Reflection #9 –
Using Research to Analyze, Inform, and Assess Changes in Instruction
November 14, 2013
The article this week as especially interesting for me as I
feel it is a summarization of one pf my Sponsor Teacher’s teaching
philosophies. Over the course of the practicum, we had many discussions
regarding her approach to using inquiry based lessons to ensure student
centered learning instead of teacher lead instruction. Students were
continuously prompted through questions, activities, and discussions to explore
the material presented in the class. The teacher intentionally made an effort
to try and prompt students to develop the conclusion of the lesson. Homework
policy, quiz marking schemes and correction opportunities along with seating
arrangements that promoted group work all worked to creating this environment
of exploration and discussion.
As a student who has had very little experience learning in
this method, I can still appreciate the value that it olds to helping students strengthen
their grasp of concepts and the relationships that exist in mathematics.
However, as a teacher, I know that I will be challenged to try and create
lesson plans that promote this type of research style thinking that show to
better assist in student success as they encounter all applications of the
curriculum material.
Reflection – Short Practicum November 10, 2013
During my short practicum, I spent my time at Lord Byng High
School with three different sponsor teachers. The math classes I observed
included Grade 9 and Grade 10 Math, and enriched Grade 10 Math as well as Grade
12 Foundations. I was able to teach three different Grade 10 lessons one of
which included a Math 10 Enriched Radical review. In doing so, I found aspects
of the experience to be what was expected in terms of a general disheartened
attitude towards math on behalf of the students, but was also surprised by how
teachers were able to generate excitement and engagement.
What I found to be the easiest aspect of teaching math are
the clear learning objectives expected of the students. How to students are
able to solve problems, word problems and apply or extend their knowledge is a
clear indicator of their level of understanding. One of my sponsor teacher’s
used exit slips at the end of most lessons to gauge students level of
comprehension and from their feedback was able to determine of more time or attention
was needed to focus on the subject, or to find general mistakes made by
students. At the same time, this also was a source of great difficulty.
Although we could identify the students that were not completely grasping the
concept, it did not help to determine why or what was the source of confusion.
Students came from a variety of experiences, math foundations, and attitudes
influenced by family, friends, and past teachers. There is still no method that
best identifies students who are having difficulties, deciding the source of
the problem, and then working towards a solution. As much of an effort we as
teachers make to help each student, there will always be students who do enough
to make it seem as if they are understanding to simply “get through it”.
At the same time, there is the issue of students who quickly
grasp the material but become disengaged since they are not continuously
challenged. I found it very challenging to try and continuously adapt
activities and discussions that ensure students of all levels of understanding
are engaged and still moving towards a new level of understanding for
individual needs.
All of this was experienced in the general Math 10 and 9
classes is taught and observed. However, the Math 10 Enriched class was a very
different environment. The majority, if not every student, was simply enrolled
because they loved and enjoyed math. Even though the class was paced together,
many of the students worked ahead, were eager to do homework, and asked for
more challenging material. What greatly piqued my interest was the question of
if the Enriched students enjoyed their Math class because the were naturally
apt with the subject, or if their understanding resulted in a more positive
attitude towards the material?
If so, is there a way to incorporate a more optimistic
environment in the classroom environment to help stimulate student
comprehension?
Reflection #8 – Reading Mathematics October 17, 2013
The language of mathematics is exactly that, it is a
language with its own set of symbols, rules, sentence structures, definitions,
connotations, and assumptions. As math teachers, before we are able to have
students’ understand the concepts, they must first be able to interpret what is
being communicated and later asked.
The first assumption I feel that is most commonly made is
that students are already familiar with common vocabulary lent from other
subject areas and can differentiate the use of its many definitions. English
language learners have an especially difficult time with this added challenge
to understanding English, never mind “math English”, and similarly with
students who have not been properly exposed to or scaffolded the needed
academic math language with each grade level.
