Chapter 8: The Teacher: Knowing and Sharing the Self
Chapter 8 focuses on why teachers need to know themselves, as well as know the strategies for metacognitive work. In knowing ourselves as teachers, the authors emphasize that we must truly know our own biases and beliefs, as these could “…have the potential to sabotage the good intentions of a differentiating mathematics teacher.” The authors state that teachers need to examine their own teaching and learning styles, multiple intelligences, talents and strengths, and thinking dispositions. The authors suggest that one way for teachers to learn more about themselves is to write their own math autobiographies, as was suggested for students in an earlier chapter.
The authors point out that it is somewhat true that teachers teach the way they were taught, which may or may not be the best method for the next generation of students. It is important that teachers examine the way that they teach, in order to determine whether or not they are teaching the way they were taught.
An important factor for teachers to consider is their Gregorc Mind Styles (work of Dr. Anthony Gregorc, 1982). His Mind Styles represent four possible extreme combinations for perceiving and ordering information. According to Gregorc, everyone “...has a natural, inherent predilection toward one or two of the combinations.”
The combinations (and when I read these I was reminded of a past principal being good-naturedly irritated with a colleague because he was so “abstract random!”) are:
1) Concrete and Sequential (perceive concretely and order sequentially)
2) Concrete and Random
3) Abstract and Random
4) Abstract and Sequential
The perceiving of information ranges from concrete (using the five senses) to abstract (naturally intuitive ways), and the ordering of information ranges from sequential (step-by-step linear approach) to random (chunking information in no particular order). Referring to a 1998 ASCD publication by Pat Guild and Stephen Garger, the authors summarize each of these styles in regards to teaching:
1) Concrete Sequential teachers are practical, hands-on teachers who manage highly structured logical classrooms. They expect students to be on-task and thorough in their work. Their classrooms are predictable, realistic, and secure for students.
2) Concrete Random teachers are practical and realistic, but are also original, spontaneous, and creative. Students are encouraged to make choices, be active, and think for themselves. This type of teacher is comfortable with change, a busy environment, and a variety of methods.
3) Abstract Random teachers are enthusiastic and sensitive, and design child-centered classroom experiences. They expect students to cooperate and share. This type of teacher is spontaneous and responsive to students’ needs and interests, but also strives to increase their own understanding of content.
4) Abstract Sequential teachers provide rich but structured environments, and encourage students to be analytical and evaluative. They expect students to develop good work habits and to provide evidence to support their conclusions. This teacher pushes for greater understanding.
Which type of mind style are you? Once you have identified that, the authors believe you can develop differentiation strategies that are comfortable for you and for the other mind styles of the students. The authors mention that a concrete sequential teacher (like me) can have highly structured centers, but that the activities in the centers can meet the learners styles by being open-ended and creative. They also suggest that menu and tiered activities, as discussed in chapter 7, can be differentiated to meet all teaching and learning styles.
In addition to analyzing their mind styles, teachers need to examine their intelligences using Howard Gardner’s Multiple Intelligence Theory (verbal/linguistic, logical/mathematical, visual/spatial, bodily/kinesthetic, musical, interpersonal, intrapersonal). The authors suggest placing teacher strengths and natural styles at the center of the planning process. Then, teachers can visualize the four learning styles as surrounding that center, and think about a particular student in each style. Planning to meet the needs of those particular students will help the teacher to meet the needs of most students.
Bottom line - teachers need to know themselves in order to most effectively teach their students.
Chapter 7: Lessons as Lenses
The second lesson that the authors describe in chapter 7 addresses two- and three-dimensional shapes and is differentiated for readiness and accessibility (rather than for efficiency, as the first lesson was.)
During the course of studying these geometry terms, the teacher observed that some students were needing more time with concepts while others were needing a challenge. The lesson was designed to address these needs and was a tiered lesson with three different activities, two of which addressed the needs of those who required more time while the third addressed the need for challenge.
The launch part of the lesson was a whole group discussion of the traits of a hexahedron, which connected previous learning and modeled one of the lesson activities. The launch also included an explanation of the lesson activities.
The exploration was the working time in the activities, which were different, but related. All groups began by describing a given three-dimensional object as completely as possible (as was done during the launch).
Group 1’s activity provided an additional chance to work with shapes and their properties. They were required to create a poster that would classify two- and three-dimensional shapes into groups of their making. They then needed to write a mathematical description of these groups.
The second group worked on identifying the faces of three-dimensional shapes- tracing them, naming them, and then drawing a net. These students also had to create a poster, with descriptions.
