Day 11--Technology in the Classroom

I have to say that I did not find the SAMR article particularly informative, it did serve as a good entry point for discussion of how to integrate technology in the classroom.

Our small group discussion, however, proved to be more engaging and bring to the fore more of the benefits and drawbacks of the use/integration of technology in our classrooms.

One of the things we discussed was the author's analogy of technology in the classroom being like grandma's house, a nice place that you would like to visit but wouldn't want to stay long term. One group member argued that technology should be the place that you go, it should be the content.

For the most part, I agree with this argument. I believe there needs to be a more seamless integration of technology in the classroom.  There is far form that can be done with it than is currently being done in most classrooms. One example we discussed was the use of the Smartboard. We have all seen a Smartboard being used more like a whiteboard than for its interactive capabilities.

One concern is the idea that technology can be too engaging. To use the Smartboard example again, students are often so excited about using the technology that they do not pay attention to the lesson. This speaks to how well the technology is integrated. Perhaps if the technology was so commonplace in the classroom that it was seen as just another learning tool like a pencil or a textbook then students might focus more on the lesson than on the technology. On the other hand, part of the appeal of technology is that it's novelty increases student engagement. I can't tell you how many students have told me that going to the computer lab to do Razz Kids or Cool Math Games was the best part of their day. The question is: Are they excited about learning, or are they excited about being on the computer? And at the end of the day, as long as the students are learning, does the cause for their excitement really matter?

Day 10--Creating a Balanced Math Program

Balance is Basic: A 21s Century View of a Balanced Math Program

As the title indicates, this article is about creating a balanced math program in the 21st century.  This article is less of a "how to" and more of an outline of the importance of and reasoning for creating a balanced math program.

According to the article, the fundamental things we want students to know and do has not really changed over time.  We want them to understand concepts, be able to perform mathematical skills, and solve problems.  The problem lies not in these broad categories but in the weight and importance placed on each. The first category is conceptual understanding.  This is the area that is most concerned with students understanding of given concepts rather than their ability to do computation or solve problems.  This can be likened to the discussions we have had in class around the big idea.  If the big idea or concept is division, when thinking about this category we would be concerned most about the student understanding that when you divide, the quantity you are dividing gets smaller.  This is similar to the example in class today where Ve demonstrated how we would help students come to an understanding that a positive exponent makes a number larger, while a negative exponent makes it smaller.

The next category is concerned with the actual skills, facts, procedures of math. Using the example above, this is where teach students how do divide, show them the symbols necessary for division, and allow them to work with the numbers to figure things out.  There is some debate as to how much computational knowledge students need.  Do they need "drill and kill" where they are doing the same calculations over and over again just with different numbers, or is it better for them to know how to do a few or specific calculations and be able to use some sort of tool to figure out things beyond that?  Marian Small advocates for an approach that does less of the former and more of the latter.  She argues that given the technological world we live in and the accessibility of devices, it is easy to find a tool to help with the calculations. This ideas is furthered by the author of the article, who suggests that students might have to know how to divide two digit numbers by using pencil and paper, but calculators could be used for larger tasks.

The last category is problem solving.  This is the area concerned with moving students out of the textbook and giving them the opportunity to apply what they know to authentic and real-world contexts.  According to the article, this category is not just solving the problem, but being able to use the right tools for the problem as well.  Problem-solving is often used as the measure to indicate how well students have grasped and assimilated what they have learned.

Beyond these three key categories is what the author refers to as the "connective tissue".  These are concepts like mathematical habits of mind, flexible thinking skills and strong reasoning skills that are necessary skills for students to have if they are to be successful in the 21 century.  Most of these skills can be related to the mathematical processes that we learned about recently. For example, the connecting process would allow for mathematical flexibility because students would learn how to apply their understanding in different contexts. Another example might be the reasoning and proving processes where students would need to provide thoughtful answers and be able to support or prove their reasoning.

One of the last things the article touches on is the idea of engagement being one part of an interconnected rope that defines mathematical knowledge.  Engagement is includes skills/qualities like persistence and willingness to take on a challenge.  This can be related to the growth mindset that we discussed needing to be cultivated in our students.  Dan Meyer seems to support the idea that this is something that needs to be developed in students because according to him, we have students who are "impatient with irresolution".  That is to say that they want quick fixes and are uncomfortable with things that do not come in simple packages.  We have to teach them that sometimes they have to stick with it and sometimes that have to just accept the discomfort.

Below is a puzzle that I made to demonstrate my understanding of the today's reading.  I conceptualized it as a three set Venn diagram with a balanced math program at the intersection of doing the math, conceptual understanding, and problem solving.  All three of the categories are bound by the connective tissue that holds everything in place. I used  a puzzle to illustrate the idea that everything is connected and without any individual piece, the picture would be incomplete.

