_0-A Framework for Procedural Understanding_final

7/31/2019 _0-A Framework for Procedural Understanding_final

1/17

1/29/2010

1

Jon F. Hasenbank, [email protected]

www.uwlax.edu/faculty/hasenbank

For the Transition Math Project Winter Retreat, Jan. 28, 2010.

` The Teaching for Understanding Guide, Blythe

et al., 1997. ISBN #0787909939, Jossey-Basspublisher. Website with a nice overview of the TfU Framework:

http://learnweb.harvard.edu/alps/tfu/info3.cfm

` Part 1 Theory and Motivation for Teachingfor Understanding

` Part 2 Implementing TfU and the Framework

` Part 3 (Re)focusing Curricula to Meet TfUGoals

` From Understanding by Design(Wiggins & McTigue, p. 39) Visiting a science class, John Dewey once asked the

students: What would you find if you dug a hole in theearth? There was no response, so he asked again: Whatwould you find if you dug a hole in the earth?

, .Youre asking the wrong question! Turning to the class,the teacher asked, What is the state of the center of theearth? Igneous Fusion! they replied in unison.

` What did the class really know?


2/17

1/29/2010

2

` When math content is not understood,

students: Experience limited transfer of knowledge

Tend to over-generalize patterns

x e. . ab2 = a2b2 and 2 a+b = 2a+2b. . , ,

x so (a+b)2 must equal a2+b2!

Quickly forget what they learn

View math as irrelevant and uninteresting

See math as a collection of disconnected rules formanipulating symbols (Kaput, 1995).

` Evidence?

Number of Students: 38

Distribution of Correct Answers:

Number of

Correct Answers

Number of

Students

5 4

4 1

3 7

2 9

1 14

0 3

Mean Number of Correct Answers: 2.03

CourseNumber of

Students

Percent

Correct

Geometry 111 1%

Al ebra II

Solve for x: x2 4x = 32

(Honors)

Algebra II 30 0%

Pre-calculus 60 2%

CourseNumber of

Students

Percent

Correct

Geometry 111 0%

Factor Completely: 27x3 + 24x2 3x

(Honors)

62 0%

Algebra II 30 0%

Pre-calculus 60 5%


3/17

1/29/2010

3

` College Readiness Process Standards

Reasoning / Problem-Solving Communication

Connections

Number sense

Geometry

Probability / Statistics

` College Readiness Standards

Geometry

Probability / Statistics

` College Readiness Standards - methods for completing a task.

to other subjects & the real world.

procedures & concepts.

Use

` College Readiness Standards of square root.

that require irrational numbers.

` Washington State K-12 Mathematics Standards An effective mathematics program balances three

important components of mathematics:

x (making sense of mathematics),

x (skills, facts, and procedures), and

x

mathematics to reason, think, and apply mathematicalknowledge).

[These standards] often ask students to

or problems being solved.

` Washington State K-12 Mathematics Standards The term fluencyis used in these standards to describe the

expected level and depth of a students knowledge of acomputational procedure (p. iii)

` Fluencyentails:x

mme a e accura e execu onx Planning knowledge (know when to use it)

x Automaticity (keeps working memory free)

` National standards and other reports also placegreat value on learning math with understanding. E.g. NCTMs PSSM(2000) E.g. The National Math Advisory Panel report (2008)

` When students learn mathematics withunderstanding, it becomes meaningfulforthem.

` As a result, they can Execute procedures intelligently (Star & Siefert, 2002)

Remember more of it, longer (Carpenter & Lehrer, 1999; Hiebert &Carpenter, 1992; Van Hiele, 1986)

Make fewer errors (Rittle-Johnson & Koedinger, 2002) Apply it in a variety of situations (Hiebert & Carpenter, 1992;

Kieran, 1992)

Re-create partially forgotten knowledge, (Carpenter &Lehrer, 1999)

Learn (and understand) new material with less effort(Hiebert & Carpenter, 1992)


4/17

1/29/2010

4

` The benefits of learning with understanding

probably arise from the way the knowledge isstored. Things take meaning from the way they are

connected with other things (Carpenter & Lehrer,, .

