Gadsden Middle School
Lesson Study Project
2007-2008
8th Grade Math Lesson
Discovering the Pythagorean Theorem
Team Members:
Danielle Arsola
Andrea Clawson
Gilbert Gonzalez
Ninfa Madrid
Laura Robles
Maria E. Sanxter
(with District support Shirley Hopson)
Gadsden Middle School
Lesson Study Meeting Dates
September 20, 2007
September 27, 2007
September 28, 2007
October 3, 2007
October 5, 2007
October 11, 2007
October 16, 2007
October 18, 2007
October 24, 2007
October 26, 2007
November 6, 2007
November 7, 2007
November 8, 2007
November 8, 2007
November 9, 2007
November 10, 2007
January 31, 2008
February 1, 2008
March 27, 2008
April 4, 2008
April 16, 2008
April 18, 2008
April 22, 2008
April 25, 2008
April 26, 2008
Team Meeting
Lesson Study Workshop
Lesson Study Workshop
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Practice Lesson/Team Meeting
Lesson Study Workshop
Public Lesson/Workshop
Lesson Study Workshop
Lesson Study Workshop
Lesson Study Workshop
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Team Meeting
Lesson Study Workshop
(facilitators)
Lesson Study Workshop
10:00 – 11:00
4:30 – 7:30
8:30 – 3:30
2:30 – 3:35
2:30 – 4:00
12:00 – 2:00
2:30 – 3:35
2:30 – 3:35
2:30 – 3:35
2:30 – 3:35
2:30 – 3:35
2:30 – 3:35
10:30 – 3:35
4:40 – 3:30
8:30 – 3:30
8:30 – 3:30
8:00 – 3:30
8:00 – 3:30
2:00 – 4:00
2:00 – 3:35
2:00 – 3:35
2:00 – 3:35
2:00 – 3:35
8:30 – 3:30
8:00 – 3:30
GROUP: Gadsden Middle School
GRADE: 8
MATHEMATICS LESSON: Discovering the Pythagorean Theorem
Date lesson will be taught: November 9, 2007
Time of lesson: 10:30 a.m. – 12:00 a.m.
Teacher Name: Andrea Clawson
Classroom #: Library
School name: Gadsden Middle School
School address: 1301 West Washington
School telephone #: (505) 882-2372
Directions to the school:
1.
2.
3.
4.
Take I-10 towards El Paso
Take Exit 162 – Anthony/Chapparal
Turn right at the top of the off ramp
Continue until you get to the stop sign at the main road and turn left (Pic Quik is on the
opposite corner)
5. Continue until you get to the stop light and turn right (McDonalds is on the opposite
corner)
6. The school will be on the left a little ways down the road
Special instructions: Check into the office to get visitor’s pass. Upon entering the parking
lot, the office will be on the left side of the school.
8th Grade Mathematics Lesson Plan
November 9, 2007
Gadsden Middle School, Anthony, New Mexico
Instructor: Andrea Clawson
1.
2.
3.
Title of the Lesson: Discovering the Pythagorean Theorem
Goal of the Lesson:
a. Draw from students’ prior knowledge that the area of the square is related
to the side length by finding the square root of the area of the square.
b. Find areas of squares drawn on grid paper, to see the relationship between
the area of a square and its side length.
c. Find the relationship between the sum of the areas of squares from leg 1
and leg 2 to the area of the square of the hypotenuse.
