Reading Journal Week 3

1. Synthesis Activity #1 A-H (p.47).  

A. Let W = {items in Casey Rudd’s working portfolio}, P = {items in Casey Rudd’s presentation portfolio}, |W| = number of items in W, |P| = number of items in P. Which one of the following is true?

a) |P| < |W|

B. Let R = {preservice secondary-school mathematics teachers}, I = {inservice secondary-school mathematics teachers}, and T = {secondary school teachers}. Which one of the following statements is true?

d) R ∩ I =

C. A definition for “PSSM” can be found in                      of this textbook.

b) the glossary and chapter 4

D. Inquiry instructional strategies are intended to lead students to

b) discover and invent mathematics

E. The assertive response style is characterized by

d) Sincerity

F. According to Kounin, a teacher who is with-it                    .

c) is aware of what students are doing

G. In this chapter’s section “The Beginning of an Eventful School Year,” reference is made to being businesslike in the classroom. In this context, a classroom is businesslike if                    .

c) learning mathematics is afforded top priority

H. During his interview for the Rainbow High position Casey Rudd indicated that                    .

a) for teaching mathematics, inquiry instructional strategies are superior to direct instructional strategies.

2. Complete Synthesis Activity #9 (pp.48-49).  

A. Write a passive response that Ms. Salzburg could make to her students in Case 1.16.

“Well, alright, I guess this one time you could have some extra time on your homework. School spirit is important I suppose.”

B. Write a hostile response that Ms. Salzburg could make to her students in Case 1.16.

“Don’t think I’m going to fall for this trick! You kids are just trying to get out of work, and I won’t have it. The assignment is due on Friday, and you’ll have to deal with it or get a zero!”

C. Write an assertive response that Ms. Salzburg could make to her students in Case 1.16.

“I understand that school spirit and social activities are important, but you still have time to adjust your schedules to make time for the assignment and the game. Time management is an important adult skill, as hard as it is sometimes; even I needed to schedule in the weekend to grade your papers, and I wouldn’t have that weekend if I gave you until Monday to turn it in. If I gave you the extra time, it would also put us off schedule in this class which might result in more work later, so I think it’s best if we stick to the schedule. Let’s still turn in our tasksheets on Friday.”

D. Compare what you wrote to that of colleagues. Discuss how passive, hostile, and assertive communications differ. (To be discussed in class.)

E. Compare what you and your colleagues wrote to the following: (see pg. 49).

(To be discussed in class.)

3.       Syllabus and opening-day tasksheet you started last week.

Click here to view draft #3 of syllabus (also listed under "Grade 7 Syllabus" in "Cognition/Instructional Strategies/Planning" section).

Click here to view draft #3 of opening-day tasksheet (also listed under "First Day Tasksheet" in "Motivation/Engagement/Classroom Management" section).

4.       Complete the Transitional Activities, pp. 49.  (Remember, the purpose of the Transitional Activities is to warm-up your brain for engaging in the next chapter.  There are no "correct" responses; just go for it and be honest and thoughtful.  No lengthy answers necessary.

1) Some teachers orchestrate smoothly operating classrooms where students cooperatively go about the business of learning mathematics with hardly any disruptions. Other teachers spend more time ineffectively dealing with student misbehaviors than conducting worthwhile learning activities. What do the teachers in the former group do to gain and maintain students’ cooperation and to motivate students to do mathematics?

            I think one of the most important components to have a smoothly running classroom like the one described in the former group of teachers is to make behavioral expectations understood on the first day of class, and to enforce those expectations in every situation. If you make your expectations clear, and a student acts out and you let it go for awhile before you do something about it, then other students will think that you didn’t mean what you said and will be more likely to act out as well. This communication of expectations is part of what it means to create a businesslike atmosphere in the classroom.

            The second and equally important component to this kind of classroom is to create learning activities, discussions, etc. that engage students and keep them interested in the material. If you present a mindless worksheet or allow much free time with nothing constructive to do, kids will become bored and misbehave. I think that the root of misbehavior is boredom.

            I’m sure there are many other things that could help create an atmosphere with no disruptions where students cooperatively learn mathematics, but I think the two components above are some of the most essential.

2) What are some of the strategies employed by teachers to build and maintain a classroom climate that is conducive to cooperation and engagement in meaningful mathematics?

