Category: Mathematics

Environmental Impact Study Lesson Plan

2/1/2016

Cradle to Grave Environmental Impact Study Lesson Plan

Differentiated and Adaptable Lesson Plan for Grades 6-12

Connect:
RATIONALE
This lesson plan was developed as a supplemental to many of the Entrepreneur/Business in the school programs that are available for grades 6-12 in BC. It was noticed that many of these programs do not consider Environmental Impact of the products the children create. This lesson plan is designed to be used with the Life Cycle Assessment Worksheet and will allow students to include a Cradle to Grave analysis of their product in their business plan. An Environmental Economics Unit Plan and marking rubric has been included as sample.

BC PROVINCIAL CURRICULUM
K-12 Applied Design, Skills and Technologies Draft Curriculum
Core Competencies:

Communication
Thinking (creative, critical)
Personal and Social (Positive Personal and Cultural Identity, Personal awareness and responsibility, Social Responsibility)

K-12 Applied Design, Skills and Technologies Draft Curriculum will focus on fostering the development of the skills and knowledge that will allow students to create practical and innovative responses to everyday needs and problems. Design involves the ability to combine an empathetic understanding of the context of a problem, creativity in the generation of insights and solutions, and critical thinking to analyze and fit solutions to the context. To move from design to final product or service requires skills and technology. Skills are the abilities gained through competence to do something and to do it increasingly well, and technologies are tools that enable human capabilities. In Applied Design, Skills and Technologies, students will grow in their ability to use design thinking to gain an understanding of how to apply their skills to problem finding and solving using appropriate technologies.

Grade 6-9

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Formative Numeracy Assessment

12/18/2015

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With the BC Numeracy Standards

Assessment standards are changing with the new BC curricula, and it’s an exciting time to be coming into teaching. I’ve been exploring my assumptions and researching the new BC Performance Standards, and am impressed with the focus on formative over summative assessment. Summative still has it’s place, of course, and there will always be a role that traditional ‘tests’ will play in our assessment of students, however the new curricula lends itself to larger projects with higher level understandings of the concepts.

For example, the Grade 2 numeracy standards state that “Relatively short questions with a single correct procedure and answer are not appropriate for performance assessment.” (Grade 2 Numeracy p 52)

Another aspect I’ve found interesting is use of the term “Numeracy”. Just as Literacy is more than, yet still includes Reading, Numeracy is not limited to Mathematics. The concept of Numeracy is broken into the following concepts and skills:

“Number (Concepts and Operations)
Patterns and Relations
Shape and Space
Statistics and Probability
Problem Solving (Grades 8-12)

Numeracy tasks and problems typically draw on concepts and skills from two or more of the curriculum organizers listed above” (BC Performance Standards Numeracy, P. 9)

These concepts are organized into four ‘big picture’ aspects of numeracy: Concepts and Applications

Strategies and Approaches
Accuracy
Representation and Communication” (BC Performance Standards Numeracy, P. 11”

The new Performance standards for Numeracy allow for a more in depth understanding of the student’s level of understanding of the concepts, beyond a simple summative test score and resulting letter grade. “Performance standards answer the questions: “How good is good enough? What does it look like when a student’s work has met the expectations at this grade level?” (BC Performance Standards Numeracy, P. 3)

The standards will be used in our teaching practice by being embedded into the curriculum. We can link our learning outcomes to the standards, and can rephrase into kid friendly “I can” statements. The standards also provide concrete examples of student work of varying quality in order to assist teachers with application. Evaluation is ongoing throughout the term, and can include observations and communication between student and teacher. It also recognizes that some students will need support and be unable to work independently, but will still be able to grasp the concepts (for example, distractable students who cannot stay focused, but who still understand). The standards also allow for assessment to be adapted by the teacher as needed, to reflect the time of year, or differentiation needs of students. (BC Performance Standards Numeracy, P. 6)

By expanding assessment standards to reflect the big picture of Numeracy, the BC gov’t allows teachers more flexibility and application of ‘real world’ mathematics to their teaching. “Numeracy involves concrete applications in which students, confidently and independently, use mathematics to address real tasks or problems in an increasing variety of situations. The ability to recognize the mathematical demands and possibilities in a situation is an important aspect of numeracy. Numeracy is based on mathematical foundations and requires the application of concepts and skills related to the formal aspects of the discipline of mathematics. “ (BC Performance Standards Numeracy, P. 9)

It is important to remember that the standards are just one piece of the Evaluation whole. A combination of formative and summative assessments must be used to create a wholistic picture of a student’s learning. “The performance standards do not address all aspects of the mathematics curricula and need to be used in combination with other forms of assessment to develop a comprehensive picture of student achievement in the BC mathematics curricula.” (BC Performance Standards Numeracy, P. 10)

I look forward to learning more about Assessment strategies in class this semester!

