SAMPLES

Analytic Journal Writing - This is provided here for you to use in practising your writing.
First 2 lines of the journal entry - What is the focus of your journal entry -> could be statement (restatement of the journal prompt), or a question that provokes the reader’s thinking a. Describe one idea in your own words that comes from the journal article (big idea) b. Substantiate idea with APA reference from journal article c. APA reference - either direct (quoted with page number (p.2 or pp. 2-3) or indirect (paraphrased) description of the author’s idea -- e.g., Davis (2005) d. ** math example to justify the idea (or reference to your own experience in learning math) ** e. unpack the meaning (to you) of the APA referenced quote a. make an inference that links to journal article idea to mathematics teaching and learning that you will plan for in your lesson b. substantiate connection between the APA reference and your math example
 * Topic sentence**
 * Describe Idea**
 * Infer**
 * Concluding Statement …**

Respond to this prompt keeping the above ideas in mind.
1. The Japanese lessons are described as “teaching through problem solving.” What does this mean? 2. What impact will the “teaching through problem solving” goal have on your lesson design?

Here is a sample of a former student's Rationale
New research in the field of education calls for a drastic change in teaching mathematics to the intermediate pupils. Ball, Hill and Bass (2008) explain that we are simply failing to reach reasonable standards of mathematical proficiency with most of our students who are the next generation of adults, and the fact that some of them will become educators worries them. These researchers argue that the quality of mathematics teaching depends on teachers’ sound mathematical understanding and skill including principled knowledge of algorithms and mathematical reasoning. Such a background enables teachers to analyze students’ errors and explain the basis for algorithms in words that the youth can understand. Ball, Hill and Bass (2008) elaborate on this idea by stating that the teacher has to think from the learner’s perspective, and to consider what it takes to understand a mathematical idea for someone seeing it for the first time.

Several ways of improving math instruction have been suggested by the new research. They vary tremendously from the traditional view of teaching and learning mathematics. Van de Walle and Folk (2008) explain that for many individuals, mathematics has been a collection of rules to be mastered, arithmetic computations, mysterious algebraic equations, and geometric proofs. I can relate this belief to my own experiences of learning mathematics because I remember acquiring the knowledge exactly as they describe it. Furthermore, Battista (1999) emphasizes that traditional methods of teaching mathematics not only are ineffective but also seriously inhibit the growth of students’ mathematical reasoning and problem solving skills. Learning mathematics in a traditional way left me temporarily with gaps in my understanding of concepts so I was not able to see relationships between them. It took me a while to rectify this by discovering these connections myself in later years of my academic education. For this reason, I support the argument of Ball, Hill and Bass (2008) that it is important that teachers know mathematics for teaching, meaning they must be able to explain, listen to students, examine students’ work, choose useful models or examples for introducing new concept, and must also have a specialized fluency with mathematical language.

The reform of mathematics education began in the mid-1980 in response to the documented failure of traditional methods of teaching mathematics. I want to be part of this process by committing to my professional development, part of which is conducting this teacher inquiry. I agree with Cohen and Ball (2001) that education will improve if teachers will learn new knowledge and skills or use more effectively what they already know and can do. My curiosity about intermediate mathematics teaching and learning is rooted in the topic of algebra. As an educational assistant, I witnessed many students struggling with solving equations. They lacked conceptual understanding of the rules for solving equations with one variable. For this reason, I chose to research the knowledge that grade eight students construct in lessons on algebraic expressions and solving equations with one variable. My secondary question supplements the primary one, as I attempt to evaluate the use of the three-part problem-based lesson design, manipulatives and graphical representations for helping students built an understanding of algebraic concepts, especially equality.

I expect that the teacher inquiry will help me understand the ways in which the intermediate students learn best, and equip me with the appropriate strategies so that I may guide them in building a solid understanding of algebraic concepts. Since I learned algebra differently, I also look forward to discovering alternative strategies for transitioning students from arithmetic to algebra as the concepts of variables and algebraic expressions need to be reviewed in the first lesson that I design for this project. Furthermore, I need to familiarize myself with alternative methods for solving equations with one variable so that I will be able to anticipate students’ solutions more readily, and make explicit connections between them during class discussions. --

Here is a sample of a part of a former student's Annotated Bibliography
Ainslie E., Erdman W., Gilfoy D., Huyck H., Lax S., McCudden B., Ryan K, Seijer J., Szeto S. & Webb M., (2004). Mathematics 7: Making Connections. Toronto, Canada: McGraw-Hill Ryerson. This grade seven textbook supports a three-part lesson design. Each chapter starts with a Chapter Problem that connects math and a student’s world. Questions related to the problem are posed throughout the chapter. The first part of each lesson helps to find answers to the key question. An activity is designed to aid in building a student’s understanding of the new concept and lead toward answers to the key question. The examples and solutions demonstrate how to use the concept. A summary of the main new concepts is given in the Key Ideas box. Lastly, the questions for practice, application of understanding and extension of learning constitute the third part of the lesson. The ideas from this resource helped me develop my first hands-on lesson on variable and expressions. The textbook illustrated how manipulatives such as a cup and counters can be used to represent variables and algebraic expressions. As a kinesthetic and visual learner, I appreciated the examples and felt that it would be beneficial to implement similar exercises in Before task of the first lesson to aid students in recalling or relearning the knowledge of these concepts. The summary of Key Ideas demonstrated what should be highlighted at the end of the lesson, so I used this information as a guide in designing my own lessons.

Friedlander A. & Tabach M., (2009, April). The Money Context. Mathematics Teaching in The Middle School, 14(8), 458.  The Money Context article illustrates how learners are expected to base the meaning of algebraic expressions in the context of savings rather than follow rules of basic algebraic operations. In the process, students learned the meaning of constants and variables using verbal, numerical, graphical, and symbolic representations, and by comparing the changes that took place. Furthermore, learners’ remarks and questions showed that they were interested and actively involved in the situation at hand. The key ideas from this article encouraged me to plan the problems for my lessons that relate to student’s life experiences, specifically money. I have also started paying more attention to the fact that solutions may be represented in many different ways, which allows for a deeper learning. Hence, in my lessons I encourage students to solve problems in different ways using various representations. The information from this article helped me establish criteria for choosing sample solutions for my Bansho and creating mathematical annotations to show students explicit connections between solutions. In closing, this resource provides me with a supportive data for making arguments that students would tremendously benefit from solving problems that are rooted in a real life context.