This week in my class we took a focus on the student's perspective when planning a lesson. Below is a script we created showing student concerns after receiving instructions for consolidating the lesson.
Teacher: Now, instead of solving an
equation given to you, you are going to create your own equation with a partner
and you must come up with a word problem that corresponds with the equation you
create.
For example we make an equation,
29 - x= 13. One example of a corresponding word problem would be "Jenny is
on the Price is Right and the price she guessed for a new toaster was $29. The
difference between Jenny's guess and the actual retail price is $13. What is
the actual retail price?"
Then you would solve for x and
show all your steps.
Student 1: Miss,
can we use any numbers?
Teacher: Yes, in your equation any
numbers can be used and you must have one variable "x" to solve for.
Student 2: So do
you have to show all the steps when you solve or can you just write "x =
..."
Teacher: When you solve your equation,
I want to see all the steps you took to isolate or solve for x, and end by
clearly stating statement "x = value".
Student 3: Can we
do this in pairs?
Teacher: Yes groups of two only. Each
group of two will hand in one paper with one word problem, equation and
solution steps.
When planning any type of lesson or assignment, it is
important to remember that explanations or instructions you hope to provide to
the students may not be as clear and concise to them as they are to you.
In today's classrooms one question
teachers ask themselves when lesson planning is "How can I incorporate
technology into my lesson?" There are numerous apps and websites available
which can be used to enhance a lesson or for student aid. The challenge for
teachers is to explore these methods of technology and decide if they are
purposeful towards student learning.
One purpose technology should serve
in the classroom is to foster math talk. During our class this week we were
working with the website Desmos, more
specifically we used the feature Polygraphs.
In this feature teachers create a class account and students use their own
electronic device to login to the class game. This game is just like an
electronic version of the board game Guess
Who where students are provided with a grid of different representations
and need to guess which picture their classmate has picked. In our case we were
working on the quadratics section and were provided with multiple images of parabolas.
In order to make an educated guess the students must ask questions (through the
app) in order to eliminate the incorrect pictures. This process forced us to
practice our math language and knowledge relating to quadratics and
translations of parabolas. For example the first question a student may ask "is the parabola concave up?" If
the response is yes then the student can eliminate all parabolas facing down. I
found the game to be fun, interactive and it tested my math knowledge not only when
I was the question person but also as the person answering.
Class having fun with Headbanz
To compare
to a more traditional lesson without the use of technology, we also did an activity
called Headbanz. In this activity the
teacher presented each of us with an equation written on a card which was attached
to a headband. Without looking at our equation we were suppose to wear our headband
and ask yes or no questions to our classmates in order to identify our equation.
This activity was also excellent for fostering math talk as students needed to
clearly explain their thinking in order to describe their equation. Both these
games are great for the "Minds On" portion of a lesson as they get
the students into the math mindset and prepare them to think critically. I'm
sure positives and negatives can be described for both methods however I would
not designate one activity to be better than the other since they both accomplish
the same goal.
Another purpose for technology in
the classroom is to make it easier for students to apply their knowledge to
real life scenarios. Using Geogebra
we were able to overlay a parabola on top of a freeze frame video of a man
shooting a basketball towards the net. From the students perspective we were able
to see what would happen to our graph as we manipulated different parts of the
equation in order to match up with the path of the ball. I enjoyed this
activity so much that for my next lesson plan I will have my students find
their our picture or video of something that resembles a parabolic curve and
have them come up with the equation of the curve using the tools in Geogebra.
