talking the talk and doing the math - … the talk and doing the math building students conceptual...
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TALKING THE TALK AND DOING THE MATH
Building Students’ Conceptual Understanding and Mathematical Vocabulary
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by Terry Walsh [email protected]
– Is it important for students in your classes to understand critical vocabulary terms in mathematics?
– Have you ever taken a course in how to teach vocabulary in math?
– So, how do we deal with the answers to these two questions?
Some Questions to Start
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Results from several years of asking these questions:
– Everyone agrees that vocabulary is very important.
– Hardly any math teachers have taken any courses on teaching vocabulary in college or graduate school.
You Are NOT Alone!
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The vocabulary tools in Math Tools, 2nd Edition can help you improve and deepen students’ mastery of critical concepts and vocabulary terms.
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The tools that I’ll be showing you today all come from the new text, Math Tools, 2nd Edition.
It contains over 64 ready-to-use tools for
• addressing the Common Core
• differentiating instruction
• keeping students engaged in the mathematics classroom
Where Did These Tools Come From?
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1. Why focus on vocabulary in mathematics?
2. How do we help students with different styles of thinking and learning master critical vocabulary?
3. What specific tools can teachers of mathematics use to help all students build deep understanding of critical terms and concepts?
Today’s Essential Questions
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• Students typically need six or more exposures to a new term before they can use it in context (Jenkins, Stein, & Wysocki, 1984).
• “During these repeated exposures, learning is greatly enhanced if students interact with vocabulary in a variety of ways” (Marzano, 2004).
How do we find the time to accomplish this while teaching all the content we need to teach?
What kinds of things are you already doing to build students’ mathematical vocabulary?
What Does the Research Tell Us?
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The CCSS Mathematical Practices stress precision in problem solving and vocabulary. They address the importance of good verbal and written communication skills. This means that teachers of mathematics must make sure their students have a solid understanding of the vocabulary they need to succeed.
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The Common Core and Vocabulary
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For example:
“Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments.” (Mathematical Practice 3)
“They try to use clear definitions in discussions with others and in their own reasoning…They examine claims and make explicit use of definitions.” (Mathematical Practice 6)
“Students who lack understanding of a topic may rely on procedures too heavily.” (CCSS for Mathematics, 2010, p.8)
Vocabulary and the Common Core
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How do we help students with different styles of thinking and
learning master critical vocabulary?
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– Based on Carl Jung’s groundbreaking research into how people learn
– Silver and Strong have taken these concepts and embedded them into strategies, tools, and lessons that can be used by first-year and veteran teachers alike.
– Two websites you can investigate to find out more about learning styles:
• www.thoughtfulclassroom.com/
The Silver Strong & Associates website
• www.mathcore2k.com Ed Thomas’ new website that contains style-based lessons aligned to the Common Core
Learning Styles and the Thoughtful Classroom
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Mastery • Facts & practical information • Set procedures and single solutions • Problem solving with step-by-step clarity
Interpersonal • Dialogue and collaboration • Real-world applications • Problem solving is an open discussion
Understanding • Emphasizes why things work • Requires explanation, proof • Problem solving by looking for patterns and the big picture
Self-Expressive • Math ideas are explored with imagination • Non-routine and “out-of-the-box” thinking • Problem solving by visualization and creativity, or multiple solution approaches
Making the Connection: From Styles to Tools
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What specific tools can teachers of mathematics use to help all
students build deep understanding of critical terms and concepts?
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Mastery • Vocabulary Knowledge Rating • Knowledge Cards/Glossaries
Interpersonal
• Vocabulary Review Games • I Know What I Know
Understanding
• Three-Way Tie
Self-Expressive
• Cinquains
Making the Connection: From Styles to Tools
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When you begin using the tools, whether they are
tools we cover today or others from Math Tools, be
sure to call them by name. As students get to know
the names, they will also become familiar with their
own role in the learning process, which saves you
time and increases students’ self-direction.
Using the Tools
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• To create a VKR, write up to 20 important vocabulary terms for the next unit in the VKR template.
• The VKR lets student self-assess their own knowledge of the essential terms for a given unit.
