# Deep Conceptual and Procedural Knowledge of Important Mathematics for All Students

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<ul><li><p>Deep Conceptual and Procedural Knowledgeof Important Mathematicsfor All Students</p></li><li><p>Deep Conceptual and Procedural KnowledgeWhat is deep? Need deep, so that knowledge is stable, long-term, useful.Our focus in this presentation an informal analysis.Examples and nonexamples:PuzzlesVideos (humorous but instructive)Classroom vignette</p></li><li><p>What is deep knowledge? (Both conceptual and procedural knowledge)</p></li><li><p>3 ExamplesPuzzlesVideos humorous but instructiveClassroom vignettes</p></li><li><p>Puzzles - Try TheseRelevance to deep knowledge?</p><p>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.</p></li><li><p> Four cards: A 2 X 3On 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?</p></li><li><p> About 20% of college undergrads get this right.</p></li><li><p> 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-MumpsTask: 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?</p></li><li><p>Many more get this correct.(Even if given separately and independently of the first puzzle.)</p></li><li><p>Compare the two puzzles:How are they different?How are they the same?</p></li><li><p>Same underlying logical structure:</p><p>A B Card 1: A Card 4: not B</p><p>Vowel Even Number Card 1: vowel Card 4: not even</p><p>Entering Cholera Vaccination Card 1: entering Card 4: not cholera</p></li><li><p>SourceFindings from the field [Cognitive Science] that are strong and clear enough to merit classroom application.</p><p>Willingham, Daniel T. How We Learn: Ask the Cognitive Scientist. American Educator, Winter, 2002.</p></li><li><p>Finding 1The mind much prefers that new ideas be framed in concrete rather than abstract terms.</p></li><li><p>Finding 2Rote KnowledgeInflexible KnowledgeDeep Structure Knowledge</p></li><li><p> Rote Knowledge: </p><p>Q: What is the equator?</p><p>A: A managerie lion running around the Earth through Africa.</p></li><li><p>Rote Knowledge:Q: Which is bigger: 100 or .001 ? </p><p>A: .001Real answer from a college student. Reason: thousands is bigger than hundreds)</p></li><li><p> 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. </p><p>Willingham, Daniel T. How We Learn: Ask the Cognitive Scientist. American Educator, Winter, 2002</p></li><li><p>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.)</p></li><li><p>Inflexible KnowledgeExample: PuzzlesA student can solve both puzzles, but doesnt understand them as essentially the same.</p></li><li><p>Deep Structure KnowledgeDeeper 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</p></li><li><p>Deep Structure KnowledgeExample: 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.)</p></li><li><p>Finding 3 How To Develop Deep Structure Knowledge Direct instruction of deep structure doesnt work. Use many examples, from many contexts. Work with the knowledge, to increase the store of related knowledge. Practice Dont despair of inflexible knowledge, and dont confuse it with rote knowledge</p></li><li><p>What is Deep knowledge (one explanation from cognitive science)Deep conceptual knowledgeDeep procedural knowledge</p></li><li><p>Conceptual and ProceduralConceptual 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?</p></li><li><p>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#)</p></li><li><p>Ma and Pa Kettle KnowledgeProcedural 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!</p></li><li><p>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</p><p>[Find this three-minute video using on online search. For example, try: http://www.youtube.com/watch?v=VnOlvFqmWEY]</p></li><li><p>Now that youve seen the movie and the math problem </p></li><li><p>Joe can paint a house in 3 hours, Sam can paint it in 5 hours. </p><p>How long does it take for them to paint it working together?</p></li><li><p>Analyze, Discuss, SolveAnswer 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?KnowledgeProcedural? Conceptual? Deep?</p></li><li><p>Knowledge and Strategies in the Movie(Ever exhibited by your students??) No idea. Cant get started.Math never did make any sense to me. There must be a formula but I dont remember it, so Im stuck.No knowledge; not procedural nor conceptualCombine all numbers every which way, hope for the best and maybe partial credit.Math is about computational procedures, Im not sure which one to use, but Ill give it a shot.Procedural knowledge; superficial. Magic formulaMath is about formulas. Just memorize and match to the right problem.Procedural knowledge; not sure about depth. </p></li><li><p>Some ProductiveSolution StrategiesThink about itMake sense of itEstimateGuess, test, and refineDraw a (useful) picture Make a table, look for patterns Draw and trace a graph Write and solve an equation Derive and use a formula</p></li><li><p>Is there any math in the task?