Well, it’s been quite a semester. I find now more so than any other year I’ve finished that I’ve actually found a lot of use in the classes I’ve been taking. I am thinking more and talking more about what I’ve been learning. So much so that I am having a difficult time finding a way to summarize everything that I have learned in my EMath class. So that is why I have decided to separate the post into five separate categories to talk about. The five things that stood out to me the most, specifically because they are the five things I immediately remember and think about EMath. Meaning that they are the five things from the class that I will most readily implement in my teaching career.
1) Open Ended Questions – Whenever I’ve considered problem solving using word problems, I always assumed that you needed to have a correct answer and a certain way of going about the question in order to reach that conclusion. With certain problems, like the infamous Handshake Problem, there is no one way to get to the answer. There is, in fact, a multitude of ways to examine a question and come up with an answer.
A student can make a chart or a diagram, a visual in order to understand a problem better so that they have a better grasp at a starting point. While it is not necessarily okay to just throw a question at students without any guidance or any detail in the question at all, there should be a semblance of creativity in the problem solving needed to answer the question.
For me, one of the biggest learning moments for open ended questions was the fact that some questions can indeed be completely open ended. There doesn’t need to even be an answer that is correct. There can be many options, many methods. With the handshake problem, there were virtually no limits to the assumptions a student could make in solving the problem: can we shake hands with someone more than once? Do we shake hands with someone we already know? Can you shake hands with yourself? The way in which a student formulates the question in their head will effect the way in which they then go about solving the problem. Structuring a question so that it allows for assumptions and wiggle room in the list of rules or stipulations without making the question too vague is a skill that I definitely do not have at this time, and I guess it will have to be something I learn as I gain more experience as a teacher. I do know, from looking at examples, some of the things that should be avoided.
When creating a question that involves visualization as a critical tool in the solving process, it is very important that a detailed and exact visual should be given to students. One of the questions we did in EMath had us calculating the circumference of a wheel using a broken piece. The visual was not exact, and wasn’t in fact a perfect circle, the shape that the full wheel would have been was more of extremely oval than was necessary. Another question we did involved a sheet of cloth being draped in a teepee, but there was no visual provided, so none of us knew where exactly to place the sheet, on the side of the teepee? Inside of the teepee? Where inside? Is it hanging down? The open ended scenario was bad because it led to confusion, not creativity.
2) Do not punish for being wrong – This sort of goes hand-in-hand with leaving questions open ended. If a teacher is going to promote free thinking and allow students to look for other solutions or methods to solving a problem, then a teacher cannot expect all students to get the “correct” answer reliably. One of the ways to promote trying new things and being creative in a math class is to not punish students harshly for getting incorrect answers. In order for students to embrace open ended questions and creativity, they need to be brave. They need to not be afraid that they will be punished for trying new things. The negative reinforcement of an incorrect answer is a very good way to ensure a student will never try what they did to get the question wrong again, and then you will be left with students who are too nervous to answer questions in class, for fear of being reprimanded.
One of the biggest hardships a math teacher has to deal with is the fact that most students have a very hateful relationship with math. The reason for this hatred is the fact that the students have consistently done poorly in their maths and so are not looking forward to failure again.
I’ve said this before in a blog post, but I do not personally believe in giving closed book tests. I believe that it is a way to promote this failure based model of thinking. Of making students scared of being wrong, of punishing students for panicking and forgetting a formula. It is less about judging students abilities, and more of seeing the way they flounder under pressure. And the problem solving approach to math is about all students having a chance to learn, and to be challenged. If a student is scared of failing then they will never try. And so we need to promote that failure is just a step on the way to success, not a setback that can never be overcome, and will always be reflected on a report card.
