Developing and Using Meaningful

Math TasksThe Key to Math Common Core

Take a moment to record on a sticky: What is ameaningful Math Task?

NormsCourtesy 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

Today’s OutcomesParticipants 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 how to facilitate a math task.

Participants will have a better understanding of how to increase the cognitive demands of a math task.

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

Let’s WatchNotice 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?

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

Helping PetsThree 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?

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?

Facilitating the TaskHow 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?

Extending the TaskWhat 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?

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?

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.

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.

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 student’s ability?

How could you adapt it to fit better at your grade level?

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

Candy ShopMelissa 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?

Candy Shop Part IIBrent 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?

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.

Report the information in a chart like this on the inside of your card.

Name of the shape Number in the Drawing

Number 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…

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?

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?

Selecting a Math TaskWhat 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?

Setting Up a Math TaskWhat 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?

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?

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?

Let’s think abut what happens when you put two groups together. Think about this problem: In Ms. Ortega’s 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?

Kindergarten

First Grade

Second Grade

GamesHow can games be used like

tasks to further student understanding of math standards?

Take a moment to turn and talk.

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 children’s mathematical thinking by playing games with them at home.

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?

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

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

Increase the Cognitive Demand of the Task

Increase complexityIntroduce ambiguitySynthesize strand of mathematicsInvite conceptual connectionsRequire explanation and justificationPropose solutions that reveal misconceptions or

common errors

Invite students to:Describe their processReflect on their decisionsExplain their vigilanceConfirm their thinkingMake connectionsPromote discourse

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

How to Assess Fluency?Take a minute to turn and talk.Must address all 4 tenets

FlexibilityAppropriate strategy useEfficiency Accuracy

Must also provide data on which facts students know from memory.

3 Protocols to Use:Protocol AAssess 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?

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

3. How can you use 3 * 7 to solve 6 * 7?

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

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.

Writing PromptsFlexibility• 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?

Writing PromptsEfficiency• 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?

Writing PromptsAccuracy• 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)?

Ideas that Address More than one Component

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.

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?

What are Math Talks or Number Talks?

Short pedagogical routinesNo longer than 10 minutesCreates number flexibility and automaticityHelps with math factsHelps develop number senseIt is a flexible, visual, creative approach to

solving mental math problems

Number Talk StepsPose problem horizontallyThumbs upShare out answers – record allDoes everyone agree with one of the answers?Defend an answer with a strategyRecord thinking with student’s nameDiscuss strategy connections, highlightDo we all agree on an answer?

Let’s Try One7 + 3

7 + 5 + 3

3 + 6 + 7

Record your thoughts1. 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.

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

Questions or Concerns?

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