Thank You and Goodnight!

Well, mathletes...

This is the end of our journey. The end of the line.  The big shah-bang. The final countdown....for now.

So where have we reached?  I came into this course fearing math and to a large degree those fears have been reduced.  I knew nothing about math congresses, gallery walks, why manipulatives are so important, never heard the term fraction plates, understand the importance of putting math into real life contexts for students, and didn't really think math was a 'group work' type of subject....and then EDBE8P29 happened. 

At first I really had difficulty understanding why we would bother using open math problems. After weeks of creating them though, I get it.  The idea is that we want students to stretch their minds and confining math to a set of rules and ideas that fit into nice neat boxes just doesn't really work.  If we want students' minds to stretch, open math problems get them thinking in ideas rather than formulas and rules.  It also boosts confidence, which I've discovered is key with math. With open math problems, any student can get started.  They have such a wide base, and by wide base I mean that students of varying ability levels can all get started on them.  Best of all, open math problems can be applied to literally anything students can relate to.

Believe it or not, I learned that math can be fun.  This semester we incorporated Hershey's chocolate, skittles, jube jubes and, my favourite of all time - Oreos.  Talk about developing positive associations.  Apparently Oreos can be more addictive than a certain illegal substance.  Which one? You'll have to do the googling yourself, but Oreos are one way to get students addicted to math!

Speaking of fun...and now for a short math-inspired musical interlude

 

The thing I feel like I benefited most from was teaching the mini-lesson on integers.  This really made me get into the big ideas and look for ways to try and relate what I came to understand to others in ways that they could understand it.  In order to do this, I went to my personal go-to: real life contexts.  I believe the best way to understand something either new or unfamiliar is to associate it with things that are familiar to us.  In my lesson I related negative numbers to negative experiences and that as we added those negative experiences, our moods became more negative.  This works the same when adding negative integers, which can often be confusing for students because they are used to numbers increasing in value when they are added, not decreasing which is what happens when negative integers are added together.  At the same time, when we subtract negative integers (or experiences), things get more positive and the numbers increase in value as a result of subtraction - again, something that can be difficult to get used to for students.

Overall, I just feel like my math aptitude became better.  I was able to understand the connections between concepts better than I did when I originally remember being put through the math paces as a student.  It was also a great refresher of the building blocks needed for more advanced concepts in math that curious students will no doubt be asking about.  Further, it was great to go through this semester surrounded by the group of students in class who all had their own struggles with certain math concepts and ideas, and weren't afraid to acknowledge that either.  I feel like we had a really supportive bunch and that the regular ability to work in pairs reduced a lot of pressure I was initially feeling and allowed me to learn directly from my peers.

And this, mathletes, ends the semester of math.  Thank you, good night...and may the schwartz be with you.

Stay mathematical, mathletes.

Back Dat Assessment Up

Hello mathletes, 

Let's cut right to it. The purpose of assessment is to improve student learning through feedback. Frequent assessment is best, although too frequently can be unnecessary. Accurate assessment can be completed after a sufficient amount of evidence has been provided from which to draw reasonable conclusions from.

Assessment can proceed in different ways. Teachers can assess student learning by way of conversations, products and observation. This triangulation of evidence can be used to provide a holistic assessment of student learning on the report card. Reporting student achievement should reflect the evidence gathered from these variety of assessment methods. It is important to provide multiple opportunities for students to reflect their learning so that results being reported accurately reflect student learning and achievement.

Assessment through conversation

 There are three purposes of assessment: 'for learning', 'as learning' and 'of learning'.  Assessment for learning uses this practice as a diagnostic in order to determine students' prior knowledge.  From there, the teacher can proceed with instruction from the appropriate starting point according to the student level of readiness.  The second purpose of assessment, as learning, is done by students themselves either by completing peer reviews or reflecting on their own work. It offers students the opportunity to reflect on their own learning. The third purpose of assessment is of learning.  This summative assessment type assesses the products of student learning. One method of assessing student products is using a rubric.  The rubric is an achievement chart based on the performance standards of the Ontario curriculum that clearly identifies the learning goals and success criteria. Teachers will use their professional judgment to indicate at what level students are achieving the learning goals.

