Kendall Brown

CAPSTONE

Dr. Santarone

November 21st

Discovering Discovery Learning

I. Introduction

Education is in the midst of great change. The role of the teacher, the common core, and

the way the content is being taught to students are just a few examples of areas that are

undergoing this change. Specifically, the concept of discovery learning is making its way to the

classroom. I was first introduced to the idea of discovery learning in my Math Education courses.

When I first heard this, it was just a new way to allow students to understand topics. It was not

until my placement for one of my classes, Project Focus, that I saw discovery learning in action.

I was interested in how the typical role of a teacher shifted, and how the cognitive demand

increased for the students. The students became more of the center. I remember when I was

shown this type of learning; I was frustrated with the teacher breaking what I considered normal

which made it difficult for me to come to a conclusion by my own opinion. I have also heard

many secondary school teachers’ thoughts on discovery learning which for the most part have

been more on the negative side. It seems necessary to explore further the effect that discovery

learning can have. My motive for doing this study comes from personal experiences which I

have previously described.

 II.RESEARCH

According to google, to discover something implies that we are bringing to light an idea,

uncovering something we have been searching for, or to simply come across a new idea.

Combining this and education leads to discovery learning. Discovery learning is a method used

in the classroom that encompasses many positive techniques. Experience, communication, and

interaction are just a few aspects of discovery learning. This type of learning allows students to

take responsibility for their own learning and understanding. It breaks the traditional norms

where the teacher is the major role, and allows the students to engage in a way they have not

before (Meece, 2002). In 2006, an article was published that believed this type of learning had a

negative influence in the classroom. Kirschner, Sweller, and Clark state, “The goal of this article

is to suggest that based on our current knowledge of human cognitive architecture, minimally

guided instruction is likely to be ineffective”(Clark, 2006, p.3). After gaining more

understanding on discovery learning and realizing how controversial this subject is, I became

more interested in exploring this concept.

“Discovery learning can have a lasting impact because learners not only experience the

content, they also improve.”(Rillero, 2013,pg.2). After multiple experiences in a classroom and

being a student myself, I know what it mean to strictly learn material for a test, and then

immediately forget about it. Experiencing content, instead of being handed content, leads to a

certain level of improvement which Weimer believes is vital. Along with experience and

improvement, students become more involved and engaged. Weimer states,

Constructivism prescribes a whole new level of student involvement with content. It

makes content much more the means to knowledge than the end of it. It and the empirical
 way of psychology change the function of content so it is less about covering it and more

about using it to develop unique and individual ways of understanding (Weimer, 2002,

p.12).

So often, it is about the content. Weimer (2002) believes that it is not about just the content, but

rather about gaining understanding.

Learner – Centered Teaching

Discovery learning falls into the category of Learner- Centered Teaching, also known as

student-centered teaching. In a traditional classroom, the teacher has most of the power and the

students do more listening. The teaching style is more lecture based. When it comes to Learner-

Centered Teaching, there are two perspectives/roles that need to be examined: the student

perspective/role and the teacher/role perspective.

The teacher’s perspective\role

It is usually assumed that when the students become the center, the role of the teacher

decreases to nothing. However, this is not true. For successful discovery learning, the teacher has

to be creative along with understand how to develop lesson plans in which the students can learn

from interacting and experiencing. Learner- Centered teaching should not be used as a means of

survival or an escape from what is considered traditional teaching, but rather a commitment to

develop an experience to remember. From a constructivist’s perspective, the teacher has to be

creative so that he/she can invent lesson plans that will further the students’ conceptual

understanding (Reinhardt, 2000, p.54). The teacher must let the students answer their own

questions and challenge each student. Reinhart shares the following statement,

When I was in front of the class demonstrating and explaining, I was learning a great

deal, but many of my students were not! Eventually, I concluded that if my students were
 to ever really learn mathematics, they would have to do the explaining, and I, the

listening. My definition of a good teacher has since changed from ‘one who explains

things so well that students understand’ to ‘one who gets students to explain things so

well that they can be understood’(Reinhardt, 2000, p.54).