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Reflections on Science

11/14/2015

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Science has always been a fascinating subject for me, but one that was beyond my reach in my formal schooling. Growing up, we moved frequently. I attended 10 schools over my 12 years, bouncing between two provinces with different curriculums. While I excelled at reading and at anything using my creativity, the repeated moves did not provide continuity or stability for my math education. I did not acquire basic math fluency or number sense. My educational challenges with math growing up always impeded my ability to engage with the sciences. While I found them fascinating, without the basic number sense that I failed to grasp in elementary school the sciences were beyond my reach in high school. I managed to graduate with the minimum requirements for both science (biology 10- the least math based science course) and a 'P' in math- achieved only through a summer school class.
I pursued a degree in social work and worked in the social work field for many years. My degree focused on psychology and counselling, along with advocacy and public policy. Science and Math were not part of my everyday world for many years.
My first positive experience with both science and math was the opportunity to take an introductory astronomy class as an elective for my degree- Elementary Astronomy for Non-science students at UVIC. My professor had crazy mad professor hair and was quirky to say the least. He took a lecture hall full of math phobic Arts degree students and Star Trek fans and toured us through the wonders of the universe, teaching us some basic physics along the way. I also had the support of my husband at the time who was comfortable with math and was able to tutor me with the basic formulas and calculations required. This year long course was the first time that I felt successful with either a science or a math subject and changed my attitude towards these topics from reluctance and failure to excitement and success.
My career path eventually lead me into show business, and I was a full time circus artist for 6 years before beginning to age out of the industry and so took the PBED program. My experience with prop manipulation was purely movement based; it was only in the past 3 years since working with a new performance partner (a math and science major) that I've been learning the math behind the movement. I've even been exploring the math of patterns and movement in my Math Blogs over the past semester. (http://www.vestaeducation.com/viu-education-program/category/mathematics)
My science project was an exciting prospect for me- starting to explore the physics of movement as it relates to circus. This was not simply writing lesson plans about concepts I was already familiar with. This was learning completely new-to-me concepts and adapting lesson plans to include circus based examples and activities. I compiled 9 lesson plans that explain the concepts that apply to circus movement. They are geared for approximately grade 6 level.
The grade 6 curriculum focuses on the laws of motion, and I have taken some liberty to include other topics in surrounding grades such as sound waves. Thankfully, the elementary curricula is focused on learning the concepts and not the math. My lesson plans do not include the math of physics.
I can easily ground my decision to exclude the math in developmental theory. Piaget states that children in grade 6 will not have the capacity to learn symbolic mathematics as they are still in the Concrete Operational stage of thinking. My own experience with high school math and science could be viewed in this lens as well- perhaps my own development was typical in that even without the fundamentals, I was not developmentally ready for the abstract maths when they were presented to me.
Reflection of my motivations shows that I have other more personal reasons for not including math. One, it is not explicitly stated in the curricula that it needs to be included. My underlying motivation is that simply I am still uncomfortable with the subject. However this leaves me with a nagging feeling that the lessons are incomplete. I am
torn between my desire to present a complete lesson and my fear of my own inability to not only understand but to explain the concepts.
This is not an isolated struggle. When we take on the role of 'teacher', we can be tempted to proclaim ourselves as 'experts' on the topics we are teaching.
As generalist practitioners however, we present the entire academic spectrum to the elementary classes. This project crystallizes a struggle that I feel we all have- how to present material to a class that we are unsure of. How do we teach what we do not know?
The answer, of course, is we learn along with our students, and remember that the children we work with may not be ready to learn everything that we have capacity to teach. This project has only introduced me to the laws of motion, and every time I teach this topic my understanding will deepen. I do not have to know everything, and while I will be able to learn more with each experience, I need to remain child centered in my approach. Teachers can take the position of co-learner instead of expert. Like my astronomy teacher from years ago, I can accompany my students on a journey of discovery. I don't need to know all the answers, I just need to model wonder and curiosity about the world around us. We can travel together.

