Week 7: Report and Reflection

This week our class focused on patterning and algebra. The main idea with patterns is that they represent identified regularities, in other words, we are able to look at representations in a way that we recognize that there is a regular consistency between them in order to come up with the next variant.

Some core ideas about patterns, from the textbook are:

• Patterns can be represented in variety of ways 
• Some ways of displaying data highlight patterns
• Much of other strands in mathematics is built on pattern foundation
• Algebra is a way to represent and explain mathematical relationships and to describe and analyze change
• Relationships between quantities can be described efficiently using variables

To have our students determine patterns we would look at a visual example or a stream of numbers and decipher what the pattern here is:

https://www.mydigitalchalkboard.org/portal/default/Content/Viewer/Content?action=2&scId=306591&sciId=18291

With this image, we can see the pattern is that a square is being added to each corner of the X and keeps expanded within the next steps.

But with algebra, we would decipher the formula which is a underlying way to describe what is happening in these images. So algebra explains the mathematical relationship to describe patterns. 

So the formula for the above image 4(Input) +1 = (Output). 4 is decided to be the constant because 4 squares are always added to each image. The input would vary depending on which step we are on and then we add 1 to get the output. 

So for example 

Step 1:  4(1)+1=5

Step 2: 4(2)+1=9

Step 3: 4(3)+1=13

Step 4: 4(4)+1=17

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For our activity presentations I found our presenters to be extremely well researched, this is probably one of the most complicated topics to teach, as we seen with a lot of confusion today, so it was great that our presenters were knowledgable with the topic and prepared. 

I thought that Brittany did an exceptional job modelling. The way she segued from introduction to activity was very clear and she got our minds on with questions as well as involving us in her example in which she clearly modelled how the activity was meant to be done and how it would be shown and written before she let us do it ourselves. I think this is so important because it is not always enough to teach a lesson and give an activity and have students go in without explicitly presenting what needs to be done. 

Brett should be given a lot of credit for choosing an activity which relates to maybe one of the most complicated subjects that we have come across in the class, algebra. He was knowledgable about his topic, which a good teacher needs to be when tackling a subject that confuses about 95% of the class. What I enjoyed most about his presentation was how much we genuinely learned from it as a class to come up with answers and better understand algebraic equations. For this activity we worked in groups to to determine the relationships between the input/output numbers to come up with the relationships and equations. What was strong about the activity he chose was that it fits into a social constructivist method of teaching, which emphasizes the collaborative nature of learning. I also liked the terminology he introduced with "input" and "output," and how it was laid out in the chart. I have never heard or seen algebra presented like this before and I think it makes everything much more clear and organized.

Brittany's Activity: Use the 3 questions to come up with the colours and amounts you will use to make up your pattern,
in each box create the next step to expand your pattern
[Chamberlain, 2015 (C)]

So the last part of the class we worked within groups to determine the pattern on our worksheet with the expanding X's (similar to the X pattern I posted above). In Figure 1 you will see my group misunderstood the instruction to represent the pattern change and we thought we had to come up with a way to represent it in a different way so we thought to do a real life situation with cars in a parking lot. After realizing we were supposed to show the different ways to represent the pattern within that same pattern we changed it so you will see to the right that we represented the pattern change with the change of colours and the red dots show the change.

Fig.1
Chamberlain, 2015 (C)

I chose to include Figure 2 below because I thought that this group represented the pattern in two very different ways that I would not have thought of when I looked at the pattern. It just goes to show you how very different everybody in a classroom may visualize the same pattern.

Fig. 2
Chamberlain, 2015 (C)

There are some questions I have, not about patterning and algebra, but about teaching math in general. I am always helping with mathematics in my placement because that takes up basically half of their day and I am finding out new things about different learners and coming in situations that I would like some advice on how to address.
For example...
If I am working with students on IEPs with modifications separately, and we are working through questions from the textbook but then they are making mistakes in basic subtraction. Now I have to teach subtraction within teaching another topic. That takes away from the main topic I'm supposed to teach. So what is a good strategy to approach this situation when you have to end up backtracking so far when they do not have these basic skills, without making them more confused than they already are?

How do you include students with needs like these in group collaborative activities when they are so behind?

Also, we learned earlier this semester on different algorithms but in my placement my teacher was going over the lesson topic and when I thought I knew how to solve it he stopped me and showed me the way that the school board wanted it to be taught (different algorithm). But we are learning that students should come up with their own algorithms and that there are many ways to solve problems. So I would like to know about the documents that tell us what and how we are supposed to teach our subjects.

I find that in class there is a lot of focus on the topic of the week and the reading and activities for the topic. But I hope we can include some discussions on difficult situations that will come up in our placement and how to deal with them. Especially when it comes to students on IEPs because the majority of these students have modifications and accommodations in math and I want to be prepared for how I can change my activities and lessons to be more inclusive towards them and in ways that will be clear to them.