The third group’s task was to create a geometry concentration game.
Students did not finish the activities during the class period, so the class summary was a sharing of what the groups had completed, as well as what work they still had to complete. The next class period included a sharing of the products. Group members that finished their tasks early worked on anchor activities.
The authors take time to explain more about tiering, which is designed for predetermined groups based on readiness, multiple intelligences, or interests. The first tier should be a basic level, the second should be an extension for students that like challenge, and the third provides scaffolds for students that need more background or support. When thinking about tiering, the authors suggest to think about what comes before and after the basic concept.
The authors offer these steps for tiering a lesson, which were adapted from Pierce and Adams (2005.) (I’m parapharasing.)
1) Identify the math standards/objectives
2) Identify the big idea/key concepts
3) Determine necessary prior knowledge
4) Determine what to tier – the content, process or product
5) Determine what to tier for – readiness, learning style, interest, etc
6) Determine number of tiers
7) Develop plan for formative and summative assessments.
The authors share another quick example of tiering for third and fourth grades, using an Array game.
Tier one is for students working on learning multiplication facts. Working with a partner, students deal the array cards with the array side (doesn't show product) faceing up. The students compare the arrays on their top card and whoever has the greatest product keeps the cards.
The second tier is for students who are ready to see relationships between multiplication facts. These students work in pairs, using the array cards as well, keeping the product side up. They start with one array and then find 2 others that will cover the original array
The third tier is for students who are ready for extension. Using the arrays with the product facing up, they write equations (with a partner) to show how the distributive property is modeled by decomposing the product in the sum of two products. Example given: 18 = (1 x 6) + (2 x 6).
These examples have gotten my brain thinking about how I can add to what I already have in place. I’ve added the creation of tiered activities to my “To Do” list, which seems to be getting longer every day!
In this chapter, the authors describe 4 different lessons that are differentiated with different purposes. Since I can't get through all of chapter 7 today, I'll just review the first lesson and continue with the other lessons tomorrow.
The first lesson is a linear relationship lesson for seventh grade, which was differentiated for efficiency. Because there were time constraints for this lesson (6 class periods to cover 5 new concepts towards the end of the year), the author used a modified jigsaw format, in which each group completed a different investigation (the class had completed a related investigation prior to this lesson). After their group investigation, groups presented to the class in the form of a report/minilesson. Each group had a group leader that was chosen based on their skill level and interest in the particular investigations. Students were assigned to groups based on their working compatibility with other group members and on the number of problems in the investigation. Because students were learning about the particular topics more independently (to simply become familiar rather than master the material), the teacher gave each group a folder with key information from the teacher’s guide. Students were assigned roles: timekeeper, materials monitor, recorder, reporter, and facilitator-pacer (leader).
The students had 2+ class periods to work on their investigation and prepare their report/minilesson. In the following 2 class periods, each group had to present for approximately 30 minutes; their presentation was to include one problem for the class to complete. The next class period was a jigsaw quiz, where students worked in new groups that had at least one member from each investigation group.
The launch part of this lesson was the explanation of the groups’ tasks and time to get started.
The exploration section was the group work time, during which the teacher met with each group to coach, question, monitor, otherwise support support the students.
The summary part of the lesson includes the group reporting and problem solving as well as the jigsaw quiz period. The procedure for the jigsaw quiz is for group members to work individually for 15 minutes, during which time they should scan the whole quiz. Then students discuss and compare their answers and complete the quiz. Each student does complete his or her own quiz.
I have never used the jigsaw quiz idea - I'd definitely like to try that.
More chapter 7 tomorrow!
Chapter 6: Planning: A Framework for Differentiating
In planning for math, the authors suggest that teachers map out the curriculum for the school year in chunks, aligned with the academic schedule. Major units should have key deadlines for beginning and ending...these dates may change, but they provide a target for the year. The authors give a specific example of the plan for a combined seventh and eight grade math class, which is designed around the trimesters they have.
The overall planning is first done by trimester, broken down by month (for me this would be by quarters). The next step is to plan the first unit, while looking at the school calendar. They use a 3-column planner – the first column lists the days/dates of math classes. The second column lists what part of the unit is planned for each day. The third column is to track the progress made each day. After the weekly planning for the unit is complete, daily planning should occur.
