Developing Mathematical Proficiency- Conceptual understanding provides representations that can...
Transcript of Developing Mathematical Proficiency- Conceptual understanding provides representations that can...
Mathematics Assessment Project
© Shell Centre, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at
http://creativecommons.org/licenses/by-nc-nd/3.0/ - all other rights reserved
Mathematics Improvement Network
© 2017 Mathematics Assessment Resource Service, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at
http://creativecommons.org/licenses/by-nc-sa/4.0/ - all other rights reserved
Developing Mathematical Proficiency The potential of different types of tasks for student
learning
Workshop Outline
• The Five Strands of Mathematical Proficiency – Conceptual Understanding – Procedural Fluency – Strategic Competence – Adaptive Reasoning – Productive Disposition
• Introducing the Tasks • Working on the Tasks • Feedback on the Tasks • Reflection
The Five Strands of Mathematical
Proficiency
Five Strands of Mathematical Proficiency
• Conceptual Understanding
• Procedural Fluency • Strategic Competence • Adaptive Reasoning • Productive Disposition
1174 THE STRANDS OF MATHEMATICAL PROFICIENCY
Box 4-1
Intertwined Strands of Proficiency
Conceptual Understanding
StrategicCompetence
ProductiveDisposition
ProceduralFluency
AdaptiveReasoning
work has some similarities with the one used in recent mathematics assess-ments by the National Assessment of Educational Progress (NAEP), whichfeatures three mathematical abilities (conceptual understanding, proceduralknowledge, and problem solving) and includes additional specifications forreasoning, connections, and communication.2 The strands also echo compo-nents of mathematics learning that have been identified in materials forteachers. At the same time, research and theory in cognitive science providegeneral support for the ideas contributing to these five strands. Fundamen-tal in that work has been the central role of mental representations. Howlearners represent and connect pieces of knowledge is a key factor in whetherthey will understand it deeply and can use it in problem solving. Cognitive
Copyright © National Academy of Sciences. All rights reserved.
Kilpatrick, Swafford, & Findell. (2001) – Adding It Up (p.116)
Developing the Five Strands Conceptual Understanding
• Enables students to connect ideas to what they already know • Supports retention and prevents common errors
Procedural Fluency
• Learning procedures can strengthen and develop mathematical understanding, while understanding makes it easier to learn skills
Strategic Competence
To come up with answer to a problem, students must: • follow a solution method and adapt as necessary • understand the quantities in the problem and their relationships • represent the relationships mathematically • have the mathematical skills required to solve the problem
Adaptive Reasoning
As students reason about a problem they can: • build their understanding • carry out the needed computations • apply their knowledge • explain their reasoning to others
Productive Disposition
Requires frequent opportunities to: • make sense of mathematics • recognize the benefits of perseverance • experience the rewards of sense making in mathematics
Conceptual Understanding
Conceptual Understanding
“Conceptual understanding frequently results in students having less to learn because they can see the deeper similarities between superficially unrelated situations. Their understanding has been encapsulated into compact clusters of interrelated facts and principles.”
Kilpatrick, Swafford, & Findell. (2001) – Adding It Up (p.120)
Levels of Conceptual Understanding
!
1 Factual Knowledge:
remember and recall factual information
2 Comprehension:
demonstrate understanding of ideas and concepts
3 Application:
apply comprehension to unfamiliar situations
4 Analysis:
break down concepts into parts
5 Synthesis:
transform and combine ideas to create something
new
6 Evaluation: think critically
about and defend a position
Adapted from: https://mathequality.wordpress.com/2012/06/25/nrcs-five-strands-of-mathematical-proficiency/
Procedural Fluency
Procedural Fluency
“Business and political leaders are asking schools to ensure that students leave high school 'college and career ready,' possessing 21st Century competencies that will prepare them for adult roles as citizens, employees, managers, parents, volunteers, and entrepreneurs. Using mathematics effectively to solve real-world problems is a critical component of those competencies, and, consequently, is a strong emphasis in the Common Core State Standards for Mathematics and other high-quality standards. Developing procedural fluency is a critical part of instruction to ensure that students are adequately prepared for their futures.”
