1 / 17

Extending K-8 Mathematics Concepts in Alternate Bases

Extending K-8 Mathematics Concepts in Alternate Bases. GAMTE Conference October 13, 2010 Dianna Spence NGCSU Math/CS Dept, Dahlonega, GA. Context. Course: Number & Operations Number systems Place value and operations Fractions. Students Undergraduates K-8 pre-service teachers.

samuru
Télécharger la présentation

Extending K-8 Mathematics Concepts in Alternate Bases

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Extending K-8 Mathematics Concepts in Alternate Bases GAMTE Conference October 13, 2010 Dianna Spence NGCSU Math/CS Dept, Dahlonega, GA

  2. Context • Course: Number & Operations • Number systems • Place value and operations • Fractions • Students • Undergraduates • K-8 pre-service teachers

  3. Background: Teaching Observation • Manipulatives are intended to help develop conceptual understanding • Pre-service teachers “know too much” • They often resist relying on the manipulatives, because they don’t perceive a need for them • Some still rely on memorized rote procedures • Then they arrange the manipulatives to match their result

  4. Motivation for Research • Question: What if pre-service teachers were given tasks for which they did NOT already have a memorized procedure? • If they were taught these new tasks using manipulatives, would the manipulatives support their understanding more fully? • Would their understanding be deeper? • Would they be more able to extend their conceptual understanding to new contexts?

  5. Setting • One section of Number & Operations course • Enrolled undergraduate students

  6. Method • Instruction in alternate bases • Support learning with Base “n” blocks • Assess outcomes • Ability to extend concepts learned beyond contexts explicitly covered during instruction • Two assessments • One question on midterm exam • One sequence of 5 questions on final exam

  7. Instruction in Alternate Bases • Topics • Counting & number representation • All 4 operations • Bases • Base 6 • Base 8 • Tools • Base 6 and Base 8 “blocks” • Paper cutouts, made by students • Techniques • Only assigned computations that could be modeled using the blocks • “Drew” block solutions on paper • Deferred conversion to Base 10

  8. Drawing Technique . . 436 + 556 Result: 1426 . . . . . . . . . . . . . . . . . .

  9. Assessing Ability to Extend All extension assessments were bonus questions (limitation) Midterm Exam (one question) Compute 241536 + 132416 and give the answer in Base 6 Students had not previously seen or discussed alternate base computations that could not be modeled with Base n units, rods, flats

  10. Results (Midterm Assessment) Compute 241536 + 132416 and give the answer in Base 6 All students attempted the computation Fully correct: 13 students (50%) Some conceptual understanding but with arithmetic errors: 6 students (23%) Incorrect: (e.g., 373946): 7 students (27%) No student who attempted this problem showed any work involving conversion to/from Base 10. Students showed conceptual work as A variety of drawing adaptations Direct computation showing carried digits, as shown at right (a technique never introduced) 1 1 24153 13241 41434

  11. Extensions: Final Exam Sequence • When we write fractions in other bases, both the numerator and denominator are given in the other base. Find the Base 10 equivalent of the Base 8 fraction • Give the Base 10 equivalent of the Base 6 decimal fraction 0.36, or • Give the Base 10 equivalent of the Base 6 number 152.36 • Compute the product in Base 6 of 4.56 2.16 • Give the Base 10 equivalent of your Base 6 result from part (d).

  12. Results (Final Exam Assessment) (a) & (b) – lower cognitive demand (c) – (e) – higher cognitive demand Most successful non-trivial item: (d)

  13. Item (d) Student Solutions “Compute the product in Base 6 of 4.562.16” Incorrect: 9.45 Correct: 14.25 Partially correct: 10.25, 13.25, 15.25,142.5

  14. Noteworthy • Items (c) and (d) on Final Exam sequence were similar level of cognitive challenge as the midterm item. • Many more students demonstrated full proficiency on the midterm item. • Final Exam sequence incorporated fraction and decimal concepts, covered with manipulatives but NOT in an unfamiliar context.

  15. Successful Extensions Most successful non-trivial extensions? • Midterm item, Final item (d) • Items whose conceptual understanding was supported by Base ‘n’ blocks, learned in unfamiliar setting • Also noteworthy: Decimal multiplication was covered with Base 10 blocks, but never with blocks in any other base

  16. Pattern • Under certain conditions, many students demonstrate ability to extend their conceptual understanding to new contexts • Situating learning in contexts where students must rely more heavily on manipulatives seems to foster their ability to extend

  17. Discussion & Questions

More Related