**Response:**

I think this is such an appropriate final blog post for me to respond on; it’s about the advice that will stick with you as you begin your teaching career. So it got me thinking… what advice/learnings am I going to remember forever and take with me into every classroom?

- Have a sense of humor.
- Remember that it’s okay to be wrong, and help your students understand that as well.
- Be yourself.
**Teach the students, not the subject.**- Use the grade book to track
**progress/master of skills**so that you are well-informed when assigning letter grades to your students. **Encourage mathematical discussion**.- Break the mold of traditional teaching – give students a chance to come up with the questions, given the answer.
- Make learning personal.
- Get to know your students and understand that there are things going on in their lives that may be affecting their school life.
- Make your classroom a
**safe and welcoming place.** - Don’t forget to make time for yourself.
**Teach students for the society that they will be entering.**- Never stop reflecting, learning, and evolving your practice.

These are the things that jump to my mind first about what I want to do when I enter my own classroom for the first time. I know that over time and with more interaction with other teachers and other resources this list will continue to grow. I think it’s important to note that a lot of these points are very simple; I don’t want to overthink things. I think that the simplicity of these ideas will go a long way in the classroom and I am excited for what’s to come.

One other thing that has really stuck out to me over the course of my practicum is the idea of teaching students to be mathematicians instead of just to do math. I fell in love with the idea of having students identify what strategies and qualities define a mathematician and have them constantly reflect on how they are developing in those areas throughout the course of the semester. I believe that these skills (be systematic, look for patterns, start small, be persistent, stay organized, describe, seek why and prove, work backwards, etc. ) are arguably the most applicable skills that students can learn in a math class and I have realized that they have become a way of life for myself. I found that I was approaching problems in my real life in ways that I would approach a math problem and so when students ask me why they need to learn something, I will always respond with something along the lines of “while you may not use this specific content in your life, I promise that you will use the strategies you are applying to the solving of the problem to something, sometime in your life.” There’s more to math than just math. And that’s the one thing that will remain with me in every classroom I enter, forever.

]]>This problem didn’t take me too long to figure out. I knew that I had seen it before (though in a different context) so I had hints of strategies floating around my brain. Before going into any math though, I thought about what conditions would leave a cell open and would conditions would leave it closed. Everything was opened first. Then, multiples of 2 would have closed them, and so on with other multiples. So, if a number has an even number of factors, it would end up closed and if it has an odd number of factors, it would end up open. This gave me a direction with which to carry out my trials. All I needed to do was figure out how many factors each number between 1 and 100 had!

As always I started off by organizing my information and doing a trial run of some numbers in hopes to identify a pattern. The idea of calculating how many factors all the numbers between 1 and 100 had would be way too time consuming so I sought out an easier and more mathematical route. I tried out the escapist’s strategy on what would be the first 20 prison cells to give me a starting point. Once I had numbers I was able to confirm my theory that an odd number of factors leaves it open and an even number of factors left it closed. There was a hint in my brain somewhere that was telling me that prime factorization would help out with this problem, so I also went about writing out the prime factorizations of these as well. I couldn’t see the connection right away, but I had remembered something of a theorem from my Elementary Number Theory class a few years ago that would tell me how many factors a number has. I looked this up and that was the last bit I needed before I was able to clearly see how to finish off this problem. The theorem states that the number of factors of a number is the product of the (exponents+1) in its prime factorization. So the next question I asked myself was: how can I use this information to create an odd number? Below is my progress of thinking to reach my conclusion:

- How can a product equal an odd number? It must have no even numbers in it
- The product is determined by the exponents+1, so my exponents can ONLY be even numbers
- SO! All perfect squares are open
- Any other case can be rewritten as the product of perfect squares, because my exponents are only even numbers
- The product of perfect squares is once again a perfect square so I can conclude that only situation in which a prison cell is open is if it’s cell number is a perfect square!

When I first read through this problem I thought it was so easy. I completed it so quickly and wondered even why we had been assigned this. Then I realized that I didn’t read the whole thing and that I needed to have 100 animals. So then I went about solving the ACTUAL problem. I started off the way I start off any problem: write down what I know. Doing this led to two equations and three variables, which I knew I couldn’t solve. There had to be something else that I could extract from the problem. Combing through it again, this time much more frustrated, I decided to try and take the constraint on having at least one of each animal and turn that into something that I could use. All that resulted in, though, was my changing my original equations so that at least one of each animal would be accounted for. The only way I could think to solve this was to start substituting guesses in for how many sheep there were, thus reducing my two equations to two variables and therefore making them solvable. I started off by guessing there were 51 sheep. I knew that I needed an odd number of sheep because I had already accounted for one and I needed to bring the cost to $100 even. This gave me a weird answer that didn’t work within the constraints of the situation so I moved on and tried 85 sheep. This didn’t work either. I had the feeling that there was something I was missing, so I took a break and went back to the problem a few hours later. That break was exactly what I needed. After looking at the algebra that I had used to solve my previous attempts I saw that I could generalize this situation to help me make better guesses for the number of sheep. My previous solutions kept giving me negative answers, so I needed to rework my guesses so that I could produce positive answers. The cost of my pigs, for example, had to be less than the remaining total after I purchased all my sheep (and one of each animal). So that meant that 3(97-s)<86.5-0.5s. If I solved this then I could find a restriction on s, thus narrowing down the possibilities for me to begin my guessing. This led me to the fact that I had to have more than 81.8 sheep, so I began the remainder of my tests there. After some time I found my solution. It was annoying to solve this through an educated guess and check method, and I wish I could have remembered a more efficient way to do so, but sometimes math is all about the long, tedious, number crunching!