NCTM President Diane Briars (2014) Students Need Procedural Fluency in Mathematics
Procedural Fluency
The ability to apply appropriate procedures: • Accurately - reliably producing the correct
answer • Efficiently – carrying out procedures easily,
keeping track of sub-problems, and making use of intermediate results to solve a problem
• Flexibly - knowing more than one approach, choosing an appropriate strategy, and using one method to solve and another method to double-check
Briars (2016) NCSM Supporting Teachers in Building Procedural Fluency from Conceptual Understanding
Strategic Competence
When faced with a mathematical problem students who are strategically competent have the ability to: Formulate • Identify accessible questions • Make simplifying assumptions • Identify significant variables and generate relationships between
them
Represent • Represent the situation mathematically, selecting appropriate
mathematical concepts and procedures
Solve • Monitor progress in a solution approach, changing direction as
needed • Interpret and evaluate results in the context of the problem • Explain why a conclusion does or doesn’t make sense
Strategic Competence
Students as Active Problem Solvers
Reproduced from https://www.oecd.org/pisa/pisaproducts/Draft%20PISA%202015%20Mathematics%20Framework%20.pdf
Relationship Between the Strands
• Strategic Competence - Provides a context for developing conceptual
understanding - Depends on conceptual understanding and procedural
fluency
• Procedural Fluency - Develops as strategic competence is used to select
procedures
• Conceptual Understanding - Develops as new skills are acquired through increased
competence at devising strategies for solving problems
Adaptive Reasoning
Adaptive Reasoning
“One hallmark of mathematical understanding is the ability
to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes
from.”
(Common Core Standards, page 4)
Relating the Strands
• Adaptive reasoning interacts with the other mathematical proficiency strands
- Conceptual understanding provides representations that can serve as a source of adaptive reasoning
- Solution strategies require procedural fluency and adaptive reasoning is used to determine the appropriateness of procedures
- Strategic competence is used to monitor progress toward a solution and and to generate alternative plans as necessary
Productive Disposition
Productive Disposition
• A productive disposition develops with the other strands, helping each of them to develop.
• Students’ disposition toward mathematics is a major factor in determining their educational success.
• Students who have developed a productive disposition are confident in their knowledge and ability.
Introducing the Tasks
Task A: Percent Change Game Use these 12 numbers to fill in the gaps below.
10, 20, 25, 35, 40, 50, 60, 70, 75, 80, 90, 100
$____ increased by ____% = $____
$____ increased by ____% = $____
$____ decreased by ____% = $____
$____ decreased by ____% = $____
You score a point for each number used only once – and in a correct expression. (Maximum score: 12 points)
Adapted from http://www.foster77.co.uk/Percentage%20Change%20TASK%20SHEET.pdf
Task B: Describing and Defining a Square
Four equal sides
Two pairs of parallel sides
Two equal diagonals
Diagonals meet at right angles
4 lines of symmetry
Four right angles
Rotational symmetry of order 4
Which pairs of statements define a square? Which pairs do not?
A lesson called ‘Describing and Defining Quadrilaterals’ based on this task, can be found on the Mathematics Assessment Project website (http://map.mathshell.org/)
Task C: Always, Sometimes or Never True?
If you add the same number to the top and bottom of a
fraction, the fraction increases in value.
If you divide the top and bottom of a fraction by the same number, the fraction
gets smaller in value.
If you divide 12 by a number, the answer will be less than
12.
If you multiply 12 by a number, the answer will be
greater than 12.
Prices increased by 20%. They then decreased by
20%. There was no overall change
in prices.
Jill got a pay rise of 3%. James got a pay rise of 2%. Jill therefore got the greater
pay rise.
Task D: Schoolteachers and Dentists
There are about 320 million people in the US. • About how many school
teachers are there?
Estimate some other facts and check them out.
• About how many dentists are there?
Adapted from http://www.bowlandmaths.org.uk/pd/pd_01.html
Working on the Tasks
Working on the Tasks
• On your own, have a go at the task. • Discuss in your groups which of the five strands
of mathematical proficiency you think your task could support, and how.
• Consider ways in which the task might be modified to address the other strands that it doesn’t currently support.
• Elect someone in your group to be the spokesperson for presenting your ideas to the rest of the group.
Feedback on the Tasks
Task A: Percent Change Game Use these 12 numbers to fill in the gaps below.