Without question teachers should be prepared to teach to a
class that has a range of learning levels however a classroom such as this is a
constant challenge for teachers. In regards to math teachers, this challenge
arises most often in grades 6-9 where each new concept heavily relies on the
use of prior knowledge. Imagine teaching area and surface area formulas in a
grade 8 class where a group of your students do not know the proper order of
operations (BEDMAS) or do not know how to perform basic operations without the
use of a calculator. This would be very difficult to teach to the entire class when
a large learning gap exists between these students and students who can apply
the new formulas to problems involving critical thinking. The goal of the
teacher is narrow the gap and one method of differentiated instruction used is
available through EduGains called "Gap Closing Resources". These
resources include topic specific diagnostic tests which can help teachers
pinpoint what concepts students are stuck on. Once a problem area is identified
there are extra work sheets for the student to practice. I have personally
worked with these resource sheets in a grade 7 math class. In a class of 26
students there were 5 students who were using Gap closing worksheets. After the
teacher had taught the lesson the 5 students would work on problems in their
Gap Closing booklet which were related to the lesson. They were required by the
teacher to finish those questions before attempting the questions from the
textbook or work book assigned that were assigned to the entire class. I
believe these booklets do help, most of the students I worked with were making
the same mistakes and after practicing questions in the booklet they were able
to understand where they were going wrong and as a result they caught up to the
rest of the class.
To assess just how wide the learning gap is in the
classroom, teachers can incorporate a variety of websites and apps into a
pre-lesson or pre-unit "warm up" as opposed to a traditional test of
existing knowledge. In my class this week we explored a website called
"Which One Doesn't Belong" where four pictures within a similar
category are shown together and students must describe how and why one picture
is unlike the rest. The website organizes the pictures into the sections; Shapes,
Numbers and Graphs and each section has pictures that range in grade level. I
think a great idea for teachers would be to combine the concept of this website
with a Kahoot presentation. Not only can teachers use the pictures from wodb,
they can create many similar questions focusing on the same specific
mathematical concept. By using this anonymous forum, students may feel more
relaxed and inclined to answer honestly instead of guessing or not answering on
regular tests. Also by customizing their own Kahoot, teachers should be able to
get a general idea of the knowledge gap in their classroom. In the end this
teaching strategy would not be a loss because for students who are struggling
this exercise could possibly help them better understand a concept and for
students who are excelling this is a great way to review and strengthen their
skills before moving on to a new unit.
References:
Gap Closing. (2016). Edugains.ca. Retrieved 11 October 2016, from http://www.edugains.ca/newsite/math/gap_closing.html
Which One Doesn't Belong. (2016). Wodb.ca. Retrieved 11 October 2016, from http://wodb.ca/shapes.html
Kahoot!. (2016). Kahoot.it. Retrieved 11 October 2016, from https://kahoot.it/#/
Image:
Learning Gap. (2016). Retrieved from https://www.quora.com/I-always-hear-people-advising-JEE-aspirants-to-get-their-concepts-cleared-But-how-do-we-get-clear-concepts
The one thing about math I have always taken solace in is
the fact that there is one answer. That one answer poses a challenge that I
feel I have to meet. It is just like reading a book, doing a puzzle or even
watching a movie I need to see it all come to one concrete end. Conversely to
these thoughts, I have also learned that math is not always straight forward,
it can be messy and complicated with multiple answers or there are those questions
that remain unanswered. Of course I still love the straight forward questions
but as I continue my work in mathematics, now through a teaching lens, I
realize that the real learning took place while working on the "messy"
questions. In order for students to work on these types of questions teachers
need to develop confident, independent and critical thinking in their students.
One way to gradually develop these skills is to implement open ended questions
in a lesson plan. I can tell you that at first trying to use open ended
questions in math is not as easy as english for example, but with practice it
becomes second nature. In my teaching mathematics class this week we discussed some
common and effective examples of open questions. Questions that leave a lot of
room for student interpretation are comparison questions; asking students how a
figure or an equation compares to another. Another key point we discussed was
how teachers should provide enough information to form a problem but not too
much information to completely restrict the question. Using words such as
"approximately", "minimum" or "maximum" can open
the question to multiple solutions. Below are two examples of open questions
where the approach was providing the answer and having the students work
backward to formulate the questions.
