Multiple choice prompts (Synthesis activity 1)
A. A, the presentation portfolio is a subset of the working portfolio, thus the number of items in the presentation portfolio will be smaller than the number of items in the working portfolio.
B. d, a mathematics teacher is exclusively either preservice or inservice, so there would be no elements in their intersection.
C. b, the definition of PSSM is found in both the glossary and chapter 4.
D. b, inquiry instructional strategies are intended to lead students to discover or invent mathematics.
E. d, assertive response style is characterized by sincerity.
F. c, withitness is characterized by awareness.
G. c, a businesslike classroom affords top priority to learning mathematics
H. c, there is considerable redundancy of mathematical content in traditional textbook driven curricula- we would have much less to teach if we focused on connections between related content.
Assertive Communication (Synthesis activity 9)
The following are a series of potential responses to students’ requests that a particular assignment’s due date be extended in light of an extracurricular activity.
A. A passive response: Well, I really want them Friday, but I suppose that the game is an important school activity, so I’ll agree to extend the due date to Monday. But I will only extend the due date this one time. In the future you’ll need to plan ahead better.
B. A hostile response: Oh, and I suppose you think that we don’t have a schedule to keep in this class or that the assignments will grade themselves? It’s not my fault you are poor planners and I certainly won’t be inconvenienced just so you can have some fun. The due date will remain Friday, end of discussion.
C. An assertive response: Supporting the school is a great idea, and I want you to be able to attend the game if you so choose. Don’t forget though that math is just as important, and we need to make it a priority as well. I don’t want our class to fall behind and I need to have them Friday so I can look over them this weekend. If you plan carefully there is time for you to complete the task sheets and attend the game, so the due date is still Friday.
D/E. I think my responses were fairly similar, although I think the book’s assertive response was perhaps a little more firm than mine.
1. I think that the biggest differences between classrooms with few and many disruptions are student engagement and proper behavior management. First, students are most likely to cause disruptions if the activity isn’t meaningful or outside of the zone of proximal development. (i.e. too easy or boring, or too difficult and not worth attempting). Secondly, students’ instinct is to push boundaries so they know what behaviors they can get away with. Regardless of how well our expectations are communicated, they learn by what we do. If we clearly communicate expectations and clearly defined consequences for misbehavior, and then follow through with these consequences immediately, they will quickly learn to behave acceptably (in most cases at least).
2. There are several strategies a teacher can utilize to build a cooperative and engaged classroom. Some of these include thoughtfully designing units and lesson plans that include a variety of inquiry-based activities as well as clear direct instruction, and helping students to do meaningful work that emphasizes connections; Using an assertive communication style in conjunction with a warm, open, and respectful atmosphere, have high expectations for students so they are motivated to try and succeed; value and utilize the strengths of every member of the class to create a sense of community.
3. Students behave well when (in addition to meaningful and engaging activities) they know what is expected of them and they know how to fulfill those expectations. Assertive communication style is essential so students feel safe to ask questions and participate, and it minimizes uncertainty on both ends.
4. Mutual respect, preparedness, participation, and always doing our best work are the guiding principles behind the expectations for my classroom. Involving student input in creation and discussion of specific rules for the class can help make them clear from the very beginning. In addition to involving students in rule creation, they should be included in enforcement as well. My first enforcement step is always a clear warning, potentially with a reminder of the consequences for the next time. Students will be expected to respectfully hold each other accountable – and they will be more motivated to participate when they understand that I am included as well. Finally, following through with consequences immediately, every time (especially for gateway behaviors) will minimize escalation.
5. All students, including students learning English, benefit from interacting with manipulatives and concrete representations and activities as an introduction to mathematical principles. Using manipulatives and visuals in addition to written materials and verbal instruction can accommodate specific student needs. Also, it is important to value the linguistic assets of our students. Taking the time to learn, personally or as a class, words or phrases in languages other than English can be enriching and validating. Finally, it is important that students have the opportunity to practice English using all four language domains – listening, speaking, reading, and writing – as an integral part of classroom activities with specific language objectives.
6. Teaching math through history is often a helpful strategy, and by including historical information from a variety of countries, cultures, and people, students will know that mathematical discovery and progress has occurred in all cultures, and that every student can identify with a personal mathematical legacy. Additionally, working within the students’ communities and experiences as well as exploring those of other demographics in activities and exercises, as well as soliciting and valuing the input of every student, will help students to better understand themselves and those around them.