d. Collect and express data in a table
e. Make conjectures based on relationships/patterns in data
About the Unit
a. Name of the Unit: The Pythagorean Theorem (Investigation 3 in Looking
for Pythagoras
b. Goals of the Unit:
i. Deduce the Pythagorean Theorem through exploration
ii. Use the Pythagorean Theorem to find unknown side lengths of
right triangles
iii. Reason through a geometric proof of the Pythagorean Theorem
iv. Use the Pythagorean Theorem to find the distance between two
points on a grid
v. Determine whether a triangle is a right triangle based on its side
lengths
vi. Relate areas of squares to the lengths of the sides
c. Plan of the Unit (5 days plus one period for mathematical reflection)
i. 3.1 Discovering the Pythagorean Theorem (this lesson – one
period)
ii. 3.2 A Proof of the Pythagorean Theorem (one period)
1. Reason through a geometric proof of the Pythagorean
Theorem
iii. 3.3 Finding Distances (one period)
1. Use the Pythagorean Theorem to find the distance between
two points on a grid
2. Relate areas of squares to the lengths of the sides
iv. 3.4 Measuring the Egyptian Way (1.5 periods)
1. Determine whether a triangle is a right triangle based on the
lengths of its sides
2. Relate areas of squares to the lengths of the sides
3.5 Mathematical Reflections (.5 period)
(Teachers guide for CM2 Looking for Pythagoras, pp. 51-72)
4. Relationship of the Lesson to the New Mexico Grade-level Standards,
Mathematics
Mathematics concepts from 7th grade:
Strand 2: Algebra
Standard: Students will understand algebraic concepts and applications.
5-8 Benchmark1: Understand patterns, relations, and functions.
 Identify and continue patterns presented in a variety of formats.
 Represent a variety of relationships using tables, graphs, verbal rules, and
possible symbolic notation, and recognize the same general pattern
presented in different representations.
Strand 3: Geometry
Standard: Students will understand geometric concepts and applications.
5-8 Benchmark1: Analyze characteristics and properties of two- and three-dimensional
geometric shapes and develop mathematics arguments about geometric relationships.
Strand 4: Measurement
Standard: Students will understand measurement systems and applications.
5-8 Benchmark1: Understand measurable attributes of objects and the units, systems, and
processes of measurement.
 Use measures to solve problems, check the units of the solutions, and
analyze the reasonableness of the answer.
5-8 Benchmark2: Apply appropriate techniques, tools, and formulas to determine
measurements.
Strand 5: Data Analysis and Probability
Standard: Students will understand how to formulate questions, analyze data, and
determine probabilities.
5-8 Benchmark1: Formulate questions that can be addressed with data and collect,
organize, and display relevant data to answer them.
 Analyze problems by identifying relationships and displaying appropriate
data to address the problem.
5-8 Benchmark3: Develop and evaluate inferences and predictions that are based on data.
 Formulate and justify mathematical conjectures based on data and a
general description of the mathematical question or problem posed.
 Analyze data to make accurate inferences, predictions, and to develop
convincing arguments from data displayed in a variety of forms.
Mathematical Concepts from 8th grade:
Strand1: Number and Operations
Standard: Students will understand numerical concepts and mathematical operations.
5-8 Benchmark3: Compute fluently and make reasonable estimates.
 Use a variety of computational methods to estimate quantities involving
real numbers.
 Use real number properties to perform various computational procedures
and explain how they were used.
Strand 2: Algebra
Standard: Students will understand algebraic concepts and applications.
5-8 Benchmark 3: Use mathematical models to represent and understand quantitative
relationships.
 Generate different representations to model a specific numerical
relationship given one representation of data (e.g.,a table, a graph, an
equation, a verbal description).
5-8 Benchmark4: Analyze changes in various contexts.
 Use graphs, tables, and algebraic representations to make predictions and
solve problems that involve change.
 Estimate, find, and justify solutions to problems that involve change using
tables, graphs, and algebraic expressions.
 Use appropriate problem-solving strategies (e.g., drawing a picture,
looking for a pattern, systematic guessing and checking, acting it out,
making a table or graph, working a simpler problem, writing an algebraic
expression or working backward) to solve problems that involve change.
 Analyze problems that involve change by identifying relationships,
distinguishing relevant from irrelevant information, identifying missing
information, sequencing, and observing patterns.
 Generalize a pattern of change using algebra and show the relationship
among the equation, graph, and table of values.
Strand3: Geometry
Standard: Students will understand geometric concepts and applications.
5-8 Benchmark2: Specify locations and describe spatial relationships using coordinate
geometry and other representational systems.
 Represent, formulate, and solve distance and geometry problems using the
language and symbols of algebra and the coordinate plane and space (e.g.,
ordered triplets).