It is important to note that the realization this teacher made is vital to student learning. This

becomes its own assessment: can a student explain the material? , can a student answer another

student’s question? Is the teacher seeing the students interact in a way that expresses the students

understand of what is going on? So, from the teacher perspective in a Learner-Centered

classroom the teacher is not the dominant figure. In fact there is a balance of power between the

students and the teacher. Some people have difficulty accepting this balance of power. Weimer

states,

Many who object the ideas of radical pedagogy do so on the ground that if faculty

relinquish control, they abrogate legitimate instructional responsibility. Students, they

say, end up running the class and teaching themselves, leaving the teacher no viable role

in the educational process. It is true that this educational ultimately dispenses with the

teacher. The goal is to equip the students with learning skills so sophisticated that they

can teach themselves (Weimer, 2002, pg. 29).

It is true that this could cause chaos, but this is where the teacher would need to have enforced

classroom policies; basically classroom management becomes another huge responsibility for the

teacher. The teacher must create an environment that allows such activities and experiences to

take place. The teacher plays a major role in the class room, even when the approach is student-

centered. Kilpatrick, Swafford, and Findell agree on the following role of a teacher,

Effective teachers have high expectations for their students, motivate

 them to value learning activities, can interact with students with different

abilities and backgrounds, and can establish communities of learners. A

teacher’s expectations about students and the mathematics they are able to

learn can powerfully influence the tasks the teacher poses for the students,

the questions they are asked, the time they have to respond, and the

encouragement they are given—in other words, their opportunities and

motivation for learning. (Kilpatrick, 2006, pg. 9)

The role of a teacher is far more than lesson planning. Another role is they must determine the

level of thinking required which is known as cognitive demand. This will be further explained in

the student perspective portion. However, it is important to note that this is another role the

teacher must take on to fully commit to partake in discovery learning. Every part of a task must

be analyzed to incorporate cognitive demand.

Student’s perspective/role

The generation of children today has grown up with technology. Nothing against

technology, but it makes things easier. Along with that, I have seen many children that are

handed everything they need/want. This has become a problem in the classroom. The teacher

hands out a list of steps, the students follow them, and then they circle their “answer”. Challenge

is a necessary part of the classroom, and sometimes there tends to be a lack of it. Listen while the

teacher lectures, do your homework, and memorize for the test. In a Learner-Centered classroom,

the student is given responsibility. They get the chance to explore and control what they learn. In

a couple of case studies, students were pleased with the way a Learner-Centered model affected

the classroom. Meece states, “Students reported more positive forms of motivation and greater

academic engagement when they perceived their teachers were using Learner-Centered practices
that involve caring, establishing higher order thinking, honoring students’ voices, and adapting

instruction to individual needs.”( Meese, 2002, p.110). When viewing the student’s role in

Learner-Centered teaching, we begin to introduce different levels of cognitive demand.

According to Stein (2000), cognitive demand is the level of thinking it takes students to

successfully engage in a task. He continues on by explaining what these difference types of

cognitive demand are. Stein states, “Not all mathematical tasks provide the same opportunity for

student learning” (Stein, 2000, p.17) Stein outlines four different types of cognitive demand:

• Memorization – Students with perform multiple different problems anywhere from 10-30

times. These problems are extremely similar. This goes along with the drill method hich

is where a teacher “drills” facts into their students’ mind.

• Procedures without Connections – This is on the same level as memorization. (lower

level of cognitive demand). The students are asked a question where they can repeat steps

that were previously given to them. They do not make connections, but rather perform

certain steps without much understanding.

• Procedures with Connections – This is where we see higher level of cognitive demand.

Students are aware of what is going on, and they are able to make connections to the

bigger picture. There conceptual understanding deepens here. We also see that they are

able to understand various representations.

• Doing Mathematics. This requires fewer problems with more exploration. Students

develop the procedures, and are not immediately given the steps. We can develop a

decent understanding of what this looks like in a classroom by what VandeWalle has to

say, “Doing Mathematics means generating strategies for solving problems, applying
 those approaches, seeing if the lead to solutions, and checking to see if your answers

make sense.”(p.13). There are a few important expectations to remember

o Persistence, effort, concentration

o Collaboration between students

o Listening to other students

o Learning from mistakes

o Making connections

These are all parts of Doing Mathematics that the students will be experiencing (VandeWalle,

2013). Each task has its own purpose. It could be to engage in reasoning or enforce a procedure.

