Data Management and Probability

Class Dojo

I loved the little class dojo video that we watched this week to kick off our class! A growth mindset is something I myself am still working to attain. I have made sure to be careful of the language I use when I am frustrated even in this class whenever I come to a difficult point in a problem or feel like I won't ever find the solution. I want to make sure I model an attitude reflective of the growth mindset when I speak to students about any subject. Recently, this has been important when teaching grade six students as they start on their instruments for the first time. It is a very steep learning curve and many students get intimidated or frustrated quite quickly. This is why you must be sure to encourage and celebrate persistence!

Data Management was my BIGGEST worry for this course. That was the very last math class I took in highschool and it sealed my fate when I decided to never again take a math course... or so I thought. 

One of the first activities we did in class this week was to guess how many Oreos were in a big clear container. I have played this game many a time at baby showers or wedding showers and I have never, ever won. My mother however, the spatial guru that she is, always wins! This time I actually got fairly close to the answer, so I was proud of that... Maybe now I have a chance of getting it!

If only Data Management and Probability had been taught with such soft language when I was a youngin! Maybe then I wouldn't have been so overwhelmed with questions and frustration as I struggled to apply the instrumental algorithms we had thrust upon us. 

We went around the class and came up with examples where we had to estimate things in our daily lives. I thought this was a great way to avoid the question - "but how does this apply to real life?" Of course it does! We estimate things all the time! Data Management is simply the process of doing something with the DATA that you have collected! 

Do kids still get things from cereal boxes?? I remember getting some pretty great prizes when we were little. One of my favorite prizes was a nintendo game, we had to save up 10 cereal box tabs and mail them away and we received the Mario All Stars games. In this case, we were trying to determine the odds for receiving all six prizes using a dice (or number cube). I was really surprised by the variety of responses throughout the room! Harpreet and I managed to get all six in only seven rolls, but our second attempt took us 31 tries! That was really interesting to explore through the Tinkerplots. The more attempts you have, the more you really get to understand the overall chance or probability of something occurring.

Assessment & Lesson Planning

So much of our journey thus far in Math class has been focused on improving our own self-efficacy as Math learners and teachers. Week after week we have learned the importance of creating lessons that are challenging but not anxiety inducing! And now we have moved into the realm of assessing our students. I don't know about you, but assessment is still something that I find challenging. How do you assess students but also continue to build them up? How can you help them to look past the grades and focus on strategies for continual learning and improvement? 

We started class this week with a clapping exercise, which I volunteered to participate in. This was a great way to kick off our discussion. I was so thankful that I got to perform my clapping rhythm second! I think we can all relate to a time where we participated in a learning activity or took a quiz that we were not properly prepared for and how frustrating that experience was! 

What not to do

Gone are the days of teachers saying "wow you're smart" or "wow that was so quick!" Um how much do I struggle to get past this myself?? Even in this class when someone has completed a question quicker than I have or they have some mind blowing method of solving it, I feel that they are superior at Math! Praise and encouragement are valuable tools, don't stop encouraging your students. But do find a way to encourage students for being hard working and persistent with a problem. 

Also, if a student just isn't getting something, don't just give them the answer! Help them in their inquiry, teach them the value of asking good questions!

Lastly, don't teach with the language that one method is "BEST." Give students the freedom to explore and practice to improve and build a relational understanding of different strands of Math.

What to do

1. find the strength/assets of a student's solution

2. use soft language when inquiring why and how a student came to a solution 

"wonder" is a great word, "I wonder why you did this," or "can you help me understand how you did this"

3. If there is a fragile understanding of a concept, how can this be improved

Knowing the Student

We have often heard how important it is to know your students. We need to create valuable assessment tools and problem solving activities that help us to understand our students' thinking! How much can you really know from an evidence-based objective test? The real learning happens when we observe students and engage in dialogue with them to really understand how they learn

Resource

The HWDSB has term by term grade based planning tools that I found this week. Great for the big picture! Each unit has suggested textbook resources to refer to when planning instruction. Not an assessment resource, but hopefully a useful one to bookmark for the future! 

Grade 6 Mathematics Plan

Measurement

Minds On

This week we began our class with a minds on activity called "I have... who has?"
We have played this game as class before for a colleague's presentation, so we were all familiar with the expectations before we had even begun. We knew we would all need to be paying attention and we had the chance to collaborate as a table to make sure we all understood our mathematical terms before we began!

I think this is an excellent way to begin a lesson as it is a good way to ensure student's are familiar with terms and gives them a chance to activate prior knowledge. I can definitely see the advantages of using it in a music classroom to reinforce that students are familiar with important musical symbols!

