Introduction
A straight line is a line which lies evenly with the points on itself.
From Euclid's Elements
Someone once said that the need for a strong foundation in mathematics was akin to a tall building having a strong foundation beneath it. You don't necessarily see it, but if it isn't there, you would surely know it, as the building would likely topple over. It's like that for our students. If they are not taught well the foundational concepts of math, they are unlikely to be able to thrive in a world that requires a substantial knowledge of mathematics more than ever. Number operations, including conceptual understanding and fluency, are often considered the core of foundational mathematics. The main goal of this unit is to present two geometric interpretations of multiplication: 1) the area model (which is a refinement of the array model for products of whole number factors), and 2) the number line. In the number line model, multiplication can be thought of as uniform stretching of intervals on the line. This supplements the interpretation of addition (taking a number and adding a fixed number to it) as translation of the number line. Thus, the number line affords coordinated interpretations of both multiplication and addition, so it can show how they interact via the Distributive Rule. Also, it is a model that works equally well for all kinds of numbers: whole numbers, fractions, signed numbers and beyond. My first job will be to determine if my students understand the number line, then I will build instruction from there.
My school district has been the fastest growing district in the state for at least ten years, becoming more affluent as we have grown. Approximately nineteen percent of the students in my district come from low income households, but my elementary school, still called the "town school" by many, has over thirty-five percent of our children living in low income households. What that means in my classroom is that a child living in poverty could be sitting next to a child who lives in a half million dollar home. Our students are a blend of race, culture and ethnicity. Many families from foreign countries have settled in the area, making my school truly diverse.
The math curriculum recently adopted by my school district is TERC Investigations in Number, Data and Space. The units emphasize conceptual understanding of a topic, rather than learning rote algorithms or emphasizing number fluency. The program philosophy is such that if mastery is not achieved by all students, those who need more time will be exposed to the concept at a later time, but I find it important that my students have some depth of understanding of the material, so I never feel comfortable forging ahead unless most of my students have some acceptable level of mastery of the material. Some of the units help with this goal by making it possible to make connections between topics, rather than students perceiving each unit as a discrete topic, unrelated to any other.
State math standards for fourth graders in Delaware directly addressing multiplication are:
- Determine factor pairs that make up a given number;
- Show how multiplication and division facts up to 50 are related, using arrays, skip counting, and area models;
- Master multiplication facts and the related division facts up to the 10s tables;
- Model situations that involve the addition, subtraction, multiplication and division of whole numbers using objects, pictures, geometric model, and symbols;
- Represent the idea of a variable as an unknown quantity using a letter or symbol;
- Develop an understanding of the Commutative and Associative Properties of whole number multiplication as a tool to solve problems.
I teach in an inclusion setting where special needs students are educated alongside their regular education peers. That might mean I have some students who possess very little number sense and struggle significantly mathematically and some children who are able to work above grade level on math concepts. The struggling math students in my classroom aren't necessarily special education students, but often some of them are. There are two full time teachers who provide instruction to students and we share the planning and execution of lessons equally. We rely on co-teaching strategies so we are both always engaged with our students. As the special education teacher in the classroom, I do minimal pull out for instruction; instead we rely heavily on flex grouping based on the strengths and needs of all of our students.
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