Reading Journal Week 4

1.      Finish reading Chapter 3

2.      Engage in Activity 3.2. Bring hard copy to class and post the initial version on your website. The revised version will be posted after you discuss the work with a colleague.

Click here to view Draft #1 of Sample HW Assignment for Inequalities Unit (this and subsequent drafts are located in the “Motivation, Engagement, and Classroom Management” section).

Click here to view Draft #1 of Sample Quiz for Inequalities Unit (this and subsequent drafts are located in the “Motivation, Engagement, and Classroom Management” section).

3.      Pick one of Ms. Culbertson's eight points about directions that you found particularly interesting, and explain why you found it to be interesting.  Elaborate as necessary.

The most interesting point to me was the following: “When teachers are giving directions, they are not conducting an inquiry lesson. Efficiently communicated directions in which transition time is minimized do not normally allow time for students to debate the pros and cons of what is to be done. Unlike Ms. Bey in Case 3.7, Ms. Culbertson did not open an opportunity to argue about the selection.”

I think this point is interesting because I have seen many teachers give unclear directions about an assignment and it significantly elongates the transition time. If a teacher has not thought through their directions beforehand to try to find points of confusion and correct them, it is likely that there will be confusion in the classroom. Even if the teacher responds to students’ questions and clarifies these points of confusion after giving directions, it is still more likely that students get sidetracked with their confusion and don’t listen to the questions, or tune out the discussion because they think the delivery of directions is over and there are distracted by the assignment, etc. I have also seen this question/answer time turn into a forum for students to express their own opinions about the assignment, usually expressing things they don’t like about it. To avoid this and hopefully ensure comprehension of directions, it is best to prepare the delivery of directions beforehand and try to communicate them clearly. It is also a good idea to write down the important aspects of the directions so that you don’t mistakenly leave out an important part while you are communicating to your students.

4.      Make up a student reasoning-level question and a student memory-level question.  Then answer each as you would in your classroom.  Finally, explain why and how you did not answer each in the same way.

Reasoning-level question:

Andrea: “I don’t get problem #4 on the homework, is says, “Show that x(3x – 2 – x  + 9) and 2x² + 7x are equivalent expressions.”

Ms. Nordin:  “This is a challenging question! Hmm… so if I were to sit down and try to figure this problem out, I would first need to know what it means for two expressions to be equivalent. We learned a couple of weeks ago that ‘equivalent expressions’ means expressions that remain equal, or have the same value, when any value is substituted in for the variable. Okay, that helps give us some direction to figure this out! So now we know that we need to show that x(3x – 2 – x + 9) and 2x² +7x are have the same value when any value is substituted in for x. That doesn’t seem like an obvious fact when we first look at these two expressions. Oh! How about we start with one of the expression and see if we can manipulate it to look like the other one. Let’s start with x(3x – 2 – x + 9). Hmm… this expression has parentheses while the other doesn’t, so let’s start by distributing to get rid of the parentheses. When we distribute we get 3x² – 2x – x² + 9x. This is an equivalent expression to the one we started with because all we did was distribute (Distributive law)! Now looking at this expression, we can see that there are some like terms, so let’s use the commutative law to switch the order of the terms, so we get 3x² – x² + 9x – 2x. Now we can just add the like terms, so we have 2x² + 7x. This looks exactly the same as the expression from our question! So we have showed that the two expressions are equivalent.”

Memory-level question:

                  Andrea: “What does the ‘commutative law’ mean?”

Ms. Nordin: “The commutative law says that any finite sum or product remains the same if you reorder its terms. This can be shown symbolically by a + b = b + a, and ab = ba.”

I answered the reasoning-level question with the think-aloud strategy. This is beneficial for this type of question because students can see your thought process and understand how to do something rather than just what to do. I answered the second question directly because it was a memory-level question, so it is appropriate to simply provide the answer.

5.      Compare Cases 3.18, 3.19, and 3.20, pointing out what you personally found compelling.

In case 3.18, Mr. Grimes presents the overhead transparency, asks his students is they can see what the blocks and circles have in common, and immediately has them answer. While this is an inquiry-learning activity, and while the discussion that ensues seems like a productive conversation, Mr. Grimes did not allow time for the whole class to think about an answer before the discussion began, which probably left many students listening to the conversation without formulating their own answers (and practicing thinking critically). The discussion also only involved four students, leaving the rest of the class potentially disengaged.

Case 3.19 represents, in my opinion, a better example of this inquiry learning activity. In this case, Mr. Smart presented the same question, but required that the students all take three minutes to formulate their own answers before he opened up the discussion. This is more beneficial to the class because everyone had time to think critically about the question and engage in the activity. Again, though, in this case only a few students were involved in the discussion (as far as we know), so it would be impossible for the teacher to gauge the understanding of each individual.