7. Knowing the kinds of consequences that encourage and discourage behaviors of students and using them appropriately can have great effect on behavior. It is also important to be aware of the motivation behind the misbehavior (avoidance, attention seeking, stimulation) so that consequences discourage rather than encourage it.
8. Before learning can begin, it is important that students are cooperative and engaged and that the classroom has an environment conducive to learning. Failure to ensure these as much as possible will usually undermine the other instructional strategies and methods we wish to employ. If you want something to grow, you need to prepare the ground and provide the right conditions before you plant it.
Unit Plan Design
The agenda in exhibit 2.5 has two major sections: Objectives and Agenda points. It is important when designing any part of the curriculum, whether it is a daily lesson plan, unit, or the entire year, to have clearly defined objectives. These objectives help focus instruction and activities as well as shape our decisions. Secondly, each of the agenda items represents a period of either allocated or transition time, with descriptions of instructions for each activity or transition. It also explicitly notes what the teacher and students should be doing at all times, and includes all the important administrative activities for the day (handing in and assigning homework, taking roll, etc.) Additionally, the agenda items state when they accomplish specific objectives. Similarly, when designing units one should clearly state the objectives of the unit, break up the unit into component parts (lesson plans or daily agendas with activity descriptions, and plan for transitioning lessons and activities between lessons and surrounding units.
Each of the cases 2.2, 2.3, and 2.4 demonstrate an underlying problem with class momentum. The first describes an ineffective transition because not everyone was prepared to hear the instructions, and not enough time was given to comply. The second describes an ineffective transition because too much time and individual attention was given during each instruction step. Finally, the third describes a loss of momentum during an activity because of an unplanned for transition to another activity that disrupted the first, and then was abandoned when unsuccessful. In all cases, improper planning and execution of transitions caused confusion and contributed to noncompliance.
Synthesis (Ch 2 #1)
A. d, students achieve higher learning objections when transition time is decreased
B. c, listening and volunteering in discussions is an example of on-task and engaged behavior
C. d, following instructions is an example of on-task behavior, but since desk-moving is part of transition time, she is not engaged in learning activity.
D. a, interrupting in a discussion is disruptive but engaged.
E. a, a reward for off-task behavior is positive reinforcement.
F. b, embarrassing a student as punishment also causes him to doubt the importance of engagement.
G. d, assertive communication is characterized by sincerity
H. d, businesslike atmosphere is characterized by purposeful activity.
I. a, using descriptive language contributes to a conducive learning environment
J. a, supportive replies communicate recognition of feelings
K. b, students tend to be most receptive to communicating expectations near the beginning of a course.
L. b, withitness refers to awareness
Descriptive v. Judgmental language (synthesis 4)
Descriptive: I notice you two are working together on this theorem, but I need you to work on it independently right now. (Thank them when they return to own work)
Judgmental: Excuse me, but what part of “devise proofs independently” didn’t you understand? Everyone else in the class understood.
The first helps students know they aren’t in trouble (yet) but need to follow instructions, and lets them know how to correct their behavior. The second is less explicit and will likely make students feel embarrassed and inferior, and could possible cause them to lose interest in deriving the theorem altogether.
Reasoning Level Questions (Synthesis Ch 3 #4)
Q: Is zero odd, even, or neither? A: (using think-aloud strategy) Good question! First of all, we need to remember what being ‘even’ means. We know even numbers are multiples of 2 as well as divisible by 2, which means that some number times two equals an even number. Also, we know that even and odd numbers alternate, so in between two odd numbers there is an even number. Based on that I think we can show that zero is even. 0 * 2 = 0, 0/2 = 0, and 0 is between -1 and 1, which are both odd. So zero is an even number.
Question-Discussion Sessions (Synthesis #5)
Both methods allowed students time to reason independently before sharing as a class. One method was verbal and required subsequent students to refute the previous argument before explaining their own. The second method involved writing responses on a task sheet (surveyed by the teacher while students are working independently) and then having selected students share their responses. These were summarized on the board and then students were asked to write comparisons.
I think as a future teacher both approaches have value and benefit. The second approach requires a bit more preparation in the production of the task sheet. However, I think incorporating both throughout the year is very important because they both target different reasoning skills, address different learning styles and meet different language objectives. The first targets on speaking and listening, is primarily auditory,and asks students to look for fallacies in arguments, and then to correctly argue their point. The second method targets reading and writing, has a visual component, and requires students to argue a point and make comparisons between arguments.