Developing and Using Meaningful Math Tasks The Key to Math
Common Core Take a moment to record on a sticky: What is a
meaningful Math Task?
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Norms Courtesy Be on time Cell phones on silent, vibrate, or
off Be mindful of side-bar conversations Focus on the task at hand
Collaborative Promote a sense of inquiry Frame meaningful questions
Pay attention of self and others Assume positive intentions Be
reflective
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Todays Outcomes Participants will have a better understanding
of what they need to expect from their students in math.
Participants will have a better understanding of how to select and
set up a challenging math task. Participants will have a better
understanding of facilitate a math task. Participants will have a
better understanding of how to increase the cognitive demands of a
math task.
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What are we asking our students to: Think about? Talk about?
Understand? Mathematics is a participant sport. Children must play
it frequently to become good at it. National Research Council
2009
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Lets Watch Notice what the teacher does to start the lesson,
what skills do students develop through daily mental math? How is
this task differentiated for every child? What do you gain as a
teacher by doing a task like the one in the video?
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Mathematical Tasks: A Critical Starting Point for Instruction
There is no decision that teachers make that has a greater impact
on students opportunities to learn and on their perceptions about
what mathematics is than the selection or creation of the tasks
with which the teacher engages students in studying mathematics.
Lappan & Briars, 1995
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Helping Pets Three different veterinarians each help a total of
63 dogs and cats in a week, but each veterinarian helps a different
number of dogs and cats. How many dogs and cats could each
veterinarian have helped?
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Setting up the Task What types of animals do people have as
pets? What do we do when our pets get sick? What is a veterinarian?
How many pets do you think a veterinarian can help in an hour or a
day? What math is in this task? How many dogs and cats do the 3
veterinarians help? Does each veterinarian help the same number of
dogs and cats? What word do we use to describe the total number of
dogs and cats? What is an addend in the problem? Will the addends
be the same for each veterinarian?
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Facilitating the Task How will you find the possible addends in
this problem? What tools could you use to be sure you are accurate?
How will you prove your solutions are correct? How will you explain
your solutions to the class?
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Extending the Task What are some of the solutions? Did we find
all of the solutions? Are there more combinations of addends that
have a sum of 63? How would our number change if each veterinarian
also helped guinea pigs? How are the number of dogs and the number
of cats related? What happens to the number of dogs when we
decrease the number of cats?
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Animals and Fences at the Zoo The Problem: A zookeeper was
promised that she could have some special animals called mathemals.
She has twenty connecting cubes to be used as fencing to build a
pen for the mathemals. What type of pen can she make to hold the
most mathemals?
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Materials: Twenty connecting cubes of one color to use as
fencing and a large supply of connecting cubes of another color to
use as mathemals when testing various solutions. Grid paper for
recording the results.
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Rules: Work in teams to use all twenty connecting cubes to
build the pen, with each cube joining another cube, face against
face. The pen must be closed, with no doors or openings, so that
the mathemals cannot get out. Mathemals cannot be allowed to stand
on top of one another in the pen. Each mathemal in the pen uses the
space of one cube.
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What learning took place? Take a moment at your table to decide
which standards this covers at your grade level. How could this
task be adapted to fit every students ability? How could you adapt
it to fit better at your grade level?
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Mathematical Tasks: A Critical Starting Point for Instruction
If we want students to develop the capacity to think, reason, and
problem solve then we need to start with high-level, cognitively
complex tasks. Stein & Lane, 1996 High achievement always takes
places in the framework of high expectation. Charles Kettering
Slide 16
Break 10 -15 min.
Slide 17
Candy Shop Melissa went to the candy store and grabbed a large
bag to fill with candy. There were 5 jars of yummy candy. At the
first jar, she put 2 pieces of candy in the bag. At the second jar,
she put 4 pieces in the bag and at the third jar, she put 6 pieces
in the bag. If this pattern continues, how many pieces of candy
will Melissa have after she visits all 5 jars?
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Candy Shop Part II Brent went to the new Candy Shop in town. He
grabbed a large bag to fill with goodies. There were 4 jars of
yummy candy. At the first jar, he put 5 pieces of candy in the bag.
At the second jar, he put 10 pieces in the bag and at the third jar
he put 15 pieces in the bag. If this pattern continues, how man
pieces of candy will Brent have after he visits all 4 jars? While
Brent was leaving the candy store his brother stopped by. Brent
walked with his brother to all 4 jars. At each jar, he ate 5 pieces
of candy from his bag that he had already collected. How man pieces
of candy does he have when he finally leaves the candy store?
