This week was the sixth and final week of our time together in Math II and to be honest, I feel as though this semester FLEW by. This weeks class was focused on assessment, but more importantly, on providing students with descriptive feedback and report card comments. One thing that Pat brought up in class this week really stuck out to me and it is something that I think will definitely influence the way that I mark and provide comments to my students. The comment that Pat made that really stuck out to me was something along the lines of providing students with just marks does not improve their performance. When you mark a child's work and only provide them with a mark or a grade, that is all they are going to care about. If the student scores high, then he or she is happy and that is all that matters. Conversely, if a student scores very low, when he or she sees their mark, they are going to be overwhelmed with anxiety or apathy, depending on the student. The student will very rarely care to think about where he or she went wrong. Instead, Patricia explained to us that it is much more effective to provide students with comments on their work as the main focal point of grading, and have the mark itself as a secondary piece of information. IF the comments are the focal point, then the student can see where they need to improve/where they made a mistake based on your comments, and thus they are able to perform better next time by applying your comments. I really think that this type of formative assessment will prove to be very successful in the classroom. We also covered writing report card comments and how we should structure our report card comments. In class we took part in a math conference that I thought was informative as each table group was given a piece of work and we were responsible for commenting on the quality of the work. We all rotated around the class, and I found this useful because it really opened my eyes to how other people in the class mark, and by seeing how others look at the same piece of work, it helps to broaden your own understanding.
This week Paddy and McKenzie were responsible for leading their webinar that was focused on report card comments and grading. I think that the two did a great job of leading the webinar, as well as providing excellent insight and research/resources links for us to access when we are in placement and in our own careers.
Reflecting on this, my second year of Mathematics in teachers college, I can confidently say that I am not the same student teacher that walked into Patricia's class two years ago. When I first started teachers college I was intimidated by teaching math. However, after Math I and my first placement, which was a math classroom, I actually was kind of disappointed to learn that I would not be in a math placement this year. I feel as though I have become very comfortable with teaching math which is a huge change from when I first started. Patricia has done an amazing job of providing us with the information, resources, knowledge, and confidence to become effective math instructors.
In this weeks
class there was a main focus on the concept of blended learning. Before
teachers college I really did not have any idea about what blended learning
was. I obviously figured it was something like mixing two or more styles
together in the classroom but I now think that I have a much better grasp of
what blended learning is, and how to incorporate it into the classroom. My
first encounter with blended learning was the first year courses in the
bachelor of education program at Brock. We were told that we would be partaking
in blended classes, which meant that the first two hours of class were in the
physical classroom, while the last hour of class would be spent online doing
activities on our own. My first understanding of blended learning was just
this, online and in class work used together. However, I have come to learn
that blended learning is much more than simply two hours in class and one hour
online.
Pat had us go
through some resources in class this week, and fill out a booklet with information
about blended learning. My personal definition of what blended learning is as
follows, both teacher led, and student led and focused activities; both in
class and online modules in which students are responsible for their own
learning. Blended learning is more than just technology, although technology
can be a major tool used in blended learning. Rather, blended learning is about
the teacher stepping back, and allowing the students to really take control of
their learning on their own, whether it be through inquiry, or technological
applications that the students use individually or group work where the
students have to regulate and govern themselves.
Our online
module this week was focused on the premise of teaching mathematical concepts
rather than memorization of formulas, as well as the importance of intuition in
math, and using visual representations and drawings to help students
understand. I have found these online modules to be very enlightening to me as
Jo Boaler does a great job of explaining these concepts. I particularly liked
the idea of teaching concepts rather than memorization. There are really only a
few basic mathematical concepts that are built upon and added to, as we advance
in our mathematical career. If we are successful in teaching our students to
understand these basic concepts, moving up in their mathematical career will be
easier for them as they have a strong base understanding of the concepts needed
to be applied in advanced math. Understanding these concepts allows students to
have enough knowledge to be able to rely on their mathematical intuition, which
is an extremely useful mathematical skill that we use almost everyday.
Lastly, this
week was the week Peter and I were responsible for leading our peers in our
webinar. We chose to tackle the issue of financial literacy and I think we did
an amazing job. Our main focuses were that times have changed since we were in
school, and there is a much bigger focus on financial literacy. This is due to
the influx of entrepreneurs entering the business sector, as well as the
general population taking more control over their personal finances rather than
just allowing the bank to take care of everything. We also wanted to focus on
fun student centered ways to incorporate Fin. Lit into the classroom (Real
World game I was sent by my T.A), as well as providing students with math that
they will actually use in their everyday life, and that will give them a head
start when they enter the real world in terms of knowing concepts such as
mortgage rates, RRSPs etc. Instead of recapping our whole webinar I will provide
a link to the video below. If you also would like more information regarding
the Real Game, feel free to comment or email me and I would be happy to send
some information along.
