Chapter 2.
Understanding Understanding
The most
characteristic thing about mental life, over and beyond the fact that one
apprehends the events of the world around one, is that one constantly goes
beyond the information given.
—Jerome Bruner, Beyond the Information Given, 1957, p. 218
Education. That which discloses to the wise and
disguises from the foolish their lack of understanding.
—Ambrose Bierce, The Devil's Dictionary, 1881–1906
This book explores two different but related
ideas: design and understanding. In the previous chapter we explored good
design in general and what the template specifically calls for. But before we
can go into depth about the template, we need to step back and consider the
other strand of the book—understanding. Bob James was a bit confused about
“understandings.” His confusion turns out to be a fairly common problem. When
we ask designers in workshops to identify desired understandings and thus to distinguish
between desired “knowledge” and “understanding,” they are often puzzled. What's
the difference? What is understanding? And so we pause to
consider a question that turns out to be essential: How well do we understand
understanding? What is it we are after when we say we want students to
understand this or that? Until now, we have written about understanding as if
we fully understood what we were after. But as we shall see, the irony is that
though we all claim as teachers to seek student understanding of the content, we may
not adequately understand this goal. This may seem like an odd claim. Teachers
knowingly aim for understanding every day, don't they? How can we not know what
we are aiming for? Yet plenty of evidence suggests that “to understand” and “to
teach for understanding” are ambiguous and slippery terms.
We see some of this conceptual uncertainty in the Taxonomy of
Educational Objectives: Cognitive Domain. The book was written in 1956 by
Benjamin Bloom and his colleagues to classify and clarify the range of possible
intellectual objectives, from the cognitively easy to the difficult; it was
meant to classify degrees of understanding, in effect. As the authors often
note, the writing of the book was driven by persistent problems in testing:
Just how should educational objectives or teacher goals be measured in light of
the fact that there was (and is) no clear meaning of, or agreement about the
meaning of, objectives such as “critical grasp of” and “thorough knowledge
of”—phrases that have to be used by test developers?
In the introduction to the Taxonomy, Bloom (1956) and his colleagues refer
to understanding as a commonly sought but ill-defined
objective:
For example, some teachers believe their students should
“really understand,” others desire their students to “internalize knowledge,”
still others want their students to “grasp the core or essence.” Do they all
mean the same thing? Specifically, what does a student do who “really
understands” which he does not do when he does not understand? Through
reference to the Taxonomy . . . teachers should be able to define such nebulous
terms. (p. 1)
Recall that when our health teacher, Bob James, was thinking about his
nutrition unit (see Chapter 1), he seemed unsure about what an understanding was
and how it differed from knowledge. In fact, two generations of
curriculum writers have been warned to avoid the term understand in
their frameworks as a result of the cautions in the Taxonomy. For example, in
the Benchmarks for Science Literacy from the American
Association for the Advancement of Science (AAAS), the authors succinctly
describe the problem they faced in framing benchmarks for science teaching and
assessing:
Benchmarks uses “know” and “know how” to lead into each set of
benchmarks. The alternative would have been to use a finely graded series of
verbs, including “recognize, be familiar with, appreciate, grasp, know,
comprehend, understand,” and others, each implying a somewhat greater degree of
sophistication and completeness than the one before. The problem with the
graded series is that different readers have different opinions of what the
proper order is. (1993, p. 312)
Yet the idea of understanding is surely distinct from the
idea of knowing something. We frequently say things like, “Well,
he knows a lot of math, but he doesn't really understand its basis,” or, “She
knows the meaning of the words but doesn't understand the sentence.” A further
indication is that, 50 years after Bloom, many state standards now specify
understandings separate from knowledge. Consider these examples from the
California standards in science, which make the distinction explicit, with knowledgesubsumed
under the broader understanding:
Newton's laws predict the motion of most objects. As a
basis for understanding this concept:
a.
Students know how to solve problems that
involve constant speed and average speed.
b.
Students know that when forces are balanced,
no acceleration occurs; thus an object continues to move at a constant speed or
stays at rest (Newton's first law).
c.
Students know how to apply the law F = ma to
solve one-dimensional motion problems that involve constant forces (Newton's
second law).
d.
Students know that when one object exerts a
force on a second object, the second object always exerts a force of equal
magnitude and in the opposite direction (Newton's third law). . . .
Scientific progress is made by asking meaningful
questions and conducting careful investigations. As a basis for understanding
this concept and addressing the content in the other four strands, students
should develop their own questions and perform investigations. Students will:
a.
Select and use appropriate tools and
technology (such as computer-linked probes, spreadsheets, and graphing
calculators) to perform tests, collect data, analyze relationships, and display
data.
b.
Identify and communicate sources of
unavoidable experimental error.
c.
Identify possible reasons for inconsistent
results, such as sources of error or uncontrolled conditions. . . .