• Students rate their understanding of key terms at the start of a unit, in the middle of the unit, and near the end of the unit.
Mastery Tool: Vocabulary Knowledge Rating (VKR)
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Vocabulary
Term
I never
heard of it
I have seen or heard it
before
I think I know what it
means
I know this term and can
explain it
Common denominator 1 2 3
Denominator 1 2 3 4
Equivalent fractions 1 2 3 4
Fraction 1 2 3 4
Improper fraction 1 2 3 4
Least common denominator
1
2
3
4
Mixed number 1 2 3 4
Numerator 1 2 3 4
Proper fraction 1 2 3 4
VKR Partial Sample:
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My Vocabulary Knowledge Rating: 32 out of 48 total.
Sample Knowledge Cards
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Student’s Knowledge Cards for Reflection and Translation
Glossaries
How do your students write definitions?
• Definitions should be personal, not copying.
• Math Tools, 2nd Edition has several examples of how to help students create meaningful definitions.
• Frayer Model
• Step Up to Writing in Mathematics Model
• What do other departments do in your school?
Knowledge Cards
Used for a variety of activities and purposes
• To self-assess vocabulary knowledge
• To deepen understanding and facilitate studying
• In a variety of review games
Mastery Tools: Glossary & Knowledge Cards
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Three-Way Tie gets students to focus on the connections or relationships between different concepts.
We often talk about these connections and are unhappy when students have a hard time remembering how the concepts are related. But do we help students discover and reinforce these connections?
Understanding Tool: Three-Way Tie
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A Non-Math Example:
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George Washington
Thomas
Jefferson Ben
Franklin
All 3 are “Founding Fathers”
of the USA
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T.J. & B. F. both signed the D. of I.
– second, minute, and hour; – fraction, percent, and decimal; – root, zero, and solution; – logarithm, exponent, and base; – fraction, ratio, and proportion; – sine, cosine, and tangent; – translation, rotation, and dilation; – numerator, denominator, and proper fraction; – rectangle, square, rhombus, and trapezoid; What others can you think of for your students?
Possible Math Threesomes
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Understanding Tool: Three-Way Tie
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rate of change
derivative slope
All 3 terms are basic concepts of differential calculus.
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The slope of the line tangent to a curve equals the derivative of the curve.
The “Ideal” Three-Way Tie
In groups, choose the best sentences and write them on a Three-Way Tie Organizer. The sentences can come from one or more students.
Three-Way Tie Extensions
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If the groups create posters for displaying their “ideal” Three-Way Tie, you can post them on the walls in your room, or even in the hall outside your room. This lets students show their work to the entire school.
Understanding Tool: Three-Way Tie
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rate of change
derivative slope
Understanding how all 3 terms give us important
information about how quickly a graph is changing helps us better understand
the function.
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The slope of the line tangent to a curve equals the derivative of the curve.
Cinquains are short, poem-like writing strategies that are similar to haikus in that they have a set pattern:
Topic
Adjective Adjective
Three Action Verbs (ing)
Simple Four Word Sentence (or phrase)
Summary (1 word or phrase)
Self-Expressive Tool: Cinquain
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(FYI: I am from Chicago originally)
Non-Math Example
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Cubbies
lovable defeatable
losing aggravating frustrating
Wait ‘til next year.
daBears
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A Sample Math Cinquain
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common denominator
numerical algebraic
adding subtracting confusing
Fraction addition needs them.
L.C.M.
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Vocabulary Games can help students review and reinforce what they know about the vocabulary they are learning. Games like Bingo, Twenty Questions, Word Search (regular or “Beehive”) Crossword Puzzles, or Jumble add a sense of play to the task of reviewing vocabulary. Since these tools represent the Interpersonal style, competition should be deemphasized. Instead, aim for fun and friendly.
Interpersonal Tool: Vocabulary Games
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In Beehive Word Search, the words are written in a hexagonal graph paper, with each letter of the words in connecting hexagons. The starting letter of the next word shares a side with the last letter of the prior word.
You might tell students the starting point, but have the word list out of order. To promote deeper understanding, provide definitions rather than a simple word list. Assuming this will be your first Beehive Word Search, the next slide has the words listed in order.