3) Inquiry all the time – When we created our own lessons to present to the class, one of the biggest notes that we were given was to make sure we were making the students give us information instead of the other way around. When we were teaching a part of the lesson, we were expected to ask the class how they think it should be done, and if they had a question about a particular part of the lesson, then they should be expected to at least try and answer their own question before being fed the facts. The reason for this is to promote inquiry based learning. Encouraging students to ask and answer questions in class opens up dialogue. The students are taught to look at a scenario and be independent. To look at the problem and think “how do I solve this”, “what steps should I take” instead of immediately sit there with their hand up and awaiting a teacher to give them all of the information. Students are not only learning the math that is being taught, they are also being taught life skills.
Different classes can bring different value to students. English gives students the ability to communicate and conduct themselves in a professional manner. Physical Education promotes healthy living, and ways to stay in shape that are fun and enjoyable, not just a chore. Math is used to help students be problem solvers, be independent. A student is taught to examine a problem and attempt it on their own before enlisting the help of others. A student is given ways in which to solve a problem if they are stuck, such as making a table, or guessing and checking. With math, a student is able to be an independent thinker, which does not necessarily imply that they will always know the answer, or how to find a solution using their skills, no. It means that a student can seek help from resources such as the internet, or a book, and can use the information that they are given to attempt to solve a problem before they simply rely on others to do all of the work for them.
I find inquiry extremely useful, because it is also a good way of assessing students as well. When teaching, if you ask the class a question and no one knows how to answer, you know that you have hit the point where student understanding is limited, and so you must go into more detail. If you have students answer questions and over reaching with an answer, then you know that the students are over prepared for a lesson and need to be challenged more.
4) Technology in the math classroom – What I wanted to talk about with respect to technology is all of the ways the internet has made it easier to give students the gift of math. YouTube is full of explanations of different types of math and mathematical theories. Google anything and there are tons of resources available in an instant to use – both as a student and as a teacher. I know that websites such as Desmos are a way for students to use technology such as a graphing calculator without having to pay the $150 or more price tag. Having technology at your disposal is also a way to make math more relevant. If your students are learning about conversions, actually make them go and research different places in the world, and the cost of certain items there, and how much they would cost in Canadian dollars. Getting students to do research, to utilize the technology that we have is so helpful because it helps being math into every day life.
It shows students that math is important past the simple fact that we need to add numbers together in order to pay a tip at a restaurant. Math is relevant because we use it every day. We use it when we are driving, to adjust our speed and figure out how much longer we have to go on a long trip. We use it to draw a perfect circle when we’re making a card for our friend’s birthday.
Technology is in the world, it is everywhere. So, using technology in the classroom makes sense because it is something all kids use and like to learn about. Giving students more reason to be excited about learning is a goal in a math classroom.
5) Habits of Mind – My final topic is the habits of mind that we had to learn about in our EMath class. The reason that this stuck with me so strongly is the fact that it seems to be the “scientific” part of the student experience. It is breaking down the complexity that is math and the thinking that is involved when a student is learning math, and putting it into smaller categories that are easier to look at and manage.
Doing a lesson about probability? Students will be learning how to recognize patterns.
It makes it simpler to see where the students are at and what you as a teacher should be promoting to the students. If you want the students to be thinking algebraically, then what are some questions you can ask to get the students into that mindset? Perhaps you can ask students to model a question using a formula, then they will be using algebra.
I guess the reason I like the habits of mind so much is because it really is the part of the lesson where you consider the students and the students only. Most parts of lesson planning involve other aspects as well – what is the space we will be learning in? What videos should I show? What do I write on the board? Do I ask this question or just give the answer? Habits of mind are solely focused on the thinking of the students. The focus is on what they are learning and how they are learning it. Giving time to consider habits of mind is giving time to consider what your students will be doing and how they will be feeling.
So that’s that then. Here’s to an awesome semester, of fond memories of our tiny class of six. I am so glad that I actually enjoyed my first EMath experience because it is what I want to do with my life – I want to teach math, I want to inspire students to love math. And to have enjoyed a class that is dedicated to just that is exactly what I wanted out of this semester. So this is Sarah, signing off for her final EMath journal.