Week 6: Report and Reflection

This week we looked at Ratio, Rate and Proportional thinking, in a nutshell the main ideas are:

• Proportional reasoning in the deliberate use of multiplicative relationships to compare quantities and to predict the value of one quantity based of the values of others
• Ratios, rates, and percents, just like fractions and decimals, are comparisons of quantities. A rate compares quantities with different units, for example, distance to time, or price to number of items. A percent always compares a quantity to 100.
• Solving rate, ratio, or percent problems generally involves representing the rate or ratio in a different form 

Here are some ideas our class had on what we know about ratios:

Chamberlain, 2015 (C)

Chamberlain, 2015 (C)

Chamberlain, 2015 (C)

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Our Learning Activity presenter Mathieu got our minds by relating some ratios to real life, which seems to be the essential goal for teaching math. An interesting point he made was that ratios are not introduced until grade 6, but they are introduced in informal ways earlier on. He gave the example of how Kindergarten teachers will say there are 2 eyes for every person. So they are using the ratio 2:1.

When we relate math to real life, students have a better understanding of how math shapes their world, and concepts are well remembered when they can be associated with what they already know to be true.

An example of ratios in salad dressing.
A good dressing is typically 2 parts oil and 1 part vinegar, the ratio would be 2:1
(Image URL: http://www.tv411.org/sites/default/files/Math16_0.jpg )

I thought the activity that Mathieu had picked out was really great. Students started to find their own algorithms within the activity. For myself, I found the points where the lines would hit an intersection on the grid, and from there you can extend and double your lines to basically connect the dots. Some students complained that it was too hard, but when you focus too much on perfecting the drawing it takes away from the actual meaning of the activity, which is to double the proportions. I thought it was unfair that he was put on the spot and justified his choice of imagery. I think that in order to avoid this type of situation in our classrooms where students are complaining about the difficulty, we as teachers would probably have to do some modelling and do an example together as a class before they did it on their own. Or like some people said to start off with a simple square and then change the shape to something more abstract. I didn't really see the shape difficulty being a factor, I thought it was a really fun way to tie a holiday like halloween into math. But again this shows that as teachers we can never expect what may come up.

Chamberlain, 2015 (C)

I liked how afterwards we talked about how we could extend this activity, or how we could present it with different ways. Mathieu's hand-out also suggested two different ways he would present the activity, for grade 4s: to enlarge the pictures so it is twice as high and twice as wide, and for grade 6s: ask questions such as, "What is the ratio of the pumpkin's eyes." I thought that this was a great addition to the activity because now we are able to start thinking about reusing activities for different grades and meeting different expectations just by asking our students different types of questions.

Don't you hate when someone has changed the aspect ratio on your TV?
(Image URL: http://imgs.xkcd.com/comics/aspect_ratio.png )

Week 5: Report and Reflection

Good job goes out to my fellow Learning Activity presenters Kevon and Zach, another assignment done! So good on us.

This week focused on integers and the majority of class today was spent on our learning activity presentations. I am glad that our text books provide us with very many activities that we can implement into our classrooms, and it is so interesting to see the ways that they are presented by our peers. 

This week since I was a presenter I want to reflect on the lesson I provided:

I put a lot of thought into what I was going to say and introduce my lesson and how it would flow into the activity. I decided I wanted to split my lesson into 3 categories: Minds On (introduction to material), Activity (let's test out what we just learned), and Consolidation (how what we just learned applies to the curriculum expectations and our overall learning). 

2 Rules For Adding and Subtracting
Integers & Cheat Sheet

When it came to presenting I decided I wanted to introduce what integers we were dealing with (positive/negative whole numbers). And ask students what they knew about describing a positive whole number, and use that to relate to what a negative whole number is. I wanted to use that as a lead way into what we were going to look at for the lesson which then lead into the rules, and examples. 

What I was hoping from my plan was that students who were not certain could have some understanding of the integers we were dealing with, and also had a cheat sheet on the board in writing and also a diagram that they could refer back to for the equations we were about to solve. 

I decided to make a larger than life number line using a relatable material, a deck of cards, which were going to be using in our Core Activity as well. I was skeptical on how this would turn out because I was not sure if the students holding the cards had the opportunity to see what was going on. But I also wanted it to be an experience so I thought it was necessary to have volunteers come up to hold the cards. I tried to differentiate for each types of learners; experiential learning, visual learning, written information on cheat sheets, and also oral information. I am hoping that it was effective but I am worried that I could have organized my number line better and remembered to walk through by taking steps to the next answer. Next time I want to make sure that there is enough space, and also that everyone can see the line and the cheat sheet well from where ever they are sitting. In a grade 7 classroom, students could change places with those sitting so that they all have the opportunity to stand up and hold a card and then sit back down and observe.

