MaTHink Conference Moving Forward Application Common Core in the Next Generation
Today’s Agenda Become familiar with why math is so important in the foundational years (K-2) Understand cardinality Introduce strategies that develop cardinality Learn more about the Next Generation Science Standards (and how to read them) Become familiar with how the Science and Math standards overlap
Norms Silence all cell phones Step outside to make or take a call Keep sidebar conversations to a minimum Have an open mind Participate fully!
Math Requires: • Flexibility to other possibilities • Logical thought • Systematic thinking • Abstraction • Problem solving • Complex thinking
Math Key Fluencies Grade Required Fluency K Add/subtract within 5 1 Add/subtract within 10 2 Add/subtract within 201 Add/subtract within 100 (pencil and paper) 3 Multiply/divide within 1002 Add/subtract within 1000 4 Add/subtract within 1,000,000 5 Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7 Solve px+ q = r, p(x + q) = r 8 Solve simple 2x2 systems by inspection
Progression: Where are we going with this? Grades 6 – 8 The Number System Grades K – 5 Number & Operations: Base Ten Grades 3 – 5 Number & Operations: Fractions • Kinder • Counting sequence • Work w/ 11-19 focusing on place value • First • Extend counting sequence • Extend place value understanding • Second • Understand Place Value • Use Place Value when adding and subtracting 14
Calendar Telephone numbers Clocks Measurement Money Concrete examples (length) What are numbers anyway?? Numbers are an infinite set that are highly systematic which makes them useful as identifiers Children’s Exposure to Number
Role Playing Don’t trust quantities Unsure about the different “looks” Unsure about quantity that is hidden Focus on “counting right” Unable to determine if a quantity is reasonable Cardinality Understand the last number counted tells how many there are in a set Can use counting to match and find a set Usually occurs by 4 to 4-1/2 Role Playing Vs. Cardinality Counting is more than reciting a rote sequence and recognizing numerals. Counting is finding out “how many.”
So what, if a student doesn’t have cardinality? What skills did Cena have? What skills did Cena Lack? Cena is probably in what grade? In what grade should we have intervened to remedy this type of misunderstanding?
What is Number Sense? “ …good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relatingthem in ways that are not limited by traditional algorithms.” (Howden)
Fluency Has 3 Components Fluency- • Efficiency- ability to work fairly quickly • Accuracy-ability to get the correct answer • Flexibility- multiple solution strategies determined by the problem Fluency is the by-product of flexibility. Assessing fluency by occasionallyusing timed tests is acceptable. Using timed tests as an instructional tool to build fluency is ineffective.
Children’s progression to make sense of the formal symbolism we use in mathematics. (Bruner) • Enactive-using tangible items to model the problem(the Rekenrek, cubes, acting it out, etc) • Iconic-representing what they did in the enactive phase with an icon (tally marks, circles, etc. on paper) • Symbolic-writing the formal signs and symbols
Best Practices A strategies review
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. Godfrey Harold Hardy A Mathematician’s Apology
Brain Research Effecting Teaching and Learning (Sousa) • Creating and using conceptual Subitizing patterns help young students developthe abstract number and arithmetic strategies they will need to master counting. • Information is most likely to be stored if it makes sense and has meaning.
How We Learn Best Memorize this eleven digit number: 25811141720 Now look for connection (relationship) within numbers 2 5 8 11 14 17 20
Brain Research cont’d • Too often, mathematics instruction focuses on skills, knowledge and performance but spends little time on reasoning and deep understanding. • Just as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisitefor succeeding in mathematics.
Carol Dweck • There is a growing body of evidence that students’mindsets play a key rolein their math and science achievement. • Fixed mindset vs. growth mindset
Essential Number Relationships • Spatial Relationship- recognizing how many without counting by seeing a visual pattern • One/Two More or Less- knowing which numbers are one/two more or less than any given number • Benchmarks of 5 & 10-how any given number related to 5 and 10 • Part-Part-Whole- ability to conceptualize a number as being made up of two or more parts Van De Walle, 2006
Using manipulatives effectively? How many cubes are there? How did you know? How does this support development of number relationships?
What does the Frame add in Building Essential Number Relationships
How did adding the frame support numeracy ? • Spatial Relationship • One/Two More or Less • Benchmarks of 5 & 10 • Part-Part-Whole
How many now? “Tasks that encourage students to think in collectionsprovide rich opportunities for them to construct abstract mathematical relationships and become powerful problem solvers.” -Wheatly & Reynolds, 2010
How did adding the pattern support numeracy ? • Spatial Relationship • One/Two More or Less • Benchmarks of 5 & 10 • Part-Part-Whole
The Rekenrek • is a tool developed at the Freudenthal Institute in the Netherlands by Adrian Treffers to support the natural mathematical development of children • in Dutch means “calculating frame” or “arithmetic rack.” • looks like a counting frame but is designed to move children away from counting each bead. • looks like an abacus but it is not based on place value.
Rekenreks • The Rekenrek has a built-in structure that encourages children to use their knowledge about numbers instead of counting one to one. • The built-in structure allows children the flexibility to develop more advanced strategies as well.
Tournaki Study “The rekenrek, in addition to being a manipulative, also acts as a facilitator of knowledgewhile students develop efficient thinking strategies. This process supports Gravemeijer’s (1991) argument that the materials themselves cannot transmit knowledge: The learner must construct it.”
Building Early Numeracy • “What do you notice?” • explore the tool • learn the built in structure • before you have them use the tool. • “Show Me” • Have them show a certain number. • Some may count one-by-one • The structure of the tool allows for more advanced strategies. • Flash forward • Show the Rekenrek of a certain number (1-10) for a few seconds • Have them determine which number was flashed. • When first starting allow enough time that children who need to can still count one-by-one. • Gradually shorten the time to encourage children to see groupings.
Growing Numeracy • Just One More • Show a certain number • Kids build theirs to show • “One More” • “Two Less” • Peek-a- Boo • Push some beads over • Cover the rest • “How many are hiding?”
Number Talks Review Read this side
Turn and Talk: What are all the possible ways children will figure out how many?
How did the rekenrek support numeracy ? • Spatial Relationship • One/Two More or Less • Benchmarks of 5 & 10 • Part-Part-Whole
Adding Context: Math Practice 4: Model with Mathematics… “mathematical meaning in their lived world” • Attendance chart • Bunk beds • Double-decker bus • Bookshelves • A parking lot • Egg cartons
Taking Attendance How many children are here today? How did you figure it out?
Math Practice 2: Reasoning Abstractly and Quantitatively… “Contextualizing and Decontextualizing” • Passenger Pairs matching game: • Moving from the bus story to a model of the context
Focus on Essential Number Relationships in the Classroom • How did the teacher keep all the students engaged and on task? • How did the teacher reinforce: • Spatial Relationships • One more/ two more • Benchmarks of 5 & 10 • Part-Part- Whole
Focus on Relationships When we focus on relationships, it helps give children flexibility when dealing with their basic facts and extending their knowledge to new task. When we build a child’s number senseit promotes thinkinginstead of just computing.