Reading Journal Week 8

1.         Read Chapters 7 and 8

2.         Engage in Activity 7.1.

“A television personality made the following comments about NBA basketball player John Stockton:

         ‘Who is more unique than a John Stockton? A 40-year-old point guard who gives a 110% every night! Most 40-year-olds are sitting in their rocking chairs with a cold one right now – not out on the court setting countless back screens on guys twice their sizes and half their ages. In spite of the unlimited punishment, he plays maximum minutes, never complaining about the bumps and bruises. He’s one of the greatest players in the history of the game.’

         Examine the quote. Discuss how the commentator’s use of the underlined words differ from how those words are used in mathematics.”

                       

·      In math, if something is “unique” that means there is one and only one of that specific thing. Something is either unique, or it is not unique. For this reason, the commentator’s description of “more unique” does not make sense in mathematical terms.

·      Viewed from a mathematical perspective, “a” John Stockton would mean one John Stockton out of many, but there is only one specific John Stockton that the commentator is referring to.

·      Giving “110%” doesn’t make sense because one only has 100% of oneself to give.

·      Describing John Stockton as giving 110% “every” night doesn’t make sense because he doesn’t play in games every night.

·      In math, “most” means greater than or equal to half of something. It’s unrealistic to believe that greater than or equal to half of the number of 40-year-olds in the world are sitting in rocking chairs with a cold one right at the moment this statement was made.

·      “Countless” refers to infinitely many of something in mathematics; it is impossible that John Stockton set infinitely many back screens.

·      If John Stockton is around 5 or 6 feet, guys “twice” his size would be 10 or 12 feet tall, and it is highly improbable (impossible?) that he is playing against other players of this height.

·      John Stockton plays against other players who are greater than “half” of his age.

·      It is impossibly for John Stockton to face “unlimited” (infinite) punishment.

·      John Stockton does not always play “maximum” minutes because he does not always play every minute of every game.

·      It’s improbable that John Stockton has never complained.

·      You cannot have “one of the greatest” in mathematics, but can only have one “greatest.

3.         Describe the stages of comprehension and communication lessons on an example appropriate to your unit.

Lesson Objective: The student incorporates the following word into their vocabulary relative to equivalent equations: “like-terms”
(comprehension and communication).

Stage 1: The Message Is Sent to the Students

For example, the definition of “like terms” (terms that contain the same variables raised to the same power; only the coefficients are different) is stated verbally and in writing. The students would have already constructed this concept in a lesson involving algebra tiles, and it would have been solidified in a lesson involving different coins.

Stage 2: The Message Is Rephrased and Explained

For example, “Like terms are the parts of an expression that have the exact same single variable or combination of variables; only the number of those variables can be different. For example, similar to our lesson involving coins, the like terms in the expression 6d + 2q + 3n + 2d are 6d and 2d, because they are both dimes, but the first describes 6 dimes and the second describes 2 dimes (they both contain the same variable and only that variable)”.

Stage 3: Students Are Questioned about Specifics in the Message

           For example, “Are xy² and y²x like terms?”

Stage 4: Students Are Provided with Feedback on Their Responses to Questions Raised in Stage 3

For example, “I agree that they are like terms! Even though the look different, we can refer to our definition and see that these terms agree with the definition; they both contain the same variables raised to the same power.

4.         Engage in Synthesis Activity 1 p. 252

A.     Which one of the following would NOT be the type of mathematical content specified by a comprehension-an-communication objective?

c) Relationship  (pg. 239).

B.     Lessons for comprehension-an-communication objectives use                           .

a) both direct and inquiry instructional strategies (pg. 244).

C.     An acceptable form for presenting the proof of a theorem is an example of a                                     .

c) convention of the Language of Mathematics

D.    Comprehension-and-communication lessons are most appropriate for incorporation into which one of the following stages of a discover-a-relationship lesson?

c) third (pg. 245-246).

5.         Design a lesson and mini-experiment for an Application objective from the Unit Draft, and modify the Unit Plan if there is a need. For more details consult Activity 8.1 p. 260, Activity 8.3 p. 263, Synthesis Activity #4 p.252.

Click here for the Application lesson plan.