Larger than Life Deck of Cards Number Line

I chose the card game activity from Making Math Meaningful called Integro, where you basically use your mental math abilities to shout out the sum of all the numbers of the cards everyone in the group has drawn. This was harder in larger groups and I think in the classroom with grade 7 students I would use smaller groups (maybe even groups of 2 for those not as strong in mental math). I would also not use this as an introduction activity, but maybe after a few days of covering examples just to make sure the students have some more experience with the material before putting them on the spot in front of their peers where they could possibly be embarrassed if some know the material and others are not sure. Nicole came up with some suggestions after the activity that it would be a good idea to group those who are stronger together and those who are not as strong together; this could help so that they are not frustrated that others are quicker than them and can give them a more fair chance in playing.

Integro: Find the sum of the numbers (+2)+(+4)+(-5)+(-7) = -6

Overall I am happy with how it went, I felt that there are a few things to be aware of when using this lesson in a grade 7 classroom and practicing it first on my peers allowed for some insight on that. I find that it is hard to be in front of the class and constantly wondering if what you are doing and saying is making an effective impact, because we can't be the observers of our own teaching. I am also aware that not everything is going to go as perfectly as it will in your own head, but it is good that our peers can give us feedback. 

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A few people asked about the large scale cards I made, here is the How To For Your Own DIY Large Scale Deck of Cards:

Where: Brock's IRC

Materials (all from IRC):

• 6 pieces of large white construction paper 
• about 3 (maybe more) pieces each of black and red construction paper
• 4" Block Ellison Dies (Numbers, Letters, Shamrock, and Heart)
• Tape or Glue
• Laminator (has a cutter on it but bring scissors to cut yourself incase)
• Paper cutter
• Ruler

How:

1. Divide/cut your red/black construction paper into 4" strips (should get about 4 strips per piece).
2. Use Ellison Dies to cut out your shapes - you can put 2-4 pieces in for one cut to save time, but press down a few times to make sure it's made the cut out.
3. Once you have all your numbers/shapes organize them on your pieces of white construction paper. Find where your middle point is and mark it, and you can lay out and tape or glue the outline for two cards on one large sheet of white paper. (I laid out my numbers 1" from the edge and then taped my symbols in relationship to that).
4. When you have laid out and taped down all your numbers and letters you can laminate all your pieces. 
5. Once everything is laminated, use paper cutter to slice excess lamination off, and now cut your white paper at the middle mark so your one large white piece is now two separate cards.

Week 4: Report and Reflection

Week 4's class consisted of the basic understanding of fractions and ways we can implement activities and strategies into teaching fractions.

First things first, a couple of things I learned about how to describe fractions:

• Fractions describe relationships between a part (the numerator) and a whole (the denominator)
• Although there are two numbers we have to think of them as one idea, and the relationship between the two numbers 
• Fractions can be used to represent parts of a region, parts of measurement, parts of a set or group, and division and ratios 
• Fractions have different meanings and we should be able to put these meanings together to compare their equivalency

Dylan began the class with his learning activity presentation, which I think set the bar high for the rest of us. He was confident and well spoken in his presentation and also provided interesting activities, and provided us with a very visually appealing handout with some colourful examples on it, which helped with engagement. An activity of his I especially liked was the grid he provided in his hand out for us to use the pattern blocks on. This allows us to visually compare the shapes to uncover the fractions. 

Comparing Equivalent Fractions: Yellow is 1 Whole : Red in Halves : Blue in Thirds

Anjali had great insight on how we can use patterns when we add and subtract fractions with different denominators. Something I was so nervous and unconfident about I now feel like I am slowly getting a grasp of. So thank you for that Anjali!

Mariska provided the class with an activity for teaching decimals and how we could relate that to fractions. She related her activity back into the strands of the curriculum for grades 4 and 5, which was a great reminder to us that we should not only be collecting these activities, but comparing them to the curriculum and what age groups they are appropriate for as well. Her activity was a useful one for those seeking to teach cross curricular, by adding some visual arts into the lesson! (I know Tim is reading this and thinking typical arts major...) Anyways... This is my weird looking heart that I made:

It is 32/100 squares or 32 hundredths.... or 16/50 .... or 8/25 ..... which divided equals 0.32..... Wow math is cool.

In conclusion, activities with manipulatives are great for teaching fractions since we use fractions to relate to real world situations, such as who ate more pizza than who, and helps us to make sense of it. 

Something I want to think about more is how the teacher differs from a mathematician, how teachers deconstruct and unpack the big ideas of the problems, lend an experience to them, and then reconstruct them. 

http://cliparts.co/pizza-pictures-cartoon