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 9th 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 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.
