We understand new things in the context of things we already know, and most of what we know is concrete.
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Thus it is difficult to comprehend abstract ideas, and difficult to apply them in new situations.The surest way to help students understand an abstraction is to expose them to many different versions of the abstraction—that is, to have them solve area calculation problems about tabletops, soccer fields, envelopes, doors, and so on.
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It’s often difficult for students to understand new ideas, especially ones that are really novel, meaning they aren’t related to other things they have already learned. What do cognitive scientists know about how students understand things?
The answer is that they understand new ideas (things they don’t know) by relating them to old ideas (things they do know). That sounds fairly straightforward. It’s a little like the process you go through when you encounter an unfamiliar word. If you don’t know, for example, what ab ovo means, you look it up in a dictionary. There you see the definition “from the beginning.” You know those words, so now you have a good idea of what ab ovo means.*
The fact that we understand new ideas by relating them to things we already know helps us understand some principles that are familiar to every teacher. One principle is the usefulness of analogies; they help us understand something new by relating it to something we already know about.
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Every new idea must build on ideas that the student already knows.To get a student to understand, a teacher (or a parent or book or television program) must ensure that the right ideas from the student’s long-term memory are pulled up and put into working memory. In addition, the right features of these memories must be attended to, that is, compared or combined or somehow manipulated. For me to help you understand the difference between ordinal and interval measurement, it’s not enough for me to say, “Think of a thermometer and think of a horse race.” Doing so will get those concepts into working memory, but I also have to make sure they are compared in the right way.
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We all know, however, that it’s not really this simple.When we give students one explanation and one set of examples, do they understand? Usually not.
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To dig deeper into what helps students understand, we need to address these two issues. First, even when students “understand,” there are really degrees of comprehension. One student’s understanding can be shallow while another’s is deep. Second, even if students understand in the classroom, this knowledge may not transfer well to the world outside the classroom.That is, when students see a new version of what is at heart an old problem, they may think they are stumped, even though they recently solved the same problem.They don’t know that they know the answer! In the next two sections I elaborate on each issue, that is, on shallow knowledge and on lack of transfer.
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Every teacher has had the following experience:You ask a student a question (in class or perhaps on a test), and the student responds using the exact words you used when you explained the idea or with the exact words from the textbook. Although his answer is certainly correct, you can’t help but wonder whether the student has simply memorized the definition by rote and doesn’t understand what he’s saying.
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Much more common than rote knowledge is what I call shallow knowledge, meaning that students have some understanding of the material but their understanding is limited.We’ve said that students come to understand new ideas by relating them to old ideas. If their knowledge is shallow, the process stops there.Their knowledge is tied to the analogy or explanation that has been provided.They can understand the concept only in the context that was provided. For example, you know that “Seize the day!” means “Enjoy the moment without worrying about the future,” and you remember that the teacher said that “Gather ye rosebuds while ye may” (from Herrick’s To the Virgins, to Make Much of Time) is an example of this sentiment. But you don’t know much more. If the teacher provided a new poem, you would be hard put to say whether it was in the style of a Cavalier poet.
We can contrast shallow knowledge with deep knowledge. A student with deep knowledge knows more about the subject, and the pieces of knowledge are more richly interconnected.The student understands not just the parts but also the whole. This understanding allows the student to apply the knowledge in many different contexts, to talk about it in different ways, to imagine how the system as a whole would change if one part of it changed, and so forth. A student with deep knowledge of Cavalier poetry would be able to recognize elements of Cavalier ideals in other literatures, such as ancient Chinese poetry, even though the two forms seem very different on the surface. In addition, the student would be able to consider what-if questions, such as “What might Cavalier poetry have been like if the political situation in England had changed?” They can think through this sort of question because the pieces of their knowledge are so densely interconnected. They are interrelated like the parts of a machine, and the what-if question suggests the replacement of one part with another. Students with deep knowledge can predict how the machine would operate if one part were to be changed.
Obviously teachers want their students to have deep knowledge, and most teachers try to instill it.Why then would students end up with shallow knowledge? One obvious reason is that a student just might not be paying attention to the lesson.The mention of “rosebuds” makes a student think about the time she fell off her Razor Scooter into the neighbor’s rose bush, and the rest of the poem is lost on her.There are other, less obvious reasons that students might end up with shallow knowledge.
Here’s one way to think about it. Suppose you plan to introduce the idea of government to a first-grade class.The main point you want students to understand is that people living or working together set up rules to make things easier for everyone. You will use two familiar examples—the classroom and students’ homes—and then introduce the idea that there are other rules that larger groups of people agree to live by. Your plan is to ask your students to list some of the rules of the classroom and consider why each rule exists.Then you’ll ask them to list some rules their families have at home and consider why those rules exist. Finally, you’ll ask them to name some rules that exist outside of their families and classroom, which you know will take a lot more prompting.You hope your students will see that the rules for each group of people—family, classroom, and larger community—serve similar functions.
A student with rote knowledge might later report,“Government is like a classroom because both have rules.”The student has no understanding of what properties the two groups have in common.The student with shallow knowledge understands that a government is like a classroom because both groups are a community of people who need to agree on a set of rules in order for things to run smoothly and to be safe.The student understands the parallel but can’t go beyond it. So for example, if asked,“How is government different from our school?” the student would be stumped. A student with deep knowledge would be able to answer that question, and might successfully extend the analogy to consider other groups of people who might need to form rules, for example, his group of friends playing pickup basketball.
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So our minds assume that new things we read (or hear) will be related to what we’ve just read (or heard).This fact makes understanding faster and smoother. Unfortunately, it also makes it harder to see the deep structure of problems.That’s because our cognitive system is always struggling to make sense of what we’re reading or hearing, to find relevant background knowledge that will help us interpret the words, phrases, and sentences. But the background knowledge that seems applicable almost always concerns the surface structure.
READ MORE:
Styles, Abilities and Multiple Intelligences
Why Is It Hard to Make Students Think Like Experts?
Why Do Students Remember Everything That’s on Television and Forget Everything I Say?
Factual knowledge must precede skill
Why Don't Students Like School?