            The answer to the question above could fit this question as well. However, going along with creating engaging learning activities, it is important to take time outside of class to thoughtfully prepare inquiry-based learning activities so that students can discover mathematics for themselves. Not only does this kind of learning entertain students and keep them involved with the activity, but it also offers them an opportunity to deepen their conceptual understanding of concepts, which motivate the direct-instructional learning activities (which generally seem more boring to students). Preparedness is also important in creating a good flow between activities in the classroom. It is important to move quickly and effectively from one activity to the other so students have more opportunities to learn, and also don’t have time to get bored and disconnect themselves from learning.

            Another incredibly important strategy to creating a cooperative learning environment is to make the classroom a safe place to speak what you think. If students feel unsafe to share their opinions without being laughed at or belittled in some way, they will not take the risk to participate in discussions. Communicating mathematics (discussions) is essential to learning it well, so if students understand that no answer is a stupid answer, then they will be more likely to participate and succeed in the class. To begin creating this common understanding, it would be useful to communicate this on the first day of class.

3) What impact does the manner and style in which a teacher communicates with students and parents - how assertively the teacher speaks, the teacher’s body language, and whether the teacher uses descriptive or judgmental language – have on how well students cooperate in the classroom?

            The manner and style in which a teacher communicates has a profound impact on how well students cooperate in the classroom. For example, a teacher must be assertive. If he/she is passive, the students will run the classroom; if he/she is aggressive, the students will push back or only grudgingly cooperate out of fear; if he/she is assertive, students will see that their teacher respects them enough to be honest with them, and they will reciprocate the respect. Assertive communication also benefits the businesslike atmosphere; students will be able to see that learning math is the priority if this is communicated and reinforced assertively.

            The teacher’s body language is also extremely important, because it communicates how involved and interested the teacher is in what he/she is doing. If the teacher is slouching, or sitting, or not making eye contact, students may think that their teacher is disinterested in the subject, so why should they be interested? On the other hand, if they can see that there teacher enjoys the subject, this attitude is contagious, and students may also come to enjoy it. At the very least, students should be able to see that their teacher is invested and devoted to teaching well, which will show them that there is value to their learning.

            Judgmental language has no place in the classroom. As I described in question two above, students need to feel safe in order to learn well. If they feel judged, they will not speak up, which will hinder their learning.

            It is important to be assertive with parents for the same reasons. How much parents value the learning of mathematics will dictate how they encourage and help their students to behave and participate fully in the class.

4) What classroom standards of conduct and procedures for safe, efficient classroom operations should a teacher establish? How should those standards and procedures be enforced?

            Teachers should establish standards of conduct that revolve around respect. They should talk about the importance of respect, and how respect would look in various situations. Some examples are how students listen to their teacher, how they communicate with each other, how they adhere to school rules, etc.

            Enforcing standards and procedures is an area that I don’t know very much about, and that overwhelms me a little bit. I do think that standards should be enforced immediately upon any sort of misbehavior so students know that you’re serious about it. I also think communicating one-on-one with a student after they have misbehaved to assertively explain why their behavior was unacceptable and to reinforce your expectations is important.

5) What are some effective strategies for teaching mathematics to students for whom English is not a first language and who are not proficient with English?

            It is important to make sure that students who are not proficient with English have equal (as equal as possible) resources to learn. To start with, it would be very beneficial for students to have a textbook in their proficient language to they could refer to it in case they didn’t understand something in class. Also, another reason inquiry-based strategies are so beneficial to students is because you can come up with activities that have very clear and simple instructions, and because they are hands-on learning activities, students could learn from them regardless of their language. I think it is also important to be deliberate about teaching things in as simple a way as possible, and not using unnecessary eloquent language that is inaccessible for students. Finally, I would investigate if there were any tutoring programs in the school designed for one-on-one help. I personally work for a tutoring program specifically for low-income students, where we try to hire tutors who speak the languages of students who are in need at our school. I don’t know how common programs like this are, but as a teacher, I would try to find them or initiate starting one.

6) What are some effective strategies for teaching mathematics in a way that reaps the benefits of cultural diversity in the classroom?

            I think the most important way to teach mathematics in a way that reaps the benefits of cultural diversity in the classroom is to teach about connections to the history of mathematics that relate to the different cultures in the classroom, and also cultures outside the classroom.

7) What are some effective strategies for dealing with students who are being uncooperative and off-task?