10, 20, 25, 35, 40, 50, 60, 70, 75, 80, 90, 100
$____ increased by ____% = $____
$____ increased by ____% = $____
$____ decreased by ____% = $____
$____ decreased by ____% = $____
You score a point for each number used only once – and in a correct expression. (Maximum score: 12 points)
Adapted from http://www.foster77.co.uk/Percentage%20Change%20TASK%20SHEET.pdf
Task A: Percent Change Game Use these 12 numbers to fill in the gaps below.
10, 20, 25, 35, 40, 50, 60, 70, 75, 80, 90, 100
$____ increased by ____% = $____
$____ increased by ____% = $____
$____ decreased by ____% = $____
$____ decreased by ____% = $____
You score a point for each number used only once – and in a correct expression. (Maximum score: 12 points)
Adapted from http://www.foster77.co.uk/Percentage%20Change%20TASK%20SHEET.pdf
Études
Études are especially written to practice particular technical skills, but are still beautiful pieces of music in their own right.
Task B: Describing and Defining a Square
Four equal sides
Two pairs of parallel sides
Two equal diagonals
Diagonals meet at right angles
4 lines of symmetry
Four right angles
Rotational symmetry of order 4
Which pairs of statements define a square? Which pairs do not?
A lesson called ‘Describing and Defining Quadrilaterals’ based on this task, can be found on the Mathematics Assessment Project website (http://map.mathshell.org/)
Task B: Describing and Defining a Square
Four equal sides
Two pairs of parallel sides
Two equal diagonals
Diagonals meet at right angles
4 lines of symmetry
Four right angles
Rotational symmetry of order 4
Which pairs of statements define a square? Which pairs do not?
A lesson called ‘Describing and Defining Quadrilaterals’ based on this task, can be found on the Mathematics Assessment Project website (http://map.mathshell.org/)
Task C: Always, Sometimes or Never True?
If you add the same number to the top and bottom of a
fraction, the fraction increases in value.
If you divide the top and bottom of a fraction by the same number, the fraction
gets smaller in value.
If you divide 12 by a number, the answer will be less than
12.
If you multiply 12 by a number, the answer will be
greater than 12.
Prices increased by 20%. They then decreased by
20%. There was no overall change
in prices.
Jill got a pay rise of 3%. James got a pay rise of 2%. Jill therefore got the greater
pay rise.
Task C: Always, Sometimes or Never True?
If you add the same number to the top and bottom of a
fraction, the fraction increases in value.
If you divide the top and bottom of a fraction by the same number, the fraction
gets smaller in value.
If you divide 12 by a number, the answer will be less than
12.
If you multiply 12 by a number, the answer will be
greater than 12.
Prices increased by 20%. They then decreased by
20%. There was no overall change
in prices.
Jill got a pay rise of 3%. James got a pay rise of 2%. Jill therefore got the greater
pay rise.
Task D: Schoolteachers and Dentists
There are about 320 million people in the US. • About how many school
teachers are there?
Estimate some other facts and check them out.
• About how many dentists are there?
Adapted from http://www.bowlandmaths.org.uk/pd/pd_01.html
Task D: Schoolteachers and Dentists
There are about 320 million people in the US. • About how many school
teachers are there?
Estimate some other facts and check them out.
• About how many dentists are there?
Adapted from http://www.bowlandmaths.org.uk/pd/pd_01.html
Developing Mathematical Proficiency
Developing Mathematical Proficiency - Handouts Page 3
Version: 9 September 2016 CD
Handout 2: Task A
Percent Change
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Developing Mathematical Proficiency - Handouts Page 4 Version: 9 September 2016 CD
Handout 3: Task B
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Developing Mathematical Proficiency - Handouts Page 5 Version: 9 September 2016 CD
Handout 4: Task C
Always, Sometimes or Never True?
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Developing Mathematical Proficiency - Handouts Page 6 Version: 9 September 2016 CD
Handout 5: Task D
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Reflection
Reflection
• Think about tasks that are already in use (either by you or well known tasks) and categorize them under the five strands of mathematical proficiency.
• Which of the five strands of mathematical
proficiency do students currently have the most opportunity to develop in your classroom?
• How do you plan to include a balance of the mathematical proficiency strands in your curriculum over the coming weeks.
Mathematics Assessment Project
© Shell Centre, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at
http://creativecommons.org/licenses/by-nc-nd/3.0/ - all other rights reserved
Mathematics Improvement Network
© 2017 Mathematics Assessment Resource Service, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at
http://creativecommons.org/licenses/by-nc-sa/4.0/ - all other rights reserved
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