Sarah Kirschman’s Feelings on EMath

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.

Bye now!

Sarah Kirschman



The last class we had for EMath was mostly a work period, but I do have one thing that I would like to discuss quickly. And that is the note that stuck with me the most from our attempted lesson plans last week. I really liked the point that was made about what we teach, and how we decide what the class needs to know.

During our lessons, I felt bad for those of us who had to go up to the front and attempt to engage our class in a lesson on math. Our class was noisy, we were rowdy, we were having too much fun. And a lot of that came from the fact that we had all taken the math lessons being taught before. We already knew the lesson, we already knew the material, and we already knew the next step to take in the next feasible lesson in the unit. As such, we were intensely distracted. Now, I know that there were other factors that were in play with our lack of focus, but it was an interesting lesson that can be brought into a classroom.

What do we teach? What do we omit from our lesson? A lot of our lesson planning should come from the skills of the students. What do the students need to know? What do they already know? If you have an advanced class that is all ahead of the game, you should not expect your students to sit silently and listen to you explain concepts that they already grasp themselves. If you have a class that is struggling with math, you cannot force them to truck on ahead because they will simply not keep up. This is another big reason that I am pro adaptation in the classroom, and strongly believe in improvisation being a key skill that every teacher needs. There is that one chart that illustrates the research of Vygotsky and his Zone of Proximal Development:


That tells us that students should be met with an equal level of challenge to their competence in an area. If we do not recognize the level our students are at, we cannot effectively teach them. This is why we cannot always rely on the same lesson plans and the same lessons to teach to every class. There is no be all, end all way to explain derivatives. There is no secret method of factoring that will be useful to every student. This is why problem solving is so effective in a classroom. It plays to the strengths of the students. Good problem solving challenges students while giving them reasons to explore without the looming threat of failure if a question is not completed in the ‘correct’ way.

For myself, I am excited to go into a math classroom someday and actually teach math. I want to see the way students learn. I want to challenge myself, learn new things. Find new ways of teaching that would never have worked for me as a student, but are immensely effective for others.


It’s not that I have a problem with technology. I love technology in fact. I love how it makes life easier, that it streamlines much of what we have to do, it provides essential information in seconds. Technology makes education and teaching easier. So it’s kind of difficult to be technology adverse when all it seems to accomplish is to make everything better. So no, I don’t hate technology, I don’t have any problem with it. My only issue with technology is that so few teachers seem to know how to use it properly.

My biggest problem with technology in the classroom is that teachers always seems to use it poorly. Technology is finicky on the best of days, and my teachers never seemed to know the tricks of the trade that helped to lessen the problems that arose when using computers or watching a video on YouTube. And I think that becoming more efficient and adept with technology is one of the first steps to being a good teacher. My reasoning is that all of our future students are growing up with this technology. They have been using it and excelling at it their entire lives. If we can’t grasp the concept of maximizing a window when we’re showing a video or know the correct terminology when we’re talking about social media, then there is an extra barrier between students and teachers that doesn’t need to exist.

I think that the issue with teachers not understanding technology is that they try to use something before becoming completely familiar with it themselves. We don’t teach a lesson in class if we don’t understand what we are teaching (well, hopefully, sometimes you do get that one French sub, though), so we shouldn’t expect ourselves to excel at a skill without familiarizing ourselves with it before trying to use it in a classroom.

Technology is essential, like I said, but it is also, in my opinion, them most colossal failure of them all if it does not work as planned. I have been in many lesson where our teacher could not get something to work as intended, and so the lesson was over, we all had to sit there and wait for the teacher to figure out what had happened, all the while silently (or not silently) making fun of them for being technologically illiterate.

So always have a backup plan. If the internet goes kaput, know where the required material is in a textbook. If you can’t get the videos to load, skip it and come back later. A teacher who is good with technology is one who is not afraid to learn all there is to know about the app or calculator, etc, and is able to teach a lesson without the technology if the need arises.


Unit Planning

Today is a day of great struggle. With only six more weeks in the semester, and only six more weeks that I will be living in Regina for this year, I find it hard to focus on the important things that still need to get done, instead of the always enticing knowledge that soon I will not have to sleep on a tiny mattress on the floor, and eat food prepared by me, uncaring and just needing something to eat before I waste away of nourishment. This time of year is always hard, because with the end of school looming, I want to be able to put all of my focus onto the most important parts of my life, as in, homework and doing well, but distractions provide so much entertainment, it is hard to tune it all out. So, struggles, because I so badly want to care, but my lack of an attention span makes every moment of focus a struggle.

