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.