State of Developmental Effec%ve Teaching and Learning ...
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Robert Cappetta College of DuPage April 9, 2010 [email protected] 1 Effec%ve Teaching and Learning Strategies for Block Scheduled Developmental Mathema%cs Robert Cappetta College of DuPage [email protected] Wm. R. Harper College Developmental Education Symposium April 9, 2010 State of Developmental Mathematics • The majority of community college students are placed into developmental mathematics courses. • Large numbers of students drop out or fail to successfully complete the sequence. • Developmental mathematics is a primary barrier to earning a degree. Why? • U.S. students do not perform well on international studies of K-12 mathematics. • Students see mathematics as a collection of unrelated procedures that must be memorized. • Developmental math teaching in community colleges tends to be similar to K-12 teaching. Attempts to improve the problem • Study skills lessons • Time-management lessons • Lessons designed to improve student attitudes • Supplemental Instruction • Learning Centers • “Yet, it is worth noting that almost none of the reforms have focused on actually changing the teaching methods and routines that define the teaching and learning of mathematics in community colleges.” • http://www.carnegiefoundation.org/carneg ie-connections/what-were-learning/what- community-college-developmental- mathematics-students-understand-about- math Today’s Students Are they different from students in the past?
Transcript of State of Developmental Effec%ve Teaching and Learning ...
Robert CappettaCollege of DuPage
April 9, 2010
Effec%veTeachingandLearningStrategiesforBlockScheduledDevelopmentalMathema%cs
Robert CappettaCollege of [email protected]
Wm. R. Harper CollegeDevelopmentalEducation SymposiumApril 9, 2010
State of DevelopmentalMathematics
• The majority of community collegestudents are placed into developmentalmathematics courses.
• Large numbers of students drop out orfail to successfully complete thesequence.
• Developmental mathematics is aprimary barrier to earning a degree.
Why?
• U.S. students do not perform well oninternational studies of K-12 mathematics.
• Students see mathematics as a collection ofunrelated procedures that must bememorized.
• Developmental math teaching in communitycolleges tends to be similar to K-12 teaching.
Attempts to improve theproblem
• Study skills lessons• Time-management lessons• Lessons designed to improve student
attitudes• Supplemental Instruction• Learning Centers
• “Yet, it is worth noting that almost none ofthe reforms have focused on actuallychanging the teaching methods androutines that define the teaching andlearning of mathematics in communitycolleges.”
• http://www.carnegiefoundation.org/carnegie-connections/what-were-learning/what-community-college-developmental-mathematics-students-understand-about-math
Today’s Students
Are they different from students inthe past?
Robert CappettaCollege of DuPage
April 9, 2010
Assumptions
• We have also become aware that studentsincreasingly seek someone to providestructure, direction, and praise in a wayprevious generations of students did not.– (Crone 2007)
• This generation of college students has beenraised on interactive technology andentertainment-style communication. We havebeen told by our students that straight lectures orPowerPoint presentations rarely hold theirattention. Experiences that involve students andrequire them to interact as a part of their ownlearning are more likely to maintain theirinterest.– (Crone 2007)
Research-Based MotivationStrategies
• http://www.vanderbilt.edu/cft/resources/teaching_resources/interactions/motivating.htm
• Ken Bain, What the Best College Teachers Do,Harvard University Press, 2004, pages 32-42.
• Linda Nilson, Teaching At Its Best: A Research-Based Resource for College Instructors, 2ndedition, Anker Publishing, 2003, pages 41-44.
• Linda Nilson, Teaching At Its Best: A Research-Based Resource for College Instructors, 2ndedition, Anker Publishing, 2003, pages 41-44.
Become a role model forstudent interest.
• Deliver your presentations with energyand enthusiasm. As a display of yourmotivation, your passion motivates yourstudents. Make the course personal,showing why you are interested in thematerial.
Get to know your students
• You will be able to better tailor yourinstruction to the students' concerns andbackgrounds, and your personal interest inthem will inspire their personal loyalty toyou. Display a strong interest in students'learning and a faith in their abilities.
Use examples freely
• Many students want to be shown why aconcept or technique is useful before theywant to study it further. Inform studentsabout how your course prepares students forfuture opportunities
Robert CappettaCollege of DuPage
April 9, 2010
Use a variety of student-active teaching activities
• Teach by discovery. Students find assatisfying as reasoning through a problemand discovering the underlying principle ontheir own.
• Cooperative learning activities areparticularly effective as they also providepositive social pressure.
Set realistic performancegoals
• Help students achieve them by encouragingthem to set their own reasonable goals.Design assignments that are appropriatelychallenging in view of the experience andaptitude of the class.
Place appropriateemphasis on testing
and grading.• Tests should be a means of showing what
students have mastered, not what they havenot.
Be free with praise andconstructive in criticism.
• Negative comments should pertain toparticular performances, not the performer.Offer nonjudgmental feedback on students'work, stress opportunities to improve, lookfor ways to stimulate advancement. Avoidcategorizing students based on performance.