“Beehive” Word Search
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• EQUATION
• GRAPH
• SLOPE
• YINTERCEPT
• XINTERCEPT
• ORDINATE
• ABSCISSA
• ORIGIN
• ORDERED PAIR
First word EQUATION, all related to graphing…
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Students write a single vocabulary term on a sticky note without letting other students see the word.
Students stick their word on another student’s back. Students try to guess the word on their backs by asking yes-no questions to other students.
Once students have figured out their word, they put it on the front of their clothes. Then, they can give hints to students who have not yet figured out their word.
20 Questions
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For Math Bingo, let students create their own 5-by-5 grid of words. Decide if they get a “Free” square or not. Read the definitions from one or more students’ Knowledge Cards (or Glossaries).
Math Bingo
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I know what I know about place value:
First, I know __________________________
____________________________________
In addition, I know _____________________
Finally, I know _________________________
_____________________________________
Now, you know what I know about place value.
Interpersonal Tools: I Know What I Know
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that two digit numbers have a 10s
that 10 equals ten 1s.
that the number 25 is made
place and a 1s place.
up of 2 10s and 5 1s.
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I know what I know about logarithms:
First, I know _____________________________
________________________________________
In addition, I know ________________________
Finally, I know ________________________________
Now, you know what I know about logarithms.
Interpersonal Tools: I Know What I Know
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the log base 10 of a number tells me
what power of 10 equals the number.
logs can be done in any positive base.
logarithms and exponents are closely related.
Why Does All This Matter?
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Robert Sternberg, provost of Oklahoma State University, former Yale
professor, and past President of the American Psychological
Association, assigned students to one of five approaches:
• A memory-based approach emphasizing recall of facts and concepts (Mastery)
• An analytical approach emphasizing critical thinking and comparative analysis (Understanding)
• A creative approach emphasizing imagination (Self-Expressive)
• A practical approach emphasizing real-world contexts and situations (Interpersonal)
• A diverse approach incorporating all approaches
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Why Does All This Matter?
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Sternberg found that when students were taught in a manner that matched their learning styles, they outperformed students who were not.
But more important…
Students who were taught using diverse teaching methodologies outperformed both the traditional and the “matched” students.
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Why Does All This Matter?
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“Most important, students in the diverse teaching condition outperformed the other students even in the multiple choice memory tests. In other words, even if our goal is just to maximize our students’ retention of information, teaching for diverse styles of learning still produces superior results. This approach apparently enables students to capitalize on their strengths and to correct or to compensate for their weaknesses, encoding material in a variety of interesting ways.”
—Dr. Robert J. Sternberg (2006)
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Principles and Practical Tools for Improving Classroom Assessment
Webinar participants
will receive 6 free sample
tools from the book!
Moving from Repetition to Reasoning: Tools for Building the Common Core
Mathematical Practices Wednesday—January 30, 2013 - 3 PM AND 6 PM ET
Presented by John R. Brunsting Author of Math Tools, Grades 3-12
Webinar participants
will receive 5 free sample
tools from the book!
TO REGISTER VISIT
www.ThoughtfulClassroom.com/events
Tuesday – February 12, 2013 – 11 AM ET Presented by Dr. Harvey F. Silver Author of Tools for Thoughtful Assessment
TALKING THE TALK AND DOING THE MATH
Building Students’ Conceptual Understanding and Mathematical Vocabulary
Presented by Terry Walsh
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Thank you for attending:
Now Taking Questions
Future questions or for more information:
800.962.4432
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Principles and Practical Tools for Improving Classroom Assessment
Webinar participants
will receive 6 free sample
tools from the book!
Moving from Repetition to Reasoning: Tools for Building the Common Core
Mathematical Practices Wednesday—January 30, 2013 - 3 PM AND 6 PM ET
Presented by John R. Brunsting Author of Math Tools, Grades 3-12
Webinar participants
will receive 5 free sample
tools from the book!
TO REGISTER VISIT
www.ThoughtfulClassroom.com/events
Tuesday – February 12, 2013 – 11 AM ET Presented by Dr. Harvey F. Silver Author of Tools for Thoughtful Assessment