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Teaching for Understanding - Part 2
Aug 29, 2010
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Understanding by design (not “teaching”)


We do not “teach” an understanding; nor do we just hope that students will somehow “discover” the understanding. Rather, we “facilitate” the understanding by (our) design. We organize work and experiences so that students will likely come to the inference themselves.

The best teachers do this all the time: Piaget famously said, “To understand is to invent”. – even in basic mathematics:

Once the child is capable of repeating certain notions and using some applications of these in learning situations he often gives the impression of understanding; however….true understanding manifests itself by new spontaneous applications…. The real comprehension of a notion or a theory implies the reinvention of this theory by the student. P. 731 in The Essential Piaget.


The same thought was more recently expressed by a Professor identified in Ken Bain’s national study of exemplary college teachers, a mathematics teacher from University of California - Irvine:


"I want the students to feel like they have invented calculus and that only some accident of birth kept them from beating Newton to the punch," Donald Saari, a mathematics professor at the University of California at Irvine, told us. Unlike so many in his discipline, he does not simply perform calculus in front of the students; rather, he raises the questions that will help them reason through the process, to see the nature of the questions, and to think about how to answer them." 

The Chronicle of Higher Education

Section: The Chronicle Review

Volume 50, Issue 31, Page B7


The challenge for you, in sum, is to make the student conclude what you wrote in the Understanding box.


How does this work in practice? Here's a simple outline of the process:


The Understanding

If that's the conclusion, what are the steps in the argument?

What experiences are required if someone is going to infer the Understanding?

What Essential Questions and other prompting questions will help them make the right inferences? 


Here is an example, using an Understanding from Part 1:


Understanding: Different expressions (e.g. equations using fractions or decimals) can represent the same quantities; the goal, context, and ease of use determine the best choice.


Steps in the argument:

  1. The same quantity can be expressed as a fraction, decimal, or percent.
  2. We have a choice as to whether to use fractions, decimals, or percentages in framing a problem and solution.
  3. The choice can be made in two ways: what is easiest for us in solving the problem, and what is most helpful to the audience for understanding the conclusion.
  4. Sometimes what is easiest for us is not best for the audience ande vice versa, so we may need to translate our result into the other kind of expression.
  5. In the real world, the context of a problem often suggests which form of expression to use.
  6. While there is no final “right” answer as to which approach to use, some approaches are easier or clearer than others.


THEREFORE THEY SHOULD SEE THAT... Different expressions (e.g. equations using fractions or decimals) can represent the same quantities; the goal, context, and ease of use determine the best choice.


Required Experiences

Students would need to be confronted with many situations in which sometimes decimals make the best sense, other times factions, and other times percentages. But it should become less and less obvious which choice would be best!

Then, they would have to solve problems in which they would make the best choice and defend it.

Essential Questions: In this context, how should the answers be expressed?



Convergent vs. Divergent Understanding

Some Understandings are relatively stable and accepted as “true” while others are often changing over time as we learn more. In other words, sometimes we often want the students to come to their own understandings and we expect the understanding to change over the course of reading, discussion and reflection. In short, while many academic Understandings require convergent work, more personal Understandings can and should demand divergent thinking. It is quite correct to say that often we disagree about our (personal) Understanding.

In the case of convergent understanding, the challenge is to help the student see and follow the reasoning and evidence that led to the original expert conclusion. So, in math, biology or economics the student challenge is not to “create” a new theorem, experimental result, or formula about prices but to follow (or even simulate) the experience and reasoning of those who achieved such understandings.

This is necessary not only to fully understand the Understanding but to avoid common misconceptions. In almost every case, the modern disciplinary conclusions in science, mathematics, history, etc. are NOT obvious. The evolution of animals is counter-intuitive. The Pythagorean Theorem and Newton’s second Law do not appear to be true when we observe the facts. It seems inconceivable now that our Founders could have limited the vote to white males who owned property. (Think about it: why would it have taken such genius to come up with modern ideas if they were so obvious?).


Though the student needs to “see” and “grasp” the logic that leads to the official understanding, it doesn’t follow that teaching is just a dull slog through claims and proofs. On the contrary, the best teaching for understanding involves putting the students in the position of working to infer conclusions themselves, with clever help on the part of teacher-designers, as we noted above. That is why, for example, students in law, medicine, engineering, business and other such fields learn via the problem-based or case-based method: students are given simplified versions of the real work in the field, to prepare them for the kind of thinking they will need and to help them understand current Understanding in those fields.


In the case of divergent Understandings, however, we want students to come to their own personal understanding, for example about the meaning of historical events, a person’s life, or the cost-benefit analysis of a scientific advance. Then, we face a new challenge: we have to impress upon the student that not all opinions reflect Understanding. The student must come to realize that a sound understanding is not just a strong personal opinion but a valid conclusion reached after a gathering and considering of many facts, woven into a logical argument that supports the conclusions reached. As I used to say to my HS students: "You need to come to realize that while everyone is entitled to their opinions, and while there is no final truth about the meaning of text, some ideas about the text are better than others. Your job is to understand that distinction!"

How should we write divergent Understandings in Stage 1? How should we teach for personal understanding? We discuss this in Part 3 next time. 


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Displaying 1 - 2 of 2 found comments.
Posted by: Julie
Oct 23, 2010
The distinction between divergent and convergent understandings is powerful. Are divergent understandings tied to developmental levels? How do you determine what divergent understandings are appropriate?
Posted by: Debbie Grafil from the Philippines
Oct 25, 2010
While it is true that everyone is entitled to his/her own opinion,I believe teachers must also help students generate or formulate responsible opinions - opinions that are based from processing of information or "meaning making" designed by teachers to happen in the teaching-learning process. The processing of information may include seeing things from the points of view of other people,empathizing with other people,weighing pros and cons,making a discernment or value judgment. Doing this will help students in making responsible decisions in their day to day life.
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