Here is a resource I found with various pre-made decks! I have.. who has?

Of course, with pre-made resources, you will need to make sure that the terminology is being used properly and that it is relevant to what you are teaching!

A disadvantage to this game is that the students whose cards come up first may quickly lose interest as they no longer have an incentive for paying attention. Once a student has answered once, they could pick up a second or third card to make sure they have a motive to stay engaged. Because our group was working collaboratively, as were many others, I noticed we did not do this. We made sure to pay attention to help our classmates not miss their opportunity to answer!

Activities

Next we worked together in pairs on a quest to find shapes with the same perimeter, but a difference in area of 6. Pat suggested that we track our attempts in a column to see what we could learn from them.

We were fairly lucky in discovering a solution that worked right off the top, but then we struggled to find another solution. We did notice that odd numbers would not work but I couldn't find any clear formula in my head as to why some of my attempts were working and some were not!

On page 502 of Making Math Meaningful the writer states that students are often surprised to discover that shapes with the same area can have different perimeters. By creating different rectangular shapes and comparing their area and perimeters, students may discover that " the perimeter of a shape increases as the are is stretched out and decreases as the parts are pushed together or become more like a circle."

My favourite part of each chapter is always the common errors and misconceptions students may have. I think this is such an important resource to consult because we can prepare prior to a lesson how to avoid these common mistakes with our students and ensure a deep understanding of material through practice.

Guided Inquiry 

The next activity we moved into was a guided inquiry activity where we helped to produce giant metal columns for our school.


We used paper rolls as our models, knowing that the actual columns would 10 times the size (the ratio given was 1:10). We were given stamps, string, scissors and a ruler to help us in our quest to determine the surface area and perimeter of the cylinder.

I remember being surprised when I was younger to discover how many shapes are so alike. When you unroll a cylinder you can easily see why you are using the formula you do to determine it's surface area. The circumference x height is essentially the same as length x width used to find the surface area of a parallelogram. By having students cut open the cone into a diamond they can then contextualize why the formula was developed and this gives them a deeper relational understanding that they can fall back on when trying to retrieve this formula in the future.

This is the magic of inquiry based discovery. Even though a concept may be challenging, giving students a chance to explore and reason how to complete a problem means that there will be greater consolidation (hopefully) of the concepts that they are hoping to learn in class! This is especially true if the problem is well developed, Pat is always careful to point out to us why she uses the examples and numbers that she does when creating her problems.

One of these things is not like the other...

The biggest obstacle for me in regards to teaching math is adopting a new relational understanding of mathematical concepts so that I can in turn teach students how to have a greater understanding as well. This means I am constantly cycling through a process of unlearning and then relearning old concepts that I thought I understood! I really did appreciate Skemp's article for this reason. There was a passage in his article on the circumference of a circle that I need to share, "The circumference of a circle (that is its perimeter or the length of its boundary) is found by measurement to be a little more than three times the length of its diameter. In any circle the circumference is approximately 3·1416 times the diameter which is roughly 3 times the diameter. Neither of these figures is exact, as the exact number cannot be expressed either as a fraction or a decimal. The number is represented by the Greek letter π." This is the first time in my life that I have EVER understood why on earth we used the formula we did to determine circumference! Well if teachers had explained it like that in the first place, I wouldn't have struggled so much to retain the formula.

I appreciated Skemp's defense of instrumental instruction though, and how both are important when teaching and learning mathematics. I agree that it is difficult to shed the instrumental instruction we were originally taught because understanding things relationally is not easy... at all! And we may encounter dissonance and push back from students when they are comfortable with memorizing a formula like we often used when dividing fractions! I was reluctant to abandon this "old tradition" as I didn't have to think about it much.

In all honesty, I have not enjoyed reading a lot of our text book material prior to our classes as I feel discouraged from the onset and I am left all on my own to understand it! When we come to class and get to use hands on materials and work together, that is when I am making the biggest strides and breakthroughs. I can then return to the text with a fresh perspective and deeper understanding. Then I see its value as a future resource for instruction because I have adopted a more relational lens on these once challenging concepts.

Skemp layed out the advantages of relational math as being:

1 It is more adaptable to new tasks.

2 It is easier to remember.

There is a seeming paradox here, in that it is certainly harder to learn.

3 Relational knowledge can be effective as a goal in itself

4 Relational schemas are organic in quality.

I strongly agree with the second point, I am remembering concepts better now because they have become deeply embedded through hands on practice and by adopting the skills and methods my colleagues have taught me as we collaborate together. This weeks class was a great example of that, we got to use hands on manipulatives! Instead of looking at pictures of geometric shapes, we created them. This certainly warrants a big thumbs up!