5-8 Benchmark4: Use visualization, spatial reasoning, and geometric modeling to solve
problems.
 Represent and solve problems relating to size, shape, area, and volume
using geometric models.
 Develop and use formulas for area, perimeter, circumference, and volume.
Strand5: Data Analysis and Probability
Standard: Students will understand how to formulate questions, analyze data, and
determine probabilities.
5-8 Benchmark1: Formulate questions that can be addressed with data and collect,
organize, and display relevant data to answer them.
 Generate, organize, and interpret real numbers in a variety of situations.
 Organize, analyze, and display appropriate quantitative and qualitative
data to address specific questions including:
• diagram and pictorial displays
• charts and tables
5-8 Benchmark2: Select and use appropriate statistical methods to analyze data.
 Use changes in scales, intervals, or categories to help support a particular
interpretation of data.
 Generate, organize, and interpret real number and other data in a variety
of situations.
 Analyze data to make decisions and to develop convincing arguments
from data displayed in a variety of formats that include:
• diagrams
• pictorial displays
• charts and tables
5-8 Benchmark3: Develop and evaluate inferences and predictions that are based on data.
 Describe how changes in scale, intervals, or categories influence
arguments for a particular interpretation of the data.
 Conduct simple experiments and/or simulations, record results in charts,
tables, or graphs, and use the results to draw conclusions and make
predictions.
5.
Instruction of the Lesson
According to the State of New Mexico Standards for Mathematics, some of the
major goals in the 8th grade strand are to: understand algebraic concepts and applications,
understand how to formulate questions, analyze data, work with area of geometric
shapes, and formulate conjectures. The standards also require students to make
predictions based on data, use visualization, spatial reasoning, and geometric modeling in
order to evaluate problems.
Prior to the 8th grade, students were exposed to patterns in a variety of formats:
tables, graphs, and expressing relationships both verbally and algebraically. They were
expected to determine a pattern from a table, use the data to construct a graph, and
determine the written and algebraic rule. From this, students made a connection between
formulas and the real-life applications of those formulas. In 7th grade, students also
learned how to determine area of various two-dimensional figures.
In 8th grade, prior to this lesson, students have been exposed to using coordinates
to specify direction and distance. They have also begun to develop the concept of square
root, which will help to lead them to the Pythagorean Theorem. Communication, both
verbal and written, are also a large part of prior curriculum. The Gadsden Math
Initiative, Connected Math, requires all students to actively participate in both small
group and whole class discussion.
Some of the challenges we expect the students to face are: accurately drawing an
askew square, not seeing the relationship between the areas of the legs and the area of the
hypotenuse, and clearly writing the relationship that they see while using the appropriate
terminology. We also expect our students to have challenges with the formulation and
application of the actual Pythagorean Theorem.
In this problem, Discovering the Pythagorean Theorem, students will collect
information about the areas of the squares on the sides of right triangles and conjecture
that the sum of the areas of the two smaller squares equals the area of the largest square.
In order to accomplish this, students will accurately draw squares, collect the information
from their figure (side length, and area), and organize their findings in a table. The
conjecture the students are expected to come up with is the basis for the Pythagorean
Theorem. In future investigations, we expect students to extend their knowledge of this
conjecture, and formulate the Pythagorean Theorem (a2 + b2 = c2).
In the first investigation of Looking for Pythagoras in Connected Math, students
review the concept of coordinate grid and are introduced to finding the distance between
pairs of points on a grid, setting the stage for developing strategies to find distances.
Students also find areas of figures drawn on dot grid and then explore the relationship
between the area of a square and the length of its side.
The second investigation introduces the concept of square root and students
develop a strategy for finding the distance between two points by analyzing the line
segment between them. Students draw a square using the segment as one side, find the
area of the square, and then find the positive square root of the area.
This activity is designed for students to explore the relationship between the area
of a square and its side lengths. They will construct right triangles and create squares off
the legs and hypotenuse. Students will collect information about the areas of the squares
on the sides of right triangles and conjecture that the sum of the areas of the two smaller
squares equals the area of the largest square.