This leads to the reoccurring societal stigma that follows
mathematics. Many students have not been properly introduced or taught how to
handle math terms in past years. The educator has the added responsibility of
checking the level of understanding of students and how comfortably they can
work with what is supposed to be assumed knowledge at the secondary level.
Teachers must then accommodate these learners needing to build their vocabulary
in addition to properly enforcing new terms introduced within the classroom. As
outlined by the article, group activities, games, and simply time put aside to
break down and explain vocabulary is required.
Next, the way in which information is presented in textbooks
provides some challenges or habits that can disrupt students understanding.
Using sources that strictly employ a single method to presentation of information,
form word problems, or sketch diagrams may constrict definitions. Students may
become attached to a single form of presentation or definition and thus become
confused when other expressions are used. (This was highlighted with division
symbols, multiplication, triangles, polygons, etc.)
Finally, especially at the secondary level, assumed
knowledge of prior experience with word problems can inhibit student
comprehension. Without stopping to break down problem solving methods or model
problem solving while demonstrating thinking patterns, some students may never
be taught how to approach problems that are so heavily used in math classrooms.
Again, putting aside the time to emphasize and practice these methods can strengthen
student abilities to confidently approach problems.
While textbooks offer a rich source of structured lessons to
introduce curriculum content, they can actually act as a barrier to student
understanding. Teachers must be familiar with how resources accessible to
students convey material, simplify or complicate concepts, present examples,
ask questions, and be cognizant of the text’s goal to promote necessary or
arbitrary learning. On top of this, I feel it is necessary for teachers to be
familiar with previous grade level textbooks/resources and following year
textbooks to know what kind of knowledge students may be equipped with, or what
they need to be prepared to encounter.
Reflection – Micro-teaching October 17, 2013
Yesterday, Miranda, Alex and I presented a micro-teaching
lesson on slope for Grade 10 Foundations and Pre-Calc Mathematics. Some of the
positive feedback we received included a good into and pace, successfully
circulating the classroom, incorporating technology through the applet,
addressing possible confusion for students, eye contact, visual aids, and using
group work. Areas for improvement were volume when speaking to the class, a
clear definition of slope at beginning or end of lesson, and having students
practice with app. General comments
concerning the planned activity favored the group aspect, providing good
guiding questions, adapting activity to challenge more advanced students, a
relevant but fun activity, and if given more time to have students draw more
slopes using different colors on laminated planes.
Overall, I feel that our group successfully worked with one
another as a team as the lesson progressed to compliment and aid each other’s
sections. I think the use of the laminated sheets to have students work in
small groups to draw slopes created good discussion. Creating a series of
questions/slopes with elements that had not been addressed in the lesson was a
great way to challenge those students who quickly understood the concept.
Personally, I feel that I should have better outlined how I
was going to use the applet to demonstrate slops and how it changed given
vertice movements. One suggestion
described using progressive slopes, 1, ½, ¾, to emphasize the change and
movements for the horizontal and vertical aspects. I feel this would have
provided a more clear and sequential explanation of the breakdown of rise and
run. Also, the original intent to formalize a slop definition was missed given
time crunch however; this could be fixed by doing so before the activity and
would also act as a guide for students as they work. Another personal aspect to
be more aware is allowing enough time for students to answer questions that I
pose – must start internally counting after posing questions to the class.
Watching other groups present was a great way to see other
approaches to presenting lessons and also inspired me with other ideas to fix any
problems or issues with our lesson, or just alternative methods to presenting
the material. This exercise also reminded me to not overwhelm a lesson with
trying to incorporate as many styles and modes of delivery as possible!
Reflection #7 – The Geoboard Triangle Quest October 8, 2013
In every course, Inquiry has been the topic of conversation
and is a highly encouraged teaching method that has been explored for the past
month. The Geoboard activity described by the author is a perfect example of
inquiry being put into practice within a classroom.