Discussion questions:
* Do you think you use one knowledge area over another? If so, which is it and how do you think you would counteract that to ensure you have a more balanced program.
* Do you think you can have an effective math program without the “connective tissue?” Why or why not? Would the program still be effective without the connective tissue?
 * Even though there is evidence to support a balanced math program, there are teachers who still favor a “back to basics” approach. What arguments do you think they might have to support this thinking?
 * Do you think you might meet resistance trying to input this program in your school? Who might you meet resistance from and how would you address the resistance?

Day 9 Feedback and Thinking Tools

Feedback

Today started with the class watching a Marian Small video.  We were asked to come up with 3 things we know about feedback, 2 things we learned, and 1 question we still had at the end of the video.  I have to admit that I watched the whole video waiting for her to say something about feedback and wondering why she kept talking about questioning instead.  I liken this this to the discussion we had the other day on success criteria.  I was given what to look for (the questions that Ve posed) and was so busy waiting to hear something that was related to what I knew about feedback that I missed that in talking about questions she was talking about feedback.

I have always seen feedback as something the teacher gives to the student in order to help the student improve upon his or her work, so using questions to prompt the students for further thinking never really crossed my mind. Ve, however, said something that changed my thinking so that I am now able to see the value of using questions for feedback.  He said that the purpose of feedback is to have students leave knowing how they are doing, and questioning allows students to nudge their own thinking so they can come up with the answers themselves and "fix" their own thinking.  This makes sense since we have discussed many times that our goals as teachers is to have our students be able to problem solve and think independently.  Rather than just simply telling the student what she has done well or what she can improve upon or where she has made her mistake, through questioning the student can figure that out for herself. 

One thing I continue to struggle with is knowing when I should delve into student thinking and when and how to let students know when they are simply wrong.  I believe a student needs to know that 3+7=10.  If he comes up with 3+7=12, that is a problem.  Of course I would have a conversation with him about how he arrived at that understanding, and it would be great if that discussion was enough.  But if it is not, when and how do I tell him that the answer is 10.  I do not want him leaving thinking the wrong answer is correct and that he can apply whatever thinking he used to get that answer to other situations.  I suppose this is something that I will understand better when I actually have a student with such a difficulty and as I gain more experience.

The Area Model and Algebra Tiles

I really liked the area model.  It made sense to me and I can see its inherent value.  I especially like that it is a visual representation that can be used in a variety of contexts as students move through the grades.

I found that the algebra tiles can be useful--like most manipulatives--in the right contexts.  During our discussion of the article, our group talked about the idea of doing 4-7 with algebra tiles somewhat problematic.  It does not make sense to add something in order to then take it away.  It also did not make sense to essentially be adding zero in order to subtract 7.  One member of our group suggested that instead of 3 zero pairs we add 7 negative tiles.  This would be a better visual representation. Students could add 7 negative tiles and then get rid of 4 zero pairs leaving -3.

Thanks for reading.

 

Assessment

This reflection has taken me a little while to write, and I wasn't exactly sure why.  I have come to the realization that it is because assessment is something very personal to me and I have long advocated for a more balanced form of assessment that did not privilege one form of evidence over another instead of giving them equal weight.

On a personal note, I am not a great writer. It's not because I was unable to formulate the thoughts or because I couldn't make interesting points, but because it was hard for me to get everything straight in my brain.  It was very frustrating for me to know the material but not get credit (or at least not get enough credit) for the things I knew because I wasn't able to fluff  my way through a ten-page paper. It was very disheartening.

That said, my past experiences with assessment make me very excited that there is triangulation of evidence in place. Now I can give equal credit to the things I hear my students say, and know that I have a document that I can point to in order to support my assessment and evaluation decisions.

Something that Ve said during this class was about professional judgement.  He said that Marbook's job is to record marks but it is his job to use his professional judgement to give a final mark.  That was something that resonated with me and made sense to me because of its simplicity.  I don't have a class of my own, but I think I would feel like I had to give whatever mark my recording system spewed out based on the information that I had input.  Now I know that as long as I have the evidence to back up my decision, I can give my students the marks they deserve based on my professional judgement.

On a different note, I find it interesting that so many of us who went to different teachers' colleges were lead to believe that assessment for learning and assessment as learning were just diagnostic and formative assessment renamed.  I am thankful that Ve made that chart for us and explained it in a way that was easy for me to understand and that made sense. 

One question I have is if assessments, or at least how they are presented, are different based on who the audience is. The chart below is from the Peel District School Board's Growing Success Monograph Series: Evidence of Learning: Conversations, Grade 1-12.  It has a space that indicates a place to check the intended audience.  This leads me to believe that the assessment might be different based on the intended audience. Thoughts?

Day 7--ngiseD sdrawkcaB and Rich Tasks

Begin with the end in mind.

This is a something we discussed quite a bit during my teacher education program but I have to admit that it was something that I still struggle with.  I think part of my problem is that I am a very (very) linear thinker and I struggle when I am forced to go in different directions and see the big picture.  I think of it as focusing on the puzzle pieces instead of seeing the big picture it creates.  So working with the end it mind and figuring out how to get there is going to take some getting used to.

