Weekly Report & Reflection Week #11!

Reflection 11

In the second last session of this math course, we started to look at formative assessment. One good question came up was that how do we assess students using the "As Learning" format. Let us give an overview first of all the assessment types. 

From the Growing Success policy document: 

Assessment FOR learning is more commonly known as formative & diagnostic assessments.  Assessment FOR learning is the use of a task or an activity for the purpose of determining student progress during a unit or block of instruction.  Teachers are now afforded the chance to adjust classroom instruction based upon the needs of the students.  Similarly, students are provided valuable feedback on their own learning.

Assessment OF learning is the use of a task or an activity to measure, record and report on a student's level of achievement in regards to specific learning expectations.  These are often known as summative assessments.

Assessment AS learning is the use of a task or an activity to allow students the opportunity to use assessment to further their own learning.  Self and peer assessments allow students to reflect on their own learning and identify areas of strength and need.  These tasks offer students the chance to set their own personal goals and advocate for their own learning.

So looking at the assessment as learning, you can see that it is all about self assessment. How can students self assess in math? Here is a simple example, give them a problem, and tell them what do you already know about this. This will allow the students to self assess and ask themselves all sorts of questions by bringing up their past knowledge in trying to solve the current problem. 

In this session of week 11, the instructor modelled for us how to self assess. He gave us the 5 fingers problem similar to the one below:

Tom likes to count on the fingers of his left hand, but in a peculiar way.  He starts by calling the thumb 1, the first finger 2, the middle finger 3, the ring finger 4, and the pinkie 5, and then he reverses direction, so the ring finger is 6, the middle finger is 7, the first finger is 8, the thumb is 9, and then he reverses again so that the first finger is 10, the middle finger is 11, and so on. Where would number 1000 land (on which finger)

So he had us solve the problem first on our own, then asked us what we go to and how, then he prompted us further and asked why we got there. Then the instructor solved the problem on the board by asking us as a class too (guided teaching).

These problems can be used from grade 2-12. Depends how complicated you can go with the solutions. The solution can be as simple as just counting, or as complicated as making algebraic expressions. 

There are such problems on the:

Collaborative Mathematics website, click HERE.

Math Counts website, click HERE. 

Important notes:

Challenge kids
Let the students do the work

Weekly Report & Reflection Week #10!

Reflection 10

In this week's session we had covered the topic of probability. Probability can be easily dismissed and sometimes be labeled as common sense, hence we do not need to study it. You can, however, think of probability as common sense put into calculations. 

"As with other beautiful and useful areas of mathematics, probability has in practice only a limited place in even secondary school instruction" (Moore, 1990, p. 119). The development of students' mathematical reasoning through the study of probability is essential in daily life. Probability represents real-life mathematics. "Research in medicine and the social sciences can often be understood only through statistical methods that have grown out of probability theory" (Huff, 1959, p. 11). Moore (1990) stated:

"Probability is the branch of mathematics that describes randomness. The conflict between probability theory and students' view of the world is due at least in part to students' limited contact with randomness. We must therefore prepare the way for the study of chance by providing experience with random behavior early in the mathematics curriculum."


From reading the quote above it is understood that students are mostly exposed to structured and organized matter around them. They are not exposed to taking risks and being random. So how as a teacher can you expose randomness?


Have students just play with dice, cards, different colours and other objects randomly. Have them explore what it means to role dice randomly and predicting what the outcome will. Probability can even be taught through other mediums that can pick up the students' interest and motivation. Use technology that can include different apps and games that can teach probability. There are many interactive probability games over the web. 


In our class this week, my classmate and I created a presentation that introduced concepts of probability through an activity using coloured marbles and Kahoot. We used Kahoot to do a quiz/survey with the whole class using probability terms such as: Likely, Unlikely, Certain, Impossible and etc. We observed that Kahoot can be a very interactive and fun way to teach students something because they can get excited in a friendly and safe competitive environment. The coloured marbles activity was also fun and simple where students get to use different manipulative to learn probability concept rather than sitting and listening to a boring lecture from a teacher. 


Here is a fun that you can teach probability to students: using M&Ms!! Who doesn't like chocolate?

Weekly Report & Reflection Week #9!

Reflection 9

This week's session we started the class with the instructor explaining to us a math portfolio that we have to create. After that our classmates started their learning activity presentations. The focus for this session is measurement. The presenters showed us the "Perimeter Around The Area" video, by the Brazillions, on YouTube. Watch below:

After we watched the video we played a game board using dice and shapes. It was a fun game but was not too sure how it related to area or perimeter. It would probably be a good game for a "Minds On" activity. The second presentations was also for the measurement unit but focused on circles (circumference and diameter).