On the contrary, the science behind this matter shows a different side. Clarke (2006) states,

We have known at least since Peterson and Peterson (1959) that almost all information

stored in working memory and not rehearsed is lost within 30 sec and have known at least

since Miller (1956) that the capacity of working memory is limited to only a very small

number of elements (p.76).

It seems that as humans we need some type of memorization to reinforce the knowledge, but

Stein sees memorization as lower cognitive demand. Throughout my study and personal

research, Stein’s cognitive demand levels will be referenced.

Learning-Centered Study- Middle School Edition

There was a previous study (Meese, 2002) that focused on Middle School students and

teachers. It involved 2,200 students and 109 teachers. The purpose of the study was to see the

benefit Learner–Centered teaching can have in the classroom. The students and teachers had to

complete surveys that allowed them to assess these practices. The study included rating scales

where the academic success, level of cognitive demand, classroom performance, and
demographics could be examined. Meese determined three different goals that were based on

prior research. The way Meese (2002) assessed the results were by the following categories:

“Mastery goals, a desire to improve ability; Performance goal, a desire to demonstrate high

ability and performances; and work avoidance, a desire to complete a task with minimum effort”

(p.112). After analyzing the student and teacher surveys, they were able to come to a conclusion.

Meese states,

The analyses revealed several interesting findings for middle school education. Both

teachers’ and students’ ratings of learner-centered practices were correlated with

measures of student motivation and achievement, but patterns of relations were stronger

for student ratings. Only teachers’ reported support for higher order thinking showed a

positive relation to student outcome (p.113).

The students’ mastery goals showed the highest positive relation. Meaning they were able to gain

and improve their overall ability. Meese was able to conclude that the students preferred

Learner-Centered, and after all the students’ voice matters the most.

III. Methods
Throughout my own personal research, I aim to discover how cognitive demand plays a

role into the classroom, specifically mathematics, explore the benefits of teacher-centered and

student- centered approaches, observe discovery learning in a college setting, and consider

student responses to this type of learning. Overall, I want to discover the advantages and

disadvantages of discovery learning, and come to a conclusion of whether or not teachers should

be using this method in the classroom. These results are important to me, since I am a future

educator.
 In fall 2015, I plan to begin my research on discovery learning. I am choosing to explore

Discovery Learning with college students from two pre-calculus classes. In my particular case, I

hope to explore these ideas by allowing students to discover a connection between the triangles

within the unit circle and the graphs of the trigonometric functions. In the end, I will be able to

come to a conclusion of which type of lesson plan gave the students more Mathematical

Proficiency. This is a term used by ___. Mathematical Proficiency is a term that incorporates

five different intertwined aspects with a mutual dependence. The book, Adding It Up, defines

each aspect individually, but also expresses how they should exist together.Kilpatrick states,

Mathematical proficiency, as we see it, has five strands:

• conceptual understanding—comprehension of mathematical concepts, operations, and

relations

• procedural fluency—skill in carrying out procedures

flexibly, accurately, efficiently, and appropriately

• strategic competence—ability to formulate, represent,

and solve mathematical problems • adaptive

reasoning—capacity for logical thought, reflection,

explanation, and justification

• productive disposition—habitual inclination to see mathematics as sensible, useful, and

worthwhile, coupled with a belief in diligence and one’s own efficacy. (Kilpatrick, 2001,

pg.5)
I will be able to examine which branches of Mathematical Proficiency they students are

demonstrating and experiencing in each lesson plan. It is important to note that the NCTM

learning principle states, “Students must learn mathematics with understanding, actively building

new knowledge from experience and prior knowledge.” As a future educator, I am curious to see

which lesson plan most satisfies this principle.

