Month: February 2014

The Teaching Gap & Five Practices

This past week as part of our home workshops for our content area seminar we continued reading The Teaching Gap as well as learning about the Five Practices for Orchestrating Mathematical Discussion. The Five Practices include first anticipating what strategies students will use while they are trying to solve the problem. Then while they are actually working you monitor what they are doing and then select different strategies for students to share. Once you have selected who you want to share then you need to sequence them in a meaningful order. As these are presented you should make sure students can see the connections between the strategies that are being used. The Teaching Gap talks about how the “teaching script” is different between the United States, Germany, and Japan. The latest chapter we read talked about the use of the chalkboard in class. They talk about how in a typical American class some sort of overhead projector is used to focus students attention. However, teachers in Japan would never use an overhead projector for that purpose, they use the board as a way to keep a record of what was talked about during the class period. The pattern of lessons was also discussed, talking about how the day was structured. All three start with some type of review of what was happening the day before. In Germany they then then give the problems and topic for the day, spend time developing the procedures for the problems, and then practice the problems. In Japan they would then present the problem for the day, and then allow the students time to work individually or in groups, then discuss the solution methods, and then summarize the day. In the US after the introduction there is demonstration on how to solve the problem, then practicing, and next assigning homework.

As I was reading I was thinking about how I see some of this in the way I teach and the way that I have learned throughout my education. I definitely have seen the American teaching script in my math classes. Reading through the routine it sounded a lot like a vast majority of the math courses I have taken. I would say typically it is the way that I teach although we are trying to get more discovery based learning activities in our classroom this semester. Before reading this I never really thought of the way math was taught was bad, I mean I’ve always done well. It’s really hard to even think about changing because this is all that I know. In my placement this semester we have worked on trying to having our students do some of the discovering on their own but after reading TTG I wonder is this really all that different from the standard American script. I’m really hoping that as I finish the rest of the book I will be able to see what it means to teach mathematics in these countries and how I can incorporate those ideas into my teaching in a meaningful way.

When they started talking about the use of the overhead projector as a way to focus students in an American classroom I was taken aback. I have a document camera and overhead projector that I use on a daily basis in my teaching and have had the thought what would I do without it. I have used it so that I can work through the same thing that students have in front of them. We work through the page and I write on the board what they need to write to write down and when I finished that portion I erase and move on to the next portion of the notes. We continue this until we are finished and even thinking back about my education this seemed to be typical. They talked about  how in the Japanese classroom the board is used as a record keeping device. I thought this was a really cool idea as I think it would be much easier to have meaningful discussion as it would be easier to refer back to things. I can’t say how many times I try refer back to something earlier in the notes and its hard because I don’t have it written and sometimes its hard for my students to find what I’m talking about. I would like to find a way to still use the overhead projector but also keep some sort of the record of what it going on. Once again I think it will be hard work to incorporate this because the American script is all I know.

Now on to orchestrating mathematical discussions…Before I read about these five practices I would say that this was something that I did or tried to do but it was really never as connected as the five practices suggest. When trying to get discussion going I do try to check with every group and tried to have an idea of who I wanted to ask to share. However, I never really thought about sequencing them in a way that made sense I always seem to pick and chose groups to share. I did try to make connections between the groups ideas but I think it could be easier to do this if I had sequenced them correctly. After reading about these and thinking about how my discussions have gone in the past I want to try and think more about these thing before we share as a whole group. As for my experiences with this I’m not really sure what my past teachers have done in terms of these and I’m honestly not sure about my CT this semester. I would say that my CT seems to be better able to do this, I’m not sure if its the five practices or more of the fact that she knows the students much better and know what they are going to have trouble with.

I think these sections helped me to start to think about how to make sure what I am doing is effective. Having the ideas that come from the five practices should allow me to make sure my discussion are meaningful and I don’t feel so scattered brained during them. I also would like to think about how to make it easier to go back to items we already talked about that isn’t just “on the last page we talked about this” so students can see quickly what I am talking about.


Math in Action 2014

This past weekend I had the chance to attend and present at Math In Action, the annual math conference held at Grand Valley. They are six, hour long, sessions throughout the day discussing a variety of topics ranging from cooperative learning, flipping classrooms, using technology effectively, the Adventures with Mathematics series, along with many of topics dealing with math education.