` Why does that lead to the benefits we justdiscussed (summarized below)? Long-lasting, flexible knowledge; can re-create

forgotten knowledge; future learning is easier;intelligent problem solving; make fewer errors.

` So understanding is about connections; thatmakes it surprisingly difficult to assess!

` an we n er un erstan ng rom s uexecution?

` How well do you think yourstudentsunderstand mathematics?

` 1) What is the value of the expression(x+1)(x+2) when x= -4?

` 2) Could (x+1)(x+2) ever be a negativenumber? Explain.

` 3) Solve this equation for x. Show yoursteps. 5x 6 = 4x + 3

` 4) Describe two ways you could check youranswer to the previous question.

` 15) If 3x + 4y = 14, and y = -(5/2)x, whatis the value of x?

` 16) What does the answer to the previousproblem tell you about the graphs of the

` You have 8 student work samples for thesesix tasks in your handouts.

` They were scored on a scale from 0 (low) to3 (high) as part of a TfU project.

` 1) What is the value of the expression(x+1)(x+2) when x= -4?

` 2) Could (x+1)(x+2) ever be a negativenumber? Explain.

` Can you find examples of correct responsesfor problem 1? How did those students doon problem 2?


5/17

1/29/2010

5

` 1) What is the value of the expression

(x+1)(x+2) when x= -4?` 2) Could (x+1)(x+2) ever be a negative

number? Explain.

` 3) Solve this equation for x. Show yoursteps. 5x 6 = 4x + 3

` 4) Describe two ways you could check youranswer to the previous question.

` 15) If 3x + 4y = 14, and y = -(5/2)x, whatis the value of x?

` 16) What does the answer to the previousproblem tell you about the graphs of the

` Can you find an example of a level 3response for problem 15? How did thatstudent do on problem 16?

` 3.15) If 3x + 4y = 14, and y = -(5/2)x,what is the value of x?

` 3.16) What does the answer to the previousproblem tell you about the graphs of the


6/17

1/29/2010

6

` So we cannot necessarily infer understanding

from skilled execution of procedures.

` Raises the question: What is the relationship

` Type: Concepts vs. Procedures

(Ideas vs. Algorithms)

` Aptitude: Novice vs Practiced.

(Arduous vs. Automatic)

` Depth: Shallow vs. Deep

(Disconnected vs. Connected)

ProceduralProcedural

ConceptualConceptualConcepts are wellConcepts are well--

memorized but remainmemorized but remainisolated, disconnectedisolated, disconnected

Concepts areConcepts arewellwell--memorized andmemorized and

wellwell--connected,connected,understoodunderstood

Procedure is executedProcedure is executedintelligently,intelligently,understoodunderstood

Procedure is executedProcedure is executedby roteby rote

ShallowShallow DeepDeep

iced

iced

ConceptualConceptual


Concepts are notConcepts are notwellwell--memorized andmemorized and

are isolated,are isolated,disconnecteddisconnected

Concepts are not wellConcepts are not well--memorized, butmemorized, butconnections areconnections are

formingforming

Procedure is not wellProcedure is not well--memorized, but ismemorized, but isbetter connected.better connected.

Execution isExecution isinformed, but slowinformed, but slow

Procedure is not wellProcedure is not well--memorized and ismemorized and isisolated; Executedisolated; Executedwith high cognitivewith high cognitive

loadload

Novice

Novice

Pract

Pract

ShallowShallow Deep, ConnectedDeep, Connected


ConceptualConceptualConcepts are wellConcepts are well--

memorized but remainmemorized but remainisolated, disconnectedisolated, disconnected

Concepts areConcepts arewellwell--memorized andmemorized and

wellwell--connected,connected,understoodunderstood

Procedure is executedProcedure is executedintelligently,intelligently,understoodunderstood

Procedure is executedProcedure is executedby roteby rote


iced

iced



Concepts are notConcepts are notwellwell--memorized andmemorized and

are isolated,are isolated,disconnecteddisconnected

Concepts are not wellConcepts are not well--memorized, butmemorized, butconnections areconnections are

formingforming

Procedure is not wellProcedure is not well--memorized, but ismemorized, but isbetter connected.better connected.

