Mathematical Measurements

How confident are you when you make guesses about someone's height?

This was a question that one of my teachers asked our class before, and our class had a range of answers from "hardly" to "pretty confident". This question was a precursor to an experiment where we had several student volunteers come to the front of the classroom, and have everyone guess their height in centimeters, then afterwards compared our guesses to their actual height. The results were interesting, with a range of accurate guesses and wildly wrong guesses. What fascinated me the most was the fact that while making their guesses several of my classmates had wrongly assumed that certain volunteers were taller than others, which completely skewed their guesses as to how tall the volunteers were.

When we did this experiment to test our observational skills, I began to think about how important accurate observational skills are, and how I had totally overlooked them before. For example, eyewitnesses are vitally important in our justice system, and they need to provide the police details about suspects so that the police will be able to take the correct person into custody. If an eyewitness gives the police a very inaccurate estimate about the height of the suspect, suggesting that the suspect was average height when he was really above average height, this misinformation can hamper the search for the suspect, or cause them to take the wrong person into custody.

I do not recall learning how to make accurate estimates about lengths when I was in elementary school. I remember learning about the importance of specific measurements, and measuring something correctly, but not how to judge lengths from a distance. As an adult, I wish that I had learned this skill when I was a child.

In order to teach students how to accurately estimate lengths, it's important to teach them several benchmarks to use to base their estimates off of. In Casey's video, she suggests having the students use their hands as one such benchmark for future estimates. One of the things that I really like about this suggestion is the fact that students always have their hands with them while they usually don't have a ruler nearby. One problematic thing about teaching students to use their hands is the fact that elementary students are still growing, and so their hands are also going to grow larger, which will skew their benchmark unless they consciously continue to remeasure their hands regularly to update their benchmark.

There are several ways to teach students about estimation. I really liked the activity on page 36 of this resource that suggested that teachers have students examine several different packaging options for 3D objects, use estimates to make educated guesses about which packaging uses less material, and then measure to test their predictions. I like how this activity goes beyond a 2D concept of distance or length, and ventures into the 3D world, and how it connects this concept with a real world context.

Shaping Mathematical Understanding

There was one single unit in math that actually made me excited for class: geometry. I've always considered this unit to be the artsiest mathematical unit, and as a self-identified artist, I was always drawn to geometry. Working with shapes engaged my inner artist, and satisfied the visual learner in my brain. Naturally, I didn't enjoy every single lesson as much as others, but overall, lessons were at the very least mildly interesting. My favourite lessons were the ones where we got to draw or physically build shapes.

This geometry teaching guide provides ideas and suggestions for activities to teach geometric concepts. One activity that it mentions are paper nets of 2D shapes that can be folded into a 3D shape (page 28). The Ontario curriculum actually has "identify and construct nets of prisms and pyramids" as required information that teachers have to teach (page 82). While I always enjoyed working with nets as an activity in math class, it fascinates me that the Ontario curriculum made nets an actual expectation that teachers are required to cover, because I never realized as a child that other children didn't have the same ease of connecting 2D shapes to 3D objects that I had. This is an area that as a teacher I need to be careful, because I could very easily teach this concept too quickly and assume that my students understand it without properly checking that they do understand, simply because I find this concept so obvious as a child.

Another activity that the guide suggests is drawing a 3D shape based off of a 2D sketch or description (page 29). One of the things that I love about an activity like this is how versatile it is for different types of shapes, and carrying degrees of difficulty, which makes it easy for a teacher to use differentiation in their lesson while using this activity. Not only can the teacher differentiate the complexity of the shape, but they can also differentiate the clues that they give the students about what the final 3D object is supposed to look like.

One of the sub-units in geometry that I found a little bit dull as a child was the concept of location and movement unit. I never had difficulty understanding the concepts that my teachers were describing, but I always asked myself, "Where would I use this?" One year of my teachers connected cartography to this unit, and when I was given a pirate map and asked to figure out how to find to the buried treasure, I found myself interested and understanding the relevance of this information.

Making Math Manageable

My strongest memory of algebra class: seeing and sharing the image above with all of my friends while my teacher rolled his eyes and tried to make us pay attention to real algebra.

Ah, yes, that age old habit of teenagers to disguise their fears with sarcasm. I know I was guilty of this quite often, but hey, it wasn't my fault that math was having relationship problems with its X.

All joking aside, algebra truly is something that many students fear. The word has built up quite the negative reputation for itself, and is often used in popular culture to reference math that is hard, confusing, and unpractical.

However, I don't think that algebra deserves this reputation. As much as I disliked math in school, and found it confusing, personally, I actually didn't have too much of an issue with algebra. It was one of the types of math that I found easier to understand. I think one of the reasons why the letters in the algebraic equations didn't bother me so much was because when I was younger my one of my teachers taught us about the relationship between addition and subtraction by replacing part of an equation with a box. We would need to rearrange the question to be able to figure out the number that was supposed to go in the box. While she used the term "algebra," besides teaching basic numeration skills, she was also preparing us for algebraic thinking. When I started doing math that was labelled as "algebra" and understood that the letters represented unknown variables, I would mentally replace the numbers with boxes, and the question became much clearer to me. Mentally replacing the variables with boxes also put me into the perspective of rearranging an addition question into subtraction, so the process of balancing an equation didn't seem so strange or arbitrary to me.

The one stereotype that I most dislike in popular culture's view of algebra is the conception that it is unpractical. On the contrary, I actually think that this type of math is one that is used on a fairly regular basis. The first example that comes to mind is ordering extra toppings on a pizza. When the base pizza costs a certain amount, and each additional topping costs a certain amount extra, and I'm aiming to keep my final bill under a certain amount, how many extra toppings can I afford? This is mental math that I have actually done numerous times, and so I disagree with the statement that nobody actually uses algebra. I really like the sample activity that the Making Math Meaningful textbook provides on page 620. In this activity, a teacher would ask students to suggest numerous situations where they would use an algebraic expression such as 5N. This activity shows that algebra is more than just sometimes practical, but that it is really a part of everyday living. I also really like the fact that this activity begins with an algebraic expression that is so simple because it works to challenge the perception that algebra is by definition complicated.

Since so many students are so afraid of simply hearing the word "algebra," it can be extremely beneficial to have them working with algebraic expressions, and then applying the label "algebra" to the exercises once they have successfully completed them. Playing games like "Guess the Number" where students have to work through an equation backwards, and creating a single grocery lists for several recipes are great examples of activities where students likely won't realize that they are actually using algebraic skills. Once you can prove to them that algebra is both practical, and that the basic processes are quite simple, you can show them that algebra isn't actually as frightening as they believed it would be, and they will be more likely to be open to learning more.