Wednesday, November 30, 2011

My (underwhelming?) 100th Post

Since I posted last and saw on Blogger that it was my 99th post, I wondered what my 100th post should be. After all, 100 seems like a rather momentous number. There is a part of me that is somewhat in awe that I am writing my 100th post after starting this blog in June 2010. I don't really consider myself a writer and I really don't blog for anyone  or any purpose other than reflection on what I've encountered as a math teacher. Most of the time, taking the time to blog about something helps me sort it out in my head and if that doesn't happen, one of my readers offers a comment that is helpful to me (or in many cases affirms whatever it is that I'm going through is not something I alone am facing). But as I sit here on the last day of November, I still have a nagging issue in my head.

I've blogged about this before (first here, then here, and most recently here). It's the whole quizzing/reviewing thing. You guessed it, today was another quiz day in my Algebra 2 classes and tomorrow is quiz day in Advanced Algebra 2. I'm not through grading the Algebra 2 quizzes. The one class I have graded actually wasn't too bad. However, today I am frustrated with time. You see, I ended up spending most of the class period with my Advanced Algebra 2 students going over questions from the review I gave them (with answers, I should add!) instead of answering a few questions and starting into the next learning target as I had planned. Yesterday I had spent significant class time going over review problems with my Algebra 2 students, but I had pretty much anticipated having several questions from them given how things are going, plus quite a few were out on Monday since it was the start of deer hunting season with guns. I expected that my Advanced Algebra 2 students would remember things better and have a better grasp on it and I was disappointed with that today.

Now, granted, I think part of the reason going over the review took a little longer was the way I did it. Instead of my explaining the problem, I pretty much just wrote what students told me was the next step. After each step, I prompted them with "what's next" or something along those lines. I actually liked this - students had to articulate what to do which helps them to review it and there was some good discourse going on while we were working the problems. One student would ask how another student got something and the students answered it instead of me. I think that helped. I think I'll do that again and bring that more into my teaching and other classes. I liked that the students who knew or mostly knew what they were doing were engaged and reinforcing what they knew as students who had questions were getting them answered.

What I am really disliking this point is that I am spending so much time reviewing. I teach 3 concepts and as we are coming up on the next quiz (which has up to 6 concepts - 3 that were previously assessed, they got some written feedback, and now are being given a score for, and 3 that are the most recently taught concepts where they only receive feedback), I give them a review sheet with answers on the 2-3 concepts that are going to be graded. Students like having the review sheets because it gives them a good idea of what to expect. I get that. But since I am quizzing every 5-7 class days, I lose pretty much 2 days every 5-7 days. Not good. I'm already behind because these students didn't cover as much ground as they should have in Algebra 1 (I've talked about that a lot here) and now I am losing more time reviewing. In my last post about this issue, Damion really hit the nail on the head - like his students, my students don't seem to be retaining well.

In addition, we are coming up on Christmas Break here. Now, mind you, we just had Thanksgiving Break (we were off Wednesday - Friday last week). We have 2 full weeks (this week and next) and the third week, the students have 4 days while the staff has a waiver day on Friday (December 16th). So when students leave on Thursday, December 15th, the next time I see them in class will be Tuesday, January 3rd. Almost three whole weeks! Now, I will be honest, I am totally looking forward to being off school that long. However, being off that long will most likely mean that anything that I teach that last week before break (or earlier) will be mostly forgotten by the time I see them again on January 3rd. I am pretty certain that I am going to need to assess everything before break, which would mean that I would not have a feedback opportunity for them.

So, at this point, I am trying to figure out what to do. I am already contemplating going back to how I used to assess - teach the concepts that form a "unit," actually follow through on quizzing for feedback only every 2-3 concepts and at the end of the unit, have a day or possibly two of review and assess. Just as I typed that, my mind was mentally calculating and maybe I'm not losing as many days as I think... I'll have to ponder that a little later on. (ooooh - squirrel!) Or, I could continue in the same manner - teach 3, assess with most recent 3 concepts getting feedback and the 3 concepts before that earning grades in the gradebook. I'm not sure what other options I have at this point without dramatically changing things and I don't want to go down that road right now. Another option I had contemplated during the summer was to give each concept 2 grades and combine them - that's definitely not something I want to change to mid-year. I'll have to think about that come Summer 2012.