Execution isExecution isinformed, but slowinformed, but slow

Procedure is not wellProcedure is not well--memorized and ismemorized and isisolated; Executedisolated; Executedwith high cognitivewith high cognitive

loadload

Novice

Novice

Pract

Pract


m = rise / run

Solve: ax+b=c

` Type (Concept vs. Procedure): Conceptual knowledge is knowledge about the

facts, concepts, and ideas of mathematics

Procedural knowledge is knowledge about the rules,algorithms, and techniques for completing

mathematical tasks

` List of Examples? Brainstorm a list of examples of procedures &

concepts from algebra


7/17

1/29/2010

7

Conceptual knowledge: knowledge of the facts, ideas, andobjects of mathematics.x Facts: Slope equals rise over run, The set of integers is unbounded, Negative

x Negative = Positive.

x Ideas: Slope, Tangent, Equality, Zero, Proportion.

x jects: T e set o integers, Zero, T e unit circ e.

Procedural knowledge: knowledge of the rules, algorithms,and techniques for completing mathematical tasks.x Rules & Algorithms: Long division, Gaussian elimination, Derivative ofax,

Addition of two fractions, Quadratic formula

x Techniques: Sieve of Eratosthenes, Get common denominators.

` Aptitude (Skill, Automation)

With repetition and practice, knowledge becomeswell-memorized & automatic

This allows rapid recall of facts, or mindlessexecution of procedures

x Algebra allows us to think less and less about moreand more. Bertrand Russell

` Aptitude is easiest to assess: Rapid recall of concepts / facts

Ability to quickly & correctly perform a procedure

` Depth (Connectedness, Understanding) With reflection and experience, the connectedness

of students knowledge increases.

Students become aware of the big picture.

They become more flexible problem solvers, aremore confident, can justify their answers, canchoose more efficient solutions paths.

This deeper knowledge (or understanding) hasmany benefits, as we will discuss later.

` Two Dimensions for Growth It seems growth can occur (independently) along

eitherthe Aptitude dimension orthe Depthdimension

in a hand-over-hand fashion (Rittle-Johnson etal., 2001)

The alliance of factual knowledge, proceduralproficiency, and conceptual understanding makesall three components usable in powerful ways(NCTM, 2000)




acticed

acticed

The seam indicates that as

procedural and conceptual

knowledge get deeper and

better practiced, they become

interconnected.


ConceptualConceptual Novice

Novice

PrPr


` At tables, select an algebraic procedure.

` Discuss:1. A titude: What would novice (vs. racticed)

knowledge look like?

2. Depth: What would shallow (vs. deep /connected) knowledge look like?

3. Finally: What sorts of activities would facilitategrowth in either (or both) of these dimensions?


8/17

1/29/2010

8

` Find the maximum of y = f(x), where

f(x) = -2x2 + 4x + 3.

` Find the maximum of y = f(x), wheref(x) = -2x2 + 4x + 3.

` Typical solution shortly after instruction &practice: - 4- , .

Now f(1)=5, so the max is (1, 5).

` Probable solution one year afterinstruction: Umm, I cant remember the formula.

-1 1 2 3

-2

2

` Try answering the following questions about theprocedure for maximizing a quadratic function:

1. What sort of outcome should I expect?

2. How do I find the max? How else could I do it?

3. y oes x= a wor ow wou exp a nit to others?

4. Is my answer reasonable? How could I check?

5. How does my method compare to others I couldhave used? How do I choose the best one?

6. What types of problems can I solve using thisprocedure?

` Try answering the following questions about theprocedure for maximizing a quadratic function:

1. What sort of outcome should I expect?

2. How do I find the max? How else could I do it?

3. y oes x= a wor ow wou exp a nit to others?

