Math Blog # 11 Probability

PROBABILITY 

This week we discussed probability as a measurement of likelihood. This measurement can be expressed as qualitative, or quantitative data! The idea is to learn to determine a theoretical probability, and possible likely outcomes. In class we had a gallery of activities presented to us one was using the clues to find the missing variables, or make connections accordingly. I really enjoyed this lesson as there were several different options to choose form and each person in the group had a chance to provide their input. It was very engaging and, hands on! When students engage in practising prediction, and determining outcomes they begin with concrete activities such as the ones above. Keeping this in mind they will be able to better deal with more abstract concepts and theoretical situations later on. Student could take interest in topics such as probability in math that leads them to take interest in jobs like forecasting, weather, sports anchor etc. 

In our daily life I use probability to make better decisions, choices, and my intuition plays a huge role in my metacognition about topics. I think that learning this at an early age and really getting the concept of mathematically probability can really effect students precise uses of it and, they can being to make connections in their everyday lives. 

Reference: 

http://www.mitaoe.ac.in/the-vast-use-of-probability-in-real-life

Math Blog # 10: Mean, Median, and Mode!

 DATA MANAGEMENT, AND PROBABILITY 

It's important to understand mean, median, and mode when interpreting data and probability for the purpose of reading and collecting data! The mean is the way to find the average between a set of data the indicates the result of putting all the data together and describing it evenly. The median is another kind of average used to describe a set of data with a single value. The median is valuable when there are one or two extreme pieces of information. The mode is another measurement used to describe a set of data's most frequent value. If there are many pieces of data the best way to determine the data is is to write all the values and look for the longest set of matching numbers!  

I enjoyed the video that was linked to our form however I felt that the content targeted a younger audience than the grades that I teach in my placement, as I teach grade 8! I thought the rendition of the video I posted above provides more engaging content. It also provides real life examples that can be used as part of an examplar to have them check their understand of the mean, median, and mode concept. 

Over all, I believe that the actives that we engage our students in should support their ideas that graphs, data, and the process of collecting and describing them should be used to solve real life problems. The Making Math Meaningful text suggests that initially students should be asked yes and no questions because it becomes much easier to collect data when there are only two obligated responses. Once comfortable students can build on these types of activities to include more choices, and options in their data! 

Reference: 

https://www.youtube.com/watch?v=IHginNwss5c 

Blog # 9 Measurement: Length, and Area

MEASUREMENT

 LENGTH AND AREA! 

Taking a look at the materials, and presentations presented on area and measurement I came across a couple terms that I wanted to clarify for myself so I could relate the ideas better with my students!

This video outlines the idea of square roots. The square root is the number value that when multiplied by itself equals the original number! Its commonly used in Pythagorean Theorem a formula introduced in the intermediate level.  It looks at both the length, and the area of an object. For example A'2+B'2=C'2, and C is the longest side of the triangle the square root of the hypotenuse is equal to the sum of the squares of the other two sides. The longest side of the triangle is called the hypotenuse. 3'2+ 4= 5'2  the calculation becomes 9+16 =25.

The third concept that I wanted to introduce is PI throughout our lesson it was suggested that before teaching students how to find the circumference of a circle its a good idea to refresh, or introduce them to the idea of PI. PI comes from the Greek Alphabet originating from the word Perimeter. It is is considered a magical number because no matter how you do the formula to calculate the circumference of a circle large, or small PI will always be the same. Here is a diagram the better articulate the idea.

I think that these would be great tools to incorporate into the minds on portion of a lesson! I would have students try out both theories to check their understanding, and provide feedback accordingly! 