Slide 19
Your principal would like your class to create greeting cards
in honor of Geometry Day. The cards will decorate the halls of your
school. The principal has put some requirements for the cover of
the cards. Design Rules: The design must include 11 polygons and at
least 44 sides. You must include at least one triangle, one
quadrilateral, one pentagon, and one hexagon. Consider using a
straight edge to draw your polygons. Work with your partner to
check and make sure you both have followed the rules on your
design.
Slide 20
Report the information in a chart like this on the inside of
your card. Name of the shapeNumber in the DrawingNumber of sides
Triangle Quadrilateral Pentagon Hexagon Totals: Use the back of the
card to complete the following sentences: I learned That Sometimes
I need to remember
Slide 21
Where is the balance? When do you do whole class tasks? How
often? Where do centers fit into the day? How do you find the right
balance?
Slide 22
Things to Think about What is the purpose of the task or
center? What are the students going to be learning or practicing?
How are you going to hold them accountable?
Slide 23
Selecting a Math Task What are your goals for this lesson? What
mathematical content and processes do you hope students will learn
from their work on this task? In what ways does this task build on
students previous knowledge? What definitions, concepts, or ideas
do students need to know in order to begin to work on the
task?
Slide 24
Setting Up a Math Task What are all the ways the task can be
solved? How will you ensure that students remain engaged in the
task? What are your expectations for students as they work on and
complete this task? How will you introduce students to the activity
so as not to reduce the demands of the task? What will you hear
that lets you know students understand the task?
Slide 25
Supporting Students Exploration What questions will you ask to
focus their thinking? What will you see or hear that lets you know
how students are thinking about mathematical ideas? What questions
will you ask to assess students understanding?
Slide 26
Break 10 15 min.
Slide 27
Where does Investigations fit in to all of this work? What
makes a number even or odd? Imagine a group of 12 students. Can
they make two equal teams? How do you know? Can they make partners
with no one left over? How do you know? What about a group of 13
students? Lets think abut what happens when you put two groups
together. Think about this problem: In Ms. Ortegas class, there are
4 students in the blue group and 6 students in the yellow group. If
we put the two groups together, could everyone have a partner? How
many pairs would there be?
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Kindergarten
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First Grade
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Second Grade
Slide 33
Games How can games be used like tasks to further student
understanding of math standards? Take a moment to turn and
talk.
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Why Play Games? Playing games encourages strategic mathematical
thinking as students find different strategies for solving problems
and it deepens their understanding of numbers. Games, when played
repeatedly, support students development of computational fluency.
Games provide opportunities for practice, often without the need
for teachers to provide the problems. Teachers can then observe or
assess students, or work with individual or small groups of
students. Games have the potential to develop familiarity with the
number system and with benchmark numbers such as 10s, 100s, and
1000s and provide engaging opportunities to practice computation,
building a deeper understanding of operations. Games provide a
school to home connection. Parents can learn about their childrens
mathematical thinking by playing games with them at home.
Slide 35
Holding Students Accountable While playing games, have students
record mathematical equations or representations of the
mathematical tasks. This provides data for students and teachers to
revisit to examine their mathematical understanding. After playing
a game have students reflect on the game by asking them to discuss
questions orally or write about them in a mathematics notebook or
journal: What skill did you review and practice? What strategies
did you use while playing the game? If you were to play the games a
second time, what different strategies would you use to be more
successful? How could you tweak or modify the game to make it more
challenging?
Slide 36
Break 10 15 min.
Slide 37
Mathematical Tasks: A Critical Starting Point for Instruction
Not all tasks are created equal, and different tasks will provoke
different levels and kinds of student thinking. Stein, Smith,
Henningsen, & Silver, 2000
Slide 38
Mathematical Tasks: A Critical Starting Point for Instruction
If we want students to develop the capacity to think, reason, and
problem solve then we need to start with high- level, cognitively
complex tasks. Stein & Lane, 1996
Slide 39
Increase the Cognitive Demand of the Task Increase complexity
Introduce ambiguity Synthesize strand of mathematics Invite
conceptual connections Require explanation and justification
Propose solutions that reveal misconceptions or common errors
Slide 40
Invite students to: Describe their process Reflect on their
decisions Explain their vigilance Confirm their thinking Make
connections Promote discourse
Slide 41
What is Fluency? Take a minute to turn and talk at your table.
