The Place for Models The Place For Models in Our Place Value System Region 10 RSS Conference Raphaella Archie
Outcomes • Putting Place Value and its associated language in perspective • Understanding four important place value concepts • Teaching and learning progressions of place value • The thoughtful use of tools/model to support and students’ learning • What are and how to face students’ challenges • Symbolic Notation and Algorithms
What’s your Number? • Please identify your number and place yourself in number order from least to greatest in front of the room. • What challenges did you have?
Now, what’s your Number? • Can you determine your number? • What can you tell about the number system? • Compare it to our number system? • Why do you think our students have difficulties understanding place value?
Understanding the Difference • Egyptian • Base Ten System • Additive numeration system (each power of 10 repeats as many times as needed) • Written left and right • Mayans • Base 20 System • Wrote their numeral vertically with one numeral above the other • The power of the base increase from bottom up
Quotes • Please read an orange card • Do you agree or disagree with the statement?
What do we know about teaching Place Value? True or False • Place value refers to the value of the place or position of a digit in a multi-digit number. • Understanding our base-ten number system is essential for students to experience success in mathematics. • Students learn about place value in a linear way. • Because the decimal number system is an extension of the whole number system, students can learn best about place value by doing decimals first. • Understanding place value requires an understanding of multiplication as well as addition. • The spoken and written forms of our numbers are consistent with each other and this helps our students as they work with place value. • Understanding the concept of a positional or place value system is fundamental to the development of a sound number sense. • Students need to be secure in their understanding of sing-digit numbers before working with numbers with tow or more digits.
Matthew Effect “The rich get richer and the poor get poorer” Similarly in Mathematics, if students do not gain an understanding of how our number system is structured and do not have the appropriate language to talk about the mathematics that they are engaged in, “The Matthew Effect” can become a reality for them as they increasingly fall behind.
Four Key Concepts – Place Value A number of sources suggest there are four key important concepts to understand when teaching and learning about place value: • Positional property - The quantities representedby the individual digits are determined by thepositions that they hold in the whole numeral. • Additive property - The quantity represented bythe whole numeral is the sum of the values representedby the individual digits • Base-ten property- The values of the positionsincrease in powers of ten from right to left. 4. Multiplicative property- The value of an individualdigit is found by multiplying the facevalue of the digit by the value assigned to itsposition.
Position and Language “Why can’t I put the numbers in order? Any order?” 487 478 874 “Why isn’t this correct?” 487 = nineteen
Position and Language The base-ten number system is a “place value” system. That is, any numeral, say 2, can represent different values, depending on where it appears in a written number: it can represent 2 ones, 2 tens, 2 hundreds, 2 thousands, as well as 2 tenths, 2 hundredths, and so forth. To understand the value system requires coordinating: 1. the way we write the numeral to represent a number (position- 487 ) 2. the way we name numbers in words
Visualization Mental images of quantities are necessary to work with quantities mentally. • Subitize – the ability to recognize dot arrangements in different patterns. Subitzing is a precursor to place value understanding. • Subitizing is a fundamental skill in the development of number sense, supporting the development of conservation, compensation, unitizing, counting on, composing and decomposing of numbers
Models and Manipulatives Teachers often say, “I do lots of ‘hands-on’ with my class.” However, hand-on and using manipulative is not sufficient. We need to know it well and how to best use it. As students are learning about our base-ten system the choice of equipment, recognizing the progression from concrete to abstract is critical. Tools and representations should be carefully selected to provide support for students’ thinking.
One by One • Tools which can be grouped one by one to make a ten and ungrouped again into the ones.
Making Tens • The grouping of ten is scaffold by the physical structure of the item but discrete ones can still be manipulated
Hundreds, Tens, Ones • Pre-grouped equipment where the ones in the ten can be seen but cannot be grouped and ungrouped.
How do I know the Difference? • Tools in which a different looking ten has no similarity to the ones and it which the ones cannot be seen. In using the tens it takes a considerable shift in knowledge and understanding to “trust” the trade of one ten for ten different looking ones.
Conceptual UnderstandingAbstract • The final shift is the one which we, as adults, now understand and that is when the tens and ones look exactly the same but it is only their place that tells us their value. This is a level of abstraction that will only be fully understood if tools used along the learning pathway carefully supports the development of this complex understanding.
Exchange Games – Base Ten • Making Tens • Trading Stickers
Place Value - Challenges • Mastering place value may take several years • Coming to terms with confusing irregularities in number words, for example, thirteen could be seen as three and ten; twenty-three reads as two tens and a three • Understanding the part zero has to play in numbers such as 702 or 3,000 • In dictation, learning not to write as they hear (125, as 100205) • Recognizing the written words for numbers (twelve, fifty) • Knowing what “ones” are and that “a ten” means one group of things.
Symbolic Notation • The real focus of modeling standard notation is helping students develop language and form images of the actions these symbols represent. • Develop over time through plenty of experience seeing, talking about, and using such symbols and language throughout first and second grade.