6. Flow of the Lesson
Learning Activates
Teacher’s Questions and Expected
Students’ Reactions
1. Introduction to the Problem
As a Class, introduce the topic, draw a
right triangle on over head or board.
Pose questions about the triangle :
 What kind of triangle
have I drawn? (a right
triangle)
Explain that in a right triangle, the two
sides that form the right angle are called
the legs of the right triangle. The side
opposite the right angle is called the
hypotenuse. Pose the questions:
 What are the lengths of
the two legs of this
triangle? (1 unit)
 How can we find the
length of the hypotenuse
of the right triangle?
(Draw a square using this
segment as a side. Then,
find the area of the square
and take its square root.)
Then Draw the squares on the legs of the
triangle on the over head or board.
 What are the areas of
the squares on the legs?
(both squares have an
area of one.)
 What is the area of the
square on the
hypotenuse (2 units).
Posing the Problem
In Problem 3.1, you will be looking for a
relationship among the three squares that
can be drawn on the sides of a right
triangle. It will help to organize your
work in a table so that you can look for
patterns.
Teacher Support
Points of
Evaluation
Make connections to
Investigations 1 and 2
regarding areas of squares and
the properties of right
triangles.
Do students
understand
what is being
asked?
May want to give students the
problem to read ahead of time.
2. Problem Solving
Work with a group of students to try
to find the answer to the problem.
Students will create right triangles
and then create squares off the legs
and hypotenuse.
Students will find the area of the
three squares.
Students will fill in a table and make
a conjecture about the areas of the
squares.
Encourage students to use
previous knowledge.
Assign the more challenging
dimensions to more advanced
students.
Provide individual and large
sized graph paper so that
students can work together
(foster more discussion)
Can each
group of
students
collect the
data and
accurately
display it in a
table?
Anticipate students’ responses about Provide students with
the conjecture/pattern:
materials:
 There is no pattern (possibly
because area is not calculated
 Paper
correctly)
 Graph paper (line and
 May see pattern in the first
dotted)
couple of rows of data, but
 Pencils
not in the others
 Markers
 Not see pattern due to
 Rulers
focusing on the length of the
 Erasers
leg information and not on
 Manipulatives (square
the area information
tiles)
 See a pattern right away
3. Discussing Students’ Solutions
a) Students will display their
information
b) Facilitate discussion concerning
patterns and relationships
c) Help students deepen their
understanding of the relationship
between the areas of the squares
built off the lengths of the legs of
a right triangle
4. Summarizing the Discussion
Pose questions to help summarize
the activity.
 Why did you construct the
table?
 What conjecture can you
make about your results?
 What led you to your
Display student work for
entire class during discussion
(give each group a
transparency to show work
on/present to group).
Help students understand that
the area of the squares built
off the legs of the right
triangle equal the area of the
square built off the length of
the hypotenuse.
Can students
explain their
data to their
peers?
Can students
examine and
justify data
presented by
their peers?
conjecture?
Pose another problem to
5. Deepening Understanding
Add another column to your table.
encourage students to test
In the column, record the hypotenuse their conjecture.
length of each triangle using the √
symbol. Approximate each
hypotenuse length.
Create a right triangle with side
lengths that are different than those
given in the table. Use your triangle
to test your conjecture.
Create a triangle that is not a right
triangle. Use your triangle to test
your conjecture.
Can you find the area without
counting individual square units?
How?
Anticipated responses:
 Students may not think that
the conjecture will work for
different triangle lengths that
the ones already used
 Students will think that the
conjecture will work for any
right triangle
Encourage students to use
6. Summing Up
a) Review what students learned
mathematical vocabulary in
throughout the lesson
journal reflections.
b) Ask students to write a reflection
in their journal; give an
explanation as to whether or not
their conjecture worked for a
triangle that is not a right triangle
Group Reflection – Gadsden Middle School
The dynamics of our group this year have greatly changed. We went from a group
attempting to work collaboratively between two schools in our District, to one group
consisting of grade level specific teachers on our campus only. We were able to meet
and plan our lesson in a much more cohesive manner, due to the fact that we had the
same prep periods and were willing to work after school and during holidays. “Research
lessons can give teachers a chance to bring up, discuss, and perhaps reconcile competing
goals or visions of education” (Lewis and Tsuchida, 1998). During our meetings, we
came to realize, that we all believe in the same overarching ideas when it comes to
teaching. Within our group, there were no competing goals or visions of education,
however there was a lot of discussion regarding our goals, our lesson, and our students.