Teacher-Centered Lesson: Plan

For my research, I taught two college pre- calculus classes. One class was taught in a

traditional manner where definitions and formulas are given. This is considered more of a

teacher-centered lecture based class. At the beginning of class, I had the students fill in a unit

circle. They were given a unit circle and 3 minutes. This allowed me to see how much of the unit

circle had memorized. We discussed the unit circle which is a topic that has been previously

discussed in class. Our discussion was a review that reminded the students of the unit circle and

how each x and y coordinate relates to

the trigonometric function. This

allowed the students to work

individually; they made tables of the

coordinates, and then graphed the

coordinate points strictly from the

table (See Appendix A for handout).

The cognitive demand in the first lesson was procedures without connections, because the

students were not making connections to the triangle within the unit circle. For example, sin(β)

=opposite /hypotenuse . We claimed procedures without connections, because the students can
aimlessly go through the steps. There must be a certain level of thinking and understanding. At

the end of the class, the students were given an assessment.

Teacher-Centered Lesson: Observations

After teaching the teacher-centered class, I made a few observations. First off, there was

less students than we expected. We expected 34 and only had 17. At the end, we had to

reconsider the way be analyzed out data. The students were quiet in this class or either on their

phone. I tried to ask probing questions, but I received little response to these questions. At the

end of class, the students were satisfied about their time in class. We will look at their responses

later on in the paper.

Student-Centered Lesson: Planning

The second class and third classes were those who participated in the Discovery Learning

Activity. They were also given the blank unit circle and three minutes. We then discussed the

unit circle. Our review looked different in this class, because they had to be prepared for the

Activity (See Appendix B. for Insrtuctions). This

discussion will incorporate a review, but also a quick

demonstration that will help them see the triangles

with three sides that are measured in distance. For

example, we discussed how we get to the first

! $
coordinate point ( , ). That required us to go over
" "

! $
, horizontally from the x-axis, and up . We also
" "

recalled that the hypotenuse is one. Then the students will participate in an activity where the
goal is for the students to discover the relationship between the graphs and the unit circle, by the

measurements of the triangles. This was considered a student-centered approach, and the

students were to experience discovery learning. The activity allowed the students to develop an

understanding of why the sine and cosine graphs look like they do. After that they were given

spaghetti noodles. The students then measured the base of every triangle with the spaghetti

noodle. They will place the measured noodles on a coordinate plane in the order of the least

angle measure to the greatest. As they do this, they will begin to actually construct the graph of

cosine. Then they repeated this by measuring the height of each triangle. This resulted in the

development of the sine graph. We classified this level of thinking as Doing Mathematics. We

claimed Doing Mathematics, because their thought had to be justified, explained, and

represented. They were able to take a circle, create the triangles, and then use those triangles to

discover the graphs of sine and cosine.

Student-Centered Lesson: Observations

When students first walked into class, they were shocked to see construction

paper, glue, and noodles. They then begun to make assumptions that this was going to be “easy”.

It was not until they started working in their groups that they realized this required thought and

focus. I observed one student, in particular, do this activity from memory. Rearranging the

noodles without participating in the activity. In thus specific case, the student took the cognitive

demand from Doing Mathematics to Memorization. When we walked around observing their

“noodle graphs” we made the following observations:

• Mislabeled axis- The location of zero was not at the intersections of x and y.

• Skipped over zero- The graphs did not go through the x-axis, but rather skipped it.
 • Extends to Infinity- After 2𝜋, the graphs went upward to infinity.

• Stops after 2𝜋- At 2𝜋, the graph did not continue.

All of these observations led to great mathematical conversations we were able to have with

the students.

IV. Data Analysis

At the end of both classes, the students received an assessment. The assessments were the

same for both classes. I was interested in only two of the five aspects of Mathematical

Proficiency, so I created my assessment accordingly. The assessment had two parts: procedural

fluency and conceptual understanding.

Assessment: Procedural Fluency

The first four questions were strictly procedural. (See Appendix C for assessment). These

first four problems were graded strictly for right or wrong. For every correct answer, the

students received 1 point; however, if they answered in incorrect, they received zero points. The

purpose of these four problems was to see if they could use their unit circle properly to apply the

procedures.