Oh, and I am making supper as well, in case you were all concerned I wasn’t multitasking right now.

My biggest struggle that I am having right now, when we are at the end of our EMath journey, and are focusing our attention on unit planning is the concept of focus. I have so many ideas and strategies that I want to try out or work with that it is so hard to just pick one concrete idea and just stick with it. It is one of the biggest reasons that I rely so heavily on improvisation when I am teaching. I have no idea what I really want to do, and how I want a lesson to go that I make up about fifty different outcomes for a lesson without giving myself any direction so that I am prepared in case of any emergency. A student is falling asleep? I have prepared a whole section of dialogue just for that occasion. I am sort of a contradiction – I hate concrete planning so much that I over plan to the nines. I create so many scenarios and options that it is so hard for me to pick what I want to do. Which is why having the assignment be a group project is both a blessing and a curse in its own right.

Blessing – I have other people to bounce ideas off of and to actually make me buckle down and commit to an idea and making the idea work. Together, we will have to create a unit plan that we all approve of and are all happy about, so, in turn, it makes it easier on all of us, because we have a person there to accomplish where we fail. Where I am bad at planning, someone in my group will be excellent at specifics. Where someone struggles with ideas, I can come up with a book full in an instant.

Curse – Whenever I did group work in high school, it was very much of the same that other students that took initiative had to face. I did the entire project by myself. So I have gotten very goof at relying on me, and knowing that I will be there to take care of the work. I have trouble with delegation, I feel bad asking others to do something when I could also do it, so I often decide to just do the work myself. Learning to give responsibility away and to not take on the entire project will be a struggle for me, but a good struggle, because it is essential to work in a team when I become a teacher. I will learn how to trust other people besides those that I have worked with for years.

So, honestly, even though group work is always a struggle for me, and focused planning is also hard, I am excited, because I love math, and I love teaching math, and creating a unit plan is going to be extremely challenging and rewarding. I cannot wait to see how our finished product works out. It will also be a load off my mind, because now is when all the homework comes up, and I can tell you now, not all of my assignments will be as fun as this one.

Making Plans

I have always been a completionist throughout my life. Now, I know that the term generally refers to trying to get 100% in a video game in relation to doing homework or something, but I am appropriating the term to use here, because I deem it appropriate.

We have finally completed a lesson plan in our EMath class, and I am feeling so much better now because of it. I hate leaving things undone, and we had been working on the lesson for a while now, without there being any real resolution, but we made it, just in time for our Reading Week break. I have two points to make before I can get to my break proper.

1) Team Teaching – I think that team teaching is a very intriguing concept because it is so contingent on the two people working together. I have team taught many times before, both as a drama teacher, and in the schools for our practicums. I find that it is not as difficult as I always think it will be, but I have also always been blessed with the honour of teaching with people I explicitly trust and now will not let me down. My practicum partner this semester, for example, is my best friend, who I have known for eight years. I am happy that being in our EMath class, I have the opportunity to work with people I do not know as well, and so I can see if team teaching does become more difficult when you do not know each other that well. So far, it seems to be going okay, however, it is still difficult when we think very alike, and so we both get stuck in the same section of a lesson plan. Ryan and I spent a great deal of time getting the first part of the lesson smoothed out because both of us didn’t quite like any of the beginnings we were forming.

2) Implementing Problem Solving – is harder than it looks. We have been given many examples of how to do problem solving effectively. We have done a few questions ourselves as well and after doing it for so long, we start to think that we have a solid grasp of how problem solving works. And we are learning the skills to do so, and we are definitely improving. But it is still apparent that we have a ways to go. I am saying “we” a lot, when I really mean “I”. I cannot speak for anyone else, I haven’t seen any other lessons. Perhaps it is just me who thinks that this is much harder than it looks. I think the part I struggle with is figuring out how to make questions that are grade appropriate that are not too hard or too simple for a person to do themselves. I think, with practice it will get easier, so I am excited to work more on our lesson plans and see how our skills develop.

Well, that’s all I have for you today, so TTFN!


I very much enjoyed our most recent EMath class, mostly because I learned a lot (and had fun while doing so!) but also because I have many an opinion on assessment, and I like the chance we get to all share our own perspectives.

I had never really considered the fact that there are three reasons for assessment, and that all three have their own purposes in a classroom.

