NUMBER SENSE M

ath & S

cience Collaborative at the A

llegheny Interm

ediate Unit

REFLECTING ON NUMBER SENSE

Go to mscsummercourses2013.wikispaces.com and then to the number sense page.

Respond to the discussion prompt “Visual Representations” on the bottom of the page.

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AGENDA

Ways to Make a Number Do these children have number sense? –

video Games to Develop Number Sense Mental Math Videos Article discussion Planning

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COMMUNITY OF LEARNERS

Building a strong and supportive community of learners is essential in making number sense routines beneficial and productive for everyone.Students share out thinking Work through misunderstandings in front

of peers Have time to think for themselves

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K1 – dot cards and ten framesWhich students are able to subitize and unitize five and which ones are not? What evidence do you have?How do the visual representations support students’ in developing number sense?

How does the ten-frame with seven dots provide opportunities for students to use five as a unit?

How does the teacher elicit and support students’ uses of different strategies?What evidence did you find of the students engaging in the Standards for Mathematical Practice?

VIDEOS: DO THESE STUDENTS HAVE NUMBER SENSE?

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K2 – rekenreks How does the warm-up with four and six provide the teacher with information to guide her instructional decisions?What examples of one-to-one correspondence and conservation of number surface in the lesson?Are there specific number combinations that seem more accessible to students? How does the teacher build on this foundation?How does the rekenrek support students’ in developing number sense?What evidence did you find of the students engaging in the Standards for Mathematical Practice?

VIDEOS: DO THESE STUDENTS HAVE NUMBER SENSE?

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IMPLICATIONS FOR TEACHING AND LEARNING

What kinds of activities do you think these children were engaged in prior to these lessons?

What evidence is there that they used these tools prior to the lessons we viewed?

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GAMES TO DEVELOP NUMBER SENSE Capture 5 Capture 10 Rack 10Play each of the games. After you play each

game, answer the following questions: What concept(s) is the focus of each game? What tools might students use to support to

solve the problems? Discuss the strategies students might use. Might

they reason with facts they know? Use landmark strategies? Use make-a-ten? Use the 5- or 10-structure? How might you promote the use of more efficient strategies?

Is there any way to adapt these games for older students to play?

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GAMES TO DEVELOP NUMBER SENSE

What evidence will you look for to see if students understand the underlying mathematics of the game?

What evidence will you look for to see if students are using number sense as they play the game?

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VIDEOS: HTTP://ILLUMINATIONS.NCTM.ORG/REFLECTIONS_PREK-2.HTML What can you say about Cena’s understanding

of numbers after listening to her explanation? Why might Cena be able to write two-digit

numerals correctly without being able to explain the meaning of digits?

What reasons might there be for the differences in Cena's performance during the whole class lesson as compared to the individual interview?

How would you describe Jonathan's understanding of place value as compared to Cena's understanding?

How would you compare Rudy's understanding of place value to Cena's and Jonathan's understandings?

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VIDEO DISCUSSION Do you think the children would have done

better if the place value interview question had used numbers above twenty instead of numbers in the teens? In English, teen numbers are especially difficult. Compare Asian languages, in which 13 is pronounced, "one ten and three," to the English word "thirteen." It is not clear how "three" and "thirteen" connect.

Do you think it was "fair" that the children were not all asked the same questions in the same way? Can we fully probe a young child's understanding without the type of freedom that a clinical interview such as these provides?

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VIDEO DISCUSSION Interviewers must be sensitive to how children

interpret requests such as "represent the '1'." Might a child be confused unless this is carefully crafted to ensure the child knows the interviewer means "the part of the initial teen number that the '1' represents"?

Would you agree that basing an assessment of a child's understanding with just standard curriculum tasks is inadequate? How might such tasks be altered so that teachers can observe individual children in the context of classroom work?

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COMPOSITION AND DECOMPOSITION

Ways to Make a Number How many ways can you make 50?When is 50 a lot? When is 50 very little?

Why might you want students to think about ways to make a number?

Math &

Science C

ollaborative at the Allegheny

Intermediate U

nit

STAND UP AND BE COUNTED

Read pages 113-126. How were the students’ strategies for making

25 the same and different from our strategies for making 50?

How does this activity contribute to developing number sense?

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MENTAL MATH

When students solve a problem in their heads, they usually use what they know about place value, number relationships, patterns, and visual images to figure it out. Problem context Naked numbers

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VIDEO: 38 + 37

Watch the video to see if any of the students used the same strategies we did.

The teacher asks students to consider previous strategies they have used? Why?

The teacher leaves Arzoo’s strategy and then returns to it. What purpose does this serve?

How is place value emphasized during this number talk?

How does each strategy foster number sense and the use of number relationships?

Is there any evidence that students were engaged in the Standards for Mathematical Practice?

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MENTAL MATH

Given that 48+37=85, what is the answer to 49+36? Show why this is true using the number line.

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MENTAL MATH

400-69. Solve and show why this is true using the number line.

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VIDEO: 70-59 How does the number line provide support for

understanding Grant’s Constant Difference strategy? In what other ways could you model this strategy to help students understand why it works?

The students share several subtraction strategies: Constant Difference, Adding Up, Removing in Chunks, and Adjusting One of the Numbers. Discuss how these strategies are similar and different?

When Andrew shared his strategy, students were unsure whether 1 should be added or subtracted. Share your observations about the student and teacher roles during the class discussion. How could you help students understand his thinking?

Is there any evidence that students were engaged in the Standards for Mathematical Practice?

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DEVELOPING NUMBER SENSE THROUGH REAL-LIFE SITUATIONS

How does thinking about numbers in real-life situations help develop and/or improve children’s number sense?

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PLANNING

Plan task focusing on one standard and one standard for math practice involving the use of visuals to develop/extend number sense.Include a mental math activity and

a game.

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HOMEWORK

Do the activity on page 3 of Developing Number Sense, Grades 3-6.

Read Get to 1,000, case 1, pages 4-16.

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