This first session I attended dealt with cooperative learning and how to do this effectively. The presenters were sharing strategies that they used in their classrooms after learning about them from Kagan Cooperative Learning. The first part of the session dealt with what exactly is cooperative learning. The first point they made that really stuck out to me that group work is NOT cooperative learning. As they talked more about this it became clear that it is not, the following quote really hit home on confirming this fact:

hog“Group work creates hogs and logs.”log

This is really true at least in my experience, students work in groups and as you walk around there is always that person who is doing all the work and those who are just piggybacking on the “hog” and they become the “log”. However, at least for me personally I never really know how to address this issue. This session really opened my eyes on how to address you can this. They said you first needed to seat your students based on their test scores for readiness for the content you are going to cover. You need to have four in a group with a “high” (1), “medium high” (2 ), “medium low” (3), and “low” (4) student. Your arrange it so (1) and (3) and (2) and (4) are shoulder partners and will work together. The key is that your students don’t know how your placing them. One activity they suggested having these pairs do, was work on the basic arithmetic skills needed (ie. Adding integers), where each person would have a different problem but it would lead to the same answer. They also suggested the use of task cards where each student would get a card with a problem on it. They would solve that problem and then have the teacher check it and then would become an expert on that problem and then be able to teach that to everyone else as they work around the room to pair up and solve the problem that everyone else has. I think these are both really cool ideas because it allows students to work with everyone in the class, instead of picking their friends, and it also makes sure that every student is doing the work. They also talked about how much easier it is to differentiate with this type of setup because the students that are identified as high achieving for the particular topic can be given material more appropriate for their level without much extra planning and the students don’t know that your doing it. This is definitely something that I want to incorporate into my future teaching.

The next session I attended was on how one Zeeland teacher flipped her classroom. She talked about how to structure the videos for the night before class so they shouldn’t take more that 20 minutes to complete while also making them available as iBooks for the iPads the students are provided as a way to get around the need for Wifi at home. She talked about how she is better able to connect with students in class about what they are doing and seemed to be able to do activities that really made them think. The biggest problem that she talked about was the issue of time to make this switch but overall she said she has really enjoyed it and her students have enjoyed it. Before this presentation I knew the basic premise of a flipped classroom but had never put much thought into it as a teaching method I would use. But after this presentation I think that if I were ever in a situation where a flipped classroom would work I would definitely give it a try.

My next session I was able to swap roles from an attendee to a presenter as I presented with other GVSU math students and one of math education professors on the activities from the K-5 Adventures with Mathematics books. I presented one game that had students work on their division skills to move up and down a pool in a “swim relay” by however much their remainder was. This session was especially fun because I was presenting games that were similar to ones that I wrote in my capstone course and and to an activity I helped create for the Algebra 2 Adventures with Mathematics book. I was able to share not only how to play these games but also talk about how engaged kids get with the math behind them as I have seen in my previous work with them.

My next session dealt with the use of technology beyond the graphing calculator. The first piece that we talked about was using Geogebra. It was nice to see that others are using it as it is something I have already started to use in my teaching. The other piece of technology that was talked about was DESMOS, which is an online graphing calculator. When they first starting talking about it I was thinking how would this be any better that a standard graphing calculator. I soon saw how it was better. As you type in functions the software starts to adjust the graph immediately. It also allows for a decent viewing window to found much easier. With these two features its easy to see how much of an improvement DESMOS is as it will allow for students to see what is happening with graphs much faster than a standard graphing calculator. I don’t know that this is something I will be able to use this semester but I will definitely be using it in the future.

The last session of the day was spent talking about the history of major mathematical topics. We were able to get through two, the quadratic formula and the Pythagorean Theorem. We talked first how an ancient Arabic mathematician was able to find formulas that are very similar to the quadratic formula we use today. This was done without all the technology that we have today and with an aversion to negative numbers. Next we talked about the Pythagorean Theorem and a couple of different geometric proofs for why this is true. These were the most interesting part of this session. (I found out that there are hundreds of different proofs!) The proofs that were shown used a square that was made up of right triangles and a smaller square and showed how to use these with things we already know about triangles and squares to prove the theorem. I think this would be interesting to share with students to show them that what they are working on, people before them figured it out without all the tools they have. I also would like to incorporate one of the geometric proofs of the Pythagorean Theorem with my students as I think this would be much better received than an algebraic proof.