After having completed the chessboard task, Allen could have
continued with the course material. However she chose to have students discuss
the possible extension activities which encouraged them to build their ideas of
patterns, counting, shapes, dimensions, ect. To then put aside time for
students to pursue these ideas exemplifies inquiry styles of teaching. Having
the students examine triangles on Geoboards allowed them to find solutions to
their own questions by developing hypothesis, recognizing patterns, identifying
triangle properties, structure classification processes, and practice working
in groups. The amount of student-lead learning that was able to take place was
impressive.
This article causes me to now consider, when going through
curriculum material, opportunities to have open discussion with students. I
would like to provide students the chance to explore their suggested extended
activities. It may not be possible to anticipate every opening or allow the
same amount of in depth studying to occur. Yet, when students express an
interest or desire to learn and become familiar with subject matter, it is
important to recognize this curiosity and to encourage this eagerness. Overall,
students will not only become more engaged in a classroom, but will also become
more confident in their abilities.
Reading #6 – Reflection on Research October 2, 2013
Instrumental and relational, necessary and arbitrary, and
skills and relationships, no matter the labels, are reoccurring topics of
discussion in mathematics education. Brahier and Hoffman’s comparison of the
American and Japanese approach to math education again emphasize the positive
and negative aspects to memorization and conceptual understanding.
It is becoming more evident that for students to overcome
the struggles of learning the math language, and for society to improve its
attitude towards math, teachers must learn to adopt a more explorative nature
in their practice. Although for some students, memorization and habits
developed from practice ensure the greatest amount of success, it is not the
case for the majority. Most are unable to fully understand through repetition.
Exploring topics while manually working through the concepts, making mistakes,
learning from and teaching peers, relating topics to one another to create
relationships of all sorts are methods that seem to deliver the greatest
results. Presenting mathematics as a world of relationships, as the Japanese
classrooms do, creates a more dynamic learning environment.
Something that struck me as comparable was cooking. It is
mindless to purchase ingredients listed in a recipe, measure out the specified
amount, and follow instructions exactly as they are stated. However, sometimes
if we take the chance to add some extra spice, substitute one vegetable for
another, use a grill instead of an over, and so on, the creation may be just as
delicious as what the recipe intended. The process of trying out different
ingredients or experimenting with the cooking methods is similar to math. Using
new strategies, taking risks, adding or taking away something that was not
explicitly instructed can still lead to the “correct” answer. The most value is
found in exploring and learning these unexplained processes as they were
self-directed. The learning and understanding accomplished will last a great
deal longer than having been simply told.
Reading #5 –
Arbitrary and Necessary September 29, 2013
Hewitt’s differentiation between arbitrary and necessary
aspects of math education emphasizes the reliance on memory by some students
that in fact hinders their understanding. Teachers that solely impart
information cause students to perceive math as one massive set of rules and
situations they need to commit to memory and are then unable to discern the
concepts themselves.
The arbitrary outlines the tools that teachers need to
explain so that students become equipped with otherwise unknown knowledge.
Without being told standard conventions, and using ways such as math history to
establish these practices, students are unable to explore mathematics in a way
that can be understood by the general community. They will also encounter difficulty
in understanding higher level concepts as they will not be able to interpret
other information and explanations that heavily employ standard expressions. By
strengthening memory capacity surrounding arbitrary knowledge, teachers move to
building awareness and comprehension when necessary knowledge is later
introduced.
As the author states “… a teacher’s role is to work within
the realm of awareness rather than memory.” (p. 5) By first establishing a firm
foundation of arbitrary knowledge, students can move to uncovering knowledge
that they have acquired the ability to discover on their own.
Arbitrary and necessary knowledge divide methods of gaining mathematic
knowledge. It serves as a specific model of scaffolding to assist students in
having a firmer grasp of how to approach the subject of math and develop the
required skills at all academic levels.
You can’t build a house without knowing how to use a hammer…
Reflection – The Locker Problem September 28, 2013
A school has 1000 lockers and 1000 students. On the first
morning:
*The first student walks along and opens every locker.
*The second student walks along and closes every second
locker.