4. Is my answer reasonable? How could I check?

5. How does my method compare to others I couldhave used? How do I choose the best one?

6. What types of problems can I solve using thisprocedure?

` Typical solution shortlyafter instruction & practice: Its at the axis of symmetry. The

formula is x=-b/(2a), so x = 1.Now f(1) = 5, so the max is (1,5).

4

a

acbbx

2

42

=

` Possible solution one year afterinstruction: I dont remember the formula...

Lets see, itll be halfway between theroots... I could find them using thequadratic formula... Oh yeah! x=-b/(2a)! (etc.)

-1 1 2 3

-2

2


9/17

1/29/2010

9

` Procedural understanding is often discussed

in the context of: Expert Knowledge

Flexible Knowledge

Procedural Literac

Meaningful (vs. Rote) Knowledge

Conceptual Knowledge aboutthe Procedure

` The

(NCTM, 2001)

` Can be used to: Define procedural understanding eepen esson con en

Select and develop homework tasks

Assess understanding through homework, quizzes,tests, journals, and interviews

` It is a student-centered guide to understandingprocedures.

` The consists of 4 parts:

(1) generative topics, (2) understanding goals, (3)performances of understanding, and (4) ongoingassessment.

` It is intended to guide teachers in designing unitsthat elicit understanding in all subject areas. Using the (TfU) Framework as a lens to look at my

teaching gives me a systematic way of making sure Imconsistently integrating all of the important elements.

` Next, well learn more about both frameworksand see how they can be used together to: Focus the curriculum,

Deepen instruction, and

Guide assessment.

Discussion Prompt: Reflect on what youve justseen: What do you think will stick with you?

` One implementation (2005) involved usingthe NCTM Framework in College Algebra.

` A second implementation (07-08) involved

8-12.

` Well focus on the implementations of TfU ineach case, as well as the primary results.


10/17

1/29/2010

10

` Teacher training

College Algebra (2005): Weekly content discussionsbased on NCTM Framework; 3 treatment & 3 controlinstructors.

8-12th Grade (07-08): Intensive summer workshopon both NCTM Framework and TfU Framework; 5treatment, 4 control teachers (in algebra).

` Curricular tweaks

College Algebra (2005): Reduce procedural HW by18%; offset with Writing in Math HW instead; 8% HWreduction overall; Lectures and weekly quizzesincluded Framework questions.

8-12th Grade (07-08): Sprinkle it in duringlessons; Identify through-lines & generativetopics; Teachers design their own examples &assessments.

` Assessments: College Algebra (2005): Exams (skill), Journal

Tasks (understanding), classroom observations oftreatment & control instructors.

8-12th Grade (07-08): Three ~16-itemassessments with skill-understanding paired items;classroom observations of treatment teachers only.

` What do you look for in a quadratic equation todecide which method will be the best way to solve?1. 6x2 3x 2 = 0

2. 9x2 24x = 0

3. (3x 4)2 = 162. =

5. x2 6x + 7 = 0

Methods: Factoring, Square Root Method, Complete theSquare, Quadratic Formula

` What would your students say? What wouldan expert say? How can we bridge that gap?

` What is the advantage of factoring first,before adding or multiplying rationalexpressions?


` How could you verify that you have donethe polynomial long division procedurecorrectly?



11/17

1/29/2010

11

` 8th grade algebra teacher:

Students work in pairs to solve linear equation. One student does a step, partner provides reason.

Then exchange roles for next step.x (Established this routine early on)

` 9th grade algebra teacher: I really like the understanding questions; it forces

my students to really think instead of being sorobotic.

I have found it difficult to implement on tests. Iguess I am scared to see what they will write & thenhave to score it as part of the test.

` All college algebra instructors emphasized

procedural skill far more than the otherFramework objectives.

` BUT: Treatment instructors managed to.