For this week’s class we were responsible for
reading two resources that were focused on creating and administering
mathematically rich tasks to the students in your math class. The resources
were found on the website NRICH and were very informative regarding what
exactly is a Rich Task, and the benefits they have when presented to students
as opposed to the more traditional tasks of the past. In Monday’s class we covered some great
concepts and went over what it is to create rich mathematical tasks. Some of
the important information that I recorded was that Rich tasks provide students
with multiple starting points, which allow for students to jump in and start
wherever they feel most comfortable. In addition, there are usually extensions
within the problems, meaning that there is usually a second part to the
question where you have to use the information found out in the first part, to
solve the second part. Another concept that we covered that goes hand in hand
with Rich tasks are open tasks. Open tasks are tasks that do not have one
exclusive right answer. These types of questions allow students to work on
their intuition, reasoning, and logical thinking skills; as long as the
students can support their answer logically in an open question, they are
right. I thought that there was one quote from the, “What is a Mathematically
Rich Task,” resource that did a great job of summarizing what a Rich Task is.
“Rich tasks open up mathematics. They transform
the subject from a collection of memorised procedures and facts into a living,
connected whole. Rich tasks allow the learner to 'get inside' the mathematics.
The resulting learning process is far more interesting, engaging and powerful;
it is also far more likely to lead to a lasting assimilation of the material
for use in both further mathematical study and the wider context of
applications.”
This quote leads perfectly into the videos
that we had to watch for this week’s online module. The main point in one of
the videos is that math is really only built upon a small number of main
concepts. And it is upon these main concepts that all other advanced math is
based off of. Therefore, if we are able to teach out students to grasp these
few main ideas or key concepts, then when they move forward in their academic
career they have a strong base and knowledge upon which they can build. The
understanding of these main key concepts, also allows students to be able to
think intuitively about math. This was another video that we had to watch
online this week. Intuition is a major part of everyday life math as well as
math in the classroom, it deals with proportional reasoning and logical
thinking, and is a skill that is essential for success in math; it allows us to
make quick judgments and estimations about math, which help us in solving the
problems. Implementing Rich Open tasks in the math classroom places the
importance on Enduring understanding. We, as teachers, need to be moving
towards implementing more open and mathematically rich tasks in the classroom,
rather than the black and white traditional problems that many of us
experienced during our early academic careers.
This
week Adriana and Kathlene ran their Webinar on differentiated instruction.
I
think that the two of them did a great job at running the webinar and providing
good insight into implementing DI in the Math classroom. However, I have been
finding that these online webinars are very unreliable and whether it be audio
or video issues there almost always seems to be some sort of issue in the
process of carrying a webinar out. That being said, the two ladies did a great
job of carrying on throughout the technical difficulties and were able to think
quickly on their feet to fix the problems.
This week in Mathematics class we were
responsible for reading a monograph issued by the Ontario Government as a part
of the Capacity Building Series entitled, Differentiating Mathematics
Instruction. As you can probably tell from the name, this monograph was focused
on the importance of differentiating instruction in the math class. The
monograph contained three main points which were: Focusing Instruction on Key
Concepts, Using an instructional trajectory/landscape for planning, and
Designing open and parallel tasks. One of the first things that the reading
states, however, is the importance of knowing your students and their zone of
proximal development. For us, as teachers, to be able to effectively
differentiate in the math classroom, we have to be aware of where each of our
students are, their abilities, and what they need to improve on/where they need
to be. This resource also gave a number of great tips on how to go about
achieving the three steps in differentiating for students in the math class.
One of the points that really stuck with me is teaching mathematical key
concepts rather than a specific curriculum expectation. For example, if you are
trying to teach how to solve a simple multiplication question, to allow for
easier differentiation, you should teach the concept behind multiplication. Such as Multiplication has many meanings (e.g., repeated addition, counting of
equal groups, objects in an array, area of a rectangle), The resource provides
some great illustrations and more key concepts behind multiplication that I
think prove to be very useful when thinking about how to differentiate.
Another point from the resource that
we spent a significant amount of time in class on was designing open and
parallel tasks. Parallel and open tasks allow for everyone to participate in
math activities regardless of where they are in their zone of proximal
development. For example, in class Patricia put up two similar but different
questions. The two questions were very similar and required the same processes
to solve it, however, one of the questions had numbers and values that were a
little simpler compared to the other. For example, one question could ask for
what 50% of 27 is and the other could ask what is 28% of 27. By doing this, the
students who might struggle with finding out 28% of 27 can still be involved
and solve a problem, and maybe after completing the simpler question, they
might move to the more difficult one. As long as the key concept behind the two
tasks is the same, this type of differentiation can be highly effective. Patricia’s
question about the jeans was a great example of this. Patricia asked us to come
up with our own price to insert into the question. This is a great open problem
as it allows the students to pick their own difficulty. For example, if you
pick an unusual price, or one with decimals, it will make solving the problem
more difficult than if you picked a whole number that is easily divisible.I think that open questions and parallel
tasks such as these are a great too to use in your classroom. Not only does
this allow students to be active participants in their own learning it also
makes the task of differentiation easier on teachers. If you can make it a
habit to provide two parallel questions or open questions in every lesson you
really don’t have to do much more differentiation as you are providing the same
information to everyone in the class, and regardless of their zone of proximal
development they will be able to participate.
