Transcript Deep Conceptual and Procedural Knowledge

Deep Conceptual and
Procedural Knowledge
of Important Mathematics
for All Students
Deep Conceptual and
Procedural Knowledge
• What is deep?
• Need deep, so that knowledge is stable, longterm, useful.
• Our focus in this presentation … an informal
analysis.
• Examples and nonexamples:
– Puzzles
– Videos (humorous but instructive)
– Classroom vignette
What is deep knowledge?
(Both conceptual and procedural knowledge)
3 Examples
• Puzzles
• Videos – humorous but instructive
• Classroom vignettes
Puzzles - Try These
Relevance to deep knowledge?
Source: “Findings from the field [Cognitive
Science] that are strong and clear enough to
merit classroom application.” Willingham,
Daniel T. How We Learn: Ask the Cognitive
Scientist. American Educator, Winter, 2002.
• Four cards: A 2 X 3
• On each card, there is a letter on one side
and a number on the other.
• You can only see the top side of each card,
as shown above.
• Task: You must verify whether or not the
following rule is true:
If there is a vowel on one side, then there
must be an even number on the other
side.
• What is the minimum number of cards
you must turn over to verify the truth of
this rule?
About 20% of college undergrads
get this right.
• You are a border official at an airport checking
passengers’ papers as they leave the airplane.
• Each traveler carries a card. One side lists
whether the traveler is entering the country or just
in transit. The other side shows exactly which
vaccinations the person has received.
• 4 travelers, each with a card that shows the
following on one side:
Entering, Transit, Cholera-Mumps, Flu-Mumps
• Task: You must make sure any person who is
entering is vaccinated against cholera.
• About which travelers do you need more info?
That is, for which travelers must you check the
other side of the card?
Many more get this correct.
(Even if given separately and
independently of the first puzzle.)
Compare the two puzzles:
• How are they different?
• How are they the same?
Same underlying logical structure:
AB
Card 1: A
Card 4: not B
Vowel  Even Number
Card 1: vowel
Card 4: not even
Entering  Cholera Vaccination
Card 1: entering
Card 4: not cholera
Source
“Findings from the field [Cognitive
Science] that are strong and clear
enough to merit classroom application.”
Willingham, Daniel T. How We Learn:
Ask the Cognitive Scientist. American
Educator, Winter, 2002.
Finding 1
“The mind much prefers that new ideas
be framed in concrete rather than
abstract terms.”
Finding 2
Rote Knowledge
Inflexible Knowledge
Deep Structure Knowledge
Rote Knowledge:
Q:
What is the equator?
A:
A managerie lion running around the
Earth through Africa.
Rote Knowledge:
Q:
Which is bigger: 100 or .001 ?
A: .001
Real answer from a college student.
Reason: “thousands is bigger than
hundreds”)
“We rightly want students to
understand; we seek to train creative
problem solvers, not parrots. Insofar as
we can prevent students from absorbing
knowledge in a rote form, we should do
so. ”
Willingham, Daniel T. How We Learn: Ask the Cognitive
Scientist. American Educator, Winter, 2002
Inflexible Knowledge
• Deeper than rote knowledge, but at the same
time, clearly the student has not completely
mastered the concept.
• Understanding is somehow tied to the surface
features.
• Meaningful, yet narrow.
• The student does not yet have flexibility .
(Knowledge is flexible when it can be
accessed out of the context in which it was
learned and applied in new contexts.)
Inflexible Knowledge
Example: Puzzles
A student can solve both puzzles, but
doesn’t understand them as essentially
the same.
Deep Structure Knowledge
• Deeper than inflexible knowledge
• Transcends specific examples
• Knowledge is flexible -- it can be accessed
out of the context in which it was learned and
applied in new contexts
• Knowledge is no longer organized around
surface forms, but rather is organized around
deep structure
Deep Structure Knowledge
Example: Puzzles
Students understand that both puzzles
are instances of A  B.
(In technical terms, a student
understands if-then implications and
that an implication is equivalent to the
contrapositive. This provides the
solution to the puzzles.)