Reference:

http://britton.disted.camosun.bc.ca/pi/piorigin/piorigin.html

https://www.khanacademy.org/math/pre-algebra/exponents-radicals/radical-radicals/v/understanding-square-roots

Math Blog 8: Geometry and Spatial Sense

                         


Today I am going to discuss symmetry in relation to 2D and 3D shapes. There are two different types of symmetry mirror symmetry, and rotational symmetry. Mirror images shapes have reflective symmetry. Rotational symmetry determines the amount of time a 2D shape can fit over itself when its rotated.  I believe the following videos will be a great way to introduce symmetry into the classroom and provide an aspect of reliability to the environment.






I learn't this week that modelling shapes allows students to understanding them. By representing certain shapes with manipulative, or through replication students will begin to visualize properties of a specific shape. IE. Using blocks to make 6 sides of a Hexagon. To the left is an example used in Eva's lesson plan where we were instructed to use the Mira Tool to complete the symmetrical drawing.

I believe her use of creativity with this lesson plan made it really effective. She used Super Mario a popular video game played among the junior - intermediate grades. In adapting this exercise into a novice level based activity I feel that students are more inclined to learn. The time constraints were even looped into the musical theme of the game by instructing that students had until the music stopped to complete their symmetric axis drawings.

She connected the idea of the chapter of thinking visually, and observing basic mathematical conventions by physically drawing the other half of the objects.

Finally, the conclusion was left open and the students felt accomplished as she linked completing the activity to completing level 1 of the Super Mario activity. Igniting a desire to want to learn advanced levels of geometry and advancing to level 2.

Math Blog 7: Patterning and Algebraic Thinking

What is Algebra?

Algebra involves a generalized thinking about relationships and number change. According to our understanding this becomes a natural process when students are ready to move from pattering to algebra. In general is said to be more about determining any terms or variables.

What is a variable?

The idea is that we teach our students to represent the mathematical relationships, and analyze the change thats occurred within them. Symbols are used to represent the unknown as specified in the video above, and students must learn to interpret expressions, and formulas. A formula is a special algebraic equation that shows a relationship between two, or more different quantities.
IE 3n+7n=10n

How to approach teaching these sorts of questions came up in the classroom and a set of helpful hints were given:

•  Use soft language:  Find two fractions that are almost but not quite equal to each other 
• Ask true or false questions: Mike says 1/3 is halfway between 1/4 and 1/2 because 3 is halfway between 2 and 4 is Mike right? 
• Ask effective questions
• Make them real
• Use open ended questions 
• Set the stage ( orient students to understand and provide the opportunity to reveal their ideas. Orient them to deal with the process and the math context 
• Encourage collaborative work by not providing all the anwsers 
• Collaborating what they've learn't, and how they got the results 

I enjoyed learning about setting up proposed questions I think this layout will be useful throughout my teaching block. Open ended questions push students further to explore the proposed problem and understand the experience further. In most cases they can begin at a level they feel comfortable addressing the variables they believe they need to solve the equation given.

With this in mind I enjoyed Adri's math lesson as it provided multi outlets for understanding the basic principles of algebra. If the student didn't quite graph the ideas of using the letter variables like "n," they also had to opportunity to recreate the pattern with the Q-tips provided. This tactical element is really useful when working with students who exhibit learning disabilities, or are visual learners.

The link below is also a great resource I used to further explain algebraic thinking. I enjoy it because it addresses all grade levels, and its very visual for those visual learners. 

http://thinkmath.edc.org/resource/algebraic-thinking

Math Blog # 6 Ratio & Proportions

This week I presented my math lesson on Ratio, and Proportions.  This was a huge learning experience for me. Its interesting how you can review a lesson so many times, yet when you present it to your class it doesn't go as planned.

The idea was that I would explain the scenario to the class and demonstrate how to measure each sample. Then I would have each group sample the next two with a partner. I feel like I should  have been more clear, and gone through the instructions thoroughly. This might have made the expectation more clear.

This is a great activity to explore as students grasp the concept of percent, ratio, and proportion, as percent is one form of ratio. 

My intent was to develop a lesson that would: 

• Investigate that proportionality is a multiplicative process and not an additive process.
  
