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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.

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Extending K-8 Mathematics Concepts in Alternate Bases

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  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

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