So, here's where you, wonderful readers, come in. I need some help. I can't shake this issue in my head. I'm on my 4th blog post about this issue and I can't work it out in my head. Please offer thoughts, experiences, advice, etc. - Do I keep my assessment system the same (teach 3 - assess 6, 3 graded, 3 feedback), revert back to what I've done before (teach the unit, quiz for feedback a few times, and give a "unit test" with however many concepts are on it), or something else? I look forward to interacting with you in the comments!

Thursday, November 17, 2011

Open Question Attempt #2

This was the open question I posed to my Advanced Algebra 2 students today:

Write a system of inequalities that has (4, 3) as a part of its solution. It should have 2 or 3 inequalities. (I had thought about leaving the last sentence off, but I was trying not to overwhelm them.)

When I asked for their answers, I got crickets again. (Recall, I had tried this with them a couple of weeks ago and pretty much got no response from them.) So, I asked them for one inequality that would work. Crickets. More crickets. That, and a "I don't know how to work backward." After a few moments, someone gave me one inequality. So we graphed it and checked (4, 3) and it didn't work - (4, 3) was on the line. I asked him how to change it and he thought about it and came back with a different inequality. This time (4, 3) wasn't on the line, but he had the wrong inequality symbol. We flipped the inequality and had a working inequality. Yay! How about a second one? Crickets.... but for a shorter time. Same student, new inequality. Worked - success!

Can we come up with 2 different ones? Just try... Different student, new inequality. Got a working one and the student came up with a second one.

By the time we were done, we came up with 5 systems (look at the first 3 pages of the pdf below - first 5 slides). The first time we did an open question, I primarily had 3 students contributing. This time I had 6 or so contributing - 4 of them who had not last time. I'll take the improvement. Maybe next time, I'll have more.

Tomorrow I'm trying this question with Algebra 2 (similar to what I did with my Advanced Algebra 2's a couple of weeks ago):
I know that (2, 3) is the solution to a system of equations. Find two equations (in x and y) that have (2, 3) as their solution. How do you know that (2, 3) is the solution?
(I think that's how I phrased it.

Wednesday, November 16, 2011

Quiz Time Blues Again

I just gave another round of quizzes over the last few days. I am not pleased with how it's going. This is nothing new. I can't decide if it's because I'm not happy with doing teach 3 - quiz (3 for feedback/3 for grade) - teach 3 or because I have a lower-ability/less-motivated/less-knowledge-of-Algebra-1 group of students this year.

As we continue with the quizzes - grading the three concepts that were given feedback only the last quiz and giving feedback only on the three most recently taught (details on my SBG system this year is here and reflections on why I wanted to make changes from last year is here), I am seeing that several students are not trying or making little effort on the feedback only questions. It is not a majority of students, but with each new quiz, I am noticing there are more students each time who are not attempting the feedback questions. This worries me and I feel it defeats the purpose of giving the questions for feedback only the first time. The students are cheating themselves of the opportunity to seeing what they know about the problem in a practice situation.

I have been giving them review sheets for the concepts that are being graded since in some cases it has been 2 weeks or so since they had the original instruction on the topic. I suspect what that is doing is focusing the students on preparing for only those problems and they are not doing much preparation on the newer concepts. I am afraid that if I give review sheets with all 6 concepts, we will be spending one day reviewing and another day quizzing every 7-8 class days. As much as I don't want to dwell on what content I am covering over the course of the year, I am already concerned that I will not get to many concepts that I should be for Algebra 2.  Last year I ended with exponentials and logarithms and I am deeply concerned that I'm not going to get there this year with their lack of Algebra 1 skills and the time I've had to spend remediating those skills - and that's not factoring in the additional number of testing days (and some review days).

In two of my three Algebra 2 classes, students expressed that they weren't ready for today's quiz. Now, I know it's not due to lack of time and energy we I've devoted to it in class. They weren't ready because they weren't prepared. Students aren't preparing well - they aren't taking the time to work through problems as they should be. I have watched as I have helped students who have wanted the help and seen several students not using the time well. They choose to talk instead of starting the problems. I'm not sure if it's due to lack of confidence in doing the problems or if they don't want to after spending class time listening and (maybe) taking notes. The last few sections I have given students a note sheet to fill in. I did this partially because we were doing graphing systems of equations and inequalities and I thought it would be helpful to have the problems with graphing spaces ready to go for them. I also did this because I thought if they had the examples in front of them, they would actually take notes. I'm not sure how well that worked.