Although we might quibble as to whether the statement “Scientific progress
is made by asking meaningful questions and conducting careful investigations”
is a concept, the implication of the standard is clear enough: An
understanding is a mental construct, an abstraction made by the human mind to make
sense of many distinct pieces of knowledge. The standard further suggests that
if studentsunderstand, then they can provide evidence of that
understanding by showing that they know and can do certain specific things.
Understanding as meaningful inferences
But how are understanding and knowledge related? The standard still leaves
the relationship murky in the phrase “As a basis for understanding this concept
. . .” Is understanding simply a more complex form of knowledge, or is it
something separate from but related to content knowledge?
Making matters worse is our tendency to use the terms know, know
how, and understand interchangeably in everyday speech.
Many of us would say that we “know” that Newton's Laws predict the motion of
objects. And we may say we “know how” to fix our car and “understand” how to
fix our car as if the two statements expressed the same idea. Our usage has a
developmental aspect, too: What we once struggled to “understand” we say we now
“know.” The implication is that something that once required a chain of
reasoning to grasp hold of no longer does: We just “see it.”
Mindful of our tendency to use the words understand and know interchangeably,
what worthy conceptual distinctions should we safeguard in talking about the
difference between knowledge and understanding? Figure 2.1 presents some useful
distinctions between the terms.
Figure 2.1. Knowledge Versus Understanding
|
Knowledge |
Understanding |
|
·
The facts ·
A body of coherent facts ·
Verifiable claims ·
Right or wrong ·
I know something to be true ·
I respond on cue with what I know |
·
The meaning of the facts ·
The “theory” that provides coherence and
meaning to those facts ·
Fallible, in-process theories ·
A matter of degree or sophistication ·
I understand why it is, what makes it knowledge ·
I judge when to and when not to use what I
know |
John Dewey (1933) summarized the idea most clearly in How We Think.
Understanding is the result of facts acquiring meaning for the learner:
To grasp the meaning of a thing, an event, or a situation is to see it in
its relations to other things: to see how it operates or functions, what
consequences follow from it, what causes it, what uses it can be put to. In
contrast, what we have called the brute thing, the thing without meaning to us,
is something whose relations are not grasped. . . . The relation of means-consequence is
the center and heart of all understanding. (pp. 137, 146)
Consider an analogy to highlight these similarities and differences:
tiling a floor with only black and white tiles. All our factual knowledge is
found in the tiles. Each tile has definite traits that can be identified with
relative precision and without much argument. Each tile is a fact. An
understanding is a pattern visible across many tiles. There are many different
patterns, some of them encompassing many or few tiles. Aha! Suddenly we see
that small patterns can be grouped into sets of larger patterns—that was not
apparent to us at first. And you may see the patterns differently than we do,
so we argue about which is the “best” way to describe what we see. The pattern
is not really “there” in an important sense, then. We infer it; we project it
onto the tiles. The person laying the tiles merely positioned a black one next
to a white one; he need not have had any pattern in mind: We may be the first
to have seen it.
Let's move the analogy closer to intellectual life. The words on the page
are the “facts” of a story. We can look up each word in the dictionary and say
we know it. But the meaning of the story remains open for discussion and
argument. The “facts” of any story are the agreed-upon details; the
understanding of the story is what we mean by the phrase “reading between the
lines.” (The author may not have “meant” what we can insightfully “infer”—just
as in the tiling example; this is one of the debates in modern literary
criticism—which view, if any, is privileged.) A well-known example from
literacy studies makes the point elegantly:
First you arrange things into groups. Of course one pile
may be enough, depending on how much there is to do; but some things definitely
need to be separated from the others. A mistake here can be expensive; it is
better to do too few things at once than too many. The procedure does not take
long; when it is finished, you arrange the things into different groups again,
so that they can be put away where they belong. (Bransford & Johnson, 1972,
in Chapman, 1993, p. 6)
As a writer referring to this passage notes in a book on critical reading
skills,
There is a point which varies depending on the individual
reader, at which readers who monitor their own understanding realize that they
are not “getting it” even though they know the meanings of all the words, the
individual sentences make sense, and there is a coherent sequence of events. .
. . At that point, critical readers who want to understand typically slow down,
sharpen their attention, and try different reading strategies. (Chapman, 1993,
p. 7)
The first passage is a vague account of doing laundry. More generally, the
goal in understanding is to take whatever you are given to produce or find
something of significance—to use what we have in memory but to go beyond the
facts and approaches to use them mindfully. By contrast, when we
want students to “know” the key events of medieval history, to be effective
touch typists, or to be competent players of specific musical pieces, the focus
is on a set of facts, skills, and procedures that must be “learned byheart”—a
revealing phrase!