Nickelcachers. (2009). Rubric1 [Online Image].
Retrieved from: Flickr.com

Providing feedback is another important aspect of assessing that is based on professional judgement.  When providing feedback, assessors should keep in mind the goal of assessment (to improve student learning). . Specific feedback allows students to acknowledge areas where they need to improve and set future goals for improvement. Feedback should be framed emphasizing the positive.  Negatively framing feedback can be de-motivating - just the opposite of what we want to do with feedback for our students. One way to structure feedback can be as 1. areas of strength, 2. opportunities for improvement and 3. recommendations for future success.

Ineffective Feedback

Stay assessable, mathletes. 

Do You Measure Up?

Hello mathletes,

Measurement.  What is? How do we do it? Why is it useful? All fantastic questions. I'm not even going to bother with a standard dictionary definition of measurement because, in the creation of our own learning, is there really a standard definition of anything? The operational definition I will create to help you understand this concept is "determining a value for a feature of something".  Vague enough for you? Good.  Now that your mind is open, we shall proceed.

Want to know how much you weigh? You could measure your weight in pounds.  You could measure your weight in kilograms. You could measure your weight in stones (God save the queen!). You could measure your weight in books or M&Ms.  All you need is a sufficiently large enough scale.

Want to know how tall you are? You could measure your height in centimetres or metres. You could measure your height in inches or feet. You could even measure your height in hands.

Want to know how far the distance to or from something is? You could measure in kilometres or miles. You could measure in car lengths. You could measure it in minutes. I think you get the picture.

A famous song about measurement? 

 

Measurement provides useful information about something and can be done in many ways. What is important about measurement is that the units of measurement should be uniform.  If you wanted to weigh yourself in books, you could say that your weight is the equivalent of 25 books. It would be most accurate if the books all weighed the same amount. Remember those thick, heavy dictionaries?...Remember dictionaries at all? If you combined books of various weights (dictionaries and paperbacks) you likely wouldn't weigh the same number of books each time.  Conversions can be made between unlike units (eg. pounds to kilograms) only if they are measuring the same thing. Differing units, however, cannot be combined. There is no such thing as kilopounds and pounds cannot be converted to minutes because one is measuring weight while the other is measuring time.

There are different types of measurement that we learned about in class and that the textbook readings discuss. Perimeter measures the distance around an object. Area measures the surface across an object. Volume measures the space within a three dimensional object. Weight measures the mass of an object. Regardless, for measurement to mean something to the person doing the measuring, they must be familiar with the units. If I told you that I weighed 16 stones (God save the queen!), would that mean as much to you if I said that I weighed 224 pounds? I don't by the way, just sayin'.

Having difficulty measuring the area of a cylinder? 

One of the most important uses of information provided by measurement is the ability to make comparisons. Assistive technology/accommodations (ladders or step stools) aside, if you needed someone to get something from a high shelf you couldn't reach, you would compare yourself to someone who was taller than you and ask them to reach the object for you. Congratulations, you've instinctively taught yourself about measurement.

Cylinders are really just rectangles after all, aren't they?

Being familiar with units of measurement allows us to estimate. We can look at someone and estimate that they are about 6 feet tall because we are familiar with this unit of length. We can estimate how long 10 minutes is because we are familiar with this unit of time. Estimation is not as accurate as strict measurement but can be used when it is not imperative or there is no way to exactly determine an exact measurement. Measurement tools like scales can be used to help determine an accurate measurement but, especially in the case of electronic measurement tools, it is important that those tools are calibrated so that they produce an accurate measurement.

Stay measurable, mathletes. 