Overall I think the whole experience at Math in Action was great and cannot wait to attend next year. I’m going to try my best to incorporate some of the cooperative learning strategies into my teaching this semester but if not definitely in my future teaching. I also want to incorporate some of the historical information as I think it is important for students to at least see where the concepts they are talking about are coming from.

Fun Classroom Management?

I recently had my first observation with my content area professor. Before this we are asked to fill out an action plan about what we would like the observation to focus on. Before I filled this out my CT and I discussed what I wanted to work on as we approached the middle of the semester. As I thought about this I realized that I wanted to improve transitions during class. I felt like things were kept pretty calm throughout the individual parts of a lesson. Don’t get me wrong we do get a little off topic every now and then but during that it is pretty easy to reign them back in. However, when we would switch from warm-up to good news, from good news to checking homework, etc. we were having a really hard time switching without them starting to talk about unrelated things. Having to reign them back it then was a lot harder and seemed to take a lot longer. I was working with I raise my hand and then the students raise theirs with the expectation that they stop talking. I had been working with this system since we started the semester because that’s what the standard was before I got there and I really didn’t know any other way. But as I worked more with this I was beginning to realize that it really didn’t fit my style as a teacher and my CT encouraged me to find a way that fit me to try out the next day that I taught. With that I took to Twitter to ask others how they dealt with transitions. I got a lot of responses from telling students to “Pause” (often sounds likes “Paws” so you get some hand gestures) to jingling keys. However there was one suggestion than really struck me they said that had a bell that they “gonged” to get everyones attention. Something about the idea of a gong really stuck out to me as a fun and effective way to get attention focused again. With this I started looking for a gong to buy. This has proved rather unsuccessful. Has anyone every looked to buy a gong? Probably not but believe me it is not as easy to find a good one as you would think. I found a virtual gong to use instead, which can be found here. Sorry I got a little long winded there but anyway back to the action plan. I decided to try out the gong for the first time while I was being observed and have him try to look for how long and how many times it took with the gong to gather attention and also to look at how often I was helping each group during their activity.

So what happened during the observation and in the discussion afterwords? I used the gong throughout the lesson and was pleased at how well it seemed to work. The first time I used it I didn’t tell my students what I was doing and to my surprise they quieted down. I explained what we would be using it for and what my expectations were when I rang the gong. During the activity for the day the first couple of times I used the gong it took a couple of times for them to quite down and focus but they did. As we got further through the lesson the gong seemed to work better and better each time. There were still chatter after the gong but it was manageable when compared to our transitions in the past, so I took it. One of the coolest things with the gong came close to the end of our day. The activity that our students were working on that day was looking at properties of some quadrilateral and then they had to present to the class. As we were getting close to the end the other groups were starting to get a little chatty and one of this students in this particular group asked for the gong to be used in order to quiet the class. As for my other goal of seeing how I was doing with making sure I was getting to every group. My professor kept a map of the room with markings for different amounts of time I spent at each group. I was able to see where I was at and came to the conclusion that a majority of the time I spent was with groups who typically need the extra support when working like this. This doesn’t mean that I left the other groups alone. I made it to every other group multiple times and it also helps that there are always three or four teachers in the room at once.

What do I do now? Since the observation I have continued to use the gong. My CT suggested that we do a little more clarifying with our students to make sure that they understand what they need to be doing when the sound goes off. We decided on the following system: 1 gong-finish up your sentence, 2nd gong- should be turning around and focusing, 3rd gong- you should be in your seat quietly facing forward. This seems to have worked really well and the last couple of days I haven’t had to use it much. I don’t know if they are finally realizing when they need to quiet with the gong helping or if they are all just having good days. I really hope its the first one. I would say that I am pretty confident with my ability to get to all groups while they are working. The next thing I need to work on is finding a way to incorporate all students in answering questions rather that the same group everyday.

Exploring Triangles

As promised here is how our triangle exploration activity went. The activity we developed used Geogebra to have students create three different types of triangles and then measure the angles. Click here to see the handout we worked from. We started out the lesson by having students recap the different triangle classifications. After we got Geogebra up and running on the laptops each pair of students took off making triangles. My CT and I talked about how surprised we were that they were picking up how to use the program so quickly. Especially when we compared it to when we used it for the first time (let’s just say it was more confusing for us then it was for them). As they made their triangles, their goal was to make three different triangles of different types (scalene, equilateral, etc.), then measure all the interior angles of the triangle, and then find the sum of the measures of the interior angles. Our hope was that they would see that it appeared likely that the sum of the measures of the interior angles for any triangle would be 180o.