• Consolidate concept understanding and procedural fluency for ratio, proportion, and percents.
  
• Solve problems involving percents, fractions, and decimals in a variety of contexts. 
  
  
  A themed activity is always a fun, and interactive way to introduce a topic. However, I realized during my presentation that this is an activity that required an entire period to demonstrate properly. 

If I were to do another micro lesson I would approach it differently. I would ask one question to draw a select number of curricular expectations. Insuring I really got the point across. 

Lesson Example: 

A concrete example would allow for the students to explore how they would individually approach the question. After they explained how they would approach it I would show them two different ways to solve the equation. They could also work in groups to present their strategies. I loved the idea of math meetings outlined on week 6. 

        MATH MEETINGS:

       - Share in groups 
       - Get clarity
       - Learn new strategies 

         

Integers Blog # 5

I want to begin this blog by point out that CLIPS is an amazing tool for math beginners! Clips takes you step by step through each lesson, expectations, with video/ animated instruction based tutorials. You can test your prior or learn't knowledge at each stage. I chose to explore Integers at the grade 7-8 level. I think this would be a great homework tool for students to test their knowledge on certain math topics they struggle with as it allows for student to see where they might have answered incorrectly. See the clips below for further explanation of my experience: In class we discussed Geometric Shapes for the purpose of understanding the root of fractions and how they play in the bigger picture of many complex math strands. Essentially, that understanding what fractions are telling us is important! How many parts to one whole? Which fraction is bigger using basic equivalent fractions? Why does 2 / 1/3 = 2X3? How many groups of 1/33 are in 2 wholes? The Fraction with the bigger denominator is made of smaller pieces! See pictures below from in class samples: The Tan Gram, and story of the elder who broke his valuables and tried to put all the pieces back together made the lesson interesting, and inviting for students. I enjoyed the tactical act of trying to put all the pieces into one whole square. Then breaking each piece down to estimate what fractions they broken down into. It provided a strong visual for comparison, and made it easier for me to break down each piece. The theatrics were engaging as well!

Blog # 4 Number Sense and Numerations Fractions

In class we explored fractions with the introduction of a book about Hershey's Chocolate & Egg Cartons! Beginning class this way was inviting, and interactive for all ages. We were instructed to pick a fraction that we found interesting, and convert it into any similar variant. I enjoyed this approach because it allowed the class to assimilate with what we were comfortable with, and can remember.

It was important to note that questions were presented with goals in mind:

Evolve and evoke thinking

Fearless speaking and listening

Open questions, student choice

Parallel problems, provide access to broad range of student readiness, reflection, and discussion

Provide appropriate scaffolding

Depth is more important then speed

 I enjoyed this visual that was given to us aimed at learners with disabilities! The original is much nicer than my drawing. However, it presented the concept that there are always more ways than one to approach a problem! We were given a T of blocks, and asked how we could represent 3/5 with 3 equal pieces! I thought that was very interesting because no matter how you come to the answer there was more than one way to represent the question to understand it.

This stood out to me more than anything else because when I was younger I would be the child crying because I didn't understand new math expectations, or assignments. The old fashion algorithms didn't make sense to me, and often I found myself frustrated and discouraged.

Blog # 3

This week discussed Number Sense, and Numeration. 

I enjoyed the video with Jo Boaler on brain crossing. She connects with the way our brain interprets, and conceptualizes information. The idea that we can reformulate how we discover math was very interesting to me that our number sense will light up when activated by the visualization ideas like estimating numbers and accessing symbols that flow across the entire brain pathway.

Developing Math Literacy:

Students at the J/I level are at a transitional stage where mathematics will help sculpt the success they have in secondary, and ultimately impact their career paths. Generally, a sound understanding will have all the variables relating to real world issues. Addressing the importance of richness to connections, “teachers must be agents of change that they did not experience as students” (Anderson, D. S. & Piazza, J. A., 1996.)

