Number Sense and Mental Math A Small but Mighty Instructional Task SIM London Region February 2015

Number Sense What does it mean to have Number Sense? How do you develop Number Sense? Person #1 Person #2 Person #4Person #3 Number Sense

Number Sense and Numeration. Page 8 Number sense refers to a general understanding of number and operations as well as the ability to apply this understanding in flexible ways to make mathematical judgements and to develop useful strategies for solving problems. In this strand, students develop their understanding of number by learning about different ways of representing numbers and about the relationships among numbers. They learn how to count in various ways, developing a sense of magnitude. They also develop a solid understanding of the four basic operations and learn to compute fluently, using a variety of tools and strategies. A well-developed understanding of number includes a grasp of more-and-less relationships, part-whole relationships, the role of special numbers such as five and ten, connections between numbers and real quantities and measures in the environment, and much more. Experience suggests that students do not grasp all of these relationships automatically. A broad range of activities and investigations, along with guidance by the teacher, will help students construct an understanding of number that allows them to make sense of mathematics and to know how and when to apply relevant concepts, strategies, and operations as they solve problems.

Best Evidence Synthesis on Effective Pedagogy in Mathematics Effective mathematical pedagogy is a coherent system rather than a set of discrete, interchangeable strategies. This pedagogical system encompasses: A non-threatening classroom environment Instructional tasks Tools and representations Classroom discourse Effective Pedagogy in Mathematics/Pangarau by Glenda Anthony & Margaret Walshaw, New Zealand (2007)

The learning of mathematics has been defined to include the development of five interrelated proficiencies that, together, constitute mathematical proficiency (NRC 2001): Conceptual understanding Procedural fluency Strategic competence Adaptive reasoning Productive disposition Effective Teaching and Learning

Number Sense – Mental Math – Number Talks Is there a connection? Four Components of the Pedagogical System Non-threatening classroom environment Instructional task Tools and representations Classroom discourse Mathematical Proficiencies 1.Conceptual understanding 2.Mathematical fluency 3.Strategic competence 4.Adaptive reasoning 5.Productive disposition

Mathematical Processes problem solving reasoning and proving reflecting selecting tools and computational strategies connecting representing communicating

What are Number Talks? - 5 to 15 minute classroom conversations - built around purposefully crafted computation problems

The problems are designed to elicit specific strategies that focus on number relationships. The problems are solved mentally.

Your Turn How would you mentally solve this problem - no paper or pencil

Key Components of Number Talks - Classroom environment & community - Classroom discussions - Teacher's role - Role of mental math - Purposeful computation problems

Classroom discussions Students clarify their own thinking, consider and test other strategies to see if they are logical, investigate and apply mathematical relationship, build a repertoire of efficient strategies, make decisions about choosing efficient strategies for specific problems

Teacher's role A shift in our role from “sage on the stage” to “guide on the side”. Much different from how we may have been taught math. Teacher is a learner alongside the students. Gain insights into alternative strategies.

Role of mental math - focus on number relationships and use these relationships to develop efficient, flexible strategies with accuracy. Solving problems mentally puts the emphasis on using what they know and understand about numbers as well as how they are related. -to help strengthen students’ understanding of place value. Initially problems are written horizontally to encourage students to think of numbers as whole quantities instead of columns of numbers Number Talks p.13

This is an essential part of number talks. The teacher’s goals and purposes for the number talk should determine the numbers and operations that are chosen. This involves thoughtful and careful planning and selection of examples of number talks. Purposeful computation problems

35 X 20 Let’s Try some Multiplication 32 X X 24

For multiplication the ability to break numbers apart in flexible ways is even more important than in addition or subtraction. Students need a repertoire of models for multi-digit multiplication. Introduce different representations (one at a time) as ways to explore multiplication until you are comfortable that the class has a collection of useful ideas. At the same time do not force students who reason very well without drawings to use models when they are not needed. Van de Walle p.114

An array model is as important to multiplication and division as the number line model is to addition and subtraction. The visual representation of rows and columns helps students as they develop their proportional reasoning. The array identifies the parts (factors) and the whole (total are of product) and can be used to demonstrate and prove student strategies. NT p.233

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From Jo Boaler Number sense is inhibited by over-emphasis on the memorization of math facts in classrooms and homes Math facts are a very small part of mathematics but unfortunately students who don’t memorize math facts well often come to believe that they can never be successful with math and turn away from the subject Math anxiety and the blocking of working memory are associated with time tests to measure number facts Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts by Jo Boaler (youcubed)

Students need to: -have the ability to reason about quantitative information -possess a sense of number -check for the reasonableness of solutions Why do we do NUMBER TALKS?

Fluency with numbers is knowing how a number can be composed and decomposed and using that information to be flexible and efficient with solving problems.