During our group meetings, we realized that we all have ongoing challenges that we face
every day: time management, making connections, classrooms disruptions, and student
reflections. Although we have 85 minute periods, we often find it difficult to complete
the entire lesson within this time frame. Time is even more of a concern when teaching
ELL (English Language Learners) students, since approximately 99% of our students are
or were classified as ELL. Through our research we have found that we need to “ask
questions and allow ample time for response (30 seconds or more for ELL students)”
(Chapin, O’Connor, and Anderson, 2003). One of our main concerns with making
connections is “students tend to think about the problem at hand, and not about how the
problem might relate to a bigger mathematical idea” (Dacey and Lynch, 2007). This was
one of the reasons that we chose our goal and the lesson that was presented.
“If time is spent solving problems, reflecting on solution methods,
examining why the methods work, comparing methods, and relating methods to
those used in previous situations, then students are likely to build more robust
understandings and strategies, and to view mathematics as a valuable tool for
making sense of and solving interesting problems” (Lappan, Fey, Fitzgerald,
Friel, and Phillips, 2002).
We really wanted students to focus on the big mathematical idea, and not just how to
solve a problem. As Dr. Takahashi mentioned, our classrooms have many disruptions
throughout the day with announcements, etc., through the school intercom. This has
become an issue that truly detracts from our classroom practices. We are not quite sure
how to alleviate the problem, since it has been brought to the attention of our
administration in the past. The last challenge we face is that of student reflections, which
really ties in with time management. Reflections are of vital importance, and we need to
come up with a plan so that they are interwoven with our lessons more frequently.
We feel that overall our lesson went really well. The students took on the problem and
ran with it. During our post-lesson panel discussion, the topic of our goals and whether
or not they were met was discussed. The overall feeling from our group members, which
was verified by Dr. Takahashi, was that yes, our goals were met. The first goal of our
lesson was met during the Launch portion of the lesson when a student stated that they
could find the side length of a square by finding the square root of the area of the square.
Goal two was addressed during the Exploration portion of the lesson where the students
discovered the relationship between the sum of the areas of the squares from the legs of
the triangle to the area of the square of the hypotenuse. The other goals were met during
the share portion of the lesson where students expressed their data in a table and made
conjectures based on the relationships in the data.
During our post lesson discussion, we were also given many ideas for changes that we
could make to our lesson for when it is taught in the future. The majority of the ideas
were things we had thought about, and a couple of them were actually planned into our
lesson, but we did not get to them because of lack of time. The first suggestion, which is
one of the obstacles we actually face daily in our classrooms, is to allow more time for
the students to work together and solve problems. Another suggestion given, to alleviate
some of the time crunch, was to assign the larger triangles to more advanced students.
That way, there would be a little less struggle time, and everyone would be successful
with the lengths they were given. A third suggestion was for the allowance of
manipulatives, or different types of graph paper. Giving students the choice as to what
they used to solve the problem could perhaps alleviate some of the time constraint, as
well as help students to better see the problem. Another suggestion was to judge struggle
time better. Let the students struggle, but don’t let the problem completely frustrate
them. Two more suggestions, that would have been part of the lesson if not for the time
situation, was to allow groups to present their work (perhaps all students do one triangle
together on large paper instead of individually creating the triangle and squares), and to
give students the opportunity to reflect on the lesson in their journals. “…Having
students write, draw, and discuss their predictions; revisit their initial predictions in light
of new observations; debate opposing viewpoints; and make predictions to new
situations” (Lewis and Tsuchida, 1998) is extremely important in mathematics. It was
also suggested that we push students further in their explanations by asking appropriate
questions that will extend their mathematical thinking.