Click here for the Application mini-experiment (also in "Assessment" section).
*Also in "Cognition/Instructional Strategies/Planning" section

6.         In your journal describe the stages of application lesson.

Stage 1: Initial Problem Confrontation and Analysis:

In this stage, the teacher will present two problems to his/her students that are very similar, but one of the problems requires the knowledge of the mathematical content in the objective, and the other does not. Students are asked to reason which one involved the mathematical content and which doesn’t. The teacher will lead them in a discussion where students explain their answer and the reasoning that led them there.

Stage 2: Subsequent Problem Confrontation and Analysis:

           During this stage, the teacher will present the students with more pairs of problems similar to the ones from stage 1 in that one in each pair will involve mathematical content from the objective and one will not. The subject in all of the pairs will be different so that students can grow in their ability to problem solve when faced with any situation (deductive reasoning).

Stage 3: Rule Articulation:

           In this stage, students will create general rules for how to decide if a given problem involves the mathematical content from the unit objective.

Stage 4: Extension Into Subsequent Lessons:

           This stage involves integrating problems from this application lesson into future lessons in the course. This allows mathematical content to be continually reviewed and integrated into new content. It also allows further opportunity for students to practice deductively reasoning.

7.         Following Ms. Ferney’s example incorporate creative thinking objectives into some of the lessons you have already created.

Creative Thinking Objectives:

·      Construct-a-Concept Lesson: To bring some creativity into my construct-a-concept lesson, I would have students pair up. I would give each pair of students a unique pair of equivalent expressions. I would ask them to use the expressions they’ve been given to describe/compare/relate two things in their everyday lives. I would then ask them to use the two expressions to describe/compare/relate themselves to each other.

·      Discover-a-Relationship Lesson: For my discover-a-relationship lesson, I would incorporate creative thinking by having students answer the following questions individually:

-How is our balance-scale activity similar to grocery shopping?

-Write a paragraph describing a connection between your family or friends and our balance-scale activity.

-Write a question you have about the balance-scale activity.

-Come up with an original name for our balance-scale activity and explain why you chose the name you did (be creative!).

8.         Engage in Activity 8.6 p.273 – bring to class so you can exchange with a colleague.

You designed a lesson for a construct-a-concept objective and a lesson for a discover-a-relationship objective. Retrieve your plan for one of those lessons. Now, design a lesson for an affective objective that you can integrate into that lesson.

Exchange your integrated lesson plans with that of a colleague who is also engaging in Activity 8.6. Critique one another’s work.

Objective: The student distinguishes between examples and non-examples of equivalent expressions, explaining the defining attributes and formulating a definition (construct-a-concept).

Objective: The student recognizes the value in being able to identify equivalent expressions (appreciation).

Affective Objective Lesson: To integrate the affective objective into my construct-a-concept lesson, I would present the class with the following problem and bring manipulatives (small pieces of paper with the picture of the item and its price) so that students could move around the objects to find a group equal to $15:

“You have a grocery list and $15 dollars to spend at the grocery store. The grocery list is: milk, eggs, apples, bananas, cookies, and carrots. Milk costs $2, eggs cost $.50, apples cost $1 each, bananas cost $.50 each, cookies cost $3 for a box, and carrots cost $2 for a bag. You want to spend all of your money. Find a combination of items (how many of each!) you could purchase for your $15.”

I would also take into account the interests of students in my classroom and I might come up with similar problems that relate more specifically to different groups in the classroom. The goal of this problem is for students to become engaged and interested in identifying an equivalent expression. After the activity was over, I would conduct a discussion and talk about the relation between the groups of items and their algebraic representations.

9.         Complete Synthesis Activity #1 p.275

A.     With which one of the following tasks do students usually have to deal when solving real-life problems, but not when solving textbook word problems?

d) Distinguish between relevant and irrelevant data

B.     Which one of the following do students usually have to determine in order to solve textbook work problems?

a) The variable to be solved as indicated by the question given in the problem

C.     Learning activities for which one of the following types of objectives are LEAST likely to be effectively integrated into lessons for other types of objectives?

c) Algorithmic skill

D.    Which one of the following strategies is LEAST likely to enhance students’ achievement of an appreciation objective?

b) The teacher tells the students how important understanding the content will be for them.

E.     Student perplexity is a critical ingredient in lessons for all BUT which one of the following types of objectives?

c) Simple knowledge

F.     Lessons for application objectives require                         .          

b) deductive-learning activities

G.     Synectics is used in learning activities for what type of objective?

d) Creative thinking