Constant Assessment Is Best

                  Several aspects and attributes are needed to be an effective teacher. You need to know your students, be able to create an inclusive environment, be able to engage students in critical thinking through lesson plans that grab their attention. What is often overlooked when it comes to key aspects to be an effective teacher is being effective at assessment. The OEM document Growing Success is the end all be all when it comes to how educators should structure how they assess their students. I always viewed assessment as its basic form, taking notes of students work, collecting their work and assigning letter grades and commenting on their performance in regards to the achievement chart. That type of learning is what Growing Success and many other educational sources deem as Assessment of Learning. What was relatively new to me is the notion that educators must actively think about incorporating Assessment as Learning within their lesson plan and assessment moves. I was aware that as a teacher, I would want to promote more and more self-regulation and editing for my students. Assessment as Learning is now something that I really want to focus on as a junior/intermediate teacher, especially if I am able to teach grade 8’s. I think focusing on teaching students at that age to self-regulate and continually assess their own works will do dividends for them as they continue to secondary school where there is much less teacher intervention when it comes to checking up on students doing work.

A great example of assessing a student with a detailed rubric but also incorporating personal feedback  that bolsters assessment as learning. Retrieved from http://bit.ly/2gFlGhH.

                   In regards to math, assessment can be easy at times, but also very hard. It is very easy for a teacher to collect work pages, quizzes and tests and come up with a grade based on quantitative data. However, just basing your assessment on those sources is not at all being close to being an effective teacher. In the text, Making Math Meaningful to Canadian Students, K-8, they state that in order to be a good assessor,  I must be able to gather information about students’ knowledge and abilities not only from a variety of sources, but on many occasions to ensure that the information is reliable and valid. The text provides a guide on how to be a good assessor in math, via 8 key paths. I think one of the most important ways the text identifies as areas to focus on is to ensure your assessment is fair to all students. In order for me to do that, I need to ensure that I am able to create strong bonds with all my students, especially students who have accommodations or individual plans that alter their learning. It is key to understand that if the way you teach needs to differ for some students that you allow students to be assessed differently too. Another interesting tactic the text presented was the idea that students benefit greatly when teachers “set high, yet realistic, expectations for students”.  The text argues, based on research that was done, that students respond well when they are presented with high expectations because teachers who do not expect a lot of their students may embrace a more negative attitude in the classroom.

                My ability to be able to create a number of different ways for students to be assessed is crucial to my future if I am presented with the opportunity to teach math. It is easy to base marks only on tests and quizzes in math, but that does not accommodate the various learning styles students have. I must not only commit to being able to different my instruction, but be able to differentiate my assessment, I believe this will result in higher achievement for both me and my students in the classroom. 

Geome-tricky

          The focus again for this past week’s math class revolved around geometry, which somewhat solidified my belief that this particular strand of math is crucial for us as future educators. We focused more on working with 3D shapes, specifically working with cylinders by working through a type of activity that our Instructor called a “Guided Learning Activity”. Like most of our activities we did in class, we were asked to team up with our classmates to take on the activity, but the nature of this type of instruction is slightly different. Guided Learning Activities are supported by a somewhat intricate back-story, a topic with a little bit more substance than usual, and the students are asked to solve a big problem by working through a worksheet. Our activity was based around the task of trying to figure out how many giant metal cylinders could be made out of a given sheet of metal. The questions started simple, asking us to use toilet paper rolls to try and determine the exact measurements of the cylinders that will be built according to the activity. The activity guided us through, gradually, but also asked us to think deeply about what it was asking and what to do with the information we were acquiring while working through the series of questions. This type of learning activity is an amazing tool for an educator. It allows educators to have their students do exactly what they intended them to do in regards to what learning goals they have set out for their class but also allows them to have their students use the key skills in regards to the achievement charts. The assessment criteria for an activity lie this is very multifaceted in regards to how much an educator can assess.

An example of the easy to get manipulatives we needed to complete the activity in class. 

                One of the best aspect about the activity we did was that it would ask students to really be able to have a grasp on the language and terms that emerges when learning geometry. Every strand of math has plenty of new words and terms that students must understand and know when to use, but geometry has so many terms that students can get hung up on and miss use. In my opinion the best way to combat confusion is to focus on having your students use physical manipulative so they can see and feel the difference between shapes, especially when it comes to 2D and 3D shapes. The text, Making Math Meaningful to Canadian Students, they identify how students often struggle with what appropriate language to use when dealing with 2D and 3D shapes. Students may have issues when they start to work with 3D objects, such as calling a rectangular-based prism a rectangle, or calling a cube a square. This misconception is very common, and is often seen when students are first introduced to 3D objects, and their confusion can be seen as them understanding that 3D objects have 2D objects within them.

The struggles students have in the geometry lessons are of course not limited to working with 3D objects. The text, Making Math Meaningful to Canadian Students, identifies the issues that students have in regards to their ability to the common belief that the orientation of a shape is what defines it. The text suggests using concrete shapes like patter blocks to illustrate that it does not matter what orientation a shape is, a triangle is always a triangle and a square is always a square.  How well students grasp the language when it comes to geometry is a good measurement about how well they will understand when they continue to work with 2D and 3D shapes.  I think the language of geometry is the huge base that students need so they can continue to explore, learn and develop geometric skills and thinking. 