I personally found Case 3.20 to be the most compelling of all three cases because the same activity was presented, but in this case Ms. Cramer had each of her students write their reaction on a tasksheet. This no only allowed each student time to formulate their own answers, but also gave them a chance to practice communicating their thought process (an important skill). The tasksheet method also provided Ms. Cramer with a means to assess the understanding and progress of each student in her classroom. After everyone had time to work on their tasksheet, a discussion was still held so that students could benefit from other ideas in the classroom, and Ms. Cramer could ensure that the purpose of the activity was conveyed to every student.

6.      Which of the nine points about cooperative-learning sessions do you find most interesting and why?

The most interesting point to me is the following: “Tasksheets direct students’ focus and provide them with an overall picture of what they are expected to accomplish in their groups.” In my experience, I have found that students stay on task better if they have something to produce, such as a tasksheet. If they are engaging in a group activity where they don’t have to write anything down or turn anything in, it seems more vague and less purposeful. But if the group is collaborating together to create something together that they will present either to the class or to the teacher, they are more likely to take ownership over the activity and engage more fully. There is a delicate balance, though, to how intensive the tasksheet should be. If it is too thorough, students may become overwhelmed and focus more on getting it done rather than learning through working through the tasksheet. For this reason, the tasksheet should be a fairly simple guide through the activity. I also think the tasksheet should, in general, not receive a grade so that students are not motivated to just find the answers to fill in to get a good grade, but can focus on the purpose of the learning activity.

7.      Respond to Synthesis Activity 9 p.129.
Swap stories with a colleague about mathematics homework assignments that include examples of efficient, productive assignments, as well as examples of useless or counterproductive assignments.

The least beneficial homework assignments I have ever seen are those that ask students to solve certain math problems where the answer correlates to a letter, where the purpose is to spell out the answer to some riddle. Or, similarly, the answers to the math problems correlate to some word on a crossword puzzle, or something of that sort. In my experience tutoring students with these kind of tasksheets, students are totally focused on finding the answer to the riddle or puzzle and don’t pay as much attention to the math part of the worksheet. It is also my experience that it is possible to solve the word puzzle without fully engaging in the math, so many times I have seen students quit solving the math problems halfway through the worksheet, fill in the remaining letters (because they have solved the riddle), and think they have successfully completed the assignment.           An example of a productive assignment I have seen dealt with finding the surface area of three-dimensional objects. The students were asked to find various 3-D objects from home and discover relationships in processes to find surface area for different objects. I remember in particular that one of my students was asked to find a toilet paper roll and measure the surface area using given formula, and then they were asked to cut the toilet paper roll and discover that the formula is just the area of the two circles added to the area of a rectangle. This was an inquiry-based learning activity that they were able to do from home, that was successful in getting them to discover a relationship but wasn’t too time-consuming or difficult to accomplish on their own.

8.      Read Chapter 4

9.    Complete Synthesis Activity #1 p.169

Look over a mathematics textbook currently being used in a middle, junior high, or high school. Select two topics from different chapters of the text. Examine how each topic is presented. Categorize aspects of each topic as to whether they originated as a concept construction, discovery of a relationship, or invention. For example, here is how I labeled some aspects of two topics from the prealgebra text whose table of contents appears in Exhibit 1.4.

                  Topic: Proportional Relationships

-Constant of proportionality (unit rate) is a concept construction of a comparison of things that exist that can be measured. The terms “constant of proportionality” and “unit rate” are invented conventions.

-Proportional relationships and comparing two ratios are relationships.

-Comparing two ratios to find an unknown amount of something, and using cross multiplication to solve for the unknown are inventions of algorithms.

                  Topic: Graphing on a Number Line

-Distance and direction are concepts that exists in nature.

-The terms “absolute value”, “direction”, “positive”, and “negative” are invented conventions.

-The number line is an invention to illustrate a concept.

10.  Read Synthesis Activity #2 pp. 169-170 and post your reactions.

I very much enjoyed reading these two excerpts. Both of these authors communicated a disconnect between real mathematics and “schoolmath,” and both argued the importance of educating kids in the art of real mathematics. The problem is how to do that. The most obvious way to me to teach students real mathematics is to use inquiry-based teaching methods that lead students to discover mathematical concepts. The order of discoveries should also be guided by the teacher in the natural progression of discovery of mathematical concepts throughout history. This would present students with an organic encounter of real mathematics.

11.    Prepare a Unit Draft -- this is the unit you agreed on with your group. Short draft, you will compare and talk about it in class. Bring a copy – for more details see Synthesis Activity #6 p.171  

Click here to view Draft #1 of Unit Plan (this and subsequent drafts can be found under "Unit Plan" in "Cognition, Instructional Strategies, and Planning" section).