We started by the string and circle diagram activity. We took the string and measured the circle with it and recorded it as the circumference. Then we measured the diameter of the circle with a ruler. We divided the circumference by the diameter and got a number very close to 3.14 but not exactly that, as it was an approximate, but we did not get a  number that was very off, so it was good. This activity introduced us to the number pi. 

Then we did another activity with toothpicks and a piece of chart paper. It introduced us to the concert of pi in degrees which is 180 degrees. It was all about the rotation of toothpicks! Awesome activity. I think these activities would definitely be useful for me as my teaching block this January would be all about circles. It would include circumference and area with some other concept. I am teaching grade 8 so these activities are definitely applicable. 

We got referred to YouTube channel: Numberphile for more ideas, by our classmates. 

This was the end of the second presentation followed by a third presentation. Also focused on the measurement strand, but is on a different topic, called time. We worked through a story that had a math problem in it. Using a timeline we figured out the answer. It was a good activity focused for the younger grades. 

After all the presentations we started a jumping activity. Which was great, I think if that is applied in a classroom it would be great to help students estimate and find actually measurements from the jumping activity. It can also help students measure using non-standard units rather than metric. Definitely a very fun activity which involves measuring too using different standard or non standard units. 

Reflection 8

Reflection-8

This week's session was very informative. We learned a lot about geometry and spatial sense. It is a great, challenging, and fun topic. As educators, we might come across this question from students: "Why do we need geometry in our lives?" We need to be prepared to answer such questions. We need to believe geometry is important first before passing on the knowledge to others. So After some research, experience and my own thoughts I came up with a list:

1. To be able to understand the wonder of the worlds shape and appreciate it, we need to be able to understand and have knowledge of spatial use. 

2. Geometry will assist us in understanding the relationship between shapes and sizes 

3. Some people think in shapes and sizes, others think with visual abilities

4. Science and technology require knowledge of geometry

5. Geometry helps bring together both sides of your brain. The left-brain is more logical, technical, whereas the right-brain is the part that visualizes and where the artist gets their creative inspiration from

6. In the fields of television, moves and even little things like puzzles or books all are influenced by geometry

7. Geometry gives us a base for students to make sue of concrete materials and activities.

There are many other reasons of why we need to study geometry, I only listed a few above. 

From our class this week, the presentations taught me something new. It is that we can teach the math curriculum only through geometry itself. We can incorporate shapes in all of our lessons. How amazing can that be? Mostly, in the early years, geometry is taught through shapes and solids. Other topics within geometry include:

1. Line and segments

2. Shapes and solids

3. Triangles and angles

4. Platonic Solids

5. Coordiniate Grids

6. Radians

7. Conic Sections

8. Circumference

9. Polygons

10. Trignometry

There were few manipulatives introduced by classmates and the instructor that we could use in our lessons. It is best to have solid shapes as examples to show and teach the students about faces, vertices, edges and angles. The instructor also put up some puzzle games and another great OSMO iPad game kit. It would be a great station to have in a classroom for the students to work with. See picture below:

OSMO Kit. Photographed by Samia Sharif.

Other games:

Puzzles. Photographed by Samia Sharif.

Weekly Report & Reflection Week #7!

  Reflection 7-

Algebra

       

            This week’s topic in math class was algebra. My classmates did two learning activity presentations on this particular topic. Today’s presentations made me realize that I can reach an answer right away without focusing much on the process to the answer. However, students need to learn the process rather than the answer.

 I as a teacher candidate need to focus on the process of math problems. I need to learn how to break down problems for students. One problem can have many ways to get to the answer. At elementary level students can have many resources to learn particular concepts, in other words these can be called manipulatives. For example, the instructor today gave us blocks to come with an algebraic expression for a particular hexagonal pattern. We found an expression for the perimeter of the blocks.

Another way I have seen before to model algebraic equations is through the balance! I think it is a great visual to use when teaching. See example below:

 

Math is Fun. (2015). Introduction to Algebra. Retrieved online from https://www.mathsisfun.com/algebra/introduction.html

I realized that using variables for an algebraic expression could be confusing for some. So how should we explain to students why we use a letter? I found a simple answer to that from Math is Fun (mathisfun.com).

·      It is easier to write ‘x’ than drawing empty boxes (unknowns) (and easier to say ‘x’ than the empty box

·      If there are several empty boxes (several unknowns) we can use a different letter for each one

·      So x is simply better than having an empty box, but any other letter can also be used

I think it is interesting how our class is grouped rather than single desks for each of us. This way is great because we get to learn a lot from the people in our group. I would definitely implement this kind of setting in my classrooms in the future as a teacher. Hence, this was the first half of our session today.