We finished up this exploration the next day by first having groups of students report out what they found out about the different types of triangles. From this we had our hypothesis that the angle sum of a triangle is 180o. But now we still couldn’t just accept this we had to prove this. We did a “proof” where our students made and cut out any triangle they could draw on a piece of paper and then label the angles A, B, and C.

Part 1.

Part 1 of the Proof.

 Next we cut the triangle into three pieces making sure each piece contained an angle.

Part 2 of our Proof.

Part 2 of the Proof.

Our students then tried to prove the angle sum of any triangle by using the three pieces. We told our students to try and think about if there was anything they had learned about previously that had an angle measure of 180o. It took us around 5-10 minutes for all the students to get the proof. The main problem that we seemed to run into was students were trying to include angles that were not part of their original triangle in their proof.

In the end though all of our students were able to use the three angles to show that the sum of the angles is 180o by showing they could be lined up in a straight line.

Part 3 of the Proof.

Part 3 of the Proof.

After thinking about how this went I was really glad that we were able to do this. First off it went a lot better than expected our students worked hard throughout the entire process. Some were trying to get the greatest obtuse triangle or working really hard to get a right triangle a different way than was stated in the directions. When we started the proof some of students were already seeing it and there were others who needed some directed questions. In the end all of our students were able to figure the proof out, which was really rewarding to see. I think it also helped them to see that doing math is more than a teacher telling them something and expecting them to remember it. We hope to be able to incorporate more Geogebra activities in our class time as we work through our geometry unit. We also have more discovery based learning ahead! This week our students are going to be working in groups to find properties of different quadrilaterals and then sharing with the whole class what they found in order to work toward creating the quadrilateral hierarchy. It’s great to see how engaged our students are when we do these types of activities rather than notes everyday!

“Give the pupils something to do, not something to learn; and the doing is of such a nature as to demand thinking; learning naturally results.”

John Dewey

Teaching to Learn

This week I began my long awaited start to teaching. After a long weekend because of mid-winter we started up on Tuesday with the plan of getting through two lessons. This was also the same day I had my first observation from my College of Education field coordinator. Looking back this might not have been the best idea because our students were a little crazy the first day back and we were trying to get through two lessons. However, after I got started I honestly forget that he was there. I was actually surprised at how comfortable and natural it felt to be up in front of the class for the first time. I did learn one thing about my teaching from my field coordinator. I have a tendency to talk to the board. As I reflected about trying to figure out why this was something I did on a regular basis. I think I have this idea that I was not really aware of where I felt like I need to talk while I write instead of talk then write and maybe repeat what was said. I know that it will most likely be more valuable when I talk to not face the board every time. I am definitely now consciously thinking about this while I’m teaching which seems to have helped so far.

I also realized how important it is to make sure you understand about giving two topics in one class period. We thought we would be able to get through two lessons in one day. Boy were we wrong about that. We got through the lessons but somewhere toward the end it was becoming clear that the second lesson we covered should have waited until Wednesday. We further confirmed this when we got the exit slips back where we could see how students responded to the prompt “I still need help in understanding…”. Reading through these it was evident that a majority of our students were lost on what we just covered and something needed to be done to correct this.

I used the exit slip information to create a re-teaching note page that we could go over the next day in class before we started our triangle activity (Post about that coming soon). We spent almost half of our class period going over this worksheet and I can honestly say every minute was worth it. As we worked through the page I could see the light bulbs coming on for students who just the day before at the end of class were staring at me with some of the most confused stares I had ever seen. By the time we got to the quick checks, students were going through them with precision and seemed to understand it from what I gathered with the questions I asked.

This experience has taught me some valuable lessons that I will keep in mind as I continue for my teaching career. I want to be able to create an atmosphere where students are willing to share struggles they had with a lesson because the exit slips really helped me to focus on what I needed to re-teach instead of trying to re-teach the whole lesson. I also realized that thinking about the time needed and what my students can handle is key. I along with my CT and partner thought that one class period was enough time to finish one lesson and get all the way through another. This obviously was not the case and we seemed to not quite nail down the fact of how confusing this would be to students. This lesson should have really had its own day in the unit. With that in mind we have adjusted our schedule so hopefully we do not run into the same problem that we did by trying to cover to much content. One last thing I want to remember is to never be afraid to spend some time re-teaching a lesson!