Finding 3
How To Develop Deep Structure Knowledge
• Direct instruction of deep structure doesn’t
work.
• Use many examples, from many contexts.
• Work with the knowledge, to increase the
store of related knowledge.
• Practice
• Don’t despair of inflexible knowledge, and
don’t confuse it with rote knowledge
What is …
• Deep knowledge 
(one explanation from cognitive science)
• Deep conceptual knowledge
• Deep procedural knowledge
Conceptual and Procedural
• Conceptual knowledge – related to a
concept, like fraction, equation, triangle,
slope, variability, … What is it?
• Procedural knowledge – related to a
procedure, like adding fractions, solving
equations, finding the area of a triangle,
computing the slope of a line, calculating
standard deviation …
How do you operate on it/with it or
compute it?
Knowledge Example?
Ma and Pa Kettle doing long division …
 Video
(Find this two-minute video by searching online.
For example, try:
http://video.google.com/videoplay?docid=710655984
6794044495#)
Ma and Pa Kettle Knowledge
• Procedural knowledge? – They know some
procedures. Do they understand division?
• Not deep procedural knowledge!
• Conceptual knowledge? – Do they
understand the concepts of number,
relationships among numbers, size of
numbers?
• No conceptual knowledge!
Knowledge Example?
• If Joe can paint a house in 3 hours and
Sam can paint a house in 5 hours …
• Video – “Little Big League” math scene
[Find this three-minute video using on online search.
For example, try:
http://www.youtube.com/watch?v=VnOlvFqmWEY]
Now that you’ve seen the movie
and the math problem …
Joe can paint a house in 3
hours, Sam can paint it in 5
hours.
How long does it take for them
to paint it working together?
Analyze, Discuss, Solve
• Answer – Solve it!
– Did they get the right answer in the movie?
– 15? 8? 4? 1 7/8??
• Strategy – How can you solve it?
– What solution strategies did they use?
– What solution strategies could be used?
• Knowledge
– Procedural? Conceptual? Deep?
Knowledge and Strategies in the Movie
(Ever exhibited by your students??)
• No idea. Can’t get started.
– “Math never did make any sense to me. There must be a
formula but I don’t remember it, so I’m stuck.”
– No knowledge; not procedural nor conceptual
• Combine all numbers every which way, hope for the
best and maybe partial credit.
– “Math is about computational procedures, I’m not sure which
one to use, but I’ll give it a shot.”
– Procedural knowledge; superficial.
• Magic formula
– “Math is about formulas. Just memorize and match to the
right problem.
– Procedural knowledge; not sure about depth.
Some Productive
Solution Strategies
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•
•
•
•
•
•
•
•
Think about it
Make sense of it
Estimate
Guess, test, and refine
Draw a (useful) picture
Make a table, look for patterns
Draw and trace a graph
Write and solve an equation
Derive and use a formula
Is there any math in the task?
(concepts and procedures)
•
•
•
•
•
•
Fractions
Proportional reasoning
Computation
Estimation
Solving equations
Multiple representations (equation, graph,
table, diagram)
• Linear functions
Lots of good mathematics!
# of
Fraction of
hours house Sam
paints (5)
1
1/5
Fraction of
house Joe
paints (3)
1/3
Fraction of
house
painted
1/5 + 1/3
< 1 ??
2
2/5
2/3
2/5 + 2/3
> 1 ??
1.5
1.5/5 ??
1/2 ??
< 1 ??
x
x/5
x/3
x/5 + x/3
=1
And the formula in the movie?
Instead of 5 and 3, use a and b:
x/5 + x/3 = 1
 x/a + x/b = 1
So, bx + ax = ab (multiply thru by ab)
and x(a + b) = ab (factor)
Thus, x = ab/(a+b).
So the player remembered the correct
magic formula!
Problem-Based Instructional Tasks
• Help students develop a deep understanding of
important mathematics
• Are accessible yet challenging to all students
• Emphasize connections, especially to the real world
• Encourage student engagement and communication
• Can be solved in several ways
• Encourage the use of connected multiple
representations
• Encourage appropriate use of intellectual, physical,
and technological tools
House Painting Knowledge
• We want deep knowledge of both –
procedures and concepts.