Pardon My Spatial Expressions

Hello mathletes,

Today's topic is the wonderful world of geometry and spatial reasoning.  For some of us older folks, the most fun we've ever had developing our spatial sense came outside the classroom while playing Tetris [thanks (original) Nintendo].  The idea was to manipulate the shapes and fit them together to create as many lines as possible.  Simple right? Tetris did for spatial sense what Pong did for hand-eye coordination. It even had the ability to self-regulate it's difficulty level. The better you became, the faster the pieces dropped.  Maybe there were some rogue teachers who assigned Tetris as homework back in the day but none of them taught me. For me, unfortunately the classroom couldn't compare to Tetris in developing my appreciation or ability to manipulate geometric shapes for a specified purpose. Maybe in today's technology enhanced classroom things would be different. For those of us who were fortunate enough to experience Tetris and for those who weren't, let's all take a moment to reflect since we're all pros at reflecting by now:

Was Tetris really just the ultimate manipulative? This game perfectly exemplifies one of the big ideas we learned about geometry and spatial sense in class:  there needs to be some kind of interaction between the student and the item.  If you want to get better at riding a bike, you ride a bike.  If you want to get better at catching a ball, you play catch.  So if one wants to develop a better spatial sense, then it's probably a good idea to interact with items that require and develop a spatial sense. With this idea in mind, enter the world's second (or first, depending on who you talk to) greatest contribution to the development of one's geometric and spatial sense....no not the guinea pig, the rubic's cube!

Markham, Tom. (2010). Guinea Pig on Rubic Cube [Online Image].
Retrieved from: flickr.com

Oh yea, and then there are these strikingly less fun, yet equally educational resources......

......Don't tell me you didn't recognize the Tetris shapes built with the pink blocks.  

The point is, trying to teach and learn about geometry and spatial reasoning without the use of manipulatives is like trying to eat jello through your nose. Make no mistake about it, it's doable but it's not going to be pleasant. These are a great way to meet the curriculum expectations that include constructing shapes and geometric figures.

Moving swiftly onward, one of the strengths of this topic is its applicability to every day life.  This topic lends itself to student relevance if they've ever thrown a baseball or kicked a soccer ball. It becomes real for them if they've ever eaten an ice cream cone or if they've ever seen common traffic signs like a stop sign.  As a text to self connection, you could ask students to take digital pictures of real world geometric applications and create some kind of collage containing their discoveries.  This could work to help meet the curriculum expectation of identifying and describing geometric figures.  In a great 'text to world' connection, students could be given the opportunity to explore the internet for prominent architectural designs and be asked to create their own ancient architectural masterpiece complete with story of the time period it was created, purpose it served and the society that it belonged to. I believe that the more you can make a topic relevant to your students, the battle is already half won because if you increase their caring, students will be more receptive to your sharing.

Stay spatial, mathletes.

Skittle Me This and Skittle Me That

Who doesn't love skittles? Seriously...

Not only do they tast great, provide a needed sugar rush and represent every colour of the rainbow, but now they have another, equally awesome function:  patterning.  Thanks to Kevin, who used skittles to help us understand patterning this week. Those were the most tasty manipulatives I've ever used.  Actually, it's a tie between those skittles and the Hershey's chocolate bar from the fractions lesson, but you get my point.  If you want to increase student engagement, consider using food.

Coloured Blocks

And now down to business.  Kevin's use of skittles was a great way to clearly represent patterning and set the stage for lessons on algebra.  After all, patterning is essential to algebra since part of algebra is recognizing patterns in numbers.  Unfortunately, I didn't get any pictures of the patterns we created using Kevin's skittles (because I ate them) but that's okay because we also used coloured blocks as manipulatives for representing algebraic patterns. 

The coloured blocks were a great way to represent the constant and the variables of a numerical algebraic expression in visual terms that made the pattern easier to recognize for us visual learners.

Another set up that can be used to try and recognize a pattern is the table of values. Among the different types of patterns are repeating patterns, shrinking patterns and increasing patterns. 


Can you guess the pattern in this table of values?

Henna used a highly relevant halloween themed activity for algebra which required us to solve algebraic equations in order to crack a code pertaining to Frankenstein's candy. This would be a solid junior/intermediate math activity. 

The main thing to keep in mind when dealing with patterns is to search for relationships.  Relationships, relationships, relationships.