Performing theprocedure

(% coded 6 Pervasive)

Other TfU objectives(% above 2 Infrequent)

Control 94% 9%

Treatment 72% 18%

Teacher 1: Focused onperforming the procedure.

Teacher 2: Nice job ofsprinkling it in.

Nov. 2007 Feb. 2008

` Treatment students scored 34%higher on the understanding tasks. t105 = 4.32, p < .001

` Treatment students had 2.8%higher average exam scoresoverall. t138 = 0.921, p = .358 (n.s.)


12/17

1/29/2010

12

` Framework students did no worse (indeed, did a

little better, but mostly n.s.) than controls.` But notice that nice gap opening up over time

Average Exam Scores

(Weighted Average of Treatment / Control)

55.0

60.0

65.0

70.0

75.0

80.0

85.0

90.0

E xa m1 E xa m2 E xa m3 F in al

Exam

Exam Number

WeightedAverageScore(%)

Treatment

Control

No differences in value-added for understanding(p = .413)

Significantly greatervalue-added for skill(p = .003).

` Be patient! The gains took time to emerge in both studies.

In fact, the assessments showed a classicimplementation dip that was later overcome.

` C&I remained largely procedure-centered. Procedural instruction dominated; Even just

sprinkling it in can make a difference.

College algebra students expected skills-orientedinstruction, but they adapted.

` Homework is a crucial component. The one college algebra class that was most

verbose on the homework did the best on theexams and journal tasks.

Lesson: Dont neglect the active homeworkcom onent ofTfU..

` Small steps can yield positive results. It is easy to supplement (deepen) existing

examples.

What if we could really embed it in thecurriculum?

` There is time to address deep questions: Reduce the number of drill exercises you assign.

Bookend examples with Framework-orientedquestions.

` Be sure to include understanding items onyour assessments Otherwise, the students may pass over the

understanding questions you assign.

` One critical (and highly relevant) differencebetween the two studies: College algebra students have seen this content

once before.

Does that make it easier to build connections?

(May explain the understanding gains in collegealgebra but not in high school).


13/17

1/29/2010

13

Discussion prompt: What implications dothese two studies have for your institution?

` Begin with Generative Topics

What topics are worth investing the time tounderstand?

` Select Understanding Goals What are your goals for student understanding?

` Devise Performances of Understanding Activities that develop anddemonstrate students

understanding

` Use Ongoing Assessment Provides students with continual feedback about their

performances to help them improve

` Generative topics are those that: Are central to one or more disciplines,

Are interesting andaccessible to students, and

Provide opportunities for multiple connections.

` These topics should form the core of thecurriculum (TfU Guidebook, p. 18).

` Generative Topics: ; ;

with

Sam le To ics: (are an of them enerative?)Slope / Rate of Change

Long Division of PolynomialsLinear Equations

Transforming FunctionsGraphing Rational Functions

Factoring PolynomialsLaws of Exponents


with

courses you teach? (To what extent can you use the College Readiness

Standards to inform these decisions?)


with

We need a tool for focusing and adapting this TfU

Framework for use in algebra.

The Framework for Procedural Understandingcanhelp by identify appropriate understanding goalsfor algebra procedures.


14/17

1/29/2010

14

1a) What is the goal of the procedure?1b) What sort of answer should I expect?2a) How do I carry out the procedure?2b) What other procedures could I use?3) W y oes t e proce ure wor ?4) How can I verify my answer?5) When is this the best procedure to use?6) What else can I use this procedure to do?

Adapted from Navigating through Algebra in Grades 9-12(NCTM, 2001)

` By setting understanding goals, we answer

the question where are we heading? Few of us would set off on a trip without first

having a sense of where we want to go. Becauseour resources are limited, we want to use themwisely. So we think carefully about where wed liketo go, and we have that destination in mind whenwe set out. It helps us decide when to stop torest, when to forge ahead, and when to modify ouritinerary. (TfU Guide, p. 35)

` Understanding goals are the destinations thatmake the journey worthwhile.

