1 0 0 0 x a f x ( ) L Dr. Margaret Adams Assistant Professor of Mathematics Northern State University Aberdeen, South Dakota NCTM Exposition April 15, 2016 Margaret.adams@northern.edu
2 Instructions • Take 3-5 minutes to answer the questions. • Answer BEFORE column now.
3 Scenario High School End of Quarter Exam Calculus Review
4 “ PLOPPPS ” Lesson Plan Model Anticipation Guide • P rior L earning: Functions & Limits • O bjectives: ▫ Reduce misconceptions ▫ Develop “appropriate schemas (Piaget)” ▫ Improve conceptual understanding • P re-test (collaborative) BEFORE column • P resent information on topic • P ost-test (collaborative) AFTER column • S ummary
Rationale • Previous Research • Unique perceptions of limits • Identify potential misconceptions prior to • instruction. • formative assessment. • Promote • appropriate schema development. • relational understanding. • Reduce teacher opportunities passing on misconceptions to students.
6 Common Core State Standards for Mathematics ▫ New emphasis placed on students’ ability to engage in mathematical practices. understanding problems. reading and critiquing arguments. making explicit use of definitions (CCSSI 2010).
7 Conceptual Understanding of Limits Shared Knowledge Structures Appear in the Intersections (Adams, 2013 Dissertation Study) Inappropriate Schemas Altered Schemas Instrumental Appropriate “Semi - Understanding Schemas Relational” or Understanding Relational Understanding No (changing action Understanding schemes) (action schemes)
8 Anticipation Guide Definition: ▫ A pre-post measure of content knowledge and conceptual understanding. ▫ A written instrument using True/False responses. ▫ Teacher-made.
9 How its used & more information • Every topic or unit. • With lesson plan. • Collaborative pairs. • Individually. • Graphing calculator. • ANTICIPATION GUIDES: A TOOL FOR SCAFFOLDING MATHEMATICS READING • http://www.nctm.org/Publications/mathematics- teacher/2015/Vol108/Issue7/Anticipation-Guides_-Reading-for- Mathematics-Understanding/
10 Implementation • In-service PD for instructors. • Any instructor, before taking a calculus refresher course. • College students after being taught limits — before a test. • High school students after being taught limits — but before a test.
11 Purpose & Goals • Identify potential misconceptions. • Assimilate & accommodate correct schema content. • Promote conceptual understanding. ▫ Instrumental Understanding (what algorithms or processes needed to problem solve) ▫ Relational Understanding (why they work)
12 Lecture on Limits • Content in Anticipation Guide • Utilize graphing calculators. • Complete the “After” column during lecture.
13 Before Given Statement After 1. A limit is a number that represents the behavior of function values. 2. A limit “approaches” a function value but never reaches it. 3. A limit can never equal a function value because limits are only about what a function is “approaching”.
14 4. When asked to “find the limit”, the limit refers to the x-value under the notation. For instance, 1 , the limit is -3 in this case. lim x 3 x 3 5. The arrow in the limit notation implies direction 1 from the left only. For example: means as lim( 2) x x 2 x approaches 2 from the left only.
15 6 . In the graph below, the limit does not exist because of the hole at (2,4). 7. The infinity symbol represents a very large number. 8. If a limit equals infinity, “ ”, the n the limit exists. (Ex: ) lim 2 x e x
16 9. The solution and interpretation to the problem below for 1 is correct and hence, a good example of an lim x x infinite limit .
17 1 10. , because left hand limit does not lim d ne . . . x 3 x 3 equal the right hand limit:
18 1 11. Given the graph of , the limit exists and lim 2 x 0 x equals infinity, because the left hand limit and right hand limit both equal plus infinity. 1 12. In the graph above for , the vertical lim 2 x 0 x asymptote at x=0 is a limit because it is like a brick wall that you can’t go past.
19 13. The graph of limcos has 2 limits: 1 and -1.
20 14. The limit is the horizontal asymptote for: 2 9 2 x lim 2 3 x 2 x 5 x
21 15. The graph below is classified as a quadratic function. 16. Above, the point (2,6) is not on the graph of the function. 17. The domain of the function above is (0,4) and range is ( ,6]
22 1 18. Even though is not defined at 0, due to symmetry of being an limcos x x 0 even function, the limit exists and converges to 0.
23 Further Exploration
24 19. The function on a finite interval domain [-1,1], the limits are pi and 0. lim arccos x , and lim arccos x 0 x x 20. , there is no limit (or hole) at pi/2 because limarccos x x 0 you can walk right over it.
25 Answers 1. T 2. F 3. F 19. F 4. F 5. F 6. F 20. F 7. F 8. F 9. F 10. F 11. F 12. F 13. T 14. F 15. F 16. F 17. F 18. F
26 Anticipation Guide • What have you learned today? • Have your answers changed?
27 Proposed Explanations for Unique Perceptions • What you teach what they learn. • Individual differences occur in perception & learning. • Math content: ▫ Assimilated into knowledge structures ▫ Organized into: appropriate altered inappropriate schemas.
28 Teaching Recommendations • Identify potential misconceptions. • Incorporate into lesson plans. • Use Anticipation Guide for Limits. ▫ Daily with 90 minute block throughout the topic. ▫ Twice during a 4 day topic in college. • Enhance quantitative literacy. ▫ Graphic Organizers (visual & KLW’s) • Utilize cooperative groups and inquiry-based learning to engage diverse learners.
29 Teaching Recommendations Use Separate Anticipation Guides • Functions • Limits at a Point • Limits at Infinity • Infinite Limits and Limits that Do Not Exist
30 Summary Anticipation Guide for Limits Instrument: What it is Develop Appropriate Rationale: Why Schemas & it is useful Conceptual Understanding Goal: Reduce Method: How to misconceptions implement it
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