Understanding thus involves meeting a challenge for thought. We encounter
a mental problem, an experience with puzzling or no meaning. We use judgment to
draw upon our repertoire of skill and knowledge to solve it. As Bloom (1956)
put it, understanding is the ability to marshal skills and facts wisely and
appropriately, through effective application, analysis, synthesis, and
evaluation. Doing something correctly, therefore, is not, by itself, evidence
of understanding. It might have been an accident or done by rote. To understand
is to have done it in the right way, often reflected in being able to explainwhy a
particular skill, approach, or body of knowledge is or is not appropriate in a
particular situation.
Understanding as transferability
It would be impossible to over-estimate the educational
importance of arriving at conceptions: that is, meanings that are general
because applicable in a great variety of different instances in spite of their
difference. . . . They are known points of reference by which we get our
bearings when we are plunged into the strange and unknown. . . . Without this
conceptualizing, nothing is gained that can be carried over to the better
understanding of new experiences.
—John Dewey, How We Think, 1933, p. 153
Baking without an understanding of the ingredients and
how they work is like baking blindfold[ed] . . . sometimes everything works.
But when it doesn't you have to guess at how to change it. . . . It is this
understanding which enables me to both creative and successful.
—Rose Levy Berenbaum, The Cake Bible, 1988, p. 469
To know which fact to use when requires
more than another fact. It requires understanding—insight into essentials,
purpose, audience, strategy, and tactics. Drill and direct instruction can
develop discrete skills and facts into automaticity (knowing “by heart”), but
they cannot make us truly able.
Understanding is about transfer, in other words. To be truly
able requires the ability to transfer what we have learned to new and sometimes
confusing settings. The ability to transfer our knowledge and skill effectively
involves the capacity to take what we know and use it creatively, flexibly,
fluently, in different settings or problems, on our own. Transferability is not
mere plugging in of previously learned knowledge and skill. In Bruner's famous
phrase, understanding is about “going beyond the information given”; we can
create new knowledge and arrive at further understandings if we have learned
with understanding some key ideas and strategies.
What is transfer, and why does it matter? We are expected to take what we
learned in one lesson and be able to apply it to other, related but different
situations. Developing the ability to transfer one's learning is key to a good
education (see Bransford, Brown, & Cocking, 2000, pp. 51ff). It is an
essential ability because teachers can only help students learn a relatively
small number of ideas, examples, facts, and skills in the entire field of
study; so we need to help them transfer their inherently limited learning to
many other settings, issues, and problems.
Consider a simple example from sports. When we grasp the idea that on
defense we need to close up available space for the offense, we can use that
understanding to adapt to almost any move members of the other
team make, not just be limited to the one or two positionings we were taught in
a three-on-three drill. We can handle entire classes of offensive problems, not
just familiar instances. Failure to grasp and apply this idea in context is
costly:
“When I got the ball in midfield and I started
dribbling,” said Lavrinenko, the [NCAA men's soccer] championship tournament's
outstanding offensive player, “I was looking to pass right away. But my
teammates opened up space, and I continued running. When I played the ball to
Alexei, 2 players went to him and opened up more space for me.” (New York
Times, December 13, 1999, sec. D, p. 2)
And because the big idea of “constraining offensive space” transfers across
sports, it is equally applicable in soccer, basketball, hockey, water polo,
football, and lacrosse. The same is true in math or reading: To get beyond mere
rote learning and recall, we have to be taught and be assessed on an ability to
see patterns, so that we come to see many “new” problems we encounter as
variants of problems and techniques we are familiar with. That requires an
education in how to problem solve using big ideas and transferable strategies,
not merely how to plug in specific facts or formulas.
Big ideas are essential because they provide the basis for the transfer.
You must learn that a single strategy underlies all possible combinations of
specific moves and settings, for example. The strategy is to get someone on
your team open, using various moves and fakes—regardless of what the other team
does or whether it looks exactly like what you did in practice. In academics,
you must learn to transfer intellectual knowledge and skill:
Transfer is affected by the degree to which people learn
with understanding rather than merely memorize sets of facts or follow a fixed
set of procedures. . . . Attempts to cover too many topics too quickly may
hinder learning and subsequent transfer. (Bransford, Brown, & Cocking,
2000, pp. 55, 58)
This is an old idea, famously framed by Whitehead (1929) almost 100 years
ago in his complaint about “inert ideas” in education:
In training a child to activity of thought, above all
things we must beware of what I will call “inert ideas”—that is to say, ideas
that are merely received into the mind without being utilized or tested, or
thrown into fresh combinations. . . . Education with inert ideas is not only
useless: it is above all things, harmful. . . . Let the main ideas which are
introduced be few and important, and let them be thrown into every combination
possible. (pp. 1–2)
In reading, we may not have previously read this book by this author,
but if we understand “reading” and “romantic poetry,” we transfer our prior
knowledge and skill without much difficulty. If we learned to read by repeated
drill and memorization only, and by thinking of reading as only decoding, making
sense of a new book can be a monumental challenge. The same is true for
advanced readers at the college level, by the way. If we learned to “read” a
philosophy text by a literal reading, supplemented by what the professor said
about it, and if we have not learned to actively ask and answer questions of
meaning as we read, reading the next book will be no easier. (For more on this
topic, see Adler and Van Doren, 1940.)