Assessment: Conceptual Understanding

The next set of questions will focus more on their thought process on how the unit circle

and trigonometric function graphs are related, and this part represented the conceptual

understanding portion. Basically, I wanted to see if the students understand the “why” behind the

concepts. Eight questions fell into this category. These questions were assessed solely on
conceptual understanding. They received somewhere between zero points and three points. A

student received zero points if they gave a blank answer or restated the question, one point if

they simply made an observation, two points if they were able to make two connections and

three points if they were able to make the connections and offer a justified explanation(See

Appendix D for rubric). They were able to receive half a point if there answer does not satisfy a

full point. It is important to notice that conceptual understanding is present when a student

received two or three points. This portion of the assessment allowed me to see if the students

who were taught a formula based lesson are aware of what is happening conceptually and if the

students who participated in a discovery learning activity are able put into words what they

discovered.

Assessment Analysis: Conceptual Understanding

After both classes took their assessments, I examined the assessments thoroughly. In four

problems, I saw a significant difference in the level of conceptual understanding. The following

problems demonstrate the different levels of conceptual understanding between my student-

centered class and my teacher-centered class:

' "
• Question 5: The value that corresponds to sin is . What does this mean
( "

Teacher-Centered: 1pt.
 Student-Centered: 2 pts

Notice that the student made an observation pertaining to the unit circle, while the student

below was able to make a connection to the triangle within the unit circle along with

applying the definition of sine properly.

'
• Question 7: Sine increases from 0 to . Why does this happen?
"

Teacher-Centered: 0 pts. Teacher-Centered: .5 pt.

Student-Centered: 2pts.

Both of the Teacher-Centered responses above demonstrate a lack of conceptual

understanding. On one hand, we have a blank response and on the other hand we have an

unspecific observation. The student-centered response shows the students actually picturing the

triangles within the unit circle.

 • Question 9: What is the range of both the graphs ? Why?

Teacher-Centered: 1 pt.

Student-Centered: 2pts. Student-Centered: 3 pts.

The answers above show us three different levels of conceptual understanding. The student

who received one point made an observation by looking at the graph. The next student, scored

two points, because he made a connection to the radius of the unit circle. The answers are

gradually building to lead to a three point answer. This student shows a full conceptual

understanding, because she is able to notice the radius and also the triangles’ hypotenuse.

• Question 11: What happens after 2𝜋?

Teacher-Centered: 1 pt.
 Student- Centered: 3 pts.

The majority of the answers for my Teacher-Centered class looked like the one above.

Most of the students knew that it continued in the same pattern, but they did not know “why”.

Since knowing why demonstrates a level of conceptual understanding, the students in my

teacher-centered class were solely able to observe the graphs. The students who participated in

the Student-Centered Class show a much greater level of conceptual understanding. This student

in particularly connects the idea that you can keep going around the circle to determine what

happens after 2𝜋 .

In all of the answers, it was evident which students seemed to have a deeper conceptual

understanding. I then chose to look at the classes as a whole.

Overall Analysis

Student Centered vs.

When I first analyzed my data, I
Teacher Centered
noticed that my teacher-centered class
90% 92%
actually has 72% of the unit circle
72%

memorized. Even though the students in this 48%

37% 32%
29% 22%
class had the unit circle memorized, it seems
UNIT CIRCLE PROCEDURAL CONCEPTUAL TOTAL
ASSESSMENT
to not have influenced their other score
Student Centered Teacher Centered
categories. While only 48% of the students
Conceptual
in the student-centered classes had the unit
Understanding- 2pts
circle memorized, they were able to answer
70%
the other questions in the assessment

without recalling from memory. Next, I 38% 38%

13%
looked at Procedural Fluency. In both
SCORED ATLEAST ONE 2 SCORED MORE THAN ONE 2
classes, they students were able to perform
Student- Centered Teacher- Centered

procedurally which was good to see.

Following analyzing memorization and procedural fluency, I examined their Conceptual

Understanding. I was surprised by the results only yielding only 6% higher in the student-

centered classes. So, I took a closer look. I noticed that conceptual understanding is not present

until a score of two points. When considering the average of one and two, it is not going to be a

significant amount. I then looked at the percent of students who scored at least one two, and the

percent of students who scored more than one two. These results showed an accurate depiction of

conceptual understanding. This chart, to the right, allows us to the significant difference between

the conceptual understandings.