One of the good things about our curriculum, Connected Mathematics, is that each
lesson, from day to day, draws upon knowledge the students have gained from prior
lessons. “Teachers… carefully design and orchestrate lessons so that students are likely
to use procedures that have been developed recently in class” (Stigler and Hiebert, 1999).
Therefore, this lesson was built on two prior investigations which expanded over two
weeks. Although we were pressed for time at the end of the lesson, students revisited the
topic in the following lessons and made the appropriate connections.
Overall, our experience this year has been really outstanding. We have learned a great
deal from the presenters and from each other. Not only have we become stronger
teachers, but the process has led to stronger friendships with our colleagues.
Bibliography
Chapin, S., O’Connor, C., and Anderson, N. (2003). Classroom Discussions: Using
Math Talk To Help Students Learn. Sausalito, CA. Math Solutions Publications.
Dacey, L., and Lynch, J. (2007). Math For All: Differentiating Instruction. Sausalito,
CA. Math Solutions Publications.
Lappan, G., Fey, J., Fitzgerald, W., Friel, S., and Phillips, E. (2004). Getting To Know
Connected Mathematics: An Implementation Guide. Glenview, IL. Prentice
Hall.
Lappan, G., Fey, J., Fitzgerald, W., Friel, S., and Phillips, E. (2006). Looking For
Pythagoras. Boston, MA. Prentice Hall.
Lewis, C., and Tsuchida, I. (1998). A Lesson Is Like A Swiftly Flowing River: How
Research Lessons Improve Japanese Education. In American Educator, Winter,
pp. 12-17, 50-51.
Lewis, C., and Tsuchida, I. (1998). The Basics In Japan: The Three C’s. In Educational
Leadership, March, pp. 32-36.
Stigler, J., and Hiebert, J. (1999). The Teaching Gap: Best Ideas From The World’s
Teachers For Improving Education In The Classroom. New York, NY. The Free
Press.
Individual Reflection: D. Arsola. Gadsden Middle School
Being a part of lesson study has allowed me to look at our districts math initiative
(GMI) more closely. I am a third year teacher in this district so working with the
program Connected Mathematics in not new but still rather challenging. I have
experienced the great potential our program has, if implemented correctly. From my
experience with Lesson Study and GMI I believe these two teaching methods
complement each other. Both techniques are centered upon “All students should be able
to reason and communicate proficiently in mathematics. They should have knowledge of
and skill in the use of the vocabulary, forms of representation, materials, tools,
techniques, and intellectual methods of the discipline of mathematics. This knowledge
should include the ability to define and solve problems with reason, insight,
inventiveness, and technical proficiency”(G. Lappan, J. Fey, W. Fitzgerald, S. Friel, E.
Phillips).
This year our group dynamics were drastically different from last year. Last year
we had two schools from the same district collaborating to towards a lesson plan. Which
made meeting times and places, as well as, the division of work, very complicated. This
year, all of the math teachers that were involved were from the same school, this includes
the Math Process Trainer, which made meeting times, and the division of work simple.
This being some of our group members second year with lesson study, we understood the
commitment; it takes for our lesson to be successful. Our group very quickly decided on
the book from which our lesson was going to come from. The trouble was deciding on a
specific lesson, we wanted to pick one that the majority of teachers were having a
problem presenting the mathematical concept.
As for the lesson itself it was taught several different times. I was able to teach
our lesson after I had seen it, in another classroom, which made it easier to evaluate. In
the panel discussion it was mention that the “Explore” part of the lesson was a bit too
long. However the summary aspect of CMP cannot take place until “when most students
have gathered sufficient data or made Sufficient progress toward solving the problem.”
”(G. Lappan, J. Fey, W. Fitzgerald, S. Friel, E. Phillips). So if we had rushed the
students to finish their tables and graphs I don’t think that they would have understood as
much or even have invested interest.
The biggest struggle as a teacher working with CMP and lesson study is the
division on time. How much time is sufficient for the Explore section of our lesson? We
actually spent a great deal of time discussing where the lesson should end to have enough
time for meaningful discussion. However, this is not as simple as it sounds, if the
students do not struggle with the mathematical concept they don’t seem to retain it as
well.