]]>**Response:**

The subject of homework has always been one that I can never agree on a strategy for. I, like, Andrew agree that there should never be incentives offered to do it. I believe this for a few reasons: I would feel more comfortable if the students were completing the practice for the class with access to my help, the students need to learn to take responsibility for their own learning and they should not be rewarded for practice; this defeats the purpose because they need to be able to make mistakes and if they’re being given marks for homework then they will not feel comfortable to do so. I would love to adopt SBG into my practice one day and so I’m open to any ideas to motivate students to do homework and value their learning. The most valuable part of this blog post I found was actually the comments section. Andrew seems to also be struggling with ways to motivate students to do homework (and he only assigns a few questions!) and there were some excellent ideas in the comments section. I absolutely love Dan’s idea of doing peer review of homework the next day. To me this seems like the perfect solution! This is a great way to incorporate assessment as learning, as well! What I love most is that this is a way to motivate students to do homework in a way that is still super beneficial for their learning. This will help students improve their mathematical communication, their reasoning, and so much more. I can’t wait to try this myself!

]]>**Response:**

I picked this specific blog post because it’s actually one that I read and tried during my own practicum. I knew that my students needed some time to work through review problems in class but wanted to do so in a way that they would feel comfortable working together and stay on task. This activity worked fabulously! It was a nice break from the usual textbook activities (even though I just took the questions from the text) but I was able to differentiate their difficulty level based on which worksheet it fell on. I also loved the idea that students had to check their own answers in a way that didn’t directly tell them what was wrong.. they instead had to go back and discuss the answers with their group and work towards identifying their mistake themselves. When I tried this in my practicum I did encounter a few problems at the start where students kept asking me if their answer was right, but eventually they became more independent with their learning. All in all, this was a fabulous review activity, it was different enough to keep the students on track the whole class, and it really optimized their thinking and review skills. I would definitely do this again!

]]>**Response:**

This post immediately caught my eye because it just sounded like a totally awesome lesson. And after reading his post about it, I definitely cannot wait until the day I can do this in my own classroom! Letting your students be “teacher for a day” accomplishes so much in one assignment. This activity really made me think of what PDP has done in shaping my own ideas about math and how students interact with it. I know that I have come so far in my own mathematical understanding while I have gone through the process of planning a lesson and to be able to give me students the same opportunity definitely excites me. I always knew that I was “good” at math, but teaching it has shown me deeper understandings into the concepts than I ever could have gotten as just a student. This is the perfect activity to go alongside bloom’s taxonomy, because it activates that highest order of thinking. Students need to evaluate the needs of other students and the work that they are going to present them and create a lesson that will accomplish their learning goals. It also gives students an excellent opportunity to participate in peer evaluation not only in their exit slips that they had to include but also in the other students’ presentations. I can’t wait to try this out!

]]>**Response:**

So this week I’m having trouble picking a single blog post to respond to, because I’m not sure that I agree with a lot that Julie is writing about. So instead I’m just going to respond to her posts as a whole, because I find that they all followed the same type of path, anyway. I went searching and searching through her blog to find some cool activities that I could critically examine and imagine implementing in my own classroom, but everything I found seems to just be more fun ways of having students complete problem drills, for lack of a better word. I got excited when I saw her idea for math stations (I love the idea of math stations: students are working at their own pace, they’re collaborating with their peers at the same station, they’re moving around the classroom, you can differentiate the levels of difficulty..) but Julie’s math stations were made up of worksheets cut into pieces. In my opinion, that is simply assigning them a worksheet but making them move around to answer it. They are working at their own pace which is great, but I don’t see a high level of differentiation in these tasks, the problems are so procedural that there is no collaboration needed, and it’s just bland. If you were going to do math stations, each station should have a unique and open ended problem that has the students sitting at the station really grappling with each others ideas, and then an open class discussion for a wrap up on what everybody discovered. Another cool addition to that would be to have problems with new concepts worked out on these stations, and have the students try to make sense of it themselves and teach themselves and each other.