The Positives & Negatives of Working with Integers

Hello mathletes,

This week I had the task of presenting an activity dealing with integers and the pleasure to watch the presentations of 2 classmates on the same topic. During my preparation I learned about some common stumbling blocks experienced by students when they are first introduced to integers and also became more aware of integers in real life contexts that hadn't previously occurred to me. We all go through positive and negative experiences every day, and carry positive and negative perspectives about certain things so isn't life just a series of integers?


 Don't Worry Be Happy by Bobby McFerrin...Is this really a song about integers?


Real Life Examples

• Credits and debits to a bank account
• Negative acceleration experienced by a moving object that is slowing down
• Negative time when you count down to the new year or blast off
• Temperatures above and below zero
• Hockey plus/minus ratings
• Golf scores
• Heights above and below sea level
• Buildings with floors above and below ground level 

Big Ideas

Operation vs. Notation
Before being introduced to integers, students will be most familiar with the + and - symbols in an operational sense being used for the operations of addition and subtraction.  Attaching these symbols to the numbers themselves presents them in a notational sense, as a way to describe the number, which students are likely to be unfamiliar with and may cause some confusion, especially when performing the operations of addition and subtraction using negative numbers.

Value and Direction
When it comes to the value of negative and positive numbers, positive numbers are greater than negative numbers. It will be obvious to students that 5 > 3 BUT when those numbers become negative their values change and -3 > -5.  Integers are also associated with a direction on the number line.  Depending on what style of number line is used (vertical or horizontal), getting more negative will result in moving further left or down and getting more positive will result in moving further right or up. Think of the coordinates on a grid.

Opposites
Every positive number has its opposite in the form of a negative (1 and -1 ; 50 and -50, etc.).  Adding  opposites will always equal zero.  This is called the zero property of integers. In addition opposite operations can have the same effect.  I will try to explain this in detail below.

What I really wanted to focus on in my presentation was to try and conceptualize the idea of adding and subtracting negative numbers. While preparing, I read that a common point of confusion for students when adding and subtracting integers is that they are used to numbers getting bigger or greater in value when they add and smaller or lesser in value when they subtract.  For example, adding 5+5 always increases in value to 10 and subtracting 10-5 always decreases in value to 5. This is not the case when adding and subtracting negatives.  Adding -5 + -5 results in -10 which is lesser in value and subtracting (-5) - (-5) results in 0 which is an increase of value.

With this point of confusion in mind, I first found it easier to not even use the words 'plus' and 'take away'. It was easier for me to use the words 'join' and 'remove'. For example, -5 join -5 made it easier for me to understand that the result was a more negative number. For subtraction,  -5 remove -5 helped me understand that I was starting with -5 and by removing it, the resulting number was going to be more positive. This was a minor change but for myself and students like myself, will help them understand what's happening when we add and subtract negative numbers a little more clearly.

This was a big one for me:

• Adding something positive has the same effect as removing something negative, and
• Adding something negative has the same effect as removing something positive

Did I add milk or remove negative milk?
Still tastes good either way.
Shannon, Adam. (2016, October 28). Milk.

The example I used in class was to think of light and dark.  They are opposites the same way that every positive integer has its negative opposite.

Since darkness is just the absence of light, another way to say this is that darkness is 'negative light'. In class we had 6 rows of lights that were on.  I asked Courtney to turn off 2 and we were left with 4.  This can be represented as 6 - 2 = 4.

BUT, if darkness is just the absence of light, couldn't we say that instead of removing 2 rows of light we added 2 rows of negative light? We can represent this as 6 + (-2) = 4.

If we started with 6 rows of darkness aka negative light (-6) and added 2
rows of light (+2), it would look like (-6) + 2 = -4...we've added 2 rows of light and still have 4 rows of darkness aka negative light.

We could also say that we started with 6 rows of darkness aka negative light (-6) and removed 2 rows of negative light (-2) and that would look like (-6) - (-2) = -4.

Stay thirsty, mathletes.