` Unit-long goals:

` Course-long (through-lines): What understanding goals span the entire course?

` It helps to prioritize by sorting yourunderstanding goals into categories: (Wiggins & McTigue, 1998)

Worth being

Important to know

and do

Enduring

Understanding

` Generative topic: Solving Linear Equations E.g. 3x + 2 = x 3

` Some possible Understanding Goals: Students will be able to use multiple solution paths

and choose intelligently among them. Related Question: Show how to solve the equation

two different ways. Which way do you think isbetter? Why?

Students will be able to interpret a linear equationand its solution as the intersection of two lines.

` Select a generative topic, and identify someunderstanding goals. Use the Framework for Procedural Understanding to

help identify important understanding goals.

Express your understanding goals as a statements(Students will) or as questions for students.


15/17

1/29/2010

15

` Broad understanding goals (those that span

the course) are called throughlines.

` Consider: What broad understanding goals do

(And how will you get them there?)

` Broad understanding goals (those that span

the course) are called throughlines.

` Consider: What broad understanding goals do

(And how will you get them there?)

` What throughlines do you see in the CollegeReadiness Standards?

` Ideas for performances of understandingemerge as we examine our understandinggoals. Want your students to recognize there are multiple

solution paths?

or to identify the most efficient ones?

or show how they might check their answer?

Then we must ask them to do so.

` Early performances are primarily aboutlearning, not necessarily assessing.

` Understanding performances progress incomplexity and build over time. Introductory performances - to gauge initial

understandings.

Guided inquiry performances when lecture justwont do (for topics you feel are especiallyimportant to understand).

Culminating performances - students are asked tosynthesize and demonstrate their understandings.

` What makes a good performance. Daily HW presentations (rotate through students).

Small group tasks, thoughtfully selected.

Journal reflections.

Individual projects

x Find a connection (in math, to life, to history) andexplain it (to me, or to the class.)

x Report may be written or oral (or both).

Quiz prompts (formative).

Exam prompts (summative).

` As students engage in understandingperformances, we should: Help students make connections by asking them to:

x explain their answers,

x give reasons for their choices,

x offer supporting evidence,

x make predictions

Listen for common questions or sources ofconfusion that should be addressed in whole-groupdiscussion or lecture


16/17

1/29/2010

16

` It isimportant for students to perform their

understanding. You will learn more about what they understand (or

misunderstand).

They will be able to sort out what they reallythink.

` Make this process visible to students!

State your understanding goals, and revisit themfrom time to time.

Post them in a prominent place, and refer to themas you teach.

` Let them know you value this kind ofknowledge.

` Ongoing assessment is the last (but notleast!) component of the TfU Framework.

` Key Points: u en s va ue w a we assess, so our assessmen s

must reflect our understanding goals.

Provide students with clear responses to theirperformances of understanding in a way that helpsthem improve their next performance.

` What are some ways you might incorporateunderstanding performances into yourassessment (and facilitation) of studentlearning?

` How to grade these novel sorts ofunderstanding performances? A simple rubric can make expectations clear and

provide for reliable scoring & feedback.

` Exam le (ada ted from WKCE-CRT 2005) 3: demonstrates a thorough understanding

2: demonstrates a partial understanding

1: demonstrates an incomplete understanding

0: demonstrates no understanding

` Example (adapted from WKCE-CRT, 2005)

3: response is accurate, complete, insightful, andfulfills all requirements of the task.

2: res onse is accurate but lacks some essentialinsights; fulfills most requirements of the task.

1: response provides some relevant information, but isto general or simplistic, or does not fulfill therequirements of the task.

0: response is inaccurate, confusing, or irrelevant.


17/17

1/29/2010

17

` Imagine if students started

asking these questions on theirown?

` are the opportunities before usfor enriching the mathematicalexperiences of our students?

` Finally

` Questions / Comments?

Documents

_0-A Framework for Procedural Understanding_final