Questioning the students within a classroom is by far one of the most important aspects of teaching to engage the students in the subject matter and keep them thinking. Not too often do you hear about discussions and questioning being done in a Math classroom. Usually Math classrooms consist of students busying themselves with problems, but rarely ever does anyone ever take them to the next level by asking them open questions and creating discussions. Or maybe that was just my math experience?

Now that I am on the other side of the desk within a math classroom, I can now understand better why I didn't have any discussions or questioning being done. It's not all that easy in a subject that is so concept based that there is much further thinking that can be done. Or so I thought.

The Ontario Government has put together a fantastic resource on Effective Questioning in a Math classroom. This is a great way to open your Math classroom up to engage the students and get them using logical reasoning and critical thinking skills. Reasoning, proving, problem solving, and communicating are all mathematical processes that are required in Mathematics, so creating discussions and questioning it allows for these processes to be used and engages the students.

Even a questions as simple as "How else could you ...?" makes the students look at every problem more logically and use their critical thinking skills to expand on the same problem in multiple ways. This uses their problem solving strategies to bring them to another level of mathematics, rather than just knowing how to solve the problem but understanding it.

Effective questioning is key to an interactive classroom, and no matter the subject there is always a way to incorporate it in the classroom. I know I feel more confident about it now!

## Wednesday, October 30, 2013

## Monday, October 21, 2013

### Tribes in the Math Classroom

Students experience a much greater level of success in their
education when they can learn in a comfortable and positive classroom
environment. When students feel compassionate
and emotionally attached toward their peers, they can feel comfortable making
mistakes, taking risks, and stating their opinion in front of their peers. Classroom communities are delicate and
time-consuming to build, but if peer relationships are nurtured, students can
learn to enjoy their education and will associate learning with positive
experiences rather than negative ones.

Tribes
Learning Communities is an educational philosophy that incorporates
socioemotional learning goals into every day teaching. In Tribes classrooms, teachers teach using
specific strategies that present curriculum material in ways that promote a
strong classroom community and provide opportunities to reflect on specific socioemotional
goals such as inclusion, social skill development, goal setting, and conflict
resolution. Here are some examples of
Tribes activities that can be modified for use in the math classroom:

**1**

**1.**

**What’s on Your Mind?**

This activity is completed after students
learn a new concept. In this activity,
students take turns sharing any concerns or point of clarification that they
require (these concerns are shared on a piece of chart paper). Next students pair up and help one another to
clarify the concerns. This activity is
great for review before a test, and is an excellent diagnostic tool for
teachers as well. In reflection
afterwards, teachers should lead a class discussion on how students felt when
their peers helped them to understand what they weren’t sure about, what
personal strengths they used to be successful in the activity, and which
communication skills were important in this activity.

**2 2.**

**Snowball I-Messages**

In this activity, students each
write an anonymous “I-message” (ex. I feel _____ when _____ happens) onto a
piece of paper, crumples it up, and tosses it to another student for him/her to
write a comment on. This activity can be
modified as an activity for practice in the math class. Students can each write down a practice
question on their snowball (piece of paper).
During the first snowball toss, students can answer one another’s
questions, and during the second toss, students can mark the answers, leaving
constructive comments. Students should
be encouraged to support one another and to clarify incorrect answers
respectfully, and should reflect afterwards on their ability to create, answer,
and mark the questions, as well as their ability to write constructive comments
to their peers.

**3 3.**

**Graphing Who We Are**

This activity is an excellent activity
to practice student’s graphing skills, and has obvious applications in the math
classroom. In this activity, students
collect data about their peers (eye colours, number of siblings, heights, etc.)
and graph these sets of data. When
students are finished, they can present their graphs to the community. After the presentations, the class should
reflect on the types of cooperative skills that they needed in order to collect
data from others, why individual differences are important, and how to approach
classmates with respect.

*All images and activities taken from:

Gibbs, J. & Ushijima, T. (2008).

__Engaging all by creating high school learning communities__. Windsor, CA: CenterSource Systems.### Connection Between Math and Music

Students can learn much more
efficiently when the curriculum material becomes relevant to their interests
and their daily activities. When
students can make connections between different subjects, their learning
becomes more meaningful and long lasting.
For all of those musically intelligent learners in your math classes,
here I will outline some of the numerous connections between music and
mathematics.