1) Assessment for Learning – I really like this form of assessment, as it is more of a guideline, or an indicator, rather than a strict feedback of student work. This form of assessment is a way for the teacher to keep track of what the students know, and what they are struggling with, in an attempt to engage students and promote their understanding. I think it is extremely important for teachers to know where the students are, especially in a math class. Math is very linear – if a student cannot grasp a concept, there is a good chance it will come into play later, and they will have trouble completing future steps because they do not have the foundation to work from. Keeping up with assessing student understanding and comprehension levels can help a teacher know how to teach, and what forms of education the students are receptive to. I like the idea of random questioning in class, but I also like the idea of grading and checking homework assignments. Not recording the mark, simply giving completion marks if anything, but it can be a bit of a guideline for understanding, keeping in mind that students have worked together and possibly copied from each other.

2) Assessment as Learning – I am not so much a fan of this form of assessment because I’ve never really trusted student evaluation. I always found that if a teacher asked me to grade my work or effort level I always marked myself lower than I thought I deserved, for fear of looking too stuck up or conceited. The reverse was that students always marked themselves higher so that they could artificially boost their marks. I have never found the process to be particularly useful or informative, and I find that it does not do any good. There are two exceptions I heard in class today that I do like, however. I do like the idea of getting group members to assess each other, and then give the group members that percentage of the grade that was given. For example, the final grade being 90%, a student being evaluated at giving 50% of their effort to their assignment would get 45% overall. I think that, like we discussed in class, the form of evaluation works best with a mature class that understands the ramifications of marking their fellow classmates. And that reasoning must be provided. The second exception is the example of giving students sample answers and essays to look at an assess to determine what is a good answer and what is not. Giving the students an idea of how they are to be assessed gives them the opportunity and the skills to review their own work effectively.

3) Assessment of Learning – basic grading, which is essential, because it is how we review what a students has taken away from the class at the end of the semester. I myself do not believe in closed-book testing, so I don’t agree with strict, no holds barred, take no prisoners style assessment, but I do like the idea of being creative with assessment, with not looking at it as a test, but rather, a true reflection of what the students know and are capable of without their intelligence being compromised by outside factors. A student who has trouble writing down information, should be given the chance to give answers orally. A student should be allowed to ask a teacher for guidance, provided the teacher does not simply do the answer for them. I think we should not be setting the students up to fail, and a way to do that is to understand your students and try and shape learning to their ways of understanding. We should be giving them the chance to succeed, the opportunity to do well, not wait for them to fall.

Drive and Determination

Today’s post is all about motivation. How do we get our students to learn? How do we get them to want to learn?

I ask this question, because one, I am already at risk of repeating myself, at least this topic gives me a way to field my comments into one area, and two, I currently have a tiny puppy on my lap, who keeps tapping me with her little paw every time I try and type words out, and when I look at her, and try to tell her I’m working, she wags her tail and nudges her nose up for pets. My determination and drive is slowing, so I must get this done quickly, for fear I will never finish.

Giving students incentive is always difficult. If a student does not care about what they are being taught, then they do not have any drive to learn the material. The key to getting students’ attention is to make your lesson about information that is partly what they know, or what they are interesting, and the other part is things that they desire to know, or figure out.

As a part of the EMath class, we wrote out activities that were examples of problems that could be used in different lessons, based on Outcomes and Indicators from the Saskatchewan Curriculum website. The first one we used was the unit on puzzles and games. The activity we created revolved around the students coming up with strategies to efficiently win different games, such as Tic Tac Toe, Mastermind and Connect Four. We also added the use of video games for students who are interested in gaming, and would be more intrigued by figuring out strategies for fighting certain bosses, or the most efficient way to progress through a more open-world game. Giving students a choice on which game they would like to look at gives them the option of picking a task that they would more readily enjoy, instead of forcing them to learn pattern recognition, and permutations and combinations through the means that the teacher finds most interesting.

The second unit we worked with was unit prices and exchange rates. The question we devised was a simple example of many questions that could be formed. It is another area where you can encourage students to learn using objects or information that is significant to them. The question involves comparing the price of hot chocolate from Costco to the price of hot chocolate at Wal-Mart, but the students could use any object available at multiple stores for money. They could compare pizza prices and sizes at various fast food restaurants. They could compare the prices of books or games at different stores. The question lets the students do research and come up with an answer that is relevant to themselves. If they do unit pricing with anything, they will know where to buy what they like to get it at the cheapest price.

Allowing choice to students, and devising questions that have real world application, and let the student utilize the information in their lives, gives meaning to the work that is expected of students. Students have drive and determination when they are given a topic that they care about, or the find to be informative. If a student can see why they are learning what they are, then it is easier to convince them to focus on the subject matter, and become informed.

Now, I shall go back to cuddling my puppy. She is very happy to see me when I go back home.