I had no idea that there were so many children's book on math! I love books and they are such a fun way of getting students engaged! This week in class we read about the Greedy Triangle. 

This was my favourite part... 

I will definitely be looking online or through the library in the future if I am teaching a new concept to students as it is a great minds on to start a lesson. 

Patterning and Algebra

I was initially skeptical about this weeks math lesson but as soon as we pulled out the pattern blocks in class everything changed...

Setting the Stage

I had attended a professional development earlier in the year where we utilized pattern blocks to represent forms in a music lesson. We diagrammed the structure of a song using the pattern blocks and it was such a useful strategy for visual learners (like myself). Although, I hadn't fully made the connection to the fact that musical form and patterning are SO similar. Which is why I probably like patterns, I like figuring out how something is working and it's exciting when that structure can be physically represented! I also enjoyed doing this in English class, when we would deconstruct a poem and represent the form to better understand what was happening.

By creating a pattern to represent our equations in colour I found it easier to visualize how you could potentially develop an equation that would represent a linear graph. This was a better reference than the cards we received for the lesson because they lacked colour.




Minds on

I love the minds on activities at the beginning of each class as we are immediately engaged in solving a problem and the collaborative nature of solving it makes the process far less intimidating!! I volunteered to "lead" the task, and try to make sure our group was understanding everything... but I was so intimidated to lead this as it was the equations I was struggling to understand. Even though I was connecting the T charts to the appropriate graph, pattern group, and equation... understanding how that equation actually worked and what it meant was such a challenge! Seeing the actual blocks represented colour was my ah-ha! celebration dance moment! It was so clear what was the constant and what was variable using that representation!

It was also refreshing to see how we each had solved the problem so we could further our understanding of the concept and be exposed to different strategies or techniques!

Providing Meaningful Context

Connecting to real life should really be a fundamental aspect of a good math question. I found this comparison to be valuable because it is something the majority of us have been exposed to. I have a parking app on my phone for my treks to Toronto that helps me find the best bang for my buck every time I go based on how long I'm going for an time of day. *yay math!*




I had never been taught how to formulate an equation based on a linear relationship by simply referring to the right hand column. I used a trial an error process in my head whenever I looked at these T-charts or sequences of numbers, but I didn't realize you could determine what a number was going up by so efficiently!

For a long time I've felt that we need greater financial literacy in schools and I think discussing investing and interest rates would be something I would want to tackle in relation to algebra. The challenge is making it engaging and using soft language! Most of us have made financial goals for ourselves even when we were small children and so students could relate to making personal goals (ie. buying a new video game or CD). One possible scenario is if our class could research the best short term investment that would give us a reasonable gain in order to save up for a new gaming system or computer!  I am bookmarking some resources here that I want to explore further in the future. Both sites offer varying lesson plans which is why I decided to include both.

Practical money skills US

Practical money skills Canada

Integers, the negatives and positives

When I saw that integers was the topic for this week I gave a sigh of relief! My simple, approachable whole numbers! I welcomed them with open arms! As I was going over the readings I was feeling pretty confident. But then I realized... wait.. I'm going to need to explain this to students without falling back on "oh use the good old rule! two negatives makes a positive!" or "use that amazing number line!" I loved doing subtraction, multiplication, division and even addition of integers as a student because I knew the rules and my favourite manipulative (ye old number line) and they gave me what I needed, the right answer! Who cared about how I got to it or why it made sense? It just did!

In class this week however I realized that I didn't really understand why I was doing something.. I was just utilizing the algorithms I had been taught, I hadn't really rationalized things on a level that meant the learning was being deeply embedded. Fortunately there were many "ah-ha!" moments in class today especially when it came to working with negative integers. 

The first activity we completed in class was a comparison of Teagan's height with a Giant's height based on their hand sizes.

The first thing we did was complete the problem individually then we moved on to comparing our methods as a small group and finally as a class. Well I had completed the question with ease and quickly too, I realized I was one of the few who had not immediately recognized that I could use a ratio to solve my problem... and my fixed mindset whispered "well you clearly don't get Math".. which is ridiculous! Because I did arrive at the answer, but I did it in a different way! I unknowingly was using ratios and decimals to get my answer! Hurrah for being creative in Math! That's what made this such a great problem, it was approachable and open to different approaches.

I'm glad we do so much discussion in our groups as I can gain insight about how else I can approach a problem and really understand how I get to an answer. How great are congresses??? This was one of my highlights for class today. Pat approached Nuha and I to help lead discussion to teach our group how another class had arrived at the unit price of each cat food can in Joel's problem and which algorithm they used to get to the answer. 