In my experience with school
music, a high number of music students excel in the areas of math and
science. I have since sought out
information on the topic and have found that several studies show a connection
between music education and the development of mathematical skills. In each the three studies listed below, the
authors found significant correlations between years of musical training and
children’s spatial-temporal processing.
Spatial-temporal processing is responsible for the development of logic
and mathematical skills. In these
studies children who were musically trained from a young age showed increased
development in the brain areas that correlate to spatial-temporal reasoning
than those who did not receive musical training. For more information on these
studies, see the following citations:

- Hyde, K., Lerch, J., Norton, A., Forgeard, M.,
Winner, E., Evans, Al,
*et al*. (2009). Musical training shapes structural brain development.__The Journal of Neuroscience, 29__(10), 3019-3025. - Rauscher, F., and Zupan, M. (2000). Classroom
keyboard instruction improves kindergarten children’s spatial-temporal
performance: a field experiment.
__Early Childhood Research Quarterly, 15__(2), 215-228. - Schmithorst, V., and Holland, S. (2004). The
effect of musical training on the neural correlates of math processing: a
functional magnetic resonance imaging study in humans.
__Neuroscience Letters, 354__(3), 193-196.

Mathematical relationships are
also fundamental to music itself. Every
piece of music is made of notes and melodic lines that are dictated by the
intervals between each note; the mathematical distances between each
pitch. Different notes are distinguished
by the differences between their frequencies.
The musical scale is made up of eight notes, which are related by the
ratio between their frequencies. For
example, a G and a D are a musical distance of a “perfect fifth” apart, which
vibrate at frequencies at a ratio of 3:2.

Musical rhythm also demonstrates a
mathematical relationship. Notes and
chords in a song make up different beats which each denote fractions in
time. Musical notation shows these
different rhythms, and each symbol represents a note as a different fraction of
time within the piece as a whole. Music
is also ridden with patterns and repetition that can be decoded and
analyzed. Musical chords are notated
using numbers and roman numerals, and all songs can be described according to a
specific functional form. Musical form
is determined according to the patterns of repetition between chords, cadences,
and musical sections.

In modern classical music,
12-tone composition has become quite popular, which is a completely mathematical
mode of composition. In 12-tone music,
the composer creates a pattern of notes using each of the 12 tones used in
Western music. This “tone row” (pattern
of 12 tones) gets repeated over and over for the entirety of the piece, which
creates a very mechanical sounding work of art.
For an example of a 12-tone composition, follow this link: http://www.youtube.com/watch?v=xrjg3jzP2uI

I have outlined several
connections between music and mathematics.
I think that math can be found in all areas of life, and should be
pointed out to students to make their education relevant. Can you think of connections between
mathematics and other school subjects?

## Sunday, October 20, 2013

### Connection Between Literacy and Math

Some of you might have
had the same experience as I did growing up where you were put into
either the “Math Brain” pile or the “Language/Artsy Brain”
pile. It is the common misconception among people that Math and
English are polar opposites and a person usually excels at either one
or the other. The problem with this mind set however, is that
Language skills affect Math skills.

As teachers, we
understand the importance of word problems in Math, and we have most
likely noticed how students who are weaker in English have more
difficulty in word problems. However, we forget that the common words
used in other math problems also take an understanding in English,
such as “estimate, evaluate, simplify, explain, prove, total, sum,
difference”.

Since as teachers, we
recognize this problem, the question becomes,

*how do we help those students with poor literacy skills, be successful in math?*
Here
are some strategies I have used while teaching math to students with
weak literacy skills:

- Focus on key words
- Read question to the student (Most students that have weak enough literacy skills to need this strategy are already on an IEP, and may already have this strategy listed as one of the accommodations)
- Reword questions
- Break question up. If the question has a few steps, I break it up into multiple questions, because shorter questions may be less overwhelming for the student.

Since
ESL students, are a common group of students that can struggle with
literacy in math, an article by Dr. Richard Barwell could be a good
read in understanding ESL struggles in Math. One of the suggestions
to teachers he makes is “learners of ESL find word problems less
perplexing if they are able to relate them to their own experiences”
(p.3). Barwell also gives the suggestions “be aware of the specific
linguistic demands of mathematics” and “students need
opportunities to discuss problems in order to make sense of
them”(p.3). You can read Dr. Richard Barwell's full article here.

Work
Cited:

Barwell,
Richard.