*The third student walks along and changes the position of
every third locker door.
And so on. Once all 1000 students have completed this
process, which lockers are open?
Briefly describe how
you solved it?
To solve this problem, I first tried reduced the number of
lockers to ten to see if there was a pattern that could be found with a smaller
set. I manually wrote out how the locker positions would change given ten
students walked along and changed the positions for the first ten lockers.
After ten students had walked along, the firs ten lockers would o longer be
changed. I circled all the locker numbers that were left open and tried to see
any pattern of this smaller set. The open lockers were 1, 4, and 9 and these
are all the perfect squares found between 1 and 10. I assumed then that all of
the perfect squares less than 1000 would be left open once all the students had
walked through. From this point, I “guess and checked” the largest number whose
square was less than 1000 - this is 31. Thus there are 31 lockers left open
after all of the students pass through all 1000 lockers.
Where do you think a
student might get stuck?
A student can be
overwhelmed when initially reading the problem. The shear number of lockers and
students to account for may present itself as an overly large counting task. If
a student does not know how to initially break down the problem into steps that
are more approachable then he/she will not feel as if they can find a solution.
Even if a student is able to simplify the problem, he/she may not make
establish a pattern between the open lockers and perfect squares. Perfect
squares are often a concept that takes time to become familiar with recognizing
its presence.
How might you assist
that stuck student?
To assist students having trouble, especially those
intimidated by the problem parameters, can suggest breaking it down into a
smaller number of lockers – counting open lockers from 1-20 is more
approachable than 1000. When acknowledging patterns, can encourage students to think
about how mathematicians describe and classify numbers – whether they are even,
odd, prime, and their possible factors.
What extensions could
you offer students ready for a challenge?
Those students who are looking to be challenged can be asked
to:
·
Find any pattern that exists between the occurrence
of perfect squares.
·
Determine
how many lockers were open after 100 students pass through? 200 students? 35o
students? Is there another pattern found to how may lockers would be left open
for any number of students passing through that is less than the number of
lockers?
Describe 2 of your memorable math teachers: September 23, 2013
Throughout my education, both early and continuing into
post-secondary, I have been fortunate to have many great and influential math
teachers. There is however two teachers that have made the most impact specific
to my appreciation of math as well as my decision to pursue education as a
career.
For as long as I can remember, active learning and practice
did not stop when the summer months began. Each weekday, whether my parents
were at work or at home, my brother and I were left worksheets and assignments
to ensure continued studies for at least an hour each day. They ranged from
reading different novels and information books, creative writing assignments,
grammar exercises and at least one math based activity or worksheet. From an
early age, math understanding and mastery was instilled as a valuable skill and
was fostered by mother. At the end of the day or when she came home from work,
we would sit together and review my work. In this way, I learned to appreciate
and look forward to perfecting and increasing my understanding of mathematics.
Since my mom had always been strong in basic algebra, arithmetic and geometry,
she even managed to prepare me in advance for lessons in the classroom until
the early years of junior high and was therefore able to solidify my confidence
in my mathematics abilities. Even though doing homework is the least appealing
activity to a child, especially during summer vacation, it was essential and pivotal
in my academic success and decisions to continue working on my exploration of
mathematics. My mother was the first, and I would also say the greatest,
influence as a math teacher.
The second most memorable math teacher that has been a part
of my life taught me both Grade 12 Math and Calculus, Mr. Baker. As has been
discussed, math has a very common reputation of being a hated and
under appreciated subject within schools. However, by ensuring his classroom was
welcoming, incorporating anecdotes into lessons, simplifying concepts, and most
importantly, getting to know his students, Mr. Baker had every student enjoying
his subject. Familiarizing himself with each student’s strengths and talents he
was able to adapt lessons to try and allow everyone comparable levels of
understanding. He also ensured that no matter the student’s academic or social
success, each was treated equally and with exceptional levels of respect
driving students to want to meet his expectations. Mr. Baker’s teaching methods
not only strengthened my mathematic skills, but inspired me to pursue education
and is a role model that I aspire to be in my teaching career path.