                        

This was such a great exercise. It was powerful to see how many different approaches our classmates had taken, there were so many different ways of thinking of the question that I hadn't realized! It also helped me to have to explain my thinking and then expand my understanding as I heard each group member's perspective. 

The final "ah-ha!" moment occurred for me during Adam's lesson on integers. I had never given negative integers a story or a real life representation to better understand them. His comparison of negative integers to negative experiences gave them a context that was easy to relate to and remember! For example, if you have had 4 great experiences this morning then you have four levels of happiness! If I prevented two of the potentially negative experiences you would encounter that day [4 - (-2)] then I have increased your levels of happiness by two by being such a great friend! That's how I see it anyways... I'm excited to read what else my classmates learned from today! 

The aftermath of our fraction lesson

 Multiplying and dividing fractions.... *shudder.* This was the lesson I dreaded! I was having nasty flashbacks of middle school math class where I sat at my desk racking my brain trying to make sense of why the algorithms we were using actually made sense... Well they didn't! Multiplying reciprocals? Dividing like a crazy person? FINDING COMMON DENOMINATORS?

 I never seemed to understand the concepts we were being taught and I wanted to cry when I saw all of the fraction questions on Elevate my Math.. more like elevate my blood pressure am I right?

I don't want to give up though! I know that math is important and that I may very well be teaching it at some point in the future. Though I do have to admit that fractions are still a concept I am challenged by. I have realized that the context in which a problem is presented does seem to make a difference, if a fraction problem is related to baking or clocks I do experience an "ah-ha!" moment! Clearly I am a visual learner...

I love baking, and had three brothers growing up, so that meant a lot of doubling or tripling of recipes! But I always added the fractions in my mind.. I didn't multiply them! ... or did I? 1/3 x 3 is 1/3 and 3/1... right... Okay, so I guess I did multiply them. OK I can do this!

My favourite activity covered during our class was building the fraction squares.

I love puzzles! I thought this was a creative and exciting way to teach students about the concept of "parts of a whole" and to make fractions fun! I know this won't appeal to every student, but there was also a chance to create different designs and shapes from the fraction pieces.

I still am in love with how much we collaborate as a class. I can't do this on my own, so why should I expect students to? Why shouldn't they be able to work on assignments in groups and share their ideas and algorithms? I do think we learn best when we work together. I love that Pat is trying to teach us what makes an effective and valuable math problem. I do want to get there! I don't' want to make students discouraged like I was when I was younger. All are welcome to learn in a math class, and everyone can do it!

Making Math Meaningful

This week I wrote a mini lesson & activity that related fractions to musical notation. I never realized before doing this activity the potential of relating a math lesson to a music lesson. Time signatures, especially 4/4 time, are a great avenue to explore fractions in your music or math classroom!

The activity I delivered broke down a 4/4 measure of music into a fraction pie. A whole pie can cover the measure, which would represent a whole note, two blue half pies each represent half notes, and red quarter pies represent quarter notes, etc. I geared the activity to grade 4 and 5, as this is when fractions in fourth are covered in the math curriculum. In the arts curriculum students are exploring rhythmic musical notation at the same time. This provides a great connecting tool between the two classes! This is also a great way to have students visually and physically represent equivalent fractions using their pie manipulatives.

I was very excited to teach the activity because as I played around with my fraction pies I was making new connections to mathematics myself! Equivalent notes is not an easy concept for students to grasp. We typically see this chart breakdown on music worksheets.

Tango Musicology. (2016, January 16). Notes and Rests: Their Duration [Online Image]. Retrieved from http://www.tangomusicology.com/wordpres/music-rudiments/notes-and-rests-their-duration/ 

Why not provide students with a hands on activity that is relating to their other coursework? This gives them another algorithm to use in their music classroom!

As I walked around the room I had a learning experiences of my own! I had not taken into account that this is old news for some of my peers and they haven't seen a bar of music since grade 9! I wish I would have taken some more time to go over clapping out the rhythms. We do learn from our own experiences, however, and now I realize I will need to get more feedback from my classes to make sure my problems have a wide base at the start!

I have included a brief excerpt from my favourite work by Maurice Ravel. This is an adaptation of his opera L'Enfant et les Sortileges. In this section of the work, the young boy is having a nightmare about his math homework. The large extending pointed finger is what got to me the most. It's so accusatory! I've included a little translation after the video.




A brief translation: 

LITTLE OLD MAN

Water out of two pipes flows into a pool!

Two cabs leave the depot right at

Intervals of twenty minutes every hour,

Hour, hour, hour!