The authors discuss Bloom’s Taxonomy and its revision to the four dimensions of knowledge (factual, conceptual, procedural, and metacognitive) with six levels of cognitive processing for each dimension (remember, understand, apply, analyze, evaluate, and create). The authors set these up in a chart, with the four dimensions along the left side and the six levels of processing along the top. This is to provide a visual to help design lessons and activities when planning.
Something to consider in planning is that all math concepts have three components of learning: the language associated with the concept, the conceptual understanding, and the skills and procedures connected to the concept. Students need to have strong language and conceptual foundations when learning math concepts – rather than simply memorizing facts or procedures.
Prerequisite support skills for learning math, identified by Mahesh Sharma are “nonmathematical skills” that are needed for math conceptualization. They include: sequencing, classification, spatial orientation, estimation, patterns, visualization, deductive thinking and inductive thinking. The authors state that these skills need to be incorporated in the planning process and should be part of differentiation.
The authors state that daily routine is important, and that an hour for math class is ideal (I have about 40 minutes). The daily routine the authors recommend is:
* beginnings (warm-ups)
* homework processing
* minilesson and launch
* homework assignment and daily reflection.
To differentiate during the warm-ups, different students can work on different problems; students could create warm-ups, have students describe strategies to build language.
Homework processing can be done in small groups, with students comparing answers, and then teacher selecting items for discussion. To differentiate in homework processing, the authors suggest organizing the groups by differentiated homework tasks or choosing discussion problems that allow for intervention opportunity with specific students.
The minilesson is the part of the lesson that focuses on the objectives for the day. “Launching” the minilesson includes connecting to previous work, discussing the math language and setting the context. If the teacher is aware of gaps, this is the time to address those needs. To differentiate at this point, the authors suggest selecting students to respond based on prior experience and assessments, design participation that reflects needs of various intelligences; include prerequisite skills practice; use think-pair-share; introduce tiered tasks.
During the exploration part of the lesson students work on assigned tasks with partners, groups, or individually. The teacher monitors, conferences, observes, supports, redirects, and asks probing questions. Students who finish tasks early are to work on the anchor activities. This is the time of greatest differentiation: students work in their own ways, at their readiness levels, with tasks designed for their needs, and students use anchor activities. During this time, the teacher lets the students do the work – lets them struggle some (not to frustration/upset level), make mistakes, and analyze their errors.
Class summary, in the authors’ view, is the most critical part of the lesson. This is the time to refocus the class on the math and “consolidate” the learning. The teacher facilitates the discussion, selects the order of sharing, asks for thinking and reasoning, questions, patterns, generalizations, etc. If the lesson is not actually “finished” at the end of math time (which happen to me quite often, with only 40 minutes!), then the summary should be a “status of the class” summary to prepare for the next class period. Differentiation during this time is the sharing of thinking, reasoning, showing evidence, making connections, testing new ideas – not about sharing “right” answers.
Homework and Reflection is recommended to be the last five minutes of class. After recording their homework assignment and asking any questions about it, students are asked to write a summary of the math they worked on during the period and record any questions they have. Prepared Exit Slips, which could be more specific to the day’s content, could also be used at this time. Differentiation during homework assignment would be assigning different homework to particular students based upon need.
I love the concept of the lesson structure presented here. As the authors stated, an hour for math is ideal, but I only have 40 minutes. I need to find a way to make this structure work…..I think about doing half of the lesson each day, which would put some exploration on day 1 and some on day 2. Then, would it be appropriate to give homework and do homework processing each day? Or should there not be homework on day 1 of the lesson? I need to brainstorm. Anyone have ideas?
Chapter 5 – The Flexibility Lens
As with chapter 4, this chapter focuses on a different "lens"of differentiating math, which is flexibility. The authors state that flexibility in the differentiated math classroom means that something is adaptable and/or able to be modified.
Before discussing flexibility, the authors identify what is not flexible – the five strands of math proficiency identified by the National Research Council’s 2001 publication, Adding It Up:
When differentiating math lessons and activities, these five strands must be addressed - no flexibility there!
The first area of flexibility that the authors address is grouping, and they describe several types of grouping:
Random groups - to create a new interactive environment. The authors describe the use of partner seating (starting at the beginning of school year), which allows for easy think-pair-share partnering. The partnering is random and changes every two weeks. The authors use random card draws to pair the partners (cards might match vocab and definition; match fraction and decimal or percent; match simple computation question and answer, etc) The pairs are also grouped with another pair, resulting in a heterogeneous group of four, readily grouped for an activity.
Readiness groups – used to provide appropriate challenge and support. Can be a short grouping (10 min) when the teacher notices a small group that needs a minilesson.
