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