Geometry, It's Fun for You and Me

              I consider myself both a geographer and an environmentalist, mostly because I majored in geography and environmental studies for my undergraduate degree. Everything about those two subjects appeals to me; I find it all so interesting. I can also confidently say that one reason why I grew do fond to those specific social sciences is that I am very spatially aware.

                In this past week’s math class, we focused on geometry, exploring the ins and outs of 2D and 3D shapes. One topic we also focused on was how geometry connect to spatial skills, specifically how the development of geometric thinking can have a profound effect on a child’s development in regards to spatial skills and awareness. The text, Making Math Meaningful to Canadian Students, K-8, summarizes the ideas behind Pierre van Hiele and Dina van Hiele-Geldof’s Taxonomy of Geometric Thinking, explain how students go through 5 stages on their way to developing geometric thinking.  Level 0 is what they explained as the “Visualization” stage where students begin to be able to identify shapes based on the fact that they a circle simply “looks like” a circle and a triangle is a triangle is because it “looks like” a triangle. Level 1 is the Analysis stage where students begin to conceive shaped in groups based on their similarities, followed by Level 2, the Informal Deduction stage where students develop the greater ability to develop their own simple logical arguments about shape properties. Level 3 and 4, Deduction and Rigor, apply to students learning at a high school level and beyond where students develop advanced understanding about shapes through traditional and non-traditional axiom structures. I found it very interesting how the stages the van Hiele’s developed, mirror many cognitive taxonomies in regards to students general development as well. This makes me believe that the fact that geometric thinking is s connected to one’s own spatial skills, the development of good understanding of geometric thought is crucial for both math students and math teachers.

                For the most part, I think some of my most favourite activities in math class during my time in elementary school revolved around the geometric strands. I recall making 3D shapes with straws and pipe cleaners. I think that exercise is used today often because it allows students to create the shapes they are learning about through the construction of their own manipulatives. Use of manipulatives while teaching geometric operations at any level is a great idea, but is also simple because there are so many different types of manipulatives teachers can use. Pipe cleaners and straws cam be used to make 3D shapes, students can also construct their own shapes with paper. The creation of shapes by students addresses the first three levels of the Taxonomy of Geometric Thinking, and most importantly, it does this in a fun and engaging way. There are many other activities students can complete with the use of manipulatives, in class we completed some “Tangram Duos”, which allowed us to use different shapes to create specific 2D shapes.

Tangram Fun!

                For me, I think my appreciation of my time studying geometry in my early years of education can be connected to my spatial awareness, but also my appreciation to geography because it is the study of space and how we interact with it. All strands of math are important , but the development of geometric reasoning and thought is somewhat more important because it can shape how individuals use and understand space. It also can be considered as one of the most important stands of math because it follows cognitive development so closely.

Thank you for reading!

Algebra is Pattern, Algebra is Fun, Algebra is Pattern, Algebra is Fun

                Out of all the math units, algebra has always been the hardest for me to complete. Algebraic equations appeared to be strange foreign language to me, and still do to some degree today. So when I realized that this week in our math class we started our conversation about teaching algebra after I completed the reading, I half excited to find out how our instructor was going to approach it through the new math lens we have been practicing but I was still nervous.

                “Algebra is just patterns”, is something that I have heard my instructor say many times already, but I did not really think about how true it way. The notion of taking a simple patter, such as number starting at 3 and increasing by 4 can be translated into a linear equation, is something that is relatively new to me. Sadly, I do not think my math teachers in the past ever taught me this way, and if they did, it certainly did not stick. We began the class by performing a “mix-and-match” exercise where we had to identify what graph, table and visual aid matched with each equation that we were given. This activity was phenomenal at illustrating how linear algebraic equations are able to be compared and visually shown in different ways.

Matching Algebra Activity

                One aspect of this activity that stuck out to me the most was the tangible use of manipulatives. I do not recall using manipulatives at all to demonstrate algebraic patterns. The use of the connect cubes allowed me to really be able to visualize how the equations we were using change as they continue through the pattern. The idea of using two different colours of cubes would really allow students to visualize the patterns they are dealing with. The image below illustrates how the equation (Number of Tiles =2s + 2. Providing manipulatives for the students to be able use the hands to actually build a model of how the values increase would help provide background for the student when dealing with missing variables in the future.

                I never specifically associated algebraic equations with patterns, but even that simple idea has improved my understanding of the subject.  Our text book states that one of the most common mistakes students make is misinterpreting equations, specifically when dealing with equations that use “x” as a variable, students commonly see “x” as the multiplicative process when first exposed to equations. It is important to combat this by not using the letter in equations or by italicizing the parts of the equation, they would appear like this for example T = ab X f.

 I think there is a general consensus in regards to students having negative views toward algebra.  I think through proper planning and structure, educators are able to remove the intimidating vail that comes with algebra by focusing on committing to the use of manipulatives and open questions. Of course, the same notion can be said with any strand of math, and it should be. However, with algebra being the foundation to higher levels of math, slow introductions that really focus on getting students to realize the pattern aspects would result in the building of a better foundation for them to better understand more complex algebraic equations in the future.

Keep on keepin’ on