__The Literacy and Numeracy Secretarat: What Works? Research into Practice__. “ESL in the Mathematics Classroom” July 2008. Web. Oct 20, 2013. http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/ESL_math.pdf## Friday, October 18, 2013

### Beyond Word Problems

One of the most common things I have heard from students from the time I was a student was, “When will I ever have to use this in life?” As adults, we know there are countless ways that we use Math in life. As teachers, we all recognize the purpose of Math and so we try to show our students with math problems that make connections to real world situations, through the use of word problems. I have found, that word problems are not always enough for students to see the purpose of math, and that sometimes students need to experience using the math in a hands on practical activity.

A lesson that I once
used with some Grade 7 students in a Numeracy Intervention class was
successful in the students experiencing using math in the real world.
This lesson wasn't based around a specific Grade 7 expectation, as
the purpose of the Intervention class was not to cover the curriculum
(that was done in their homeroom) but to bridge a gap so that they
could be more successful in their classroom. The gap I was trying to
bridge in this case, was for the students to see a purpose in math
and see that math can be enjoyable.

The activity the
students did was to create checker/chess boards on four picnic tables
for the school yard. Throughout this activity they used math skills
such as measurement, division, and symmetry. The activity was student
lead, where with very little guidance, the students figured out what
calculations and measurements they would need to make the squares on
the boards equal and fit in the centre of the picnic table. They
discovered throughout the activity, how important math skills were.
As the students calculated, measured, and painted that tables, the
students had a blast. For one of the first times that term, I
witnessed at-risk students who normally complained about school,
enjoy themselves and look forward to class. This was a ground
breaking moment in the term, which changed their view of math from
that day forward.

For those that work in
a Catholic board, you might also appreciate how this activity also
worked towards some of the Ontario Catholic Graduate Expectations. By
creating the game boards for the school yard, the students learned
about being a responsible citizen, and they became very proud of
themselves that they did something for the school community.

## Tuesday, October 8, 2013

### Math About Me

One of the Grade 7 teachers at the school I work at gave her students a Math About Me assignment at the beginning of the year. The students needed to describe themselves using math equations, or other mathematical descriptions. For instance, I have worked for the school board

for nearly

for nearly

__5x3__years.
2

The level of difficulty of the equations (or other concepts) could be adjusted depending on the Grade/Skill level of the students. Perhaps they could describe their house using geometrical terminology, or show a graph of the genders of the people in their house to extend the skills. They could plot the ages of their family on a scatter plot. There are no set guidelines so each teacher could adjust the assignment to their own needs depending on the curriculum being addressed. Here are some pictures to show examples of the students' work.

The mathematical and creative possibilities are endless.

### Expectations, Accommodations, and Modifications, Oh My!

In a system where we strive to meet each student's individual needs, we may feel overwhelmed when trying to figure out the best way to accomplish this goal. All teachers share these experiences, but it becomes even more difficult at the intermediate Math level since there is a drastic difference between the intermediate and junior level Math programs. The intermediate expectations address topics which are not examined at lower levels so outside resources are often needed. Now the challenge is to locate high-quality, age/skill appropriate resources. Many of the technological tools we examined in this course will go a long way towards meeting these expectations.

Other resources and ideas may be obtained from colleagues. Some amount of experimentation may be necessary to determine the student's skill level, which can seem daunting at times. Again, collaboration with previous years' teachers, Special Education teachers, and Educational Assistants will often alleviate some of this pressure by offering ready made tools and ideas. There is no need for each teacher to "reinvent the wheel" if suitable resources already exist.

No matter what level a student is working at, our goal is to encourage their success by providing appropriate learning materials. In most cases, a teacher's best resources are their colleagues.

## Sunday, October 6, 2013

### Robert Lang: The Math and Magic of Origami

Robert Lang does a TED talk on The Math and Magic of Origami. This interested me because my Dad taught me to make an origami cube and a bird that flapped its wings when I was 8 or 9 and I have been fascinated with origami ever since. I liked how Lang showed how math was responsible for the leaps and bounds that have been made in origami. The ways this technology can be used in real life was very impressive from being able to fold panels in order to get them into space or fold a stint to get it into the human body. This could be an intersting way to show students the connection between math and the real world. Lang also talks about

**Tree Maker**, software he developed to create a crease pattern for the base of an origami creation. I would like to try using this as an exploration in a Geomotry and Spatial Sense unit. I think students could get very creative in making their own origami creations.**Robert Lang: The Math and Magic of Origami**### Wrong Answers are Okay