CCSSM describes procedural fluency as skill in carrying out
procedures flexibly, accurately, efficiently, and appropriately.
Fact fluency as the efficient, appropriate, and flexible
application of single-digit calculation skill and an essential
aspect of mathematical proficiency Baroody 2006
Slide 42
How to Assess Fluency? Take a minute to turn and talk. Must
address all 4 tenets Flexibility Appropriate strategy use
Efficiency Accuracy Must also provide data on which facts students
know from memory.
Slide 43
3 Protocols to Use: Protocol A Assess fluency 1. Write 4 + 5 on
a card. What does 4 + 5 mean? 2. What is the answer to 4 + 5? 3.
How did you find the answer to 4 + 5? Can you find it another way?
4. If your friend was having trouble remembering this fact, what
strategy might you suggest to him or her?
Slide 44
3 Protocols to Use: Protocol B Assess flexibility and strategy
selection 1. What is 8 + 5? 2. How can you use 8 + 2 to help you
solve 8 + 5? OR 1. How can you use 3 * 7 to solve 6 * 7?
Slide 45
3 Protocols to Use: Protocol C Assess use of appropriate
strategy (Henry and Brown 2008) Probes: What is 7 + 8? How did you
figure it out? Codes: R = recall A = Automatic (within 3 seconds)
M10 = Making 10 Strategy ND = Near Doubles Strategy D = Some other
derived fact strategy CO = Counting On CA = Counting All MCA =
Modeling and Counting On
Slide 46
Writing Prompts to Develop Fact Fluency Appropriate strategy
selection: Explain how to use the count on strategy for 3 + 9 What
strategy did you use to solve 6 + 8 A friend is having trouble with
some of his times 6 facts. What strategy might you teach him? Emily
solved 6 + 8 by changing it in her mind to 4 + 10. What did she do?
Is this a good strategy? Tell why or why not.
Slide 47
Writing Prompts Flexibility How can you use 7 * 10 to find the
answer to 7 * 9? Solve 6 * 7 using one strategy. Now try solving it
using a different strategy. Emily solved 6 + 8 by changing it in
her mind to 4 + 10. What did she do? Does this strategy always
work?
Slide 48
Writing Prompts Efficiency What strategy did you use to solve 9
+ 3? How can you use 6 + 6 to solve 6 + 7? Which facts do you just
know? For which facts do you use a strategy?
Slide 49
Writing Prompts Accuracy Crystal explains that 6 + 7 is 12. Is
she correct? Explain how you know. What is the answer to 7 + 8? How
do you know it is correct (how might you check it)?
Slide 50
Ideas that Address More than one Componet Develop a Face the
facts or Ask Bulldog column (like Dear Abby) for the class.
Students send a letter about a tough fact. Rotate different
students into the role of responder. The responder writes letters
back, suggesting a strategy for the tough fact. Create a strategy
rhyme. Make a facts survival guide. Children prepare pages
illustrating with visuals of how to find tough facts. Write a
yearbook entry to some facts.
Slide 51
Reflecting on What You Do for Fluency With your current
assessments, what percentage of emphasis might you assign to each
of the four categories we have discussed? Is this balance what you
would like it to be? If not, how might you alter your assessments
to equitably address the four areas of fluency?
Slide 52
What are Math Talks or Number Talks? Short pedagogical routines
No longer than 10 minutes Creates number flexibility and
automaticity Helps with math facts Helps develop number sense It is
a flexible, visual, creative approach to solving mental math
problems
Slide 53
Number Talk Steps Pose problem horizontally Thumbs up Share out
answers record all Does everyone agree with one of the answers?
Defend an answer with a strategy Record thinking with students name
Discuss strategy connections, highlight Do we all agree on an
answer?
Slide 54
Lets Try One 7 + 3 7 + 5 + 3 3 + 6 + 7
Slide 55
Record your thoughts 1. Are there any aspects of your own
thinking and/or practice that our work today has caused you to
consider or reconsider? Explain. 2. Are there any aspects of your
students mathematical learning that our work today has caused you
to consider or reconsider? Explain.
Slide 56
Last Thought One thing is to study whom you are teaching, the
other thing is to study the knowledge you are teaching. If you can
interweave the two things together nicely, you will
succeed...Believe me, it seems to be simple when I talk about it,
but when you really do it, it is very complicated, subtle, and
takes a lot of time. It is easy to be an elementary school teacher,
but it is difficult to be a good elementary school teacher. Quote
from Tr. Wang, Ma 1999