Keep It Simple Students

Hello mathletes,

The biggest thing I took away from last class was that you don't need to complicate things any more than they are already perceived to be with math. In particular, I'm thinking about the discussion we had about using common denominators when dividing fractions and the story we were told about a previous student who ended up in tears because they were, at one point in their own school days, taught that they were wrong for wanting to use common denominators and that the way to divide fractions was to cross multiply and then reduce.  That student later found out (years later) that either way, the answer is still the same and more importantly, they weren't wrong.

KISS: Keep It Simple Students...not these guys
Shannon, Adam. (2011, July 4). Kiss. 

The unfortunate outcome for that particular student is what I'll call a #teacherfail in that it resulted in the teacher closing off a pathway for the student rather than encouraging the exploration of it. One of the things that has been freeing about this class is the discovery of and encouragement to explore  many pathways to arrive at a particular destination. Certain math truths still exist  (2+2 is still 4 as far as I know) but how we arrive at those truths is up to us. If it makes more sense to you to add 1 + 1+ 1 + 1, go nuts.  If you're more comfortable doing 2 x 1 ÷ 2 + 3, have at it.

           Math truth: 2+2=4:

Keeping it simple doesn't have to mean finding an answer in the least number of or most basic steps. It can also mean doing what you're most comfortable with or what makes the most sense to you. If you're okay with flipping numbers and multiplying as the main operation to find the answer to a division question, power to you. I think we and our future students will gravitate to what is simplest for them to do in order to answer a math question or problem, but it's up to us as teachers to give them the opportunity to figure out what that is for them by providing them with multiple pathways while encouraging and supporting their exploration of them. 

If old school math used 'drill and kill' as a teaching method, consider referring to the new way of math as 'seek and destroy' - seek out negative math preconceptions/experiences and destroy them by encouraging the discovery and exploration of multiple pathways, and teaching math in a way that is doable, sensible, useful. In other words, keep it simple students...and it's okay to make it fun too.

                Seek and Destroy...negative math preconceptions and experiences

Make Math Great Again

Hello mathletes,

One thing we are regularly reminded that educators need to do is work on changing attitudes towards math, and including chocolate is a good place to start.  Our most recent class focused on fractions and I became considerably more engaged when the Hershey's bars were placed on our tables.  The Hershey's book and chocolate bar activity would be a great introduction to fractions for students since it was engaging and memorable. I thought the coloured plates were another great idea from one of our presenters for including manipulatives. 

1 half + 1 half = chocolatey goodness 

Fraction Plates

Another thing we touched on in class was that it is important to teach kids about the big ideas around a topic in addition to just the topic itself.  I personally think this makes sense because, for me,  knowing about what I'm doing makes it more relevant to me and therefore more interested in doing it.  Imagine if you were given a shovel and just told to just dig  without being told anything about the hole you're digging or why you're digging it.  If you were told that the hole you're digging is for the purpose of laying the foundation of a house and that your hole is going to be the start of what will become the basement of the house, you'd understand the purpose of why you're digging and likely be more willing to keep going.  Knowing more about something in general makes it more relevant to me and automatically I become more interested in it.  If i can relate to that thing in some way, my interest increases.  We've talked about it before but I feel like this is so important to do with math, especially because it isn't something that comes easily to everyone.  For those who it doesn't come easy to and have a hard time with it, helping them to understand not just how to add or subtract fractions but also things about fractions - what they are, what they're not, how they are relevant to the students and how they can be applicable in the real world - are ways to engage students more than they would be engaged learning math the traditional way where we all sat in our seats and learned 'the rules' and formulas of math.

A graphic organizer for fractions

One of the ways to teach kids about a topic is the graphic organizer.  In the middle is the topic, fractions in this case.  At the top left is where a usable definition can be determined.  At the top right, characteristics of fractions are identified.  The bottom left is for examples so students can see what a fraction looks like and the different types.  The bottom right is for non-examples.  I think knowing what something is not is just as important as knowing what it is.  This graphic organizer is a good jumping off point for students to become familiar with the math topic they will be learning.

