Number Sense

1. Building Number Sense K-2Math ExtravaganzabreakoutOctober 8, 2013Debbie Schraeder, ESU#3dschraeder@esu3.org

2. Number Sense A “good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989).

3. Number and Operations StandardGrades Pre-K-2 • Understand numbers, ways of representing numbers, relationships among numbers, and number systems • Understand meanings of operations and how they relate to one another • Compute fluently and make reasonable estimates

4. Understand numbers, ways of representing numbers, relationships among numbers, and number systems • Count with understanding and recognize “how many” in sets of objects • Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections. • Connect number words and numerals to the quantities they represent, using various physical models and representations

5. Prenumber Concepts • Patterning • Sorting • Classifying

6. Rational Counting Stages • Rational counting uses the ability to rote count, but goes one step farther. Rational counting by ones requires the child to make a one-to one-correspondence between each number name and one-and-only-one object. In addition, the child must realize that the last number said is the total number of objects in the set. Children must also be taught to use partitioning strategies, that is, to systematically separate those objects counted from those that still need to be counted. • Rational counting is an important skill for every primary-grade child.

7. Counting Strategies • Counting On & Counting Back: In Counting On, the child gives correct number names as counting proceeds and can start at any number and begin counting. For example, the child can begin with 7 pennies and count “eight, nine, ten” or begin with 78 pennies and count “79, 80, 81). Counting on is an essential strategy for developing addition. ***Many children find it difficult to count backward, just as many adults find it difficult to recite the alphabet backward. The calculator provides a very valuable instructional tool to help children improve their ability to count backward. • Skip Counting: In skip counting, the child gives correct names, but instead of counting by ones, counts by twos, fives, tens, or other values. In addition to providing work with patterns, skip counting provides readiness for multiplication and division. (Counting change…start with the largest value coin and then continue skip counting by the appropriate value).

8. Writing Numerals • Start with very clear, very strong models. • Focus on one number at a time. • Provide maximum guidance at first. • Be accepting of initial efforts. • Gently reduce the amount of guidance. • Reward correct performance. • Review previously-learned material at regular intervals.

9. How I Boost My Students’ Number Sense • Link school math to real-world experiences • Model different computing methods • Ask students to calculate mentally. • Have class discussions about computing strategies. • Estimate, estimate, estimate! • Question students about how they reason. • Be sure to do plenty of measuring activities. Burn, Marilyn. How I Boost My Students’ Number Sense. Instructor Magazine. April, 1997.

10. Real-World Experience1. Link school math to real-world experiences "When students are engaged in a 'balance' of mathactivities, they can succeed where it counts - in applying their skills and reasoning ability to solve real-life problems requiring mathematical solutions," (Ainsworth, 2000).

11. Math Connections/Integration Greg Tang books Math Matters books

12. Modeling 2. Model different computing methods Standards of Mathematical Practice The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.

13. Standard 4: Model with Mathematics Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

15. Typical first grade place value problem: Write two hundred fifty-three as a numeral. 253

16. What does 253 look like? • Two hundred fifty-three • Two hundred + fifty + three • 200 + 50 + 3

17. What does 253 look like? • My father told me that he has been saving money for Christmas. Right now he has $253 saved. 2 one hundreds + 1 fifty + 3 ones.

18. $253! 2 hundreds + 1 fifty + 3 ones 2 hundreds + 2 twenties + 1 ten + 3 one. .

19. What does $253 look like? • Using the most number of bills? • Using the least number of bills? • How many different ways can we make $253 using U.S. bills? • What about coins…..? .

20. What does 253 look like? The sum of the ages of everyone who lives in my house is 253 years. What could each person’s age be if the following people live in my house: • My grandmother • My grandfather • My mother • My father • My three brothers and sisters. • Me

21. What does 253 look like? • I have $253 to spend at the mall. Here are some things I like. What can I buy and still have more than $50 left? Video games$19.99 each Manga books$6.99 each T-shirts$9.98 each Jeans$15.98 each Movie Tickets$10.00 each What else does 253 look like?

22. Relationships Among Numbers • Spatial Relationships: Spatial relationships – Children can learn to recognize sets of objects in patterned arrangements and tell how many without counting. Prior to counting, children are aware of small numbers of things: one nose, two hands, three wheels on a tricycle. Research shows that most children entering school can identify quantities of three things or less by inspection alone without the use of counting techniques. • One and Two More, One and Two Less: One and Two More…The two-more-than and two-less than relationships involve more than just the ability to count on two or count back two. Children should know that 7, for example, is 1 more than 6 and also 2 less than 9. • Number Benchmarks: Benchmarks or anchors give students a reference point. Since 10 plays such a large role in our numeration system and because two fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 to the important anchors of 5 and 10. (e.g. 8 is 5 and 3 more or two away from 10) • Part-part-whole Relationships: To conceptualize a number as being made up of two or more parts.

23. The meanings for the number five suggested by young children…

24. Calculate Mentally3. Ask students to calculate mentally. • Real life requires mental computation. Students need to be able to move numbers around in their heads and discuss their strategies. *Try playing Deep Sea Duel from Illuminations. Very challenging!

25. Classroom Discussions4. Have class discussions about computing strategies. Improving Participation with Talk Moves Teaching Channel Video clip • The first talk move- repeating. • Another talk move- adding on • One of the talk moves that we created is the silent signal (I am thinking the same thing as you are) • Revising our thinking

26. Numerical problems that have more than one possible answer Students must be able to explain their reasoning. This not only will give you insight into how they think, but also will help the children to cement their own ideas and reevaluate them. *nrichhas free activities (new ones each month) that you can use in the classroom. The focus in on multiple ways of thinking to arrive at an answer. Students can even submit them to the site and a best strategy is chosen each month.

27. Estimation5. Estimate, estimate, estimate! • Estimation • Guess it Game…

28. Estimation Jar #1 What kind of questions do you have about the jar? What kind of mathematical information can we get from the data?

29. Estimation Jar #2 What kind of questions do you have about the jar? What kind of mathematical information can we get from the data?

30. Mathematical Questioning6. Question students about how they reason.

31. Measurement7. Be sure to do plenty of measuring activities. As People Get Older, They Get Taller Illuminations Lesson/Activity

32. 5 Misconceptions in Elementary Mathematics Gallery Walk • Select a cut-out from the center of your table. • Move to the designated chart. This will be your “touring group”. • Read and discuss the charts at each station. • Every 2 minutes, you will be chimed to move to a new chart.

33. Take 2 Take 2 minutes to process your gallery walk discussions with your table.

34. Numeracy Rich Environment • Look at your classroom through the eyes of a new student. Walking into your class, what would he or she see that would indicate the importance of mathematics? Environments rich in numeracy for teaching mathematics might provide the following: • Calendars • Manipulatives • Problem of the Day/Week • Word Walls- mathematics vocabulary • Math Journals • Graphic Organizers • Class Charts • Tools for Measuring • Math-Related Children’s Literature • Math Books by Student Authors • Math Connections to Other Curricular Areas Star/ Identify the 2 attributes you see as your strengths related to a numeracy rich environment. Check/ Identify the attribute you will work on to create a richer environment.

35. Symbaloo Resources/Exploration • Jot down 2 sites/resources you’ll use. • How will you use the resource(s)? • Share after exploration time…

36. Resources • slideshare by Dr. Harriet Thompson. Developing Number Sense and Early Number Concepts • How I Boost My Students’ Number Sense • Teaching Number Sense • NCTM News Bulletin

37. Today’s Learning: Exit Ticket • What worked or went well for you in this session? • What will you change because of this session? • What are you wondering? • What do you need now?