the correction afterward. I do make minor edits for grammar, spelling, or clarity without notification.
I'm going to give chapters 4,5, & 6 relatively short summaries. The information is, I think, pretty well known and reasonably uncontroversial. In chapter 7, on the other hand, the author explodes some myths about "multiple intelligences" and "learning styles" that will surely raise some eyebrows. Chapter 8 addresses differentiated instruction head on, and chapter 9 deals with teacher self-reflection and professional development (from a personal standpoint as opposed to an outside mandate).
Willingham sums up chapter 4 as follows:
We understand new things in the context of things we already know, and most of what we know is concrete.In science education this idea has been the cornerstone of virtually every program and class that I've been involved with. The whole constructivist approach is in part built upon (and perhaps takes a little t0o far sometimes) the idea that our traditional way of teaching science is wrong precisely because we typically start out teaching abstract ideas first and then use concrete experiences only later and sporadically to illustrate those abstract ideas.
For example, to take the concept that Willingham uses, Newton's laws of motion are sometimes taught first as a series of abstract statements (an object at rest tends to stay at rest, an object in motion tends to stay in motion, etc.) or even a more abstract mathematical expression of the ideas (F=ma). Maybe later if the students are lucky they will be given a lab activity to illustrate the idea, and maybe the lab will have the intended effect or not, depending on how well it is set up and how good the equipment is and how seriously the students actually think about the consequences of the lab.
The constructivist approach, and the part of it that is more or less supported by Willingham's research, suggests a better way might be to turn this model on its head and begin a unit of study on Newton's laws with a series of concrete experiences that students can then think about and relate to the abstract concepts of motion described by Newton. Where Willingham might part ways with the strict constructivist approach (not that he discusses it, I'm just inferring here) is in allowing that we can simply use previous concrete experiences, tap into the prior knowledge that students have stored in memory, rather than having to come up with a novel hands-on, concrete experience for every new idea we present. F = ma is an abstract concept that doesn't make intuitive sense until you use a couple of examples; compare hitting a baseball with a bat and hitting a car with a bat - obviously the car will not move much (the a or acceleration in the formula) compared to the baseball because of the different masses of the two objects. Stated that way it is "intuitive" because we all have the concrete experience of trying to move objects of different masses.
The point is that in order for students to be able to understand the abstract laws, they must relate them in some way to concrete experiences. And this is itself a universalized law - ALL abstraction is built upon a foundation of the concrete world and physical experience.
Chapter 4 also deals with the related difficulty of knowledge transfer. Having described a situation above (the baseball and the car example) and hearing the familiar chorus of "ohhhh, I get it," you might think it would be a simple matter to then have the students apply the law to a similar problem, let's say throwing a baseball versus throwing a softball. It is entirely possible, however, that a student with limited experience would not recognize that the key element of the first scenario is the mass of the objects. Instead the student might get hung up on the fact that a ball is a small spherical object whereas the car is a vehicle with wheels, or the use of a bat in the first scenario might make them think that the use of a throwing arm in the second scenario requires a completely different set of rules. In other words, a student with shallow knowledge might not know which elements of the scenario to generalize or transfer.
Implications for teaching
Recognize that for many students a single concrete example will not suffice to allow them to generalize a rule or concept. Provide as many different examples as possible so the student begins to see the pattern and can identify key elements.
Make sure that "understanding" (deep knowledge) is incorporated into every aspect of your teaching, from homework to class activities to assessments. Especially assessments. If your assessments are testing shallow knowledge, that's what students will focus on.
Be realistic. At any particular level of education, there will be limits to how deeply a student will be able to achieve understanding. Sometimes we have to accept that we are simply planting the seed of an idea that
Next: Chapter 5, Is drilling worth it? picks up and expands on the question of how to help students achive deep understanding.
This is a topic that I am also struggling with in math.Willingham's notion about concrete and abstract ideas is quite true. Students understand better when they do or see something rather than when they are told about it. But how about those situations when there is not much to do or see? Will drilling do and how long will we drill?I have a felling that with experience sudents will be able to overcome their inability to understand something and uplift the curtain between concrete and abstract ideas(Ofcourse ,when examples are proved for students to see the pattern)This is what I am waiting to read next.
ReplyDelete"Drilling" is just a derogatory term for practice. There's no question that some practice is essential for success in any field - but more on that later in Chapter 5.
ReplyDeleteFor now, I'll stick with the concrete/abstract theme. I don't have much experience with math teaching, but I do know there was a movement a while back to make math more hands-on and concrete, at least in the lower grades (see "math wars"). It may be that the difficulties many student have with high school mathematics (and high school academics in general, really) stems in part from lack of exposure to the concrete foundations in elementary school.
As for what to do now, you can make the abstract concrete through analogies or models when the actual substance is impossible to see directly (think of the structure of an atom). Of course, it might be helpful to have this discussion around a "concrete" example from your experience :-)