Heterogeneous groups – to represent a broad range of styles, intelligences, and abilities.
The next aspect of flexibility discussed is time. Because students can work at such different paces, the authors believe that anchor activities are a “major strategy” in accommodating for those paces and maintaining the flexible use of class time. The anchor activities were mentioned in chapter 2 as options for students when they have completed assigned tasks before the class is ready to come back together; they include things like math challenges, activities, games, centers, or books. It seems that the anchor activities can really be anything, so long as they have a purpose, are challenging, are engaging, and build math knowledge. (So, I have a lot of planning and organizing to do to get these ready!)
Content flexibility is a third area the authors address. All students need to be able to access the content, but the way in which it is presented can be adapted. Content flexibility is closely related to student readiness, and tiered lessons/activities are helpful in meeting student needs within the study of a particular topic.
Process flexibility refers to the ways in which students work through math – paper and pencil, manipulatives, calculator, mental math; strategies such as using models, guess and check, looking for a pattern, and solving a simpler problem. Allowing students to choose a particular process is a motivator for the students, though students should be encouraged to expand their process choices.
Product flexibility allows students some choice in what product they might produce when working on a project .
Assessment flexibility – the authors discuss the fact that assessment can be formal or informal, but that in math, assessments are often “casual,” as teachers are always observing students as they work through concepts. The authors offer a partial list of about 30 assessment tools, including: tests, rubrics, skill performance, pop quizzes, exit slips, checklists, partner quizzes, and logs.
The authors discuss rubrics, their purpose as tools to guide assignments and their evaluations, and the flexibility in the variety of rubrics that can be used – scoring rubrics, instructional rubrics, and student self-evaluation rubrics. Rubrics that are created by students and teachers together can be more effective because students are more vested. Self-reflection rubrics help students to focus less on a score and more on the types of mistakes they may have made, as well as on the math that they showed an understanding of. For example, when a unit test was returned, a teacher gave a self-assessment rubric listing the math concepts and the problems that addressed each concept. Students had to look over their test, look at teacher corrections, and analyze their own performance, to determine if they “did not meet,” “partially met,” or “met” each standard. These self-reflections help guide teacher instruction and grouping.
While I am using flexibility in some of these areas, I do need to increase my flexibility in others (more planning and creating needed!)
Chapter 4 A Problem-Solving Platform
I was very interested in reading this chapter, as I am always trying to improve my use of problem solving.
According to the authors, in a differentiated classroom, the problems presented to the students will allow students to differentiate for readiness for themselves.
The authors define a “problem-solving platform” as “a math curriculum that consistently draws students into mathematical inquiry through stories, situations, or scenarios that challenge students with intriguing problems.” These types of problems allow students to “grapple” with the math at their own readiness level while also collaborating and listening to other perspectives.
“Good problems” are needed for this type of thinking to occur; they must be open problems, can be a variety of forms, and they invite persistence (which, in my opinion, needs to be greatly encouraged – I have noticed a general decline in persistence over the years...anyone else?).
An example of a good problem that one of the authors likes to use at the beginning of the year: holding up a chessboard, ask students how many squares are on the chessboard. Many students will begin by counting the 1 x 1 squares, but others will help to expand their vision by counting the very largest square, or the 2 x 2 squares, and so on.
A second example is called “Petals Around the Rose,” from an article by Marie Appleby. Five dice are rolled, and each time, an answer to the five dice is given by the teacher. The teacher continues to roll and give answers to each set of 5 dice. Students must find the connection between the numbers on the 5 dice and the answer. The authors suggest doing this activity periodically over several days, giving students time to think, brainstorm, and discuss throughout this time.
To incorporate this type of problem solving into your program, authors suggest that each unit could begin with a problem that has the upcoming content “embedded” in it. Good problems could be situations, puzzles, games, or questions.
A “good problem”:
*leads to significant math
* is open-ended; can be approached with various strategies
* is accessible to students with different strengths, needs and backgrounds
* has various solutions
* leads to higher-level thinking and discussion
* has constraints that provide direction but don’t limit thinking
* strengthens conceptual development
* allows teachers to assess how and what students are learning and to identify needs
The authors discuss the idea that the amount of time used for such problems will vary according to the needs of the classroom. The problem could be a warm-up or an introduction to a specific topic. It could be in the beginning or middle of a unit…it doesn’t necessarily need to take a great deal of time.