The Third International Mathematics and Science Study
(TIMSS) evaluated math achievement in several countries. Specifically the study
looked at differences in how Japanese and American math teachers conduct their
classes. This study was especially interesting because it looked at variables
associated with high levels of achievement in math and science. What I discovered
was that mistakes are more tolerated by Japanese math teachers and that students
weren’t as embarrassed by them. The wrong answers were seen as part of the
process of learning and not discouraged. I feel this is important if one is
trying to create a positive classroom atmosphere that promotes participation,
discussion, experimentation and risk taking. Mistakes are an opportunity to see
how students think and to help them, a process which often helps other students
who are in a similar situation. I learned not to avoid or be anxious
about wrong answers but to use them as an assessment and teaching tool to
consolidate learning.

### Declining EQAO Math Scores

A Hamilton Wentworth Catholic District School Board (HWCDSB)
release on Sept. 18, 2013, stated that the EQAO literacy results continue to
improve but the math scores have worsened in all the grades tested (grades 3, 6
and 9). These worsening EQAO math results are actually reflected throughout the
province. This raises several questions and areas for reflection. I have seen
several opinions and discussions put forth as explanations. One suggestion I
found interesting is the notion that the teacher’s own knowledge of math has
become weaker and this is reflected in the EQAO math scores. I think that in
the past several years most board have undertaken professional development in
math, purchased new resources and manipulatives and implemented identified best
practices. If so, I think this underlies a point made by several students in
this class in their discussions, namely that there are many wonderful resources
and aids available as we’ve seen but the bottom line is that the “teacher”
still has to teach. Another point to consider is a review of the math
curriculum in general. Are we teaching to get good EQAO scores as this implies
good knowledge and understanding? Are we teaching to develop life-long problem
solving skills that will extend beyond math and help students in their future
career? Are we doing both? A can of worms to ponder.

### Adam Spencer

I really like Adam spencer. Maybe it's because he has a Phd in math but chooses to be a morning radio DJ. Or maybe because he is really good at taking complex ideas and making them understandable by most humans (usually in a humorous way). Here is his TED talk on prime numbers. Another great offering is his Book of Numbers where he looks at all the mathematical properties (and oddities) of the first 100 numbers.

http://www.ted.com/talks/adam_spencer_why_i_fell_in_love_with_monster_prime_numbers.html

http://www.ted.com/talks/adam_spencer_why_i_fell_in_love_with_monster_prime_numbers.html

From YouTube

From the TED Site

## Saturday, October 5, 2013

### Graphing Battleship

I was looking online for some activities to use for middle school math and one of the things I found was Graphing Battleship. This activity is on education.com. This link will take you to the activity:

**Play Graphing Battleship**. Basically, it is the game of Battleship but each player has Cartesian coordinate planes for the playing boards and use coordinates for looking for hits. So instead of A4, a player asks for (4,-3). This would be a good activity to fit into the Grade & Geometry and Spatial Sense curriculum. It would fit under the Overall Expectation to describe location in the four quadrants of a coordinate system, dilate two-dimensional shapes, and apply transformations to create and analyse designs and the Specific Expectation to plot points using all four quadrants of the Cartesian coordinate plane. I think this would provide good practice in plotting and naming coordinates. It could be an activity students could do together if they had some extra time or as a centre activity. It gets students using the language associated with coordinates and plotting points by doing an activity with purpose.## Wednesday, October 2, 2013

### Overextension

I love that there are so many different techniques and ways to teach to different students. It's important to make the effort to try and get thru and be able to communicate with each and every student. However, spreading yourself too thin is a huge concern. A teacher is only as competent as their ability to facilitate their techniques. So if you are constantly trying new things and never mastering, or trying to juggle 10 different methods at once to appease every single student, you might be counteracting your own efforts. In my experiences being taught, the best teachers are the ones who have mastered their delivery. They have their methods, agree with them or not, but they are very confident and their method is refined. So remember to keep yourself current and open yourself to new methods, but ensure that your goal is to master your techniques.

### Math in Every Day Life

Such an important part of teaching any class is being accessible to students. An important extension of this lies in your ability to immerse your students in the content they are learning. A great way to do this is to have them use the differing technological mediums to find examples of math. It can be as simple as an common app that incorporates math, to a TV show where the plot is determined using mathematical processes. Making students look for the content you teach in their every day lives helps them make the link between school material and methods used in life.

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