Tuesday, July 22, 2014

The Differentiated Math Classroom - Chapter 6: Planning: A Framework for Differentiating



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
* exploration
* summary
* 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?


Thursday, July 17, 2014

The Differentiated Math Classroom - Chapter 5, The Flexibility Lens




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:
* understanding
* computing
* applying
* reasoning
* engaging
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!)


Wednesday, July 16, 2014

Workin' on it Wednesday, July 16

This week, I have continued to work on reading and posting about The Differentiated Math Classroom. (I've finished chapter 4 - only 7 more to go!)
I've been babysitting my nephew a couple days a week this summer (he's 4 and loves numbers); last week, he (and the book) made me remember that I really wanted to work on creating color by number products for my classroom. He made me remember this because he's really into color by number right now, and we spent time coloring together last week. The book made me think of it because in the first chapter there's a scenario with students coming in and getting to work on different activities....this reminded me that I needed to make more activities! I've been planning to work on color by numbers since January and had even started a Valentine's one, but never got myself to finish, and went on to other things. I finally worked on one this week!

What have you been working on?









Tuesday, July 15, 2014

The Differentiated Math Classroom: Chapter 4, A Problem-Solving Platform



Section II The Essentials: The Lenses of Differentiating Mathematics

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
·         * leads to important math ideas for future
·         * is open-ended; can be approached with various strategies
·         * is accessible to students with different strengths, needs and backgrounds
·         * is interesting and engaging to many students
·         * has various solutions
·         * leads to higher-level thinking and discussion
·         * has constraints that provide direction but don’t limit thinking
·         * strengthens conceptual development
·         * provides chance to practice key skills
·         * 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?



Friday, July 11, 2014

Making Protein Bars

While I waited to see if the swelling in my lip would go down (see today's earlier post), I decided to try this recipe for peanut butter protein bars. I did alter the recipe a little bit  - instead of vanilla protein powder, I used chocolate peanut butter protein powder, and I added a handful of chopped up peanuts and a small handful of chopped up chocolate chips. I liked the idea of rolling out the dough while it was in a  gallon-sized baggie. (I've never done that before!)

I also cut the dough into bars while in the baggie, using the back side of a sharp knife (didn't want to cut the baggie). You can tell from the picture that I didn't worry too much about rolling the dough into  a perfectly-shaped rectangle.....next time!

I put them in the freezer for about 2 hours. They weren't completely firm at that point, so maybe they needed to stay in longer; but after the 2 hours, I cut the baggie open to take the bars out and then I put them into individual baggies. I put them back into the freezer and may keep them there....I'll have to see how hard they get. I had little tastes as I was making these, and I had one bar after the 2 hours in the freezer. I did make them pretty thin; next time I will make them a bit thicker. But, they were yum!


A couple weeks ago, I made protein balls using this recipe, again using chocolate peanut butter protein powder (instead of chocolate). These were very easy and tasty as well.

I think either of these recipes are good for making protein bars or protein balls, whichever you prefer. I'm so glad I tried them. They are simple to make and taste great  (and a bit healthier than the candy I love)!






My Garden Attacked My Lip!!

I decided to weed in my "garden" this morning (it doesn't have much planted this year but berry bushes and a couple fruit trees, so the cleared space I used for vegetables last year is now weeds. Booo.).
When I decided I was finished, I walked into the house, noticing that my lip felt a little funny and puffy. I looked in the mirror, and the left half up my upper lip was swollen! There was a little black dot on it, which I got off (or out - I'm not sure which) with a tweezers. I took some Benedryl (only one though, because I don't want to fall asleep) and am waiting to see if it gets better (hasn't gotten worse yet and it's been an hour). I don't remember anything touching my lip, but maybe a tiny thorn or bug got to me....

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