It is critical that students learn to represent numbers in as many ways as they can. Each of these representations broadens the meaning of the number for the student. Numerals are only one form of representation. Marian Small, Making Math Meaningful p. 139

Conservation of number and one-to-one correspondence are essential foundations in building number sense for young children. Conservation of number is the understanding that the quantity of a given number of objects remains the same regardless of how it is spatially arranged. Number Talks by Sherry Parrish p.37

Even when the arrangement has been changed, a child who has conservation of number will still recognize that he still has five marbles without having to recount. Number Talks by Sherry Parrish p.37

How many dots do you see? How do you see them?

These were the student guesses. Grade 1 FI Classroom June 2013

How many dots do you see? How do you see them?

Examples of other Dot Images to make 7

SUBITIZING - to immediately recognize a collection of objects as a single unit. e.g. looking at 5 pips on a die and immediately recognizing it as 5 Using five- and ten-frames, or rekenreks provide opportunities for children to build recognition of numbers and their parts. Number Talks by Sherry Parrish p.39

Numbers are most meaningful to a student when they are related to anchor or benchmark values that are well understood. Marian Small, Making Math Meaningful p. 139

FIVE FRAMES TEN FRAMES

Counting All Counting On

enrek/REKENREK/ How many beads do you see? How do you see them?

How many beads do you see? How do you see them?

How many beads do you see? How do you see them?

CHECKING BACK - REFLECTION Number Sense and Mental Math How does it connect with: the curriculum expectations? the pedagogical principles? the mathematical proficiencies? the task features? the mathematical processes? developing a positive mindset? Number Sense and Numeration page 8?

Full Day LearningGrade 1Grade 2Grade 3 -demonstrate an understanding of numbers, using concrete materials to explore and investigate counting, quantity, and number relationships – solve problems involving the addition and subtraction of single-digit whole numbers, using a variety of mental strategies (e.g., one more than, one less than, counting on, counting back, doubles) -solve problems involving the addition and subtraction of whole numbers to 18, using a variety of mental strategies (e.g.,“To add 6 + 8, I could double 6 and get 12 and then add 2 more to get 14.”) – solve problems involving the addition and subtraction of two-digit numbers, using a variety of mental strategies (e.g., to add , add the tens, add the ones, then combine the tens and ones, like this: = 50, = 13, = 63) -multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies (e.g., doubles, doubles plus another set, skip counting) What does the Curriculum Say?

Number Sense – Mental Math – Number Talks Is there a connection? Four Components of the Pedagogical System Non-threatening classroom environment Instructional task Tools and representations Classroom discourse Mathematical Proficiencies 1.Conceptual understanding 2.Mathematical fluency 3.Strategic competence 4.Adaptive reasoning 5.Productive disposition

Mathematical Processes problem solving reasoning and proving reflecting selecting tools and computational strategies connecting representing communicating

What Teachers are Saying about Number Talks I think number talks has for me been the best idea I have come across since teaching. This has been a whole new way of learning for the students and myself! It has opened up so many doors for the students to gain confidence, access more strategies, and make Math more understandable. This ripple effect has made me look at all the Math curriculum and begin to teach it in a variety ways (not just the standard way I was taught). I see students more engaged and ready to take on more tasks. Derek Bouma

What Teachers are Saying about Number Talks "Since implementing number talks, I have seen increase engagement and confidence in my students. All students have an entry point, regardless of their ability level, and I see participation from students who have a negative attitude towards math and low confidence. Our staff has begun using number talks on a school wide basis and it is so invigorating to hear the other teachers share their excitement and the wonderful things they are also seeing with their students. We are looking forward to seeing what happens after our students have been exposed to number talks for several years in a row and how that will change their attitudes and skill levels." - Jennifer Pedersen

What Students are Saying about Number Talks "Number talks are fun and they help me learn easier strategies to make the questions easier." - Aiden "Number talks are fun but sometimes a little challenging but I am learning math easier because of it." - Jaxon "Number talks help me learn new strategies that help me with other math." - Kate "It helps with my facts, it makes multiplying and dividing easier. It also improves what I think about math." - Melia "Number talks are amazing. I think I am getting better." - Jadyn "I like number talks because it helps me build my mental math skills and it is helping with my math." - Tyler

What Students are Saying about Number Talks I think number talks are good for us because it helps us with our learning, it helps us become better at math. I have noticed myself grow and become better at math than I was before. I like that the questions get harder each time and build on each other. Number talks helps me grow mathematically. I always found that I do better with a visual and a voice tutorial and number talks incorporates both. I think that number talks are great! They help learning and they don’t make people feel bad by not getting the answer right or quick enough. I found myself using new strategies for getting the answer since we have done number talks.