(concepts and procedures)FractionsProportional reasoningComputation EstimationSolving equationsMultiple representations (equation, graph, table, diagram)Linear functionsLots of good mathematics!</p></li><li><p># of hoursFraction of house Sam paints (5)Fraction of house Joe paints (3)Fraction of house painted11/51/31/5 + 1/3< 1 ??22/52/32/5 + 2/3> 1 ??1.51.5/5 ??1/2 ??< 1 ??xx/5x/3x/5 + x/3= 1</p></li><li><p>And the formula in the movie?Instead of 5 and 3, use a and b:x/5 + x/3 = 1 x/a + x/b = 1So, 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!</p></li><li><p>Problem-Based Instructional TasksHelp students develop a deep understanding of important mathematicsAre accessible yet challenging to all studentsEmphasize connections, especially to the real worldEncourage student engagement and communicationCan be solved in several waysEncourage the use of connected multiple representationsEncourage appropriate use of intellectual, physical, and technological tools</p></li><li><p>House Painting KnowledgeWe want deep knowledge of both procedures and concepts.All too often we achieve only superficial knowledge of one procedures.</p></li><li><p>Fraction interview T: 4 = ?</p></li><li><p>S: 4 =</p><p>What do we conclude about this students knowledge?Got it right! He understands division of fractions by whole numbers.Or, maybe we need more evidence </p></li><li><p>Fraction interview T: 1/2 4 = ?S: 1/8T: How did you get your answer?S: invert and multiplyT: How does that work?S: Turn 4 into 4/1, then flip it so its 1/4, then multiply across the top and bottom to get 1/8</p><p>Now do we have enough evidence to make a judgment about the students understanding?</p></li><li><p>T: Why does that work?S: Because thats how you divide.T: What about long division that youve done before?S: ummm, thats different, I dont 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 dont do pictures.</p></li><li><p>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 </p></li><li><p>Lets 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?</p></li><li><p>Procedural Knowledgeof this studentCan successfully carry out the procedure of invert and multiplyCant explain why it works.Cant explain how it relates to another division procedure he has learned long division.Not deep procedural knowledge.</p></li><li><p>Deep Procedural KnowledgeSuccessfully carry out the procedureExplain the procedure (it makes sense)Reason about the procedureConnect to other related proceduresChoose appropriate procedures for the task at handFlexibly use proceduresConnect procedures to concepts</p></li><li><p>Deep Conceptual KnowledgeConcept: FractionStudent must understand (among other things):Fraction consists of top, bottom, wholeWhat 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.</p></li><li><p>Deep Conceptual KnowledgeExplain the concept (it makes sense)Reason about and with the conceptConnect to other related conceptsChoose and apply appropriate concepts for the task at handFlexibly use conceptsConnect concepts to procedures</p></li><li><p>GoalWe want deep knowledge of both procedures and concepts.All too often students achieve only superficial knowledge of one procedures.</p></li><li><p>Deep Conceptual and Procedural Knowledge Some ReferencesWillingham (cognitive science research, 2002)Deep structure knowledgeContrast with rote and inflexible knowledgeSee earlier slidesBlooms Revised Taxonomy (Anderson, 2001)Level 5: Evaluatee.g., Judge which of two methods is the best way to solve a given problem.Relates to deep procedural knowledge</p></li><li><p>National Research Council (review of research)How Students Learn, 1999Students must develop procedural knowledge along with conceptual knowledge and understand the connections between the two.Liping Ma (mathematics education research)Profound Understanding of Fundamental MathematicsTopic: FractionsSee book with this titleStar (mathematics education research)Its not that conceptual knowledge is good and procedural knowledge is badBoth are valuable Both must be deep [and connected]Reconceptualizing Procedural Knowledge, Journal for Research in Mathematics Education, November 2005</p></li><li><p>Deep Conceptual and Procedural Knowledgeof Important Mathematicsfor All Students</p><p>**************************** * ** *Mention USEFUL picture re: picture in movie**Check fraction sense* *Ask why engagement and communication?Mention linear function re: mult repsY = 3x equation, graph, tableDoes task fit criteria? In movie, no. For us, yes.***************</p></li></ul>