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

Reading Journal Week 7

1.       Read Chapter 6

2.       Engage in Activity 6.2.  You are highly encouraged to share with a colleague, but it is not necessary.

Information I’ll never forget, but don’t specifically recall learning:

            My older sister’s name is “Jarah.” I don’t remember learning her name; I have always known it and I’ll never forget it. I am sure that I was exposed to my family continually referring to her and saying “Jarah,” so at a young age I drew a connection between my sister and her name. I would imagine my family also continually urged me “say ‘Jarah’” and pointed to my sister. The repetition of saying her name myself and hearing others say it has caused me to “overlearn” her name.

Information I discovered for myself and remember well (am unlikely to forget):

            I remember discovering that when my closet doors are open, my bedroom gets extremely cold in the winter. I had a specific problem to solve (my room was frequently cold) and figured out that the attic connected to my closet, which led me to discover the connection between the temperature of my room and the openness of my closet. Similar to simple-knowledge stimuli, feeling cold in my room stimulates my mind to check if my closet doors are open.

Information I can recall being taught and remember well (am unlikely to forget):

            I remember my mom teaching me my phone number and home address when I was young, maybe in kindergarten. I remember her having me repeat them to her often in case I ever got lost or needed to call her. I don’t think I will ever forget either of these because I’ve used them so often. I’ve used the phone number for years to call home, and I use the address on letters, or applications to things, or in giving people directions to my house. The repetition has made me remember.

Information I recall learning but have difficulty remembering:

            I recall memorizing the essential components of the immune system (T-cells, Killer-T-cells, etc.) and the general process by which the immune system protects our bodies in 9^th grade Biology. Because I have only used this information a handful of times in seven years, I can’t remember it. I would know how to find this information though.

Description of a process by which people acquire and retain information:

            The process by which people acquire and retain information begins with people learning a piece of information either by someone informing them or through them discovering the information for themselves. If someone taught them the information, it is very helpful for them to draw a connection between this information and something they already know (they can be taught this connection). And for something to be solidified in a person’s memory, it must initially be repeated and “overlearned,” and usually that person also must periodically be exposed to the information to retain it.

3.       In your own words, summarize the five stages in which students receive and retain a simple-knowledge objective.

Stage 1: Exposition

            Students are introduced to the information, which requires simple exposure.

Stage 2: Explication

            The teacher concisely explains the stimulus and the information and how they are connected.

Stage 3: Mnemonics

            The teacher provides a mnemonic device for the information, which is something that helps students remember the information. It is particularly helpful to provide a mnemonic device that links the information to something the students are already familiar with; otherwise students are likely to forget both the mnemonic device and the information. This step is optional but beneficial.

Stage 4: Monitoring and Feedback

            The teacher monitors how well students have learned the information. Students’ errors are corrected and correct understanding is reiterated.

Stage 5: Overlearning

            Students are given opportunities for repetition of the information. Through this repetitive practice, their retention of the information is solidified.

4.        In Case 6.7, Ms. Ray creates Exhibit 6.3 and 6.4.  Which does she decide to use and why?  As a teacher of young people, which do you prefer and why?

           

            Ms. Ray decides to use Exhibit 6.4, which describes the six trigonometric functions in the following terms: the “opposite” side of the angle, the “adjacent” side to the angle, and the “hypotenuse”. She decides to use this exhibit because she hasn’t yet introduced right triangles in relation to the coordinate plane and circles.

            I think I would start with Exhibit 6.3, even though it may initially take some extra time to introduce the connection between right triangles and the coordinate plane/circles. I chose this exhibit because I have tutored many students who struggle drawing the connection between right triangles and the coordinate plane and circles, and who view the “adjacent, opposite, hypotenuse” terminology as separate from the “x, y, r” terminology. I think it would help solidify this connection if students initially learned this simple knowledge in terms of x, y, and r, because they would then more naturally view triangles in relation to the coordinate plane and circles. If the teacher wanted to, it would be easy to teach students the “adjacent, opposite, hypotenuse” terminology once students have grasped the “x, y, r,” but teaching “x, y, r” first would better serve students’ conceptual understanding of right triangles and trigonometric functions.

5.       How is a construct-a-concept objective different from a simple-knowledge objective?  How are they similar?  How is a discover-a-relationship objective different from an algorithmic-skill objective?  How are they similar?  How are simple-knowledge and algorithmic-skill objectives different?

            The goal of a construct-a-concept lesson is for students to draw connections and recognize distinctions between examples and non-examples of a concept, whereas the goal of a simple-knowledge lesson is that students will remember a specific piece of information when presented with its corresponding stimulus. Construct-a-concept objectives require students to use inductive reasoning, while simple-knowledge objectives should not require any reasoning but only stating information from memory. They are similar in that both may involve concepts. The information that students are to remember in a simple-knowledge objective may be the definition of a concept, and it is likely that construct-a-concept objectives will be followed up with simple-knowledge objectives so that students remember the concept and can build on it.

            The goal of a discover-a-relationship lesson is for students to find for themselves that a certain relationship exists or why it exists. The goal of an algorithmic-skill objective is for students to remember and be able to carry out a certain succession of steps that make up a procedure. Discover-a-relationship objectives require students to use inductive reasoning, whereas algorithmic-skill objectives don’t require reasoning but only memorizing and executing a process. These two objectives are similar in that many times a relationship is introduced, and later the algorithm connected to the relationship is introduced. One of the objectives on pg. 167 is to discover the following relationship: A=(P+(r/t))^(kn). This is a general relationship, but when solving specific problems dealing with relationships, students will use algorithmic skills to solve the equation for an unknown. Relationship form the foundation for algorithms.

            Simple-knowledge objectives only require students to remember certain information, while algorithmic-skill objectives require students to remember information (a series of steps) and be able to execute those steps.