Student Experience

After observing this data we concluded that a teacher copying and pasting information

into a student’s brain is proving to be less beneficial for the students. Even though the

assessment demonstrated a difference that supports a Discovery Learning, Student-Centered

approach, I wanted to consider student reaction. At the end of their assessment, they were asked

how their experience in the class was. From my teacher-centered class, I received responses that
either complemented me as a teacher, expressed desires to know more, or contentment because

class was easy to follow. Some examples of those responses are as follows:

These responses were not all negative, but they definitely displayed confusion or desire to know

the reason why we can do what we do. Next, I looked at responses from my student-centered

classes. These responses expressed excitement that they had retained information, understood a

connection, or enjoyed being interactive. They are as follows:

These are a few examples that demonstrate how beneficial this type of learning was to the

students. It was encouraging to see that the students gained so much for an experience. It is clear

that the students left class with a lasting impression. It is also encouraging to see the student

make such an important connection to the reasoning of why sin corresponds with y and cosine

cosine corresponds with x.

V. Conclusion

In conclusion, this research experienced proved to be beneficial to my future experience

as a teacher. The results of my research, in fact, demonstrate that when the students are the center

of the classroom and have a chance to participate in an experience, they will have a deeper

conceptual understanding of the concepts. When students have this deeper understanding it helps

with their future. Students leave each class with a lasting impression which allows them to

continue to recall what they have learned. It also allows the students to have a firm foundation,

and allows them to make connections to other mathematical concepts. It was evident by the data
I collected that the students responded better to the activity in which they were the center. If

students prefer this type of learning, it is important that we strive to enact this type of learning

into our classrooms especially since the assessments show a deeper conceptual understanding.

Benjamin Franklin says this quite well. He stated, “Tell Me, I forget, Teach Me and I may

remember, Involve me and I learn”. Applying these results to a classroom influences students in

a positive manner. After all of my research, I am able to conclude that when students involve

themselves in a lesson; they not only take the responsibility for their learning, but they leave

each day with a lasting impression.

 VI. Appendix
Appendix A. Lecture Worksheet
Appendix B. Discovery Learning Activity

Today, you will be discovering the graphs of sin𝜃 and cos𝜃. To complete this activity

you need the following:

• A group of 3 people

• Construction paper & glue

• Spaghetti noodles

• Paper copy of the unit circle

The Instructions to this Activity are as follows:

1. Make sure your group has all materials needed to complete the Activity.

2. On your piece of construction paper, create a graph with an x and y axis.

3. Label the x- axis with the values of theta on the unit circle. (begin with 0, end with 2𝜋.)

4. Beginning with the graph of sine, we are interested in the side length

5. opposite of 𝜃 of the triangles within the unit circles (or y coordinate on the unit circle).

This means the distance a given point on the unit circle is from the horizontal axis.
𝝅 𝟏
Ex. At , the y value is . So your group would measure the vertical distance with
𝟔 𝟐

𝟑 𝟏
the spaghetti noodle from the x axis on the unit circle to the point ( , ). Break the
𝟐 𝟐

𝝅
noodle to represent the distance and then place the noodle vertical at , on the graph
𝟔

you created.

6. Repeat this process for the rest of the angles to create the sine graph.
 7. Once you have discovered the sine graph, repeat this process on a new sheet of

construction paper with the cosine function. (hint. For cosine, we are interested in the

adjacent side length of the triangles within the unit circles (or x coordinate on the unit

circle)

Class Discussion:

Key features sin𝜃 cos𝜃

Intercepts

Maximum

Minimum

Other notes:
Appendix C. Assessment

Compute the following:

' 1' "' 2'
1. sin = _________ 2. cos = _________ 3. sin = _______ 4. cos =_________
0 ( ! (

Answer the following:

' "
5. The value that corresponds to sin is . What does this mean?
( "
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
' $
6. The coordinate point ( , ) has been marked in red on the graph of cosine.
! "
Explain the meaning of this coordinate point.
________________________________
________________________________
________________________________
________________________________
________________________________
________________________________
________________________________

