Teaching Proportional Reasoning Concepts and Procedures Using - - PowerPoint PPT Presentation

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Teaching Proportional Reasoning Concepts and Procedures Using - - PowerPoint PPT Presentation

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation 1/50 Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation Topic Study Group 1: Quality Mathematics Curriculum and

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation 1/50

Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation

Topic Study Group 1: Quality Mathematics Curriculum and Materials

Joel R. Noche1 Catherine P. Vistro-Yu2

1Ateneo de Naga University, Naga City, Philippines 2Ateneo de Manila University, Quezon City, Philippines

7th ICMI-East Asia Regional Conference on Mathematics Education Cebu City, Philippines May 11–15, 2015

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation Synopsis 2/50

Synopsis

We describe worksheets created for a study that aimed to provide empirical evidence on the causal relationships between conceptual and procedural knowledge in mathematics using an East Asian perspective. The supplemental self-paced instructional materials are intended to help young adults improve their performance in tasks designed to assess their proportional reasoning understanding and skills.

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 3/50

The Worksheets

Students are to individually answer one written-response worksheet each day without using books or calculators. They answer the worksheets at their own pace (taking around 15 to 30 minutes on the average to do so), prioritizing performance

  • ver speed.

Each worksheet is a booklet eight half-letter-sized pages long and includes a short discussion of the concepts or procedures involved, with examples and tasks arranged in a slowly increasing level of difficulty. The tasks are to be done in a strictly sequential order and are to be repeated until mastery is attained. Each of the eleven worksheets has a conceptual version (with nonnumeric tasks) and a procedural version (with numeric tasks).

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 4/50

A Locating numbers on a number line B Identifying points on a number line C Using a linear scale to represent ratios and proportions D Comparing ratios without context E Finding the mass of a liquid given its density and its volume F Conserving linear speed (the product of a gear’s number of teeth and angular speed) G Dissolving grains in a liquid to get a solution with the same volume as the liquid H Comparing the volumes of liquid in identical containers but with different orientations I Decomposing and composing a solid with uniform density J Conserving volume (the product of a liquid’s height and area in a container) K Review of worksheets D, E, F, G, H, I, and J

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 5/50

Locating numbers on a number line: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 6/50

Locating numbers on a number line: Conceptual

conceptual (relationships) nonnumeric tasks (no arithmetic operations needed)

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 7/50

Locating numbers on a number line: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 8/50

Locating numbers on a number line: Procedural

procedural (techniques) familiarity with symbols and syntax rules or procedures numeric tasks (arithmetic operations needed)

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 9/50

Identifying points on a number line: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 10/50

Identifying points on a number line: Conceptual

concept of a fraction as a quotient

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 11/50

Identifying points on a number line: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 12/50

Identifying points on a number line: Procedural

procedure to identify a point on a number line

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 13/50

Ratio and proportion using a linear scale: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 14/50

Ratio and proportion using a linear scale: Conceptual

concept of a ratio as a relationship

  • n a double-scale number line

concept of a proportion as two relationships on the same double-scale number line

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 15/50

Ratio and proportion using a linear scale: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 16/50

Ratio and proportion using a linear scale: Procedural

double-scale number line is new to students so conceptual and procedural versions have similar first two pages

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 17/50

Ratio comparison problems: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 18/50

Ratio comparison problems: Conceptual

comparing fractions with common denominators (no arithmetic

  • perations needed)

fractions as points

  • n a number line

common numerators

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 19/50

Ratio comparison problems: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 20/50

Ratio comparison problems: Procedural

comparing fractions (arithmetic operations needed) to emphasize procedure, student is asked to show calculations

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 21/50

Mass of a liquid: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 22/50

Mass of a liquid: Conceptual

no procedure given concepts taught through repetition with variation, highly sequential presentation hints on expected form of answer (fraction, not decimal) conservation of density

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 23/50

Mass of a liquid: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 24/50

Mass of a liquid: Procedural

procedure given and illustrated proportion as an equality of two rates (density is the quotient of mass and volume)

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 25/50

Interlocking toothed gears: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 26/50

Interlocking toothed gears: Conceptual

conservation of linear speed

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 27/50

Interlocking toothed gears: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 28/50

Interlocking toothed gears: Procedural

proportion as an equality of two products of measure (linear speed is the product of a gear’s number of teeth and angular speed)

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 29/50

Sugar and water: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 30/50

Sugar and water: Conceptual

dissolving grains in a liquid to get a solution with the same volume as the liquid

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 31/50

Sugar and water: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 32/50

Sugar and water: Procedural

procedure: express sweetness as a fraction

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 33/50

Water rectangle: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 34/50

Water rectangle: Conceptual

novel task: same container in different orientations

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 35/50

Water rectangle: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 36/50

Water rectangle: Procedural

rectangular grid allows numeric comparison

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 37/50

Masses of chocolate bar pieces: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 38/50

Masses of chocolate bar pieces: Conceptual

decomposition and composition of a solid with uniform density

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 39/50

Masses of chocolate bar pieces: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 40/50

Masses of chocolate bar pieces: Procedural

calculation of mass allows numeric comparison

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 41/50

Volumes of liquids in different containers: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 42/50

Volumes of liquids in different containers: Conceptual

conservation of volume most difficult task is given last

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 43/50

Volumes of liquids in different containers: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 44/50

Volumes of liquids in different containers: Procedural

proportion as an equality of two products of measure (volume is the product of a liquid’s height and area)

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 45/50

Review of worksheets D, E, F, G, H, I, and J: Conceptual

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 46/50

Review of worksheets D, E, F, G, H, I, and J: Conceptual

highly sequential order builds upon previously learned concepts ( 2

5 < 3 5 , 3 5 < 3 4 , 2 5 < 3 4 )

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 47/50

Review of worksheets D, E, F, G, H, I, and J: Procedural

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation The Worksheets 48/50

Review of worksheets D, E, F, G, H, I, and J: Procedural

hints for earlier topics

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation Results and Recommendations 49/50

Results and Recommendations

Undergraduates assigned the procedural version of the worksheets had significantly longer completion times than those assigned the conceptual version of the worksheets. These worksheets were used to find empirical evidence on the causal relationships between conceptual and procedural knowledge, so the two sets of worsheets should differ only in the type of instruction and not in the duration of instruction. The worksheets need to be further revised and tested to finally determine whether or not there is a significant difference in completion times between the two versions. Additional worksheets are also being planned.

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Teaching Proportional Reasoning Concepts and Procedures Using Repetition with Variation Results and Recommendations 50/50

End