The videos in the Online module this
week were also very informational and of value to us as aspiring math
educators. This week the videos were focused on mistakes, and how making
mistakes is such a crucial part of learning, especially in the classroom. In
the videos, Jo actually points out that when we get a question right, we are
not growing our brains at all. However, when we make mistakes, our synapses
start firing and we create new connections in the brain which leads to better
understanding and thus, learning. I also thought it was interesting that Jo
stated her mathematician colleagues are some of the slowest math thinkers that she
knows. I always associated being fast in math class with understanding the
material the best and being the most efficient at solving problems. However, Jo
states that there is no evidence that states that the faster you are at solving
a math problem; the better you are at math. Rather, those who take long to
solve a problem are thinking deeply about the problem. We should be encouraging
our students to take their time and to think deeply about the concepts behind
the equation/problem and to not worry about how fast they are able to solve the
problem
This week Jessica and Danielle
hosted the first of the webinars. Their focus was on inquiry in the math
classroom, and the two of them did an amazing job. While all of us are still
trying to figure out the technology, the two of them handled all the glitches
well and were able to present a clear and concise webinar. What I liked most
about the webinar was their use of the interactive Google docs where everyone
worked on the same sheet. I think Peter and I may have to make use of that
great idea in ours. Overall it was a massive success and they definitely set
the bar high for the other presentations.
This week we were responsible for reading
three different articles regarding mathematics and different mathematical
concepts. The first document that we were responsible for reading was the “Paying
Attention to Mathematics Education, K-12” which was issued by the Ontario
Government. In this document 7 foundational principles for Improvement in
Mathematics are laid out as follows:
1.Focus on Mathematics
2.Coordinate and Strengthen
Mathematics leadership
3.Build understanding of
effective Mathematics Instruction
4.Support collaborative
professional learning in Mathematics
5.Design a responsive Mathematics
learning environment
6.Provide assessment and evaluation
in Mathematics that supports student learning
7.Facilitate access to
Mathematics learning resources.
These principles were issued serve as a
guide for planning and implementing improvements in mathematics teaching and
learning and as support for Math programs and changes that school boards are
already implementing. The first principle states that teachers must focus on the
curriculum in terms of what they are teaching, as well as what the students should
know and what they will learn in the upcoming years. The 2nd
principle focuses on ensuring that, the teachers, as well as the board members,
principals etc are all aware of effective mathematics instructing techniques
and can engage in conversations with the teachers about how to improve etc.
This leads to the third principle, which is building an understanding of
effective math instruction for students to become successful in the 21st
century. The 4th is to ensure that you and your fellow teachers are
working and learning together when planning math units etc. The last three
principles are focused on creating a responsive learning environment for the
students, providing students with assessment and feedback that supports their
learning and lastly ensuring that students have access to varying learning
resources. I thought that this was a valuable resource because not only did it
provide principles for improving your mathematics pedagogical ability, but it
also gave specific examples regarding how you could go about achieving this
improvement.
The second document that I read was, “Paying
Attention to Proportional Reasoning K-12.” This document was focused on
proportional reasoning and why it is so important to Mathematics education. I
always knew that proportional reasoning was important, but it wasn’t really
until I read this article that I realized that we really do use proportional
reasoning in so many different situations, in, as well as outside of, the
classroom.Teaching a student to really
understand proportional reasoning gives them a logical lens through which they
are able to look at the world. Proportional reasoning involves a number of
different aspects and concepts and this document does a great job of providing
examples of these concepts and explanations as to why they are so important to
the greater concept of proportional reasoning. The document also lists the strands within the
grades that are related to proportional reasoning as well as sample questions so
you can focus your teaching. One last piece of this document that I found
useful was the inclusion of sample EQAO questions
for grades 3 and 6.
Strand of Math. Proficiency. 2001, Journal. http://bit.ly/2cAdzUY
Lastly we were responsible for reading, “The Strands of Mathematical
Proficiency.” This document describes the 5 strands of math proficiency as
Conceptual Understanding, Adaptive Reasoning, Strategic Competence, Productive
Disposition and Procedural Fluency. The document describes these different
strands in detail whilst providing examples of each. The most important piece
of information from this document is that fact that it is stated that these
strands of mathematical proficiency are not stand-alone concepts. Rather, they
are intertwined, much like a rope, and when put together they result in
mathematical proficiency.
There were three things from class this
Monday that really stuck with me. I
think that the personality colour test that Patricia briefly touched on would
be a great activity to lead your class through at the beginning of the year to
establish the different learning styles and thus be better able to tailor your
lessons to meet your specific class/student needs. I also thought that the hate
example was a great refresher on the concept that everyone looks at math
differently. While one person may prefer to think of the problem algebraically
the other student may want to think about it visually etc. Finally, the video
about the shepherd’s age really resonated with me. I was interested in this
video because it showed me the problem with much of the math instruction in today’s
classrooms. Students are taught to really think and understand the
problem/question they are being asked. Rather, they instantly start looking for
numbers, which they can use as clues, to plug into an equation and reach an
answer. This is the difference between Understanding vs just Doing. We have to
be teaching students to really think about questions they are being asked and
to try and understand the problem/question first, and then try and solve,
rather than just looking for information to plug into an equation that does not
make sense.