The other half of our session today was lesson planning. We took our drafts of lesson planning today to the instructor for him to look it over. I have some areas where I need to touch up on. Lesson planning is an exciting part of teaching. It is the core of our teaching service. If the lesson plan is good than teaching, classroom management, and outcomes can be great as well. I want to gain the confidence to lesson plan and I am looking forward for it to become the intuitive part of me as a teacher.

According to the instructor, after much experience, the implementation of lesson planning will be become engraved in me as a teacher that I wouldn’t even realize that I am taking those certain steps. These thoughts are however for the future only, but what about the present? Our time for internship is coming up fast and I am nervous to teach because I want to put my best out there rather than getting confused.

Weekly Report & Reflection Week #6!

Reflection 6- 

Witch's Recipe for Proportions

I can definitely say I enjoyed this week's class a lot as I learned a lot and got to do few challenging activities. I want my students to feel the same way when they are in my classroom. Each student in my classroom needs to feel that yes I am being challenged, but that challenge is at my level only. 

That takes us to differentiating instruction, which means to engage students in instruction and learning in the classroom. All students need sufficient time and a variety of problem-solving contexts to use concepts, procedures and strategies and to develop and consolidate their understanding. When I as a teacher would be aware of my students’ prior knowledge and experiences, I can consider the different ways that students learn without pre-defining their capacity for learning. So it comes all back to me as a teacher, I want to be the best math teacher out there. 

So, this week's class we learned mostly about proportions, what better way to learn about proportions than to engage students' in a witch's recipe? Yes, one of our activities was to mix a brew for the witches. I think that is a perfect way to engage students. They can have fun and be interested with the problem but also learn the core concepts. Concepts in the problems included multiplication, division, estimation, calculation, and proportions. 

Proportional thinking can be hard. It is not an easy concept for students to grasp. However, the different activities that we were shown in class by classmates was amazing and can be used for my practicum block that is coming up soon. That takes me to lesson planning. Today finally I had a practical experience with lesson planning. I had fun lesson planning and I think I was on track with it. I wasn't too sure about my lesson planning until the instructor showed us a video and explained step by step how one lesson plans. I have to thank that instructor for making it easier for us. 

After the video, I reviewed my lesson plan and thought I only need minor changes rather than major. That tells me that I am somewhat ready to lesson plan. Our instructor had divided our class into two and asked us to lesson plan in a large group. I got the witch's brew recipe problem and I got the perfect idea of how I would start my lesson if I were to teach that to my students. I would bring a large test tube with me and show them if I have 20 mL of water and I want to multiply it by number how would the amount of water change. 

That takes us back to the witch's problem because in that problem the recipe serves 3 people but we want it to serve 9, so we will have to multiply 3 by 3 to get nine. Hence, if water is an ingredient and we need to 20 mL for 3 servings, we can multiply 20 mL by 3 to get total of 9 servings. Therefore, this procedure applies to all the other ingredients on the list of a recipe. 

I think that assumptions and precisions matter when studying proportions!

Weekly Report & Reflection Week #5!

Reflection 5-

 The Mystery of Zero

The title of my reflection blog is 'Zero' today because that came up a lot in this week's session. What is the number zero? When you are a kid you learn that you cannot divide by the number zero. As we grow older we start to learn something different, we get the hint that if we divide by the number zero a lot of 'crazy' things happen. Today in class we were told that these things are against the laws of nature that we are so used to. So that's why we stay away from dividing by zero. 

But why?  Have you ever thought exactly why you cannot? I searched up to find some answers and I found this one by Dr. Tom:

"Because there's just no sensible way to define it.

For example, we could say that 1/0 = 5. But there's a rule in arithmetic that a(b/a) = b, and if 1/0 = 5, 0(1/0) = 0*5 = 0 doesn't work, so you could never use the rule. If you changed every rule to specifically say that it doesn't work for zero in the denominator, what's the point of making 1/0 = 5 in the first place? You can't use any rules on it.

But maybe you're thinking of saying that 1/0 = infinity. Well then, what's "infinity"? How does it work in all the other equations?

Does infinity - infinity = 0?
Does 1 + infinity = infinity?

If so, the associative rule doesn't work, since (a+b)+c = a+(b+c) will not always work:
1 + (infinity - infinity) = 1 + 0 = 1, but
(1 + infinity) - infinity = infinity - infinity = 0.

You can try to make up a good set of rules, but it always leads to nonsense, so to avoid all the trouble we just say that it doesn't make sense to divide by zero.

What happens if you add apples to oranges? It just doesn't make sense, so the easiest thing is just to say that it doesn't make sense, or, as a mathematician would say, "it is undefined."
Maybe that's the best way to look at it. When, in mathematics, you see a statement like "operation XYZ is undefined", you should translate it in your head to "operation XYZ doesn't make sense.""