12.    Engage in Transitional Activity 1 p.171

What strategies should teachers employ to lead their students to construct mathematical concepts for themselves? How should lessons for construct-a-concept objectives be designed? What are some strategies for monitoring students’ progress during lessons for construct-a-concept objectives?

Teachers should employ creative, hands-on activities for their students to participate in that will lead the students to construct mathematical concepts for themselves. For example, the activity that our class did on the first day to work with grid of nails on a wood board and use rubber bands to find all of the possibly different sizes of squares. This was an engaging activity that allowed us to discover through experimentation how to find the different sizes of squares, which solidified our conceptual understanding.

Lessons for construct-a-concept objectives should be designed similarly to the example described above. First, students should participate in an engaging activity that leads to discovery and motivates direct instruction. Second, the teacher should employ direct instruction, ensuring that each student understood the concept involved in the hands-on activity and introducing algorithms.

One strategy for monitoring students’ progress during lessons for construct-a-concept objectives is to walk around the room while students are engaging in the discovery activity and listen to discussions or look at written papers, and guide students who may be struggling. It is also a good idea, as I talked about earlier in this week’s journal, to have a simple tasksheet to act as a guide and to give teachers an idea of individuals’ understanding.

13.   Interview with a teacher (Activity 1.4)

Kaitlyn: For how many years have you been working as a professional mathematics teacher?

Mrs. Lindsay: This is my fifth year, so I have been at West High for four and half years.

Kaitlyn: How, if at all, do you teach differently now than you did during your very first year as a mathematics teacher?

Mrs. Lindsay: I collaborate a lot more with other teachers than I did when I first started teaching. This has changed and improved my teaching style quite a bit. I have also integrated more investigative learning than I did when I started. Something else I’ve discovered is that projects that students take home take a huge amount of time to grade and students don’t really get enough out of them for me to invest that much time in grading them. I would rather spend my time more efficiently in investigative learning activities in class. I also used to bring in a lot of guest speakers to my College Prep. Math class to give students a chance to hear from people working in various professions that are using mathematics in their careers, but I haven’t done that once this year. It is so time consuming to plan the speakers, and I just haven’t had the time this year.

Kaitlyn: Do you feel like you spend less hours working now than you did when you started?

Mrs. Lindsay: A little bit less... but it’s not that different. I have taught many different courses each year I’ve been here, so I always have to spend a lot of time formulating new lesson plans. It’s hard for me to work from home, though, so I always get it done here.

Kaitlyn: How late do you usually stay after school to get things done?

Mrs. Lindsay: Well I used to come in at 6:30 every morning, now I usually come around 7:30. I generally leave before 5 every day.

Kaitlyn: Reflect back on that first year and talk about your experiences. What were the major surprises - both the satisfying ones and the disappointing ones?

Mrs. Lindsay: One positive surprise that I experienced my first year is that there were actually quite a few students who worked extremely hard in my class despite their low-income circumstances and bad family situations. I was expecting these students to be the most difficult and least driven, but many of them were very dedicated students. One difficult surprise I experienced was how difficult it is to manage all of the different classes I teach. It can be really challenging when you’re teaching a lot of different subjects to create engaging lesson plans for all of them, and at this school we have red and black days, so I will often get confused between my classes! Another disappointing surprise was that there were some students that I just didn’t have the capacity to help. I came in with the attitude that I would get every students excited about math, and at least engaged in the material, but I learned that sometimes there’s only so much you can do to make kids interested.

Kaitlyn: How is your professional life different today than it was then?

Mrs. Lindsay: My first couple of years I decided I wouldn’t take part in any extracurricular activities at all because I’ve heard that it’s really easy to burn out, and I didn’t want that to happen. Now, though, I participate in more extracurricular activities; mainly I lead a competing math team. It becomes easier to be involved once you get to know the administration better and you’ve had a couple of years of teaching experience.

Kaitlyn: I am in the process of learning how to be a professional mathematics teacher. What advice do you have for me at this stage of my career?

Mrs. Lindsay: This is a hard question because there are so many things that you just can’t learn until you get thrown into the classroom, but something I would strongly advise is to absorb everything you can from your student teaching. While you’re in student teaching, learn how to deal with discipline and classroom management problems. Then incorporate these skills that you’ve learned into your own classrooms on the first day. Start the year off right. Your main battle will be managing the classroom, but once you’ve learned those skills teaching becomes much easier. Another piece of advice I have for you is to teach them something everyday. Your first couple of years as a teacher will be immensely challenging and some days you will probably feel so overwhelmed and unprepared that you’re not sure what to do, but no matter what state you’re in or how you’re feeling, just make it a goal to teach them something each day.