Reading #4 – Battleground Schools September 20, 2013
The history of the math curriculum development due to
societal movements and political motivations, provides great insight into how
types of instruction and understanding have become polarized. Continued
arguments about how to teach and what to teach within the schools has resulted
in an overall fear and intimidation of the subject since we seem to never be
able to “get it right”. This separation of method memorization and
understanding itself disrupts complete appreciation of math but explains the
dominant opinions and outlooks of the subject. In my own personal experience, I
continuously receive remarks of exclamation and almost awe of choosing to study
mathematics, as I am sure every post-secondary math student can attest, yet y
experience has only lead me to embrace math as another language I have become
fluent.
A possible contributing factor to the attitude towards
mathematics, offered by the article, is something that happens within the
classroom in the early years of education. Many elementary teachers do not feel
comfortable or confident with the subject and therefore often unconsciously
pass on these insecurities to their students. Methods of diverse instruction
and presentation are shied away from and heavy reliance on textbooks and
worksheets is introduced. This results in the common opinion expressed in the
article on page 349: “Anyone who does not do well under such a regime [lots of exercise and drills] may be written
off as not meant to do math.” It is no wonder then that insecurities have
thrived. This lack of confidence is carried forward and becomes increasingly
difficult to reverse as students move through higher levels of education.
If we can identity as early education as an important factor
in mathematic confidence, then it is essential for the teaching community to
come together to develop a solution. Perhaps teachers in early years who have
encountered success with specific techniques and lesson pans can begin
communicating these amongst not only colleagues, but with teachers in various
schools across districts. A shortage of math specialist exists in all levels of
education, however perhaps with the support of school administration, those
with experience and knowledge of the subject can attempt to outline their
approaches, develop workshops, share living documents, and teach content to
colleagues to increase their confidence when teaching students.
A general dialogue and exchange of practices need to be
encouraged within the teaching community to increase overall knowledge base of
those directly involved in inspiring mathematics understanding and content
delivery to students. The attitudes and stresses students bring to the
classroom evidently affects their academic success – so it is now time to
consider the impacts the teacher’s confidence and levels of stress in turn
impacts the students not only in the classroom, but as they continue on through
their education.
Reflection – How many squares in an 8 x 8
chessboard? September 20, 2013
A chessboard itself
is aesthetically appealing due to its repetition of a simple colour and square
combination. From a student perspective, my immediate reaction is to then see
how I can use this pattern to assist me in counting the number of squares. I
tried to first use the obvious, and to me simplest approach, before trying to
explore more in depth and complex solutions. Breaking down the pattern into one
simple sequence of shaded and unshaded squares then allows me as a student to
build another pattern after first completely understanding the base. Start by
counting all squares of size 1 x 1 that line the width and length of the board,
follow by counting all squares of size 2 x 2 that line the length and width
again (this gives 7 squares by 7 squares), and so on for the larger squares
until we arrive at a single 8 x 8 square. (A pattern emerges that the
chessboard is composed of the sum of the squares of square sizes inclusive of 1
and 7.)
However, that being
said, it is important to acknowledge from a teacher’s perspective that the
problem itslef may be overwhelming. Students may become stressed and not even
feel as if they are able to begin the problem since the idea of having to
imagine and count various square sizes appears to be such a large task. For
these students, it is important to then simplify the problem and make it more
relatable. Breaking down the beginning of the solution into simple steps and
presenting it in a few different ways may help to get those troubled students
to become comfortable with exploring further on their own. A teacher could
begin by drawing the chessboard and highlighting how different square sizes can
appear along the board. Another approach could be to remind students about
previously discussed topics such as perfect squares and areas and see if they
can make connections between the concepts. Thirdly, a teacher could suggest
writing tables that record the size of the square found in the chessboard and
how many squares are found (starting with the two largest being 8 x 8 and 7 x
7). Each of these approaches appeals to slightly different learning styles and
can then help students to be able to more confidently find a solution.
