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?