Extending K-8 Mathematics Concepts in Alternate Bases

GAMTE ConferenceOctober 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

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

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?

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

ECE Middle Grades

Total

Male 2 0 2Female 20 4 24Total 22 4 26

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

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

Drawing Technique436 + 556

Result: 1426

... ...............

..

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

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 as1. A variety of drawing adaptations2. Direct computation showing

carried digits, as shown at right (a technique never introduced)

1 1 241531324141434

Extensions: Final Exam Sequencea) 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

b) Give the Base 10 equivalent of the Base 6 decimal fraction 0.36, or

c) Give the Base 10 equivalent of the Base 6 number 152.36

d) Compute the product in Base 6 of 4.56 2.16

e) Give the Base 10 equivalent of your Base 6 result from part (d).

8

1231

6

310

Results (Final Exam Assessment)

Not Attempted

Attempted, Incorrect

Partially Correct

Fully Correct

(a) 8 30.8% 4 15.4% 1 3.8% 13 50.0%(b) 8 30.8% 6 23.1% 1 3.8% 11 42.3%(c) 7 26.9% 10 38.5% 5 19.2% 4 15.4%(d) 6 23.1% 9 34.6% 7 26.9% 4 15.4%(e) 9 34.6% 10 38.5% 5 19.2% 2 7.7%

ALL 5 2 7.7% 1 3.8% 22 84.6% 1 3.8%

(a) & (b) – lower cognitive demand(c) – (e) – higher cognitive demand

Most successful non-trivial item: (d)

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

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.

Successful ExtensionsMost 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

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

Discussion & Questions