The videos that we watched in class helped clarify how to begin the process.  The first video, "An Interesting Take", was helpful in illustrating how students and teachers can be at cross purposes if we don't have the same goal in mind.  It also served to show how getting to the end may require different things for different people (ex. one person had to get used to wearing contacts, another had to take swimming lessons.)

''Aligned' is Key" outlines the importance of ensuring our assessments are actually aligned with our expectations and learning goals.  I have tried to keep this in mind, but it is hard.  It is easy with an example of making a pie, and the evaluation of that makes sense.  I find it a little more difficult to create  those kind of aligned assessments in math, especially if I want the assessments to be interesting and still speak to the learning objectives. This may be where a better understanding of rich tasks come in.

I have to say that the I found the article "Rich Tasks and Contexts" rather dry and somewhat repetitive.  Still, there were some good points.  One salient idea that came out of the article was the idea of persistence.  I feel that persistence is a skill that has to be taught.  I think that kids may be uncomfortable with some of the things and ways they are learning.  It is important to feel the discomfort and move past.  

The other important point that the article makes is that tasks in and of themselves aren't inherently rich. It is what the teacher does with the task and the questions that she asks that make it rich.  I think this is something important to remember.  I know that when I am preparing a lesson, I troll the interntet for something "cool" for my students to do.  But the task is only cool because of the way I am able to implement it.  If I do a task similar to the glass task that we did in class to day but then give the students all the information they need and tell them to calculate the volume, the task is no longer rich.

Day 6--Questions about Questioning

At first thought, it would seem that asking effective questions might be among the easiest parts of our job.  After all, we have been asking questions since we could talk: "But why?" "Why not?" "How come"  What today's reading and the group discussion highlighted for me is the fact that there is a lot more to asking effective questions than the who, what, where, when, why, and how than I previously thought.

First, asking effective questions needs to be thought of as a skill like any other.  The pitfall with learning this particular skill, I feel, is the ease with which we can easily fall back into our old way of thinking and asking questions.  If asking effective questions is something that we are going to endeavour to do, it is going to have to be something we are cognisant and mindful of.

One of the most fruitful parts of our small group discussion was around the use of rhetorical questions.  With a question like the one from the article where the author asks, "doesn't a square have four sides?", it is easy to see that this question does not actually require a response.  The question did, however, lead us to a discussion on the use of sarcasm in the class.  More importantly, though, what I took away from our talk is the idea that questions that we might have can be reframed in such a way that the answer shows more thinking on the student's part.  For example, the question about the square having 4 sides might have been used as a question to guide the student to a calculation mistake they might have made.  Changing the question to ask the student to tell you what he knows about a square not only achieves the same goal of getting the student to check his work, but it also gets him thinking about the properties he knows about a square and which of those properties is necessary to answer the question.

Another thing to consider is not only the questions we are asking, but when.  I know from observation and my own practice that as teachers we are often eager to jump into a conversation with students and ask them to talk about or explain what they are doing in that moment.  The article suggests that although it is important to ask good questions, they are not a replacement for thoughtful listening.  I feel that if we are just patient and listen to the conversations that are going on, our questions may be answered without having to interrupt the learning process.

Finally, I think that one thing I have to be mindful of for myself is to take myself out of it and, as Ve says, "don't get my feelings hurt."  I think that as teachers we have the best interest of our students at heart and so we come very invested in the things and ways we teach.  I need to learn to let that go and understand that if I ask a question that does not work, I will just learn from that for next time.

Day 5--What is Mathematical Thinking?

One of the greatest things I took from today's reading on Mathematical Thinking by Barbara Ball was the idea of mathematical thinking vs. mathematical knowledge. From our previous class discussions, I think that many of us have been hung up on our lack of mathematical knowledge and we were concerned about how we can teach math without that strong foundation.  Today's reading and class discussions have served to make me feel more comfortable with the not knowing.   I am not saying that we as teachers should just "wing it" and not know how to do the math that we are trying to teach.  What I am saying is that if we are trying to promote mathematical thinking, then we can worry less about the answer and more about the thinking and process to get us to the answer.  In this digital age of Google and the internet, answers are but a few clicks away: If I want to know almost any formula, all I have to do is put it in a search engine and wait for the information to be displayed on a screen.  However, the thinking that goes into knowing when and how to use that formula is more difficult to know. That is mathematical thinking: not just trying to get the answer, but thinking about ways to arrive at an answer.

I feel the the mathematical processes that we discussed today go hand in hand with mathematical thinking.  It is through the use of the processes that we enhance our thinking and strengthen our understanding of math and how we become better able to use that thinking in larger contexts, which is the "independent thinking" that Polya speaks of.

My goal going forward is to not only challenge my students to think mathematically, but to also challenge myself, to learn to be comfortable with the discomfort of no one right answer and no one way of arriving at an answer.