You can find the above information and more from this website:
http://mathforum.org/dr.math/faq/faq.divideby0.html

You might also want to check this video out for more information:

That being said, let me briefly cover what we did today in class. We were introduced to the topics of integers and exponents. I learned that exponent is just a way to represent a given equation in order to shorten and simplify it. For example it is better to write 12^200 rather than to write 12 multiplying by itself 200 times. A lot more convenient.

In every class I learn something. This class I learned that sometimes to simplify math concepts you as a teacher have to 'lie'. For example, you have to say you can't divide by zero but in some dimensions of different mathematics you can. I also learned that we should watch out what we are saying as a teacher. For example it would be good to avoid saying "this is really easy" when in fact it is not easy. Not every student finds what you find easy is easy.

Weekly Report & Reflection Week #4!

Reflection 4

"Fractions, fractions, fractions!" Did we not hear that phrase when we were in elementary school? That was because we used to dread the topic. I am pretty convinced that is how students still feel about fractions if they have just starting the topic.

Not everyone feels comfortable with fractions, it can be a very complicated one because we are not working with whole numbers anymore. This is where I want to come in as a teacher candidate. I want my students to feel confident with their number sense skills whether it be a whole number or not. 

Phillip Martin. Phillip Martin Clip Art. Retrieved from
http://math.phillipmartin.info/home_fractions_01.htm

Today in class we focused on two topics: Fractions and Decimals. My three other classmates and I presented our activity learning task, a 10 minutes presentation plus 5 minutes for discussion. One person focused on decimals. He gave us out a really useful tool called the hundredths wheel. I think this is a tool I would definitely use with my students. 

Hundreths Wheel. Retrieved from
http://www.eworkshop.on.ca/edu/pdf/Mod27_representing_hundredths.pdf

Second really interesting method of calculating fractions is called the Macarena Method. I learned that from another presentation and it came to me as a complete surprise as I never heard of it before. I am convinced this method will help a lot of students, I do wish I knew that method myself when I was studying! If you want to check out the method see the video below:

I realized today in class that what math students really need is a tool and a platform. They need resources from which they can learn. The traditional way of teaching is no more and it does not teach anything. Students need to explore and discover. They need to make mistakes, fix and learn. This is math. The more variety of tools they have in their toolbox the more options they can have at problem solving. Just like how if a carpenter has more tools in his/her box he/she can do a lot more work than just one thing. 

Kids are capable to learn if they have, very much like the TEDx we saw in the TECH class. The speaker talks about how if you provide children with one computer and an access to internet they can teach themselves. I think that talk is worth to share it here too, please click HERE.

I think with that talk it would be a great place to end my post for the day. Stay tuned for next week's post. 

Learning Activity- Fractions

Learning Activity Summary

Topic: Fractions

Grade Level: 4

Mathematics Curriculum Strand: Number Sense and Numeracy Fractions

Content and Process Expectations: You can find in the Guide to Effective Instructions in Mathematics- Number Sense and Numeration Grades 4-6 Volume 5- Fractions. You can also find it in Chapter 11 of the course textbook ‘Making Math Meaningful.’

Source of Activity: www.eworkshop.on.ca

Approach

At a camp, campers stayed in 3 cabins. In cabin A there were 4 campers, in cabin B there were 5 campers and in cabin C, there were 8 campers. One day the campers were treated to pizza in the following way:

Cabin A- 3 pizzas, Cabin B- 4 pizzas, Cabin C- 7 pizzas

Pose the Question: Did some campers get more pizza than others, or did all the campers receive the same amount of pizza?
Clarify that:

·       all the pizzas are the same size;

·       the pizzas can be cut into any number of equal pieces.

Ask students to think about how they might solve the problem. Have students share their thoughts with a partner, and then invite a few students to share their ideas with the whole class. Provide an opportunity for students to ask questions about the problem or about possible approaches to finding a solution.

Pose questions that help students think about what they found out:

·       What strategy did you use to figure out the amount of pizza the campers in each cabin

·       How much pizza did each camper in each of the cabins receive?

·       Which campers received the most pizza? How do you know?

·       Which campers received the least pizza? How do you know?

Tools 

 You can use: Area model, fraction number lines or fraction bars

Assessment – things you need to check with the students

·       how well they understand the problem and whether they are applying an appropriate strategy;

·       whether they are dividing the pizzas into appropriate fractional parts (e.g., dividing a pizza into fourths for 4 campers);

·       how well they are relating division to fractions (e.g., 3 divided by 4 is 3/4);

·       how well they are comparing fractional parts (e.g. a fifth is larger than an eighth);

·       how well they are comparing fractions (three fourths is less than seven eighths).