deep conceptual and procedural knowledge of important mathematics for all students
TRANSCRIPT
Deep Conceptual and Procedural Knowledge
of Important Mathematicsfor All Students
Deep Conceptual and Procedural Knowledge
• What is deep? • Need deep, so that knowledge is stable, long-
term, 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 BCard 1: ACard 4: not B
Vowel Even NumberCard 1: vowel Card 4: not even
Entering Cholera VaccinationCard 1: enteringCard 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: PuzzlesStudents 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=7106559846794044495#)
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 ProductiveSolution Strategies
• 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 functionsLots of good mathematics!
# of hours
Fraction of house Sam paints (5)
Fraction of house Joe paints (3)
Fraction of house painted
1 1/5 1/3 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: ÷ 4 = ?
1
2
S: ÷ 4 =
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 …
1
8
1
2
Fraction interview …
T: 1/2 ÷ 4 = ?
S: 1/8
T: How did you get your answer?
S: invert and multiply
T: How does that work?
S: 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 Knowledgeof 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
• 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 KnowledgeConcept: 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 Mathematicsfor All Students