'
7. On the interval 0 to radians, the sine graph is increasing. Why?
"
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

8. On the interval 0 to 𝜋 radians, the cosine graph is decreasing. Why?

_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
 9. What is the range of the sin graph and how is it determined? The cosine
graph?
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

Predict the following:

10. What happens to the graph of sin ϴ after 2𝜋? cos ϴ after 2𝜋? Why?
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

11. What happens to the graph before sin ϴ before 0? cos ϴ before 0? Why?
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
12. What do you think the graph of 2sin ϴ would look like? Why? (Draw the
graph).

____________________________________

____________________________________

____________________________________

____________________________________

Evaluate your Experience.

How was your experience in today’s class?

_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________

Have you seen the graphs of sin and cosine before? If yes, when?
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
Appendix D. Rubric

Question # 0 pts 1 pt 2 pts 3 pts

Question 5 Blank or Mention the y State the State the

restatement of coordinate & connection of the connection of the
question. reference the unit definition of sine definition of sine
circle. according to the according to the
triangle within the triangle within the
unit circle unit circle and
thoroughly explain
in this specific case.
Question 6 Blank or Stated proper x State the State the
restatement of value in reference connection of the connection of the
question. to the unit circle definition of cosine definition of cosine
according to the according to the
triangle within the triangle within the
unit circle unit circle and
thoroughly explain
in this specific case.
Question 7 Blank or Mention the y Mention the State the
restatement of coordinate & increase in distance connection of the
question. reference the unit from x –axis or opposite side
circle. mention the length lengths of the
of the opposite leg triangle within the
is increasing. unit circle and
thoroughly explain
how this makes the
graph appear
increasing
Question 8 Blank or Mention the x Mention the State the
restatement of coordinate & decrease in length connection of the
question. reference the unit of the adjacent adjacent side
circle. side. lengths of the
triangle within the
unit circle and
thoroughly explain
how this makes the
graph appear
increasing
Question 9 Blank or State the correct State the correct Correct maximum
restatement of maximum and maximum and and minimum
question. minimum and/or minimum with followed by a
reference that the reasoning of the reasoning of the
graph visually unit circle. unit circle. Further
 justification
necessary including
a reference the
triangles
Question 10 Blank or Realizes it Realizes it is Realizes it is
restatement of continues in the continues in the continues in the
question. same manner, no same manner, same manner,
explanation. because the unit because the unit
circle is continuous circle is continuous.
Proper justification,
and possibly
predictions of
exactly what
happens after 2pi.
Question 11 Blank or Realizes it Realizes it is Realizes it is
restatement of continues in the continues in the continues in the
question. same manner, no same manner, same manner,
explanation. because the unit because the unit
circle is continuous. circle is continuous.
(understands the Proper justification,
negative concept) and possibly
predictions of
exactly what
happens before 2pi.
Question 12 Blank or States stretches Realizes there is a State the
restatement of with a proper stretch because it is connection of the
question. graph. taking the length of definition of sine
the opposite side according to the
and doubling it. triangle within the
unit circle and
thoroughly explain
in this specific case
how this would
change the graph.
 VII. References

Kilpatrick, Jeremy. Adding It Up: Helping Children Learn Mathematics. Washington, DC:

National Academy, 2001. Print.

Paul A. Kirschner , John Sweller & Richard E. Clark (2006) Why Minimal Guidance During

Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery,

Problem-Based, Experiential, and Inquiry-Based Teaching, Educational Psychologist,

41:2, 75-86, DOI: 10.1207/s15326985ep4102_1

Meece, Judith L. "Applying Learner-Centered Principles to Middle School Education." Theory

Into Practice: 109-16. Print.

Stein, Mary Kay. Implementing Standards-based Mathematics Instruction a Casebook for

Professional Development. New York: Teachers College, 2000. Print.

Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. 5th ed.

Boston: Allyn and Bacon, 2004. Print

Weimer, Maryellen. Learner-centered Teaching: Five Key Changes to Practice. San Francisco:

Jossey-Bass, 2002. Print.