Give students as muchcontrol over their owneducation as possible.
• Let students choose paper and project topicsthat interest them. Assess them in a varietyof ways (tests, papers, projects,presentations, etc.) to give students morecontrol over how they show theirunderstanding to you.
My Developmental Math Class
• College of DuPage - Math 0481• 5 semester hours• First half of a combined
elementary/intermediate algebra class.• Only recently have students needed a
minimum placement test score for enrollmentin this class.
• Retention and success rates are belownational norms.
Robert CappettaCollege of DuPage
April 9, 2010
Perceived Problems
• Late Start Class– Students who register late tend to perform
more poorly.– Class meets for 12 weeks rather than 17
weeks
Perceived Problems
• Class meets twice a week for 3-hours each.– Can students maintain the necessary focus for
three hours?– If a student is absent can the material be “made
up”• Class meets at a regional center instead of
main campus.
Theoretical Framework
• Piaget’s assimilation and accommodation– In Assimilation, what is perceived in the outside
world is incorporated into the internal world withoutchanging the structure of that internal world, butpotentially at the cost of "squeezing" the externalperceptions to fit — hence pigeon-holing andstereotyping.http://www.learningandteaching.info/learning/assimacc.htm
Piaget
• In Accommodation, the internal world has toaccommodate itself to the evidence withwhich it is confronted and thus adapt to it.When you try to manage information whichdoes not fit the pre-existent fields andcategories, you have to develop new ones toaccommodate the new information.http://www.learningandteaching.info/learning/assimacc.htm
Reflective Abstraction
• Constructs– Interiorization– Coordination– Encapsulation– Generalization– Reversal
Novelty
• Students in developmental algebrahave studied the same material multipletimes.
• How can developmental math bedifferent from high school math?
Robert CappettaCollege of DuPage
April 9, 2010
Novel Questions
• Guess my rule• Explain why• Given a solution, construct a question• New ways of looking at old questions• Coordinating arithmetic and algebraic
strategies
Guess my rule
x f(x)
0 2
1 5
2 8
3 11
4 ?
5 ?
Follow ups
• What number comes next?• Describe the rule in multiple ways.• What does the graph of the points look
like?• How are the ideas of “slope” and “y-
intercept” relate to the “guess my ruleproblem?
Guess my rule
x f(x)
0 1
1 2
2 5
3 10
4 17
5 ?
6 ?
Follow ups
• What are the next numbers?• What are f(-1), f(-2), f(-3) ?• Describe the rule in at least three
different ways.• If we graph the points what does the
graph look like?
Explain why
• Explain why 5/0 is undefined.
Robert CappettaCollege of DuPage
April 9, 2010
!
6
2= 3 because
!
3 " 2 = 6
!
28
7= 4 because
!
4 " 7 = 28
!
0
2= 0 because
!
0 " 2 = 0
!
5
0= [ ] so
!
[ ] " 0 = 5 . This has no solution.
Explain why
Explain why
!
70
=1.
!
1=73
73
= 73"3 = 70, so
!
70
=1.
Given a solution, construct aquestion.
• Construct an equation with solutionx = 4.
• Construct an “interesting” system ofequations with solution (3, –2).
• Construct an inequality that has nosolution.
• Construct an equation of a line thatincludes the point (4, 2).
New ways New ways
Robert CappettaCollege of DuPage
April 9, 2010
Coordinating arithmetic andalgebra
Simplify
!
3
4+1
6+5
8+1
7
!
3
4=3
2 " 2=3 " 2 " 3 " 7[ ]2 " 2 " 2 " 3 " 7[ ]
=126
168
1
6=1
2 " 3=1" 2 " 2[ ] " 7[ ]2 " 2[ ] " 2 " 3 " 7[ ]
=28
168
5
8=
5
2 " 2 " 2=
5 " 3 " 7[ ]2 " 2 " 2 " 3 " 7[ ]
=105
168
1
7=1
7=1" 2 " 2 "2 " 3[ ]2 " 2 "2 " 3[ ] " 7
=24
168
=283
168
Simplify
!
2
x+3y
x2
+1
y
!
2
x=
2
x=
2 " x " y[ ]x " x " y[ ]
=2xy
x2y
3y
x2
=3y
x " x=3y " y[ ]x " x " y[ ]
=3y
2
x2y
1
y=
1
y=
1 " x " x[ ]x " x[ ] " y
=x2
x2y
=2xy+ 3y2 + x2
x2y
Lesson Structure
• Daily “Quiz” - 30 minutes• “Quiz Review - 10 minutes• Lecture-Discussion - 50 minutes• Break - 15 minutes• Collaborative Work Session - 50 minutes• Q and A session - 15 minutes• Preview the next lesson - 5 minutes
Daily “Quiz”
• Challenging Questions that review theprevious lesson or pre-requisites.
• Transition questions that will preparestudents for the upcoming lecture.