Image 2
Image 3
In one
of my placements during my undergrad, I taught the unit on area and perimeter to
the grade 7 class. One assignment I gave to the class was similar to one of the
unit problems from the textbook however I tweaked it making it more of an open
ended problem. The students were required to create a patio design under some restrictions
including using only the five specific shapes. (Prior to completing the problem
I had marked out the five shapes on the floor using tape and the students
calculated the area by measuring the floor tiles.) In the end each student had
submitted a completely unique patio design. As long as the design included the five
shapes and met the minimum area requirements then each of the designs were 100%
correct. I think this assignment was a good balance between open ended and
closed questions. After asking the students to formulate their own solutions to
the design, I then asked all of them the same closed questions which they had
to answer in relation to their own individual design area. I believe problems such
as these are beneficial to all students because at the very least everyone can
make an attempt and as I mentioned before this is a step towards building their
confidence when dealing with math.
Geometric Containers can be filled with water, rice etc.
Ropes and Ruler for tying knots activity
"Math
Manipulatives" This is common vocabulary for present
day math education and at the same time it should not be unfamiliar to teachers
and students in older generations. In planning a math lesson, today's teachers are
encouraged to be thinking "manipulatives, manipulatives,
manipulatives!". Instead of having that box of tiles or cubes in the back
corner on a shelf, they are up front and centre, used in almost every lesson or
at least it is explained to the students how manipulatives could be used in the
lesson. As a student I remember having manipulatives in the classroom however
they were mostly used as an aid for learning instead of a tool to build
learning concepts. What I mean by this is teachers would not immediately
incorporate manipulatives in their lesson and when students would struggle with
concepts the teacher would refer them to the manipulatives station. This could
have been just from my own personal experience but there seemed to be some type
of stigma surrounding students who required manipulatives in math. It was
almost as if the students who understood the concepts and got the answers right
away without the use of any further visual were somehow stronger than the
others. This rationale of course is not true; manipulatives are used far beyond
the use of a visual aid. The first and most obvious use for manipulatives in
math is to engage students in their learning. There are so many forms of
manipulatives such as apps, games, and the traditional tiles or household
objects all of which make math fun and learning worthwhile for the students. In
our "Teaching Mathematics" class this week our instructor had brought
in sample activities that make use of a variety of manipulatives. One activity was
used for teacher linear relations. It involved ropes which the students had to
tie knots in order to determine the type of relationship between the length of
the rope and the number of knots. No matter what the manipulative being used
is, the teacher has to ask themselves is it useful, is it fun, is the activity
memorable enough for future student reference.
Algebra Tiles
Manipulatives are also used to
develop a student's critical thinking skills similar to the skyscraper problem
in my last post you can have students at all different levels try a problem
with manipulatives and see where it takes them. For all students the
manipulatives should help to visualize their problem and where they need to go
from here. For some of the stronger math students, having the manipulative
might allow them to come to a conclusion alternate to their original thoughts
and they are able to learn there are multiple ways to facing a problem. In
class we experienced something similar when our teacher handed us each a card
with a number or word and in a group we were to create one sentence from our
cards. This activity was used as more of an icebreaker however you could repeat
this process by having different variables on the cards and asking students to
create a function or have descriptive words such as max/min and have students
describe a modeled situation.
Word Cards
Another use for manipulatives in the classroom comes as an advantage
for both teacher and student. In my class we had discussed an article on the
difference between Relational vs. Instrumental understanding. In a mathematical
context instrumental understanding is following rules and applying formulas
without fully understanding the reason why and relational is the latter. In
order for students to have more of a relational understanding teachers can use
manipulatives to make that connection for students between the "what to do"
and "why are we doing it".
Below
I have attached a link to an article titled "Why Teach Mathematics with Manipulatives?". I believe the
article does an excellent job arguing why manipulatives are critical in the
classroom and it clearly explains that through the use of manipulatives
students facilitate their learning by making the strange become familiar.
