Numbers and Angles

I was preparing for part of the next unit in grade 8 math that I am teaching, Geometry and Spatial Sense and came across this....

Angles and Numbers!   

The maker of this illustration used math in a creative way.  That is great, because that's how we want our kids to think as well... out of the box....being creative and creating new ideas/solutions to problems.  This illustration could be a starting point for students to use angles in a different way.  I could combine this exercise with art, and have students create a piece of art with angles.  It's neat when we can do cross-curricular projects, because that allows our students to see the value of all subjects, and realize how inter-twined they really are.

Pictorial Integers for Visual Learners

I was going through a resource that I had saved, and was wondering why I saved it, so I opened it up and I found some very resourceful ideas.  One of these ideas was pictorial integers for Visual Learners.  This would be apart of our differentiated learner piece, teaching a concept in a different way for all our learners to understand it.

Looking at the table above, a student who works well using pictures can make good sense of the tiles being used.  You can see that  red tiles are positive and blue tiles are negative.  The student would just put or draw these tiles on the placemat, and allow for the natural process of cancellation to take into affect.  Once cancellation happens, cancellation meaning the same amount of different colours cancel each other out; which ever is left over would be counted.  Even though the idea of cancellation is a very simple one, it is very effective as well.  I could picture 4 students in my grade 8 Math class who would benefit from this type of activity.  I guess its good to look at resources you've saved, its usually for a reason!

Math is Beautiful

'When will I ever use this?', is a question that may get thrown around quite often in a classroom.  Our first response may have something to do with a connection between the subject matter and a future task, job or career.  We may try to connect it to something fun that our students may encounter and try later on in life, or we may talk about how it has changed the world thus far.  Either way, it is always nice to sit back and appreciate the work others have done in the field.  Math is all around us and it really is the universal language (just ask Jodie Foster in Contact).

We may not realize how much of the world around us is governed by different mathematical principles...so here's a quick little montage of some of the beauties of math.

Games and Math

I have long had a little bit of an obsession with logic games.  Since I was young, I was 'forced' (read: tactically convinced) to try them out.  This definitely sprung from my aunt and uncle who were both teachers and had engaged themselves in teaching the entire spectrum of students.

This interest keeps itself ingrained in my life and surfaced once again with the wonderful trend that was sudoku.  I collect the puzzle everyday form the paper here at work and keep a nice pile that now goes back to 2011 for those rainy days.  I am always on the lookout for a newer, different, perhaps more interesting/challenging type of logic game, and have even recreated some as emergency lessons or as intros to try and foster the critical logical thinking in my students.  The 2 that I have now challenged other with are Yubotu and Nonograms (which may commonly be called Pix-A-Pic).  I stumbled upon this beauty at a small town bookstore and have trying them ever since.  The main creator that I have found is Conceptis Puzzles, but there are plenty to be found online

Both have a similar type of goal, that is to determine what spaces in a grid need to be filled in.  Yubotu more resembles a pen and paper battleship, while the other often creates a pixelated image.  I enjoy challenging my students to try them without any trial and error and guessing and show them how there are many ways to attack a problem, while working through it logically.  Students are always going to play games, why wouldn't they, it's much more fun that work.  The trick is to get them to be working their brain and not just clicking away aimlessly.

In the end, it's something extra that isn't a specific curriculum expectation, by the transferable skills they develop should stay with them much longer.

Questioning in the Math Classroom

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!

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?

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

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.

 

                                            

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 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.

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

From YouTube 

From the TED Site

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.

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.

Access to Mathematics E- Learning Resources

Through my ABQ Math class I learned about Ontario E-learning resources at HTTP://resources.elearningontario.ca/d2l/home. I was very excited and spent many days exploring the database of very use full and student enriching activities and lesson plans that are very impressive. In my excitement I wrote to the Ontario Ministry Of Education E-Learning contact person asking for access to this remarkable source of information. Besides being informed that it is only for provincially funded school systems, I was also told that “Teachers and students in private schools are not deemed to be eligible users.” At this point a question kept flashing in my mind "Should knowledge be only accessible to “eligible users” or all Canadian Taxpaying Citizens should be allowed to access this resource.” Should teachers and students of private schools and students who are home-schooled, be allowed to benefit from Mathematics E-Learning resources such as available at OERB? I believe that all taxpaying Canadian Citizens should be allowed access to knowledge especially children and youth as they are very important members who contribute to strengthen their communities that they grow up in. No matter which school system they go to, the common goal is to educate our next generation of Canadians. I would like to initiate an intellectual discussion, if as educators we agree or disagree to this approach.