One general strategy that I’ve already mentioned to deal with students who are uncooperative and off-task is to try to create material that is engaging and even fun to allow them to learn mathematics in a hands-on way.

Also, during classroom discussions, you can single out that person and ask them a simple question like “What do you think about that,                ?” The purpose of this question is not to point out that they were off task, or embarrass them because they don’t know the answer, but the purpose is to bring them into the conversation.

If students are doing individual work and are off task, it may be helpful to walk around the classroom. If students know you are paying attention and will see that they are off-task, they will be more likely to do their work. It is also helpful to ask questions to students as you walk around the classroom. It is helpful to ask stimulating questions to students who are off-task, and dialogue with them to get them involved started on the assignment.

8) Chapters 4-11 of this textbook focus on the principles, strategies, and techniques for developing and delivering mathematics curricula to middle, junior high, and high school students. Why do you suppose that, prior to those chapters, a chapter with the title “Gaining Students’ Cooperation in an Environment Conducive to Doing Mathematics” and a chapter with the title “Motivating Students to Engage in Mathematical Learning Activities” are inserted?

            These chapters precede chapters on learning about developing and delivering curricula because knowing how to formulate curricula, develop good lesson plans, write effective assessments, etc. is all useless unless your students are involved and engaged in what they are learning. Knowing how to get your students to cooperate and knowing how to motivate them to learn must come learn how to teach them.

5.       Read Chapter 2.

6.       Study Exhibit 2.5.  Why might it be helpful to organize a unit in this way?

It might be helpful to organize a unit in the same way as Exhibit 2.5, because this organization clearly lays out each activity that will take place and what materials are needed. This kind of preparedness makes it clear what kind of transitions will take place so that, as a teacher, you can set aside the proper amount of time for transition time and allocated time.

In this example, the teacher clearly thought out things that could become time wasters and he/she eliminated them. For example, there is something planned for the students to do immediately as they walk into class that is relevant to the lesson. The teacher also uses this time to take role while the class is engaging in a learning opportunity so that no time is wasted calling out names. While the teacher is walking around, he/she is also paying attention to who did the homework assignment the way he/she intended it, so that no one would come up to the board and feel embarrassed that they did the assignment the “wrong” way. Calling up the students who did the assignment correctly also prevents confusion because the other students will only see one example, and it saves time when you’re not calling up multiple students trying to find one who did the assignment as you intended it. Also, having a student write on the board while the rest of the students take out graph paper is a purposeful use of time.

I like that this teacher allowed enough practice (the homework at the beginning and the sets of equations near the end) to give the students a chance to start discovering concepts, but the times were cut short and the rest of the assignments given as homework. The fact that the homework assignment was integrated into the lesson is good because it allows students to begin the homework with help, and hopefully have a good understanding of it by the time they get home to finish it. Students are less likely to be intimidated by their homework because they won’t be starting an entire new assignment that they haven’t even looked at, but instead will be finishing something they’ve already started and already have an idea of what to do. In addition, if the students were engaged and stimulated with the beginning of their homework assignment, they will already have a positive attitude toward it and will be more likely to finish it at home. I like that this teacher left time at the end for homework, for the same reasons, but also because this allows a time buffer. If things don’t go quite as planned in the lesson and it takes more time than anticipated, then the time for homework at the end could easily be sacrificed. However, if the lesson is finished early, the time designated for homework is still useful for students’ learning.

7.       Compare and contrast Cases 2.2-2.4. 

In Case 2.2, Mr. Rudd is much more prepared for the lesson (not that it flows well); he has a clear agenda, had a tasksheet ready for the students when they came in, knows exactly what materials students will need, and knows which page number they should turn to. Conversely, in Case 2.4, Ms. Lem seems rather unprepared; she could not remember if the class had covered a certain topic, and fragmented her directions with trying to find the page to turn to.

The teachers in these cases also seem to have different teaching styles. For example, Mr. Rudd introduces an activity and has one of his students read something aloud for the rest of the class to follow along with, and it seems implied that he will continue his explanation for the activity. Ms. Lem, on the other hand, vaguely tells her students to read something on page 197, assuming that they will understand what they read. (This may not be the correct interpretation of what happened, but from the short descriptions, I gathered that their teaching styles were different in this way.)

            In Case 2.2, as described above, Mr. Rudd abruptly explained directions while students were unfocused, and expected them to know that page to turn to; he then immediately had someone start reading. On the other hand, In Case 2.4, Ms. Lem ensured that her students were all opened to the relevant page before she assigned directions and had them read.