“In learning you will teach, and in teaching you will learn.”

Phil Collins

Planning to Incorporate Engagement and the Math Practice Standards

This week as we continue our unit on geometry we will be introducing types of triangles, quadrilaterals and other polygons From our understanding our students have seen some of this information before but we are hoping to extend on this by using both Geogebra and a group exploration activity. We are going to first use Geogebra to explore a “proof” (we’re using that term loosely) to prove that the sum of the measures of the interior angles of every triangle is 180 degrees. Our plan is to have each student create 3 different types of triangles (equilateral, scalene, etc.) and then find the sum of the measures of the interior angles of their triangles. With this we should have around 75 unique triangles in hopes to show students a different area of mathematics and to stress that they should not always take our word as absolute fact.

We are also going to incorporate a quadrilateral discovery activity for groups of students to do. We have adjusted a quadrilateral project our CT had already created to make it work so groups can explore one type of quadrilateral to explore what properties every quadrilateral of that type has. Once they find all of these the plan is to allow them to prepare some type of short presentation so they can teach the class all about that type of quadrilateral. We think this type of activity will be meaningful because the relationships between the different types of quadrilaterals will come up again and again throughout the rest of their mathematical careers.

My two goals for these two activities are to increase engagement and to promote the Standards for Mathematical Practice. I think by doing having our students doing a proof of the Angle Sum Theorem they will be able to see the fun exploration side of math and I think will allow them to take ownership of their learning. Maybe we will be lucky enough to get them to start thinking about why this actually works and wanting to explore that on their own. I think the self exploration that take place between the two activities will also tie in the Standards for Mathematical Practice well rather than just taking notes down without really paying attention to what I am saying. We hope to be able to show our students that doing math is more that remembering a rule in order to complete the homework assignment. I hope to post more about how these activities went and about how our Geometry Fold-cabulary (I didn’t forget) went this week.

Planning my First Unit

What does planning look like for a teacher? As a student this was something I was always aware that teachers did this. I mean they had a planning hour, right? It had never occurred to me how much hard work actually goes into planning a unit. During the last couple of weeks Kelsie and I have been working on our plans for our introductory unit on geometry for our 7th graders and I am beginning to understand what my teachers were doing behind the scenes.

We started by looking over the note pages that our students will be using. The first thing we noticed was the tremendous amount of vocabulary that is contained in the chapter. I counted yesterday and there are 49 vocabulary words! Seeing that we wondered if having the vocabulary spread throughout the notes would be beneficial or if it would be better to create some form of a separate vocab packet. We tossed around various ideas. Should we just do a separate packet in a table format, a Frayer model packet, keep it like it is, or look at various foldables with different components? Our goal in the end was to have a compact “packet” that could hold all the vocab well and also allow it to be helpful for studying if our students choose to use it for that (hopefully they do!). With that we decided upon a foldable that allows for the word and a picture on one side and the definition and a non-example picture (for words it applies to) on the other side of a page. I will post an update later on how these seemed to work, but I’m excited about the possibilities, as this is also something our CT hasn’t tried before.

As for the planning for the actual teaching of the lesson the planning hasn’t been as difficult as I think it could have be. We are using the same note pages that our CT would normally use. They do have their problems and we decided where adjustments could be made. These have been extremely helpful tools as it already lays out how we will work through the lesson and it keeps a level of consistency for our students as we start out teaching. By having this already planned out I have been able to focus more on finding real world examples and trying to look at the conceptual understanding of the topics we will be teaching. As I work through this I feel like I have a pretty good understanding of what my lessons will look like. The next step will be to get this down on paper.

I have also spent some time looking into some accompanying activities and resources. Finding these has been much harder than I thought it would be. A lot of what I seem to find doesn’t seem like it’s worth the time to do. I have looked at a couple of Geogebra activities that I think might be worthwhile but I still need to look more into those. I am hopeful that I will be able to try out an angle game (designed by a classmate last semester), where teams compete against each other by finding different kinds of angles on a “game board” set up on the floor with painter’s tape. I hope that is able to pan out and I can share it later in this unit.

So where does all this planning leave me? I feel much more confidant about teaching it (especially skew lines after I learned about that today). I would still like to work on developing some sort of exit pass/self-reflection piece with Kelsie that we can use after each day so we can try to understand where our students are at. I am excited to start teaching on Thursday and I look forward to sharing the highs and the lows as I encounter them.