Another activity that I came across was her Draw It game. In this game students are racing to answer the questions projected on the board in their given space. I was watching the video and noticed that there were two girls racing to draw and label the area of a parallelogram. By the time the one girl got it right, the other one hadn’t even finished drawing a parallelogram. How do we know that she knows what that is? How can we help her to better understand? Was it just time pressure that messed her up? The problem was never addressed, because the other girl had put the right answer up on the board. Point for that team, next contestants, next question, done. Perhaps this would be a good activity to use for preparation for a review class so that you have an idea of what students need to work on, but I’m a little concerned as to how and what would happen to the students who go up there for their turn and have no clue how to answer the questions.

I also read a lot about quizzes and tests and summative assessments and I want my classroom to be drowning in formative assessment. I want my students to stop caring about collecting marks and I want to ease them into the world of learning for what it is. I think that classroom time shouldn’t be spent drilling students. Nor should I be assigning them 30 questions on one concept out of the textbook. I want them to embrace the struggle that is playing with mathematics in my classroom. I would be more than happy to provide them with some extra questions if they feel that they need to brush up on the procedural before jumping into a rich problem, but I think that needs to be their call. I want to teach them problem solving, not step memorization.

]]>This was an interesting problem to work on. I only watched the video twice, and both times I spent pausing and staring to see how much liquid was being transferred in that eye dropper. Right away, I knew that would be my key to solving this problem. Every time I watched the video though, I failed at figuring it out. How frustrating! I knew exactly how I wanted to solve this problem, and yet I had a key numerical value missing and therefore couldn’t really go anywhere. So I started googling how much the average eye dropper would hold.. I finally agreed upon the fact that they hold anywhere between 5 and 15ml so I knew I had a few calculations ahead of me. Now I had everything I needed to know to solve the problem! Being the avid pop drinker that I am I knew that each can contained 355ml. My method was simple: assuming the eyedropper held x ml of pop, I would do a step by step percentage calculation of the original pop and the new one, based on the new total quantity of pop and how much of each were mixed in. My instincts told me that it would probably end up being the same (assuming we transferred the same amount both times) but another part of me was dead set on the fact that that was way too obvious of a solution and that couldn’t be true. Another part of me was tempted to try it out and taste them to confirm my results, haha. In the end I did the calculations a ton of times with the eyedropper holding 5 ml, 10 ml, 11 ml, 12 ml, 13 ml, 14 ml, and finally 15 ml. They all gave me the same answer: the percentage of original pop was identical in both of them. I redid all the calculations with more decimal points because I just didn’t want to believe that they had equal concentrations! In fact I still don’t want to believe that they have different concentrations, but I have no clue how to prove that! So while the math is right in front of me, I still find myself pondering other ways to do this problem in hopes of getting a different answer, but I still can’t think of anything! Frustrating!!

]]>**Response:**

I absolutely love Geoff’s idea of creating a taxonomy of problems and I think that this is a really great idea to tie into all of the things that we learned during our assessment class with Peter Liljedahl. I think that this definitely goes hand in hand with tracking the progress of the students by skill and classifying our questions and this taxonomy helps us to create the problems that we can use to get to that point where we can track their progress. I absolutely love the idea of problem based learning and think that it gives the students a great sense of ownership of the learning because ultimately they are the ones coming up with the ideas and that’s an empowering feeling! I also think that more often than not, math educators forget that there is a difference between problems for learning and problems for confirmation and that they let the lines blur between what those really mean. I think that we need to watch that there are questions that help students to practice skills, but ultimately we need to see that they can think their way through a problem and apply the skills correctly. I think that by using these ideas to guide us towards conceptional understanding questions for confirmation and assessment questions for confirmation then we are giving our students the skills that they need and assessing them for the right reasons. Just recently I was helping a cousin study for his foundations 11 final exam and he was given booklet upon booklet of multiple choice review questions. While talking him through a lot of the content in those questions, it was clear that he was just able to take the multiple choice answers, plug them into a great conceptionally worded question, and get the correct answer without understanding any of the concepts that he was actually testing. So when, as teachers, we think about the problems we are assigning and where they fall on this taxonomy, we are giving ourselves a more accurate reading of the students skill set as well planning well rounded lessons that lead to the students further developing their critical thinking skills.

]]>**Response:**

This post, to me, was magical. I would have given anything for a high school teacher to have done that for me and I cannot wait to write letters to my own seniors one day. While he may complain that his writing was rushed, I think it was perfect. It spoke to my high school self, venturing off into university to seek out knowledge in the only subject I ever enjoyed. I wish somebody had given me his advice though, to live in the moment and to just enjoy *learning* things and *knowing* things. I always saw my degree as a path I had to go down to get where I wanted to be and having that mindset probably caused me to miss out on a lot of opportunities. I became immersed in the “work hard, play later” lifestyle and probably didn’t get enough joy out of the content and the learning. Things changed when I got into PDP. I found myself talking about the things I loved and was learning about teaching with other PDP students. I found myself buying books on assessment and differentiation because I just wanted to *know* everything that I could about them. Now I can totally relate to everything that Sam Shah is telling his seniors. I absolutely love this idea and I love his words of advice and I will definitely keep them in the back of my mind as I continue on my own personal learning journey. Sometimes you just need to take a step back and smell the roses!