Retrieved from https://pbs.twimg.com/profile_images/602027362247516160/mc_7Jfjv.jpg
This week in my "Teaching in Mathematics" class
we focused on a student's ability and process in problem solving. Problem
solving
questions are most often the hardest questions for student to complete
and/or understand. The most common phrases from students during problem solving
are "I don't get it!", "What are we doing!", "What's
the answer!" and the worst one "I give up!". My teacher
presented us with "Skyscraper" math problems and let me tell you, the
atmosphere in our classroom full of adult Teacher Candidates was not much
different from a class full of students. To give a brief summary of the
Skyscraper problem, the students are given a handout with a square grid (can be
2x2 and up) and on the sides of the grid each square is labeled with a number
as pictured here in this 4x4 example:
Retrieved from https://solvemymaths.files.wordpress.com/2015/01/ss.png
Retrieved from https://solvemymaths.files.wordpress.com/2015/01/ss1.png
We were given linking cubes by our teacher and were
instructed to find the number of cubes needed on each square in order to
represent the numbers on the side of the square. Huh? Exactly my thoughts! Well
our only clue was that this problem is similar to Sudoku puzzles where a single
number can only be represented once in each row and column. Our group started
to build our towers for each square and right away I noticed our different approaches
to solving this problem. I had taken the blocks and built the required number
of buildings to fill the grid. Based on the Sudoku clue I knew that for a 4x4
grid we needed 4 buildings of each height (4 cubes tall, 3 cubes, 2 cubes, 1
cube). I needed to see what we were working with before I made decisions about where
to place these buildings where as my group members were placing cubes right on
the grid linking new heights and taking cubes away as they filled the grid. Once
our group had reached the point of frustration, we asked our teacher for help
and she explained that we should be looking to count the number of "pop
out" cubes from the top. We attempted from this perspective for a while
with no progress. In the end we did solve the problem by taking on a different
view and counting the visible faces of the cubes. Once we knew the key to this
puzzle we jumped from the 3x3 grid straight to the 5x5 and by applying our accrued
knowledge completed it with ease. Now that we fully understood the problem we
also realized that we could add one block to each building and have the same
correct answer.
The same
process and results can be seen in any problem solving situation, the goal for
us teachers to is change the negative phrases I mentioned earlier to more
positive and hopeful questions such as "What can I do next?",
"What would happen if..?". The issue we should be focusing on is not
that the student is struggling; it is how they react and overcome the struggle.
My university math professor would say to us "struggle is good" and
we would all laugh and possibly cry at the thought of working through some of
those calculus problems but in the end I know that he was right. Whenever I had
a question I found impossible, it challenged me to think creatively, try new
things and not give up. Also for the people reading this saying "I
hate math I would have given up eventually" rest assured I did not
solve all of those impossible questions and I did get the answer from my
friends or the professor but it was because of my initial struggle and failed
attempts that I understood the
problem and the solution. Know that this can be applied to any subject or any
problem in general. Before students give up or are told the answer they
continue to learn as long as they make an attempt, success or fail they are
gaining understanding through experience. This message I am sending is deeply
related to the Constructivist Learning Theory in which "the core idea is
that learners actively construct their own understandings, rather than absorb
what they are told or copy what someone shows them" (The Problem-Solving Cycle: Professional Development for
Middle School Mathematics Teachers The Facilitator’s Guide, 2003).
This week I leave you with this video that explains
the answers are never easy, we have to grow in our experiences and whether it
is in something you hate or something you love it is possible for everyone
learn.
References:
The Problem-Solving Cycle: Professional Development for
Middle School Mathematics Teachers The Facilitator’s Guide. (2003) (1st ed.).
Retrieved from
http://www.colorado.edu/education/staar/documents/FacilitatorsGuide.pdf
Khan Academy,. (2014). You Can Learn Anything. Retrieved from https://www.youtube.com/watch?v=JC82Il2cjqA
My name is Stefanie
DiSimoni I am a Teacher Candidate at Brock University and I am a "Mathemaddict"! I truly
love math and hope to spread the love to students, teachers and parents through
this portal. In this blog I will be posting everything related to math that I
find exciting and/or awe-inspiring. I
also will be posting my opinion on teaching certain topics in math and I will
share my experiences with math in the classroom.