The authors reference the work of Sullivan and Lilburn’s Good Questions for Math Teaching (2002) to explain the steps for creating a good question for any math topic:
1. identify the topic
2. think of a closed question
3. open the closed question by including the answer and working backward to situations that might give that answer.
Or, rather than follow those steps to create your own question, adapt a standard question from something like – What is the volume of a 2 in by 3 in by 4 in box? To something like – The capacity of a box needs to be 24 cubic inches. What are the possible dimensions?
Another given example – instead of asking students to solve a fraction multiplication question that results in the answer of 2 ¾ - give the answer of 2 ¾ and ask students to find what two numbers would result in that product.
An example I came up with – instead of asking students the perimeter of a rectangle that is 9 feet long and 6 feet wide, pose this question: “Ben is getting a puppy and wants to build a rectangular pen in the yard for the puppy to play. Ben’s parents told him that there is 54 feet of fencing in the garage. If he uses all of the fencing, how long and how wide can the pen be?”
To me, this question is accessible to all students – if they know how perimeter works, they can use that knowledge to determine possible lengths and widths; if they do not understand perimeter yet, they can draw the rectangles, guess the side lengths and then add the sides to check. For those students who quickly find a possible answer, they can extend to find all of the possible answers and then determine which one would give the puppy the greatest area in which to play. This allows all students to solve, but allows for differentiation based on student strengths and background.
I have done quite a lot of problem solving in my classroom, giving situations like the puppy one above, but many of my actual questions were closed; a few were open. I need to take some time to go back to these and see how I can revise them to make more of them open.
Do you have any favorite "open" problem solving scenarios, puzzle, games?
Chapter 3: Knowing and Understanding Students as Learners
The differentiated math classroom is learner-centered and attempts to tap into individual student strengths and styles since each student’s background is different. This chapter reviews the importance of learning styles, individual characteristics, and brain function in the learning of math.
The authors begin by sharing an example of a “line design” lesson that is used by sixth-grade teachers. The lesson is designed to reveal student characteristics like math disposition, work habits and learning styles. The lesson involves minilessons that lead to a product. First, they learn how to create paper and pencil drawings, using straight lines (but giving the illusion of curved lines.) Then they create their own designs and stitch them onto cardboard (I used to do an activity like this years ago, and I found this example in my closet - I never throw things away!)
The authors explain that this project can be completed by all students, but the resulting product will demonstrate the range of learners, as well as the way in which they approach math work.
As students work on this project, the teachers observe to see how students work, types of interactions between students, how students deal with making mistakes, how well they use their time, who is motivated, who is resistant, quality of work, and so on. The project helps students to learn a bit about themselves as learners and helps them to understand that they need to be responsible for their own learning.
The authors reference Caine and Caine (1992) and Jensen (1998) to point out that teachers need to be aware of the following facts about the way the brain operates: 1) “brain functions are significantly affected by social relationships and emotional responses; 2) the brain responds with positive intensity to appropriate challenge but shuts down when threatened; and 3) like fingerprints and DNA, the subtle organizational working of each and every individual brain are different.”
In order to be ready to adapt lessons and to plan effectively for the entire class, it is important to know the characteristics of students - their interests, cognitive attributes, emotional/social characteristics, intellectual needs and learning styles. The authors reference the Addressing Accessibility in Mathematics project that proposes that a teacher select three students who are representative of the range of learners in the room and create a “focal student” profile of those three students, showing their strengths and weakness in the following areas: conceptual, language, visual-spatial, organization, memory and attention. The thought is that by being sure to address the needs of those 3 learners when planning lessons, the needs of most other students will be addressed as well.
Students’ dispositions in math are important as well – are they quiet, withdrawn, outspoken, motivated (and by what), eager to think and learn, or are they discouraged?
The authors take a good deal of time to discuss learning styles (visual, auditory, kinesthetic), as well as four typical styles that have common threads.
They also reference the styles set forth by the Human Dynamics program by Dr. Sandra Seagal, and Gardner’s Multiple Intelligences. All of these are important to consider as teachers are working to create the differentiated classroom.
There is, again, quite a lot to think about here. I will have much larger classes this year, with a greater range of interests, strengths, and needs, I'm sure. Much that is offered in the chapter will be very helpful!
A quick review from the first part of chapter 2 – the differentiated classroom environment is developed through expectations, norms, and management. I reviewed the expectations in the last post, so we are on to norms (the values, customs, habits for how things are done) and management (organizing the students, materials, time and space).
The authors state that norms should be reviewed regularly and should be clearly displayed, and they include several norm categories.