These strategies are a departure from the traditional methods of learning math and I think would be effective to make math more fun, engaging, doable and sensible to students. They help communicate what's important about fractions in different ways and allow students to experience the same principles from different vantage points in an attempt to develop for them a more rounded and full understanding of the topic at hand. 

Good start, math, at trying to become great again.

I Don't Always Do Math but When I Do, I Use Alternative Algorithms

Hello mathletes, 

Last class was a pretty solid lesson on alternative algorithms for addition, subtraction, multiplication & division. Just to quickly recap, there was skip counting to add or subtract, partial addition/subtraction, compensation addition/subtraction, constant sum addition and constant difference subtraction.  I have no idea how I went through my entire years of schooling without being taught these methods. All I could think during class was "why am I learning this for the first time?".  

My favourite concept was the idea of 'friendly' numbers.  Numbers have not been friendly to me in the past so my first thought when I heard this term was "whatchu talkin' bout, Willis?".  After being enlightened, at one point in class we were told to multiply 28 x 25 mentally.  At first, I looked at that and had a brief panic attack.  Once composed, I first turned 25 into the friendly number of 100 by multiplying it by 4.  From there I knew that 28 x 100 was 2800.  After that, I just needed to divide by 4 since I multiplied by 4 earlier in the question and bam! I arrived at the correct answer of 700.  Mind...blown. 

A live action shot of skip counting to add

What we are being encouraged to do, and I completely agree with it, is to encourage the use of alternative algorithms.  They can be thought of as tools in a toolbox.  Some equations lend themselves to being solved more easily with certain algorithms and it's up to us to recognize which tool fits the situation.  For me, trying to do 28 x 25 was most easily done by finding the friendly number and going from there.  In class, each person who spoke up used a different algorithm to arrive at the answer.  So, you could say...'different strokes' for different folks. 

This point leads into something that we also talked about this week in class and which I think is important: respect for diversity of learning.  Not every student will be able to comprehend the world of math with the same clarity.  For some it will be crystal clear & for others it will be clear as mud. Encouraging the use of alternative algorithms gives students the choice to use the method(s) that make the most sense to them.  Teaching them principles like these will do more for them than trying to drill into their heads "the correct formula".  It also inspires a creativity of thought that math isn't especially known for.  It can be a great confidence booster for them once they learn to use these strategies effectively and the confidence that comes from being able to succeed in math is will be what keeps them from running away from it.   

Stay hungry, mathletes...hungry for numbers.

Week 1

Hello fellow mathletes and welcome to my blog where I will be blogging about my experiences learning and learning to teach junior/intermediate math. Math has never been my strong suit and I was pretty apprehensive about what to expect from this course, but the first two weeks have demonstrated to me new ways of thinking about math. For starters, this week’s reading showed that your strength in math isn’t necessarily related to your ability to teach it and there are other important aspects of math worth knowing like making connections, representing the same thing in different ways, making real world applications and arriving at the same answer in multiple ways rather than the traditional straight line methods of just plugging in numbers and following a formula to arrive at an answer.

One of the biggest eye openers for me was the idea of open math problems. In an open math problem the learner is presented with a pretty abstract idea/question and needs to come up with their own set of questions about the problem. The answer is that there really isn’t an answer but it forces the learner to think about the problem in different ways in order to come up with questions that would shed light on how to arrive at an answer.

It was evident from class this week that the overwhelming majority of us learned very traditional algorithms for solving math problems growing up. I think it’s pretty safe to say that the same majority had varying degrees of ‘ah ha’ moments when learning some alternative algorithms that were shown this week and I expect we will have many more as this course progresses.

Overall, it’s been an eye opening first couple weeks so far and it’s at least made me question some of my previous ideas about how evil math can be so hopefully things will keep trending up.

A simple example of arriving at the same answer using multiple algorithms.  The first shows a left to right method, which can be considered a more logical approach since that's the way the brain wants to interpret the information presented.  The second method shows the more traditional algorithm of adding from right to left.  This way can be problematic, especially when the numbers get more complicated.