6.       In your own words, and perhaps using your own examples, summarize four types of algorithms commonly used in mathematics (see p. 217).

Arithmetic computations: Performing a process or using a method that employs simple arithmetic operations. An example is adding 891.4578 and 17¼.

Re-forming symbolic expressions: Manipulating expressions using the associative, commutative, and distributive properties along with basic operations to form equivalent expressions. For example, using distribution and addition we can manipulate f(x) = 4(3x+2) + 7 to get it into slope-intercept form: f(x) = 12x +15.

Translating statements of relationships: Simplifying equations to solve for unknowns (or to rewrite the equation in a way that shows the same relationship in a different way). For example, x^2 + 8x + 7=0 can be factored and then solved: (x + 7)(x + 1)=0  => x=-7, x=-1.

Measuring: Physically finding a quantity of something. An example is measuring the side lengths of a triangle.

7.       In your own words, summarize the seven stages of an algorithmic-skill lesson.  

Stage 1: Explanation of the Purpose of the Algorithm

            Because mathematical relationships are the foundation of algorithms, you should start an algorithmic-skill lesson by introducing the relationship connected with the algorithm (note: you may have already done a discover-a-relationship lesson to introduce the algorithm).

Stage 2: Explanation and Practice Estimating Outcomes

            In this stage, students will estimate the outcome of their algorithm before performing it (or after to check). The purpose of this is to ensure that students understand what they’re doing before they begin executing the steps of the algorithm. This is an extremely important step, as it connects the students’ algorithmic skills to the foundational mathematical ideas behind the algorithm.

Stage 3: General Overview of the Process

            In this stage, the teacher will give the students a general overview of the algorithm.

Stage 4: Step-by-Step Explanation of the Algorithm

            In this stage, the teacher will walk through each step of the algorithm with the students. For each step, the teacher will explain it and then allow the students’ to try the step themselves. This step may involve checking for and correcting errors at the end of the algorithm.

Stage 5: Trial Test Execution of the Algorithm

            In this stage, the teacher will assign exercises for the students to practice with. The teacher may choose to assign exercises that contain opportunities for common mistakes so that the comprehension and execution of the algorithm may be more thoroughly analyzed.

Stage 6: Error-Pattern Analysis and Correction

            The teacher uses the assigned exercises from Stage 5 to gauge how well his/her students understood and performed each part of the algorithm. Considering the results of the analysis, the teacher will provide more exercises that address any error-patterns.

Stage 7: Overlearning

            Students are given repetitive practice exercises so that they will better remember the algorithm. The teacher will continually analyze how well his/her students understand and can perform the algorithm.

8.       How is a miniexperiment for an algorithmic-skill objective different from a miniexperiement for a simple-knowledge objective?

            Miniexperiments for simple-knowledge objectives simply assess if students can respond with the right information when presented with specific stimuli. On the other hand, miniexperiments for algorithmic-skill objectives assess more than a final outcome. Each step of an algorithm serves as a stimulus for the next step of the algorithm. For this reason, the student’s response for each step should be assessed and analyzed. A student may not produce the correct final answer, but also may have only missed one step in the algorithm and performed the rest correctly. It is important to assess which steps your students are missing and correct them, especially if there is an error-pattern.

9.       Design a lesson and mini-experiment for an Algorithmic Skill objective from the Unit Draft, and modify the Unit Plan if there is a need. For more details consult Activity 6.6 p. 223, Activity 6.8 p. 226, Synthesis Activity #3 p.228.

Click here for the algorithmic-skill lesson plan.
Click here for the algorithmic-skill tasksheet.

Click here for the algorithmic-skill miniexperiment (also in "Assessment" section).
*Also in Cognition/Instructional Strategies/Planning Section.

10.    Engage in Activity 6.7.  

            It seems that John is missing the step in the algorithm where you manipulate the equation so that one side is equal to zero. He is separating the two possibilities of “roots,” but they are not roots because one side is not equal to zero before he factors the polynomial.

            It seems that Jane thinks that whenever there are exponents and variables multiplied in any way, all of the exponents should be added, and each different variable should be raised to the power of that sum.

            Pat thinks that squaring the numerator and denominator of a fraction produces an equivalent fraction to the original, when that is really the same as squaring the entire fraction. He also displays inconsistency in his work, because in questions one and two, he tries to get a common denominator and appears to understand how to add fractions. However, in questions three and four, it appears as though he is trying to get the variables outside of the square roots, and then it looks like he tries to “cross-multiply” across addition and subtraction signs.

            Pete’s work shows that his understanding of calculating perimeters is to add all of the labeled sides. He does not seem to realize that there are unlabeled sides that contribute to the sum.

11.    Engage in Transitional Activity #1 on p. 228. 

            What mathematical shorthand notations (e.g., those listed by Exhibit 4.5) should students learn to comprehend and use in their own communications?

            This depends on the level of the student. There are some examples of shorthand notations in Exhibit 4.5 that I didn’t know until I came to college, and I don’t necessarily think that hindered my ability to do math. In my time tutoring, though, I have seen some students who don’t know the meaning of some very basic shorthand notation, like square roots, cube roots, etc. I have also had a lot of students who are very confused by function notation, such as f(x) and F:A -> B such that A,B ⊂ , D={Reals}, etc. I also think the notation of the number sets is not emphasized very much, and my students have struggled with it:. Set theory notation is something I’ve seen many students get confused about as well.  There are too many important notations to write down. I think if common shorthand notation exists, that notation should almost always (unless it is particularly confusing and not very useful) be introduced to students so that they can learn communicate in many different mathematical environments. I think it is important to allow students to discover concepts and relationships, etc., before introducing notation so that they are not caught up with or confused by the shorthand.