Minds on: Suggests how to get students mentally engaged in the first minutes of the class and establishes a positive classroom climate, making every minute of the math class count for every student.
Action! suggests how to group students and what instructional strategy to use. The instructional strategy chosen outlines an effective way to support students in learning the specific concept or skill. These strategies reinforce the connections with literacy and learning for life, as appropriate.
Consolidate/Debrief suggests ways to ‘pull out the math,’ check for conceptual understanding, and prepare students for the follow-up activity or next lesson. Often this involves whole class discussion and sharing. Students listen and contribute to reflections on alternate approaches, different solutions, extensions, and connections. 

These are the types of lesson plans we reviewed in class! This was an eye opening experience for me as I did not fully understand the expectations of broader based learning until today.

We explored this question and were asked to find multiple ways to solve it. At first I was frustrated because I couldn't think of an answer. Then another group started giving examples like, area, money etc. I began to analyze the question further as follows: 

If 4 Quarters equals $1 then 28 Quaters is $7. Therefore, 25 X 28 = 700.  Where as before my mind would of directed to finding out what X 25 equals 28 with no reasoning for that logic, or formula. 

Dan Meyer's Talk is where I discovered my reasonsing for the process rather than the answer.  He showed a pyramid of pennies, he has them ask questions as to what factors they needed to know to get the results. He introduced the formula at the end. However, the questions was not assessed based on correct, or incorrect. The question was based on closest answer. Proving to me that the process of inquiry was far more valuable than the outcome. On my first observation day I learn't that the mathematics curriculum is no longer based on a scoring guide, but rather a rubric of best suitable response. I really enjoy this as it allows for students to feel more confident, that their way of solving a problem has a safe space to resonate. 

To See Dan Meyers Talk: https://www.youtube.com/watch?v=YG9oqlQdVp0#action=share

Blog # 2

Today we touched on pedagogical content knowledge. We learn't to unpack, make connections, mathematical ideas, sensibilities, critical ideas, and play driven lessons. We touched on grades 1-8 in the Ontario curriculum suggesting that the central idea for lesson plans should be to make a plan, do the plan, and generally see how you did naturally. Solving the plan should model the natural process to build the same schema for students. 

We touched shortly on brain plasticity, and neroplasticity where the hippocampus grows after inquiring efficient information. I found this information really interesting in discussing the original algorithms we learn't in the early 90's. The potential that everyone can learn math if its taught in a more tactile and diverse way stood out to me the most. Using group lesson plans, open-ended questions, multiple entry points, and a high ceiling mentality.

I thoroughly enjoyed this youtube video on Brain Crossing:

The example of London Taxi drivers really stood out to me. Comparing learning complex routes, by repetition, and the study of brain growth and its ability to adapt. That there isn't difference between high- low ability learners that all students can change in 3 weeks, or a year. Every child has the ability, and potential to grow and each learning experience changes their fixed ability. I am excited for the potential to rewire our society and provide resources for new generations of learners to think out side the box! 

An Introduction to Math

Hey Everyone,

I would like to take a second to introduce myself, and my relationship with math. I have a learning disabilitiy with math that highlights a difficulty with cognitive understanding and my working memory.  With this in mind I have always struggled with math throughout my academic career.
In elementary school I spent many nights crying because I didn't understand the expectations, and I felt very overwhelmed with how I was taught to learn. In high school I spent many math exams in a separate room, with formula sheets, and the use of a calculator. Needless to say, when I entered the classroom on day one I had many fears regarding math.


Although, I am highly optimistic that I will rediscover passions for math that I never knew existed. I enjoy the open concept, and inquiry based learning. I also enjoy that the curriculum doesn't implement right or, wrong. I love that the curriculum has introduced level based learning.  I hope to explore these adaptations over the next two years so I can successfully alleviate the fears of my prospective students in the near future!

Looking forward to it!