            In both cases, some students in the classroom were not caught up with the class or engaging in the material. In Case 2.2, this is because Mr. Rudd moved too abruptly from one activity to another and did not allow enough time for students to act on his directions before he started the lesson. In Case 2.4, this happened because Ms. Lem continued with her lesson after she found that some of her students didn’t understand an important concept.

           

8.       Complete Synthesis Activity #1A-L on pp.85-86

A.     Time for students to achieve learning objectives increases as                         .

d) transition time decreases

B.     Duane listens intently to his classmates and occasionally volunteers his own opinions during large-group discussion sessions about application of combinations and permutations in real-life situations. Duane’s behavior appears to be                     .

c) on-task and engaged

C.     Pat quietly rearranges her desk as directed by her teacher as the class moves from a large-group to a small-group session. Pat’s behavior appears to be     .

d) on-task but not engaged

D.    Dawn speaks out, interrupting Erin during a class discussion session on the origins of certain mathematical terms. Dawn’s behavior appears to be           .

c)    off-task and disruptive

(It is possible that she accidentally interrupted with a comment about what they were talking about. Even though interrupting is not following the teacher’s directions, she might be considered “a) disruptive and engaged” because she was attempting to follow the teacher’s instructions and discussing the subject.)

E.     When a student is rewarded for an off-task behavior, the reward serves as    .

a) positive reinforcement for an off-task behavior

F.     Larry interrupts his teacher in class. The teacher says, “Larry, what makes you think we care what you have to say? Sit and listen without interrupting!” Larry feels so embarrassed he silently vows not to interrupt the teacher again. He begins doubting that his peers really want to hear his opinions. In this example, the teacher’s remarks serve as            .

b) destructive punishment

G.     An assertive communication style is characterized by            .

d) sincerity

H.    A classroom with a businesslike atmosphere is characterized by              .

d) purposeful activity

I.      Which one of the following contributes to a businesslike classroom conducive to learning and doing mathematics?

a) use of descriptive language

J.      A supportive response to a student tends to communicate          .

a) recognition of feelings

K.    Students tend to be most receptive to communications about your expectations for them         .

b) near the very beginning of a course

L.     By “withitness,” Kounin refers to how           .

b) aware teachers are of what is going on in their classrooms

9.       Complete Synthesis Activity #4 on p.86.

Imagine having just directed your students to devise proofs independently for a theorem you have just stated. Although you were quite clear that they were to work silently by themselves, you notice that Haywood and Howard are talking. You walk over to them and realize that they are discussing how to prove the theorem. State an example of a descriptive comment you could make to them. State an example of a judgmental comment you could make to them. With colleagues discuss the relative advantages and disadvantages of making the first instead of the second.

Descriptive comment:

“Excuse me, Haywood and Howard, I asked you to work on these proofs by yourselves, your talking may be making it hard for others to concentrate.”

Judgmental comment:

“I asked you to do this assignment individually; you two are cheaters, and you will both be getting zeros on this assignment. Now be quiet!”

*Discussion in class

10.    In light of your reading of the section ``True Dialogues” from pp. 64–67, write scripts for two different dialogues in which you engage students in a conversation about a mathematical problem (you need to create a problem). Write the scripts so that IRE cycles dominate the first conversation; write the second script so that you engage the students in a naturalistic conversation with true dialogue that is free of any IRE cycles.

Problem: This problem is for 7^th grade mathematics. They are given the following equation and told to graph it on a number line: −x – 3 ≤ 2.

IRE Cycle conversation: The student’s (Ben) solution had an open dot at −5 with the bolded line going left.

Ben: Here is what I got.

Ms. Nordin: I’m glad you found that −5! You’ve almost got this right. Let’s walk through the steps together. So what did you do first?

Ben: Well I plused the 3 to both sides.

Ms. Nordin: Excellent, that’s the correct first step. What comes next?

Ben: Next I timesed −1 to both sides.

Ms. Nordin: Good job. The correct way to say that is, “I multiplied −1 by both sides,” but that is the right step.

Ben: Okay. So then I put the dot on −5, and went left because it’s less than.

Ms. Nordin: I’m happy that you know that less than is to the left, because the numbers get smaller the farther left we go. Why did you do an open dot?

