Introduction
I teach seventh grade mathematics in an urban school located in Charlotte, North Carolina. My school was classified as a full International Baccalaureate magnet school two years ago and this has changed the type of students enrolled in the school. All students must enter a lottery system and then be selected to attend our school. They must be on grade level as it relates to their end of grade scores in mathematics and reading. The school is diverse in its student body makeup with students from all over the world present in my classroom.
I have been teaching seventh grade mathematics for eight years and I still feel like I am a new teacher. I constantly seek out better strategies and a deeper level of understanding of my content area. I typically have few to any discipline problems in my classroom and the students work well in groups. I always keep in the back of my mind however that students are effective for only short periods of time in small groups. I have to refocus and direct them on a periodic basis.
Headlines often tell us how behind students in the U.S. are in mathematics. Should we believe them? What are the tests? Why is it that our colleges are full of students from these countries that are supposed to be ahead of us? Are the grade schools behind and the colleges ahead? Do other countries teach differently than we do? I can provide answers from my own personal experiences as a teacher. Most of the students that I have taught know mathematics only on a superficial level. They know how to plug and solve, they remember formulas and they remember rules told to them by their previous mathematics teachers. They do not understand mathematical concepts on a deep level. The length of time during which they remember these things varies and is often indicative of a successful versus a non-successful student.
My unit will address an area of mathematics that is often not fully understood by students, place values. My students are scheduled in honors level classes and they are expected to take high school Algebra the year after I teach them. It is important in my instruction that I prepare them for a high school level course by giving them a better understanding of mathematical content that they should already be familiar with. The title of my unit "Algebra Really Is Just Arithmetic" is my way of saying that if you understand basic arithmetic then you will have a better understanding of Algebra. Exploring place value and operations as they relate to place values should help clear up many issues that students have with Algebra.
The activities and strategies that I will use in my unit will take the basic concept of place value and using expanded form allow students to better understand operations and why they have to borrow or carry. Students will then see the direct relationship that exists between arithmetic and algebra. The method of performing operations with expanded form and exploding dots will seem like a lot of unnecessary work to some but it is the underlying knowledge of numbers and place value that is my objective. Lack of that, in my opinion, is what got us in the boat we are in now. Mathematics is a way of compressing things to make them easier to work with. My point here is there is no sense in compressing something if you do not understand it to begin with. If students do not understand what they are doing with numbers other than going through the motions of what teachers have told them for years then how will they know what to do when these numbers are now variables?
The unit is designed to be taught over a twenty day period. Students are expected to work in small groups of three or four and present their discoveries to the class on an as requested basis. While teaching this unit I have to be careful not to reveal too much information. I expect students to learn through investigation. The teacher should be a facilitator and guide not a lecturer or revealer. Think of teaching this unit as farming. A farmer plants seeds and then nourishes the seeds into a grown crop that can be harvested. I will plant the seeds and the students will provide the nourishment. The overarching concept here is that, the students must obtain a deep level of understanding of place value and arithmetic in order to be successful at algebra.
Rationale
I have students who seem mystified by fractions and the concept of decimals and percentages. When I introduce variables into problems this further confuses them. I have often had students not be able to correctly say the number 4,567,878 (four million five hundred and sixty seven thousand eight hundred and seventy eight. This may just be an issue with vocabulary or it could be an indication that they truly struggle with the concept of place value.
I have chosen to take my students back to what they may consider elementary school content. I want them to learn place value and operations with numbers so that they will be successful in Algebra. Teachers may find this somewhat difficult in execution. Most students will want to take the shortcuts that they know and love. This is not the unit for that. To make sure that students stay on task I have included problems that will require a good to great working knowledge of place value to solve.
It is my goal to have students that participate in this unit be more successful in Algebra than those who are not exposed to it. By understanding arithmetic they will understand Algebra.
Curriculum Objectives
In June of 2010 the state of North Carolina along with most states in the United States adopted the Common Core Standards. These standards were developed by the states themselves and should be adopted by most if not all states. This unit will address all mathematical practices of the Common Core Standards. The unit addresses seventh grade expressions and equations standards. Because of its nature the unit will address many different standards on many different grade levels. I encourage teachers that are trying to do this unit to look at their standards related to operations and algebra. The scope of the unit is so broad that teachers should have no difficulty tying into either an existing state standard or Common Core Standard.
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