Once a country peasant,

Peasant, peasant, peasant,

Carried all his eggs to market!

Then an advertiser,

Iser, tiser, tiser,

Bought up sixty seconds airtime!

CHILD

freaking out

Oh, God! It’s arithmetic!  LITTLE OLD MAN
fascinated, repeating

acquiescingly

Tickle, tickle, tickle!

(He dances around the Child in a more harassing way)

PUPPETS

rising up and shrieking

Tickle, tickle, tickle!

LITTLE OLD MAN

in falsetto

Eight and eight, twenty,

Twelve and six, thirteen,

Eight and eight, twenty,

Nine from three, sixteen.

CHILD

Nine from three, sixteen?

It's clear the boy doesn't understand a lot of his math problems as his teacher in the dream is giving out the wrong answers! I think most of us can relate to that overwhelming feeling of confusion in certain lessons but thankfully math educators are making changes to ensure their students don't get left behind.

My favourite fraction manipulative we used in class this week was the fraction clocks! When we had to add 3/4 and 1/3 the answer came to us so much faster than converting the fractions and then adding! I have always been a strong visual and kinesthetic learner and knowing this I will need to incorporate strategies in my classroom to make sure that the auditory learners are not excluded.

I am excited to see the next group of math activities in our class. I love seeing how my peers approach tasks so differently. There is so much to learn from the variation of strategies and delivery of each task.

Dancing to the beat of my own Algo-rhythm

This week in Math was enlightening, in positive, and very surprising ways! When I was going through Math class as a youngster, we received a mish-mash of metric and imperial measurements. One moment we used a meter stick and another a measuring tape with inches and feet! I'm going to leave a few resources here for myself (and maybe others too.. hopefully I'm not alone in this). All three address metrication in our glorious nation! Clearly I have a bit of a learning curve ahead of me...

Metric System in Canada

Canadian Metric Association

Metrication Matters!

Moving on... This week was focused on the various algorithms available to you when completing addition, subtraction, and multiplication problems. We also explored differentiated models and manipulatives that can aid various learners when tackling these topics! I remember using number lines in school but the compensation and constant sum addition were SO useful! That is what I do in my head! Who knew someone had given it a name.. which we will never share with children... Note to self.



Constructing a multiplication array was mind blowing. It was so easy for me to understand, and it broke down multiplication to a simple and basic level. I'm pretty sure my jaw dropped in class when we did this exercise.

Quick meme. (Unknown date). Not Sure if Math Test Was Easy [Online Image]. Retrieved from http://www.quickmeme.com/meme/3r7rz2.

The best part about learning so many different methods is that you can check your work after you have completed a problem using one strategy. It also levels the playing field in the classroom. Each child can hopefully latch onto a different strategy that they comprehend.

As a class we discussed what makes a "good" Math problem. We decided on:
-It has a level playing field (wide base)
-It is relevant
-It can be expanded
-It is doable
-It is engaging
-There is more than one way to solve it

The problem must also be sensible to your students or apply to real life situations. I mean.. how many times have you seen a watermelon farmer?

Ifunny. (Unknown date). #Watermelon [Online Image]. Retrieved from https://ifunny.co/tags/watermelon/1441065677

I never would have believed that differentiated learning was possible in a Math class. Ironically, we watched a differentiated Math instruction video in our Psychology class this Tuesday afternoon. It was incredible watching kids that would have struggled in my childhood Math class, tackle difficult problems that were seemingly beyond their academic scope.

This has encouraged me to adopt a growth mindset about my own capabilities, not only as a Math learner, but as a Math teacher! I can now take steps towards a way of teaching Math that I understand and can confidently participate in. My students will be able to experiment with a Math question, twisting, and turning it around until they find an algorithm that they actually understand not just a formula they are going to memorize??? Maybe they will even come up with their own formula?? That would be incredible!

What lurks for us in the shadows of division I wonder... I don't have to worry! Is it lame I'm kind of excited?

Oh hey there, Welcome to my Math Journey!

Not quite an example of an open problem, but I'm getting there...

 It is so foreign to me to be embracing creativity in math, but more than that... there can be more than one correct answer? Students have a voice in finding the answer?

Whenever I successfully solved a math problem in school I would experience a thrill of genius! It was empowering to finally understand how to get a correct answer. Going over the four 4s task in class brought back that feeling.

It reminded me of the scene in Hook where Peter is finally able to use his imagination at the never-feast and play with the lost boys! You feel a sense of inclusion when you can finally participate and understand what is going on! I didn't know that was always possible in math class but I have to say, I'm excited with where this is going...