There are quite a few management factors offered in this chapter….I don’t want to skip many because they are all critical, but I will attempt to shorten them here:
* establish routines
* define student responsibilities
* give clear directions for all learning styles
* establish indicators of quality
* establish acceptable noise levels
* create signals to ask for help
* define record keeping
*establish lines of communications
It is critical that a considerable amount of time is spent at the beginning of the year, practicing routines and setting expectations.
Anchor activities, which were mentioned previously, are explained as being tasks that are options for students once they have completed their assigned tasks. The anchor activities are critical to the management aspect of the classroom, and they need to be challenging, big idea activities. Anchor activities can also be used when students are waiting for help. These activities could be general or individualized and could be enrichment, extension of current or past units, or skills practice. (The authors state that they use problems from the Mathcounts program (www.mathcounts.org), and that more anchor activity examples are found in chapter 10). What is important is that students understand the directions and the proper use of the activities, so that they can work independently. The authors practice using anchor activities in the beginning of the year.
Several scenarios/activities are given as examples of ways to help establish the classroom environment that will allow for successful differentiation:
* survey students’ needs
* observe students as they are working cooperatively for the first time, to
understand their strengths and needs
* get students to listen and respond to each other
One activity that the authors discuss is having the students write a math autobiography (although I didn’t finish this book last year, I did read this part, and I did have my math students write their math autobiography this past year – I thought it was a very worthwhile activity). In their writing they should include ideas like:
* earliest memories of learning math
* school experiences with math
* how they use math every day, outside of school
* math triumphs
* who has helped them most in learning math and how
* how they best learn math (group, alone, with manipulatives, discussion, writing,
* one concept they know well and one they’d like to know more about
* something they don’t like about math, if there is anything (and why)
The authors include a teacher reflection guide, with 8 questions based on Ron Ritchhart’s work in his book Intellectual Character: What It Is, Why It Matters and How to Get It.
On to Chapter 3!
Chapter 2 is fairly long and has quite a bit of information to process, so I'm dividing my summary
into 2 days - I don't want it to get too long for one post!
This chapter begins with a classroom scenario (a middle school class), with the teacher explaining what students will be working on during the class - various activities based on their errors/misconceptions on their preassessment. Students had already received a list of concepts covered in the unit, and in their notebooks, they had highlighted concepts (highlighted by the teacher) on which they needed to focus...different students may have had different concepts highlighted, according to their preassessment results.
Among the activities was a mini-lesson with the teacher, for those who need to work on regrouping when subtracting fractions. The remainder of the class chose the activity they wanted to work on (according to their own list) and they got started. A few of these activities were: designing an I Have Who Has activity; a fraction game; practice problems with answer key provided; and cooperative group problem solving. Teacher checked to see that the rest of the class was working; the authors identified examples of positive student interaction during this work time; the teacher circulated once her group was working on practice problems. At the end of class, students filled out an exit ticket to reflect on their work and to explain what they need help with and what they will work on the next day.
The authors explain that this differentiated math class is a result of the learning environment that the teacher has established, in which “respectful attention” is given to all of the students’ needs. Their needs are determined by the preassessment and by teacher observation. Because the teacher has developed specific expectations early in the year, the classroom runs very smoothly, with students clearly understanding their purpose, the routines, and what behavior is appropriate during their work time. The following “classroom structures” were used in this classroom and, according to the authors, are critical to differentiation ( I will paraphrase some):
1. Class attends to general and mathematical needs of all students with consideration and respect.
2. Students know routines and expectations, such as: assignment posting; entering and moving about the room; getting to work; acceptable voice and noise levels; asking for help; minilesson rules, etc.
3. Students are clear about the location, use and care of materials.
4. Students have different work assignments, due to interest, skill level, learning style.
5. Everyone in room is a teacher and learner, so think-alouds are practiced by all, in partners, with teacher, in small group, and in whole class.
6. Students are expected to stretch their thinking.
7. Independence and self-starting are nurtured and valued.
8. Respectful listening skills are critical.
9. All work is documented.
The authors explain that the differentiated environment is developed through expectations (what you believe and want to occur in your class), norms (values, customs, and habits for how things are done), and management (organizing the students, materials, time, and space).
Among the expectations that should be set are:
* expect students to work toward independence, responsibility for own learning,
* expect students to think about how they think and learn;
* clarify performance standards;
* prepare students to expect different assignments based on need;
* expect students to understand that all in the class are teachers and learners;
* expect students to respect and value each other.
Next post will pick up with norms and management:)
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.