After completing my second year with Lesson study, I have found the process to
be invaluable. As an educator, I feel I will always struggle with certain aspects of
teaching; time management, learning what a teachable moment is, and asking questions
in a matter that does not devolve too much information about the problem. However, I
will continue to be a reflective educator.
Personal Reflection – Andrea Clawson
This is the second year that I have had the opportunity to participate in Lesson Study.
The experiences I have had, ranging from workshops presented by Akihiko Takahashi
and Tad Watanabe, to watching actual lessons, to presenting the overview of our lesson
in Chicago a year ago, have given me a lot of insight into the teaching of mathematics.
These experiences have also given me validation that the curriculum used in my District,
Connected Mathematics, is on target with how students should be learning mathematics.
Mathematics is not about learning a new equation and how to manipulate the numbers for
the equation, but it is about investigation and struggle which lead to a more meaningful
and thorough understanding of math concepts.
As I reflect on the past two years, I must say that the second year was a much better
experience all around. My group this year was much easier to work with than the group
last year. I can probably attribute that to the fact that last year the group was a
combination of teachers from two different campuses that had not worked together
before. This year, my group was made up of teachers from my own school, and
amazingly enough, it consisted of nearly the entire eighth grade teaching population. The
group really did work well together, and I feel like the experience made our bond as
colleages and friends that much stronger. I also think that I learned so much more from
the presenters this year. Perhaps it was my own personal growth that helped me tune in
to them more; but whatever it was, the overall experience was much more profound this
time around.
As the teacher of the public lesson this year, I was excited about that lesson that was
chosen to fit our goal. I feel that the students need to know the background as to why
something works before they simply plug numbers in and solve problems. The lesson
that was chosen allowed students to explore the reasoning behind the Pythagorean
Theorem before the Theorem was ever introduced. The A-Ha! moments that students
have when they figure out the relationship between the areas of the squares and then
make a conjecture based on the relationship is priceless. The students did an amazing job
during the public lesson, and their hard work helped my Lesson Study group reach the
goals we had set forth for this lesson.
As I said before, this year has been a great experience for me. I look forward to next year
when the group from my campus will have some new members, as well as some that
have been part of the experience for the past two years as well. I also look forward to
expanding my own knowledge regarding the implementation of lessons and how to better
serve the students on my campus.
Lesson Plan Reflection
Submitted by Gilbert Gonzalez
In looking back at the lesson that we did here at GMS. I feel that the lesson went well and
only needs minor tweaking of the lesson. I think that the assignment of the different right
triangles that vary in size and complexity should be based on group ability. This is due to
the time constraints of the lesson and because some groups may finish much sooner than
other groups. Also, the results should be posted on one large continuous poster so that
students have a better chance of seeing the pattern in the areas of the squares built from
each side of each right triangle.
The time element needed for the reflection by the students is critical and therefore we
need to make sure there is ample time to get into the final discussion of the findings and
the pattern that is seen once the lesson is in its final stage of reflection and discussion.
There should be an opportunity for the students to make a journal entry for their
reflection of what was learned during the lesson. This communication portion is vital and
lacking in most lessons done in today’s math classrooms.
In conclusion, the overall the lesson went well and only needs minor tweaking as we
continue to learn through this lesson study process.
Personal Reflection: Ninfa Madrid - Gadsden Middle School
Participating in the Lesson Study for the first time was a great experience for me.
I personally believe that I should never stop learning, and Lesson Study was a great
opportunity to understand how valuable it can be to work together and how we can learn
from each other.
In our lesson I feel that students accomplished the goals. Some students were
struggling and some came out with ideas that were not expected. We as group took the
time to examine the effectiveness of the lesson. I taught the same lesson after the
presentation and it was amazing how students came out with ideas.
Out of the many good things that I learned from meetings I attended with Dr.
Takahashi and Dr. Watanabe was when they emphasized that if a teacher helps too much,
nothing is left for the students; and if left alone students may not make any progress. So, I
also learned that as a teacher I need to find a balance as to how much help I give the
students, and that is okay to let them struggle.