Transfer is the essence of what Bloom and his colleagues meant by
application. The challenge is not to “plug in” what was learned, from memory,
but modify, adjust, and adapt an (inherently general) idea to the particulars
of a situation:
Students should not be able to solve the new problems and
situations merely by remembering the solution to or the precise method of
solving a similar problem in class. It is not a new problem or situation if it
is exactly like the others solved in class except that new quantities or
symbols are used. . . . It is a new problem or situation if the student has not
been given instruction or help on a given problem and must do some of the
following. . . . 1. The statement of the problem must be modified in some way
before it can be attacked. . . . 2. The statement of the problem must be put in
the form of some model before the student can bring the generalizations
previously learned to bear on it. . . . 3. The statement of the problem
requires the student to search through memory for relevant generalizations.
(Bloom, Madaus, & Hastings, 1981, p. 233)
Knowledge and skill, then, are necessary elements of understanding, but
not sufficient in themselves. Understanding requires more: the ability to
thoughtfully and actively “do” the work with discernment, as well as the
ability to self-assess, justify, and critique such “doings.” Transfer involves
figuring out which knowledge and skill matters here and often
adapting what we know to address the challenge at hand.
Here's an amusing transfer task to illustrate the point one more time. See
if you can use your knowledge of French pronunciation and English rhymes to
“translate” the following song. Say it out loud, at a normal speaking speed:
Oh, Anne, doux
But. Cueilles ma chou.Trille fort,
Chatte dort.Faveux Sikhs,
Pie coupe Styx.Sève nette,
Les dèmes se traitent.N'a ne d'haine,
Écoute, fée daine.*
* This is a fake song in which French words are used to make up a familiar
rhyme about numbers—in accented English. Hint: the book is called Mots
d'Heures: Gousses, Rames. (Mots d'Heures: Gousses Rames, by Luis
d'Antin Van Rooten [Penguin Books, 1980; first published by Grossman
Publishers, 1967]).
All of the cases we've discussed here illustrate the importance of
confronting students with a real problem for thought if understanding is to be
called for and awakened. This is very different from giving students lessons
and tests that merely require taking in and recalling from memory, based on
highly cued exercises in which learners simply plug in what is unambiguously
required. (See Chapters 6 through 8 for further discussions on crafting
understandings and meaningful assessments.)
The failure of even our best students to transfer their learning is
evident in many areas but is most striking in mathematics. Consider the
following examples of test items, all of which are testing the same idea (in
each case, approximately two-thirds of the tested students did
not correctly answer the question):
From the New York State Regents Exam:
To get from his high school to his home, Jamal travels
5.0 miles east and then 4.0 miles north. When Sheila goes to her home from the
same high school, she travels 8.0 miles east and 2.0 miles south. What is the
measure of the shortest distance, to the nearest tenth of a mile, between
Jamal's home and Sheila's home? (The use of the accompanying grid is optional.)
From the NAEP 12th grade mathematics test:
What is the distance between the points (2,10) and (-4,
2) in the xy plane?
6
8
10
14
18
From a Boston Globe article on the Massachusetts MCAS
10th grade math scores:
The hardest question on the math section, which just 33
percent got right, asked students to calculate the distance between two points.
It was a cinch—if students knew that they could plot the points and use the
Pythagorean theorem, a well-known formula to calculate the hypotenuse of a right
triangle if the lengths of two legs are given. The sixth-hardest math question,
which only 41 percent of students got right, also required use of the
Pythagorean theorem. “It seems applying the Pythagorean theorem was a weakness
for kids,” said William Kendall, director of math for the Braintree public
schools. “These weren't straightforward Pythagorean theorem questions. They had
to do a little bit more.” (Vaishnav, 2003)
All three problems require students to transfer their understanding of the
Pythagorean theorem to a new situation. It is likely that most students in the
United States could not do it, despite the fact that every set
of state standards identifies a grasp of the Pythagorean theorem as a key
desired result.
We can apply our understanding to this news without too
much difficulty, based on what has been said thus far. We surmise that the A2 +
B2 = C2 theorem is taught as a fact, a rule for
making certain calculations when confronted with a known right triangle and
simple tasks. Remove a few blatant cues, however, and students cannot transfer
their learning to perform with understanding. Is it any wonder, then, that
students do not understand what they supposedly know? And
what few educators seem to realize, therefore, is that drilling students for
state tests is a failing strategy.
Understanding as a noun