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Mathematical Patterns in Juggling

11/11/2015

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Math is Beautiful part 3

Mathematics of Juggling

MATH IS BEAUTIFUL: Mathematical Patterns in Juggling
A basic primer for learning Siteswap

Juggling has been defined as manipulation of more objects than there are hands. I've also heard it described as just learning to drop things in more and more complicated ways. The classic definition involves balls moving through the air, with the standard 3 ball cascade involving one ball in the air, and a ball in each hand.

Balls are thrown one at a time alternating which hand does the throwing: first
one hand and then the other. Right, Left, Right, Left, Right, Left...

The basic 3 ball pattern doesn’t take long to figure out – every time a ball is close to landing in a hand you throw another one up to replace it. Timing is also important; a given ball spends about half of its time in the air and the other half in one of the hands. This movement pattern (the 3 ball cascade) can be adapted to produce more interesting patterns by varying each ball's height and timing.

3 ball cascade - the basic juggling pattern

As with all patterns, these movements can be expressed mathematically.However, it was not until 1985 that a model for juggling heights was developed by Bruce Teimann. Prior to this jugglers described the height of the throws either by the estimation of distance (2 feet) or by even looser terms such as "a bit higher".

The mathematical model, called Siteswap, uses numbers to represent the time between a ball being caught, thrown, and caught again. Siteswap denotes time in between throws, and this time can be broken down into beats. The numbers in siteswap represent the number of rhythmic beats until a ball is thrown again. The larger the number the higher the throw as it will take longer for a ball to complete it's parabola (more beats).

Some common three ball patterns are represented by the siteswaps 3, 441, 531, 6316131, 52512. Some common four ball patterns are 4, 741, 534, 6451, 7441, 7531.

Because the patterns repeat, we just write all the numbers once. Instead of 3333333333333, we can simply say "3" and know to repeat it. Instead of 423334233342333, we just write 42333 as the pattern's name.

441

534

Fun Math Fact of siteswap is that the average throw must be equal to the number of balls being juggled. This allows for us to figure out how many balls are being thrown in any given pattern. For example:

441 is 4+4+1= 9 /3 (3 numbers) = 3 ball pattern.
6316131 is 6+3+1+6+1+3+1= 21/7=3 ball pattern
534 is 5+3+4= 12/3= 4 ball pattern

This simple math can be used to engage children as a mathematical 'trick' that can be demonstrated in practice by a juggling teacher of moderate skill or by a siteswap pattern generator online. Children may have fun with a pattern generator online trying different combinations of numbers to make different juggling patterns. Juggle clown juggle!

http://jugglinglab.sourceforge.net/

Siteswap gets more complicated the further down the rabbit hole you go. Juggling mathematicians have been calculating siteswaps and mapping patterns in ever-increasing complexity since it's creation in the 80's. However the scope of this blog is to demonstrate not only my own rudimentary learning on the subject but to explore some ways I could integrate Math into my circus teaching practice. Children who are learning juggling may be more motivated to explore the mathematical representation of the patterns they are creating beyond 3 ball cascade. The availability of online pattern generators allows for them to play with patterns in theory that may be beyond their ability to juggle in practice. Many incredible patterns come only from the direct study of the math behind juggling. The ability to think about patterns in mathematical terms allows us to link numbers with our own artistic expression and to realize that math is, in fact, beautiful.

Kat of Vesta Entertainment Juggling on Lantzville beach at Sunset

Many thanks to Katlan Irvine of Vesta Entertainment for the hours of conversation, demonstration, and laughter in the creation of this 3 blog series, Math is Beautiful.

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<<Previous

Environmental Impact Study Lesson Plan

Cradle to Grave Environmental Impact Study Lesson Plan

Differentiated and Adaptable Lesson Plan for Grades 6-12

Formative Numeracy Assessment

With the BC Numeracy Standards

Reflections on Science

Mathematical Patterns in Juggling

Math is Beautiful part 3

Karina Strong

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