While it is
important to prepare for the possibility of students experiencing challenges
with the activities, it is equally important to account for those students who
are not challenged enough. To increase the level of reasoning and promote
further investigation into the topics introduced through the activity, a
teacher could add an increased challenge of counting rectangles with specific
width to length ratio. Having students count the number of cubes found in a
large cube, a 3-D version of a chessboard, such as a Rubik’s cube, could also
push students to now think through volumes. To bring the activity to an even
more challenging level, students could attempt to count the number of squares
found in a cube – requiring an in depth understanding of surface areas of 3-D
shapes.
When providing
activities for students, teachers must be able to prepare for the possible
troubles their diverse students may have and the ways to assist them through
these problems. They must also prepare for the lack of challenge that some
students may experience and again have additional suggestions to ensure
continuous learning. Overall, this activity reinforces the idea that although
we may be working on developing skill sets to become effective and meaningful
teachers, we must never lose the part of us that remains a student.
While reading the Thurston article, I was struck with the realization of two essential aspects that were missing from the general mathematics high school classroom: socialization and familiar language.
While studying mathematics at the undergraduate level, I can remember being intimidated by the familiar language used by graduate students and professors that caused me more difficulty to understand and relate to the topics being discussed. Thurston comments on the social aspect of math and its important role in the transmission of ideas, however one must already be at a certain level of understanding to continue exploring ideas.
If we apply the same observations to a high school classroom, we might be able to gain some insight into how our students can feel intimidated about approaching the subject. If some students have learned the academic vocabulary and gained the confidence in its application, their peers may feel intimidated and shy from ever trying to understand the topics. The academic language has always presented difficulty for students. Accepting the challenge it presents in social situations, how it can assist learning or be a detriment to learning, and how it can isolate students must now also be considered.
Reading #2: Benny's Conceptions of Rules and Answers
After presenting the case of Benny and his grasp of
fractions and decimals using the IPI system, Erlwanger’s observations on what
Benny absorbed reminded me of our reading last week. The emphasis on rules and
memorizing certain methods for specific problems presents itself as encouraging
a strong instrumental understanding of the concepts. The lack of communication between
the students and the teacher, the seeming discouragement of it entirely, caused
for serious misunderstanding or ability to grasp a concept by the student to be
unaccounted for. Thus, this type of instruction prevents any sort of support
system to be put in place. Instead of ensuring that a student understands the outlined
topics, the method is in fact checking if the student has understood how to
master guessing the correct answers. There is certainly no emphasis of
understanding the rational behind the correct answers.
It seems to present a valid, although extreme, case as to
why teachers should be conscience of the level and extent that instrumental
understanding is incorporated in their lessons.
Reading #1: Instrumental vs. Relational Understanding
Reading this article caused me to associate instrumental
learning with memorization where as relational understanding is the ability to
grasp a concept and associated and applicable “rules”. As a teacher, it may be
less involved to draft a set of rules for students to remember and the required
topics can be completed. However, as students explore more difficult and
reaching ideas, they are not able to rely on a firm foundation with a
relational background of the material to enable them to explore the new
information.
In my past experience as a tutor, I often found that my
students were either of an instrumental understanding or a rational
understanding (but far less so). The difficulty with having “instrumental
students” was the increased difficulty in solving new problems that they had
yet to memorize the solution. I soon discovered that if I could teach them the
basic ideas and concepts that they could apply to any mathematical problem, in
essence fill their toolbox, their overall skill and confidence in their
abilities grew. They then became able to break down a problem into conceptual
ideas they understood to build a solution rather than memorizing a “type” of
problem.
The extra time and individual attention needed to take on a
relational approach to teach mathematics can be overwhelming, but a student’s
overall long-term success could be greatly impacted. Thus, I feel it is
important to consider opening that line of discussion with colleagues and
exploring the idea of including these practices at all levels of learning.
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