I enjoyed working with the Study Lesson group because I feel more confident on
helping students become more independent learners, and I also acquired skills and
knowledge.
Personal Reflection-Laura Robles
This is my second year doing Lesson Study. I was looking forward to
working with my colleagues and creating a new lesson. At our school we are
using the Connected Mathematics which focuses on that “all students should be
able to reason and communicate proficiently in mathematics. They should have
knowledge of and skill in the use of the vocabulary forms of representation,
materials, tools, techniques, and intellectual methods of the discipline of
mathematics. This knowledge should include the ability to define and solve
problems with reason, insight, inventiveness, and technical proficiency” (G.
Lappan, J. Fey, W. Fitzgerald, S. Friel, and E. Phillips, 2002) This gives us a
head start in how we conduct class and how we plan and present a lesson.
After selecting the lesson we discussed on how to present it and how to
identify the mathematical concept which we wanted to address. We had to focus
on addressing only one math goal and not making the lesson too long. This year
the whole group works in the same school and most of us have the same
planning period since we all teach the same grade. This made it easier for us to
meet and to work together.
This lesson was taught four times before we did one practice lesson before the
public lesson, after we taught the lesson we all discussed how the outcome of
each individual lesson. This helped the teacher that was going to do the
practice. After we observed that lesson then we agreed on what changes had to
be done to the public lesson. After the many discussions and observing other
teachers presenting the same lesson it has helped me to anticipate for problems
and how to solve them as they occur.
One issue that I have is how to schedule the appropriate time for each of the
parts of the lesson. In Connected Mathematics the investigations are set up into
three areas: Launch, Explore, and Summarize. My students take the longest
time during the Exploration part. This is an area that we have to plan for
because in our class we can always do the summarizing the next day.
During the debriefing we always address the facto of asking the right
question. This is an area in which I have been working on since doing Lesson
Study and the Connected Mathematics. I now do more self-reflection as I am
planning and conducting class. These questions and ideas will help me to
become a better teacher.
Reflection – M. E. Sanxter
Lesson Study has been a wonderful learning experience. From the beginning of
the lesson study process to the end, I learned a lot about the way I teach and how to make
lessons better for my students.
We began by selecting a skill that we felt students were scoring low on in the
New Mexico Standards Based Assessment. We felt that the eighth grade students needed
to better their skills on the Pythagorean Theorem. We had a lot of discussion about what
we should focus on. This really helped me to realize how much planning a well
developed lesson takes. We looked at assessment, and then we selected a lesson from our
curriculum that fit our needs.
When we found our lesson there was a large amount of discussion about our focus
again. Some believed our lesson was not at the level that it should be, and we began
modifying it and making it too difficult. We were brought back by our original goal, and
determined that the original lesson was sufficient because future lessons would bring in
all of the additional skills that we wanted the students to learn. I really learned from our
discussions. It is really important to stick to the goal of the lesson and not make it to
complex. It is really easy for us to begin adding too much to our lessons because we
already have the knowledge that we need to do the work. Our students on the other hand
do not have our mathematics background and I needed to learn that sometimes keeping
things simple is better.
Our practice lesson went really well. I was surprised at how well the students
received our lesson. We each made a few adjustments when we taught the lesson in our
own rooms and learned that the littlest changes to a lesson can make a big difference in
the outcome. I changed the introduction a bit because I saw that during our practice
lesson the students needed a lot of guiding questions to get to the point that they needed
to be at to begin the lesson. I learned that the way that we phrase things and the pictures
that we draw while we are teaching really make a difference. These questions really need
to be thought out because they can either help the students or confuse the students more.
The changes that we made from our practice lesson to our presentation really
helped and the students really did a wonderful job. I had never done lesson study before
and I was worried that the about of people in the room and the video camera would really
distract them, but they did a great job and they achieve our lesson goal.
I have also learned by going and observing other lessons. Working with other
teachers has given me more ideas about lessons, and how to present them. I have tried
some of the strategies that I observed, and have found that while some strategies work for
some teachers and students, some may not. Lesson Study has been a great experience
and I have made lasting friendships with my collegues.