N UMBER S ENSE Math & Science Collaborative at the Allegheny Intermediate Unit

R EFLECTING ON N UMBER S ENSE 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. Math & Science Collaborative at the Allegheny Intermediate Unit

A GENDA Ways to Make a Number Do these children have number sense? – video Games to Develop Number Sense Mental Math Videos Article discussion Planning Math & Science Collaborative at the Allegheny Intermediate Unit

C OMMUNITY OF L EARNERS 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 Math & Science Collaborative at the Allegheny Intermediate Unit

K1 – dot cards and ten frames Which 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? V IDEOS : D O THESE STUDENTS HAVE NUMBER SENSE ? Math & Science Collaborative at the Allegheny Intermediate Unit

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? V IDEOS : D O THESE STUDENTS HAVE NUMBER SENSE ? Math & Science Collaborative at the Allegheny Intermediate Unit

I MPLICATIONS FOR T EACHING AND L EARNING 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? Math & Science Collaborative at the Allegheny Intermediate Unit

G AMES TO D EVELOP N UMBER S ENSE Capture 5 Capture 10 Rack 10 Play 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? Math & Science Collaborative at the Allegheny Intermediate Unit

G AMES TO D EVELOP N UMBER S ENSE 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? Math & Science Collaborative at the Allegheny Intermediate Unit

V IDEOS : HTTP :// ILLUMINATIONS. NCTM. ORG /R EFLECTIONS _ P RE K-2. HTML HTTP :// ILLUMINATIONS. NCTM. ORG /R EFLECTIONS _ P RE K-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? Math & Science Collaborative at the Allegheny Intermediate Unit

V IDEO D ISCUSSION 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? Math & Science Collaborative at the Allegheny Intermediate Unit

V IDEO D ISCUSSION 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? Math & Science Collaborative at the Allegheny Intermediate Unit

C OMPOSITION AND D ECOMPOSITION 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 Collaborative at the Allegheny Intermediate Unit

S TAND U P AND B E C OUNTED Read pages 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? Math & Science Collaborative at the Allegheny Intermediate Unit

M ENTAL M ATH 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 Math & Science Collaborative at the Allegheny Intermediate Unit

V IDEO : 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? Math & Science Collaborative at the Allegheny Intermediate Unit

M ENTAL M ATH Given that 48+37=85, what is the answer to 49+36? Show why this is true using the number line. Math & Science Collaborative at the Allegheny Intermediate Unit

M ENTAL M ATH Solve and show why this is true using the number line. Math & Science Collaborative at the Allegheny Intermediate Unit

V IDEO : 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? Math & Science Collaborative at the Allegheny Intermediate Unit

V IDEO : Watch the video to see if any of the students used the same strategies we did. How is number sense used to determine whether an answer is reasonable or not? What purpose is served by the teacher accepting both correct and incorrect answers? Cole’s strategy does not involve removing or taking away an amount. He adds to solve the problem. Why does this work? Is there any evidence that students were engaged in the Standards for Mathematical Practice? Math & Science Collaborative at the Allegheny Intermediate Unit

V IDEOS smentsOverview After each video we will stop and discuss: Which of the students’ strategies matched our thinking? What was new, what was different? Based on strategy students used, which properties did they use? Analyze student’s errors Math & Science Collaborative at the Allegheny Intermediate Unit

D EVELOPING N UMBER S ENSE THROUGH R EAL -L IFE S ITUATIONS How does thinking about numbers in real-life situations help develop and/or improve children’s number sense? Math & Science Collaborative at the Allegheny Intermediate Unit

P LANNING 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. Math & Science Collaborative at the Allegheny Intermediate Unit

H OMEWORK Do the activity on page 3 of Developing Number Sense, Grades 3-6. Read Get to 1,000, case 1, pages Math & Science Collaborative at the Allegheny Intermediate Unit