Ben: Oh! I forgot the equals sign thingy. It’s a closed dot, my bad.

Ms. Nordin: That’s right, do you know why that is?

Ben: Because… that’s how you told us to do it?

Ms. Nordin: Well, I did tell you that, but there’s a reason it’s that way. We color in, or bold, the possible values of x right?

Ben: Right.

Ms. Nordin: So if x is less than or equal to, that means x could be equal to negative five. That’s why we color in the dot there. And how about this sign? Do you remember talking about what happens when we multiply by a negative?

Ben: Kinda…

Ms. Nordin: Well we have to flip the direction of the inequality when we multiply or divide by a negative.

Ben: Oh yeah, I forgot, okay so I color in the other way? Toward the right?

Ms. Nordin: That’s right! Very good job, Ben.

True dialogue:

Ben: Here is what I got.

Ms. Nordin: So it looks like you added a 3 to both sides to start.

Ben: Yes, because I wanted to get it away from the x.

Ms. Nordin: Oh I see, that makes sense. And then you had the x by itself?

Ben: No because there was a negative out in front still.

Ms. Nordin: Okay, and it’s not by itself until there’s absolutely nothing around it, I see. And so you multiplied both sides by −1, so you went from

–x ≤5 to x ≤ −5. I don’t see how you got here, could you explain it to me?

Ben: Sure, I multiplied −x  by −1 to get x, and 5 by −1 to get −5.

Ms. Nordin: So –x ≤ 5 and x ≤ −5 mean the same thing?

Ben: Well, I guess the same numbers have to work for x for every step, that means we have to plug into –x ≤ 5 and x ≤ −5 and get the same answers for both. But for the first one -2, -3, -4, -5 work for x… oh yeah so we have to flip the sign.

Ms. Nordin: What do you mean, flip the sign in front of the negatives?

Ben: No I mean the sign between the x and the number.

Ms. Nordin: Oh, yeah, you mean flip the direction of the inequality, I see what you mean! Try this one (−6 – x ≥ 8) at your desk and let me know if you come to the same conclusion about changing the direction of the inequality.

11.    Read Chapter 3, pp. 88 – 117

12.    Read Synthesis Activity #2, pp.128-129.  (You do not have to write anything down.)

13.    Complete Synthesis Activities #4 and #5, p. 129.

Synthesis Activity #4: Suppose Sharyce, one of your prealgebra students, asks you, “Is zero odd, even, or neither?” Formulate an answer to Sharyce’s question using a thing-aloud strategy for responding to reasoning level questions. Discuss your answer with a colleague who is also engaging in this activity.

“That is a very challenging question. Well, let’s think about our definition of even and odd numbers. We know that for a number to be even, it must be equal to 2n where n is an integer. And for a number to be odd, it must be equal to 2n+1 where n is an integer. So lets set 0 equal to 2n+1, and solve for n to see if n is an integer! If n is an integer, then all parts of the definition are true so 0 would be odd. So let’s see, 2n +1 = 0, subtracting 1 from both sides we have 2n=-1, so now we divide by 2 on both sides and have n=-1/2, that’s a fraction not an integer! So since one part of the definition doesn’t work, 0 can’t be odd. What about even. Let’s set 0 equal to 2n, so 2n=0, now let’s divide both sides by 2, and we get n=0. Well, 0 is an integer! So since all parts of the definition for an even number work for 0, we’ve figured out that 0 is even.”

Synthesis Activity #5: Compare Mr. Smart’s method of engaging students in a question-discussion session in Case 3.19 to Ms. Cramer’s in Case 3.20. Which one of the two methods do you expect to use more often? Explain why. Discuss your responses with a colleague.

Mr. Smart’s method required less time and the main challenge was thinking about the problem; Ms. Cramer’s method require more time and the main challenges were thinking about he problem and communicating the answer through writing. Ms. Cramer’s classroom environment seemed a little more serious, while Mr. Smart’s seemed safer to communicate ideas. I expect that I will use Mr. Smart’s method more often, because, while communicating ideas is very important, the majority of the class will be learning math concepts. The way Mr. Smart presented the question and led the following discussion was much more focused on the problem at hand, and allowed students to communicate their ideas aloud, with guidance and encouragement. Ms. Cramer’s method seemed a little more intimidating, as if there was one right answer and you had to have it completely thought out to be able to write it down. I like the discussion environment better because students can learn from the thought processes of other students.