• Students are encouraged to worktogether
• Students may use any resources
“Quiz” Review
• Instructor asks students to describehow to solve problems.
• Instructor solves each problem.• Instructor attempts to solve problems in
more than one way.• Students are asked to do problems
similar to quiz questions.
Robert CappettaCollege of DuPage
April 9, 2010
Lecture - Discussion
• Instructor introduces new material• Instructor asks questions to clarify
conceptual understanding• Instructor solves a problem• Students solve a similar problem• Students do problems that require
generalizing ideas into new domains
Break
• Individual students discuss issues withthe instructor
• ALL STUDENTS SEEM TO PULL OUTCEL-PHONES AND START TEXTING
• Other typical break activities
Collaborative Work Sessions
• Students practice on the dailyassignment. (Typically 50 questionsper day.)
• Students share ideas on the previousassignment that is to be submitted atthe end of class.
Q and A Session
• The students direct attention to the teacher.• Students ask questions about the daily
assignment.• Students and the instructor answer student
questions.• The instructor asks questions to clarify
concepts.
Preview the Next Lesson
• The instructor introduces the conceptsto be discussed in the next lecture.
• The instructor tells the students whichsections of the text book to read.
• The instructor discusses pre-requisiteskills that students may want to refreshbefore the next class
End of Class
• Students submit their assignments asthey leave the class.
• Students who have not completed theassignment must stay after class todiscuss plans to submit it.
Robert CappettaCollege of DuPage
April 9, 2010
Daily Assignments
• Usually 50 questions per day.• Challenging questions (fractions, word
problems.)• Questions that students miss on
previous assignments or quizzes willappear on future assignments.
• All assignments are “comprehensive” -spiraling.
Daily Assignments
• Procedural questions that promoteinteriorization
• Solving problems multiple ways topromote coordination
• “Explain why” questions to promoteencapsulation
Daily Assignments
• Extension problems to promotegeneralization
• Construct questions for a given answerto promote reversal
Interesting Questions
Explain the different meanings
of !24
and !2( )4
.
At Home Run Inn, a fourteen-inch pizza costs $16 and an eight-inch pizza costs $7. What is the cost per square inch of each pizza?
Simplify 1+3
410t ! 3( ) +
5
8t +
1
10
"#$
%&'
Robert CappettaCollege of DuPage
April 9, 2010
You are making pies to sell at a fundraiser. It costs $3 to make each pie, plus a one-time cost of $20 for baking tools. You plan to sell the pies for $5 each.
a. S e t up the equation used to find the number of pies you need to sell to break even.
b. F ind the number of pies needed to break even.
Express the answer in scientific notation.
3!106
8 !10"4
George has a piggy bank filled with quarters and dimes. The total value of the coins is $18.60. There are twice as many quarters as dimes. How many dimes and quarters are in his bank?
Hint 1: Number of quarters( ) ! 25 + Number of dimes( ) !10 = 1860
Hint 2: Number of quarters( ) = 2 ! Number of dimes( ) H int 3: Let x = Number of dimes( )
.
Find the equation of the line containing the
points 1,7( ) and -2,11( ) .
!y Rise( ) = ________ !x Run( ) = ________
Slope = ____________
Solve for b by plugging 1 in for x , 7 in for y ,
and your slope value in for m into y = mx + b
y-intercept = ____________
Equation of the line in slope-intercept form: ________________
Find the equation of two parallel lines. Let one line
go through the point 2,5( ) and let the second line
go through the point !1,4( ) . Note: There are many possible answers.
Equation of line above: _____________________________
E q u a tion of the line below: __________________________.
Grading
• Tests every 3 weeks• Test 1 - 100 points• Test 2 - 100 points• Test 3 - 100 points• Quizzes - 100 points• Daily Assignments - 100 points• Participation - 100 points• Departmental Final Exam - 200 points
Robert CappettaCollege of DuPage
April 9, 2010
Results-Trends
• Increased student levels of confidence.• Increased retention rates and success
rates.• Improved performance on departmental
final exam.• Many students re-enroll after failing and
then succeed.
News
• Carnegie Foundation– A one semester developmental class will
prepare students for a basic statisticscourse
• Core Standards– How should newly adopted K-12 standards
for mathematics influence developmentaleducation?
Illinois
• Identify model developmental mathprograms
• Establish statewide cutoff scores thatdefine college-readiness. ACT,COMPASS, Accuplacer
• How much “similarity” should IllinoisCC’s have in developmental mathcurricula?
Ban the word SHOULD!
• Students should . . .• High school teachers should . . .• The part-time faculty should . . .
The Buck Stops Here
• We must do our best with the studentswe have. We must continually strive toimprove as curriculum designers,teachers and motivators. Complainingwithout devising solutions is useless
Questions?
Robert CappettaCollege of DuPage
April 9, 2010
Thank You!
• Robert Cappetta• Professor of Mathematics• College of DuPage• Glen Ellyn, IL• [email protected]
