Chapter 1: Guidelines and a Differentiated Unit
This chapter includes guidelines to follow when differentiating, as well as a specific, detailed example of a differentiated unit. I tried to make this “short,” but wanted to include the necessary unit info to provide a complete picture:)
The authors state that when differentiating, teachers must first decide why the differentiation is necessary – is the purpose to make the content accessible for all students, to motivate students, to make learning more efficient? Next decide what part of math needs to be differentiated – the content, the process, or the product. Third, determine how the math will be differentiated.
Before designing a differentiated unit, the essential questions and unit questions must be identified. The essential questions are the key understandings the students should have as a result of the unit. The unit questions are specific elements of the essential questions.
For example, an essential question for place value is:
“What are the characteristics of a place value number system?”
A corresponding unit question might be:
“What place value number system do we use every day?”
The authors share the following example of a place value unit that was created for an accelerated third grade class, and then explain how this unit was “transformed” when the place value unit was taught to a combined class of third and fourth grade students.
Original unit: Students compared number systems (they worked with Roman numerals and our base 10 number system, as well as bases five and two) and then created their own place value number systems (their systems were designed for different planets, where the inhabitants had different numbers of fingers). Students had to display their number symbols, their counting numbers to thirty, and create a diorama to show how life would be different using their number system. A later revision to this project was to have students use only base three or base four place value systems for their projects, to keep the process more manageable.
Transformed/differentiated unit (as briefly as I can summarize!): Students discussed and compared Roman numerals and our base 10 number system (as a whole group). Students then individually recorded similarities and differences between the systems. Next, using an activity found in Math Matters by Chapin and Johnson, students “worked” for a truffles candy factory that packages truffles in single boxes, three-packs, nine-packs (trays) and twenty-seven packs (cartons). The packaging of certain amounts of truffles into these place values resulted in specific truffle numbers. For example, 38 truffles would be packaged in 1 carton, 1 tray, 0 three-packs, and 2 boxes, resulting in the truffle number of 1102. This activity was completed with both whole group and partner aspects. Students looked for patterns in the truffle activity, compared their truffle charts to the base ten system, and wrote about the similarities and differences between the two systems. The various activities were conducted with various student grouping methods– individual, partners, whole class.
After these concept-building activities, the assignment of creating their own number system was given, this time to be based on a planet called “Quarto,” where the inhabitants have two fingers on each of their two hands. Students had several options of products to create, individually or with a partner, but needed to include these components: design, define and name each number symbol; use the symbols to count to at least 30, and display a four-column table to compare the base 10 and Quarto systems; describe the worth of each place value in the number system and explain how place values relate to the place values next to them. There were also additional project elements, like: creating an addition and subtraction chart, writing three addition/subtraction fact families; creating a multiplication and division chart and writing three fact families; creating a diorama or mural to show how daily life would change if we converted to the Quarto system; explore what fractions might mean, using illustrations and models.
In this unit example, both the process and product were differentiated (this is the what). The activities used to build the content ideas had open-ended aspects, allowing for both student and teacher differentiation. The project was differentiated by level of difficulty and choice.
The how aspect of differentiation addressed the readiness activities – the activities (truffle candy packaging) allowed for different challenges, depending upon how many truffles the students attempted to package – while some students might attempt to figure out packaging for 10 truffles (1 tray and 1 unit box, which results in the truffle number of 0011), others might attempt to package 100 (2 cartons, 2 trays, 2 three-packs, resulting in the number 2220).
The reason the unit was differentiate (the why) was readiness, accessibility, and to accommodate various learning styles.
Before closing the chapter, the authors identify Tomlinson’s Key Principles of a Differentiated Classroom (1999):
1. The teacher is clear about what matters in subject matter.
2. The teacher understands, appreciates, and builds upon student differences.
3. Assessment and instruction are inseparable.
4. The teacher adjusts content, process, and product in response to student readiness, interests, and learning profiles.
5. All students participate in respectful work.
6. Students and teachers are collaborators in learning.
7. Goals of a differentiated classroom are maximum growth and individual success.
8. Flexibility is the hallmark of a differentiated classroom.
The authors close the chapter by referencing the problem solving standard in the Principles and Standards for School Mathematics (NCMT 2000), which states that instructional programs should enable students to build new mathematical knowledge through problem solving. Using engaging problems “with embedded worthwhile mathematical tasks….can help all students to reach their full potential in math.” This concept is clearly demonstrated in the unit example in the chapter.