• All too often we achieve only superficial
knowledge of one – procedures.
Fraction interview …
T:
1
2 ÷4=?
S:
1
1
÷4=
2
8
What do we conclude about this
student’s knowledge?
• Got it right! He understands division of
fractions by whole numbers.
• Or, maybe we need more evidence …
Fraction interview …
T:
S:
T:
S:
T:
S:
1/2 ÷ 4 = ?
1/8
How did you get your answer?
invert and multiply
How does that work?
Turn 4 into 4/1, then flip it so it’s 1/4, then
multiply across the top and bottom to get 1/8
Now do we have enough evidence to make a
judgment about the student’s understanding?
T: Why does that work?
S: Because that’s how you divide.
T: What about long division that you’ve done
before?
S: ummm, that’s different, I don’t know, this is
just how you do it with this problem.
T: OK, and you did it really well and got it right.
Can you tell me why you flipped the 4 and
multiplied?
S: Not really, it just works that way.
T: Can you draw a picture of 1/2 ÷ 4?
S: I don’t do pictures.
T: I need to see it though. Can you draw me a
picture?
S: OK… [circle, cut in half horizontally, then
draw 2 perpendicular lines through the top
part, points to one subsection, but then
hesitates, says should be 1/8 so draws 2
more lines vertically in the top part, points to
one of the new subsections, and says 1/8.
T: Well, so that is 1/8 of what?
S: ummm …. [no response]
Time ends. Now what do we conclude …
Let’s debrief …
• Is there a procedure involved in this
problem?
• Is there a concept involved in this
problem?
• S has procedural knowledge? Deep?
• S has conceptual knowledge? Deep?
Procedural Knowledge
of this student
• Can successfully carry out the
procedure of “invert and multiply”
• Can’t explain why it works.
• Can’t explain how it relates to another
division procedure he has learned –
long division.
• Not deep procedural knowledge.
Deep Procedural Knowledge
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Successfully carry out the procedure
Explain the procedure (it makes sense)
Reason about the procedure
Connect to other related procedures
Choose appropriate procedures for the task
at hand
• Flexibly use procedures
• Connect procedures to concepts
Deep Conceptual Knowledge
Concept: Fraction
Student must understand (among other things):
• Fraction consists of top, bottom, whole
• What does the top number mean and how does
it relate to bottom number and the whole?
• What does the bottom number mean and how
does it relate to top number and the whole?
• What is the underlying whole? How does it
relate to top and bottom numbers?
Student does not have deep conceptual
understanding.
Deep Conceptual Knowledge
•
•
•
•
Explain the concept (it makes sense)
Reason about and with the concept
Connect to other related concepts
Choose and apply appropriate concepts for
the task at hand
• Flexibly use concepts
• Connect concepts to procedures
Goal
• We want deep knowledge of both –
procedures and concepts.
• All too often students achieve only
superficial knowledge of one –
procedures.
Deep Conceptual and Procedural
Knowledge – Some References
• Willingham (cognitive science research, 2002)
– Deep structure knowledge
– Contrast with rote and inflexible knowledge
– See earlier slides
• Bloom’s Revised Taxonomy (Anderson, 2001)
– Level 5: Evaluate
– e.g., “Judge which of two methods is the best way to
solve a given problem.”
– Relates to deep procedural knowledge
• National Research Council (review of research)
– How Students Learn, 1999
– Students must develop procedural knowledge along with conceptual
knowledge and understand the connections between the two.
• Liping Ma (mathematics education research)
– Profound Understanding of Fundamental Mathematics
– Topic: Fractions
– See book with this title
• Star (mathematics education research)
– It’s not that conceptual knowledge is good and procedural
knowledge is bad
– Both are valuable
– Both must be deep [and connected]
– “Reconceptualizing Procedural Knowledge,” Journal for Research in
Mathematics Education, November 2005
Deep Conceptual and
Procedural Knowledge
of Important Mathematics
for All Students