I'm at a bit of a loss tonight. I cannot continue to go backwards. Students have to learn they have to pick up the slack at some point. I feel like I am wasting time quizzing every 5-7-10 class days if they aren't preparing for all of the concepts. Part of the reason I changed to the quizzing concepts twice system was that I thought they would do better on the concepts since they had a chance to get some written feedback from me as a part of the process. If students aren't even going to attempt the feedback-only problems, then the purpose of quizzing this way is moot. I could go back to the way I assessed last year (at the end of a "unit" with a couple of days of review in front of the assessment) and end up with the same results with hopefully less wasted time, not to mention less stress since I would only have to prepare an assessment every 2-3 weeks. Part of what also appealed to me about the teach 3 - quiz 6 - teach 3 - quiz 6 system was that it would be smaller chunks for students to focus on and hopefully they would find that to be more manageable.

So what to do, what to do? At the moment, my plan is to put the How to Study Mathematics article that crstn85 referenced in her recent blog post in front of my Algebra 2 students tomorrow. I am going to do it a little differently. I gave the article to the Advanced Algebra 2 students today (since some of them were still finishing their quiz) and had them read the introduction (through the middle of page 3). Then I counted them off and had each pair read one section instead of one page. Each pair shared with the class what they thought was important. I did not have a chance to share the Cone of Learning with them - I am going to do that tomorrow. My hope is with my Algebra 2 students to get through all of that in class tomorrow. I also had my husband print out in color 11" x 17" Cone of Learning posters that I am going to put up around the room. I want a visual reminder to my students that they need to take an active part in mathematics.

I need to continue to reflect to see if this is how I want to continue to assess, or if I want to go back to what I was doing last year, or if there is a different way I want to try. If I am going to make changes, ideally, I would do it after the end of the 2nd 9 weeks in January. I just don't know what the "right" answer is. If anyone has some thoughts to share that would be helpful, I sure would appreciate it. I am having a hard time separating myself emotionally from this decision. There is a part of me that feels that I am failing my students here. I feel like I am teaching them (not intentionally) to prepare for the short term rather than learning the material thoroughly. Intellectually, I know that this group of students does not have good study skills and that is part of it, not to mention that there are several students who are not strong mathematically. Their experiences in Algebra 1 isn't helping the situation either. Regardless, I have to work with the students I have, where they are at, and make the best of it I can.

Thursday, November 10, 2011

Struggling & GReader Cleaning

I feel like I am struggling at the moment. I really wanted to make some changes in how I teach this year and I have made some changes. I have incorporated some different things into my teaching, but I am still at the forefront of my classroom. I struggle with this mentally at least weekly, if not more so. I wanted to be incorporating Reasoning and Sense Making into my classes. I'm not. I did bring in an open question last week into my Advanced Algebra 2 class and it didn't go as well as I hoped.

I did partially use this approach today that John Scammell blogged about last week. I think the next time I use it, I will follow the approach as it was blogged about - when I was thinking about it this morning on my way into work, all I could remember was showing which points made the inequality true. In my first class, I had students choose any point on our grid (both x and y values go from -6 to 6 on it) and they all chose points with two positive coordinates. None of those points worked, so I had them choose a coordinate that had at least one negative coordinate and we got some points that worked. Also with the first class, I didn't have them graph the points we came up with. We just graphed the shaded area that did work. For my second and third classes, I had them plot the points we found that worked on their grid before we graphed the inequality on the grid. They then graphed the line and we talked about testing a point to determine which half of the plane to shade. Below is the pdf of my SMARTNotebook file from my last class.

In that last class, one of my students who has struggled with math spoke out as we were working through the problem. We had graphed the individual points that we found that worked and had just finished graphing the boundary line and right as we finished drawing it in, he said "I get it now!" I hadn't even gotten to the point where we talk about testing a point in one of the half-planes - he already saw where the answer would be and "got" why that was going to be the answer. That was worth it. #win

Getting back to the opening... I know I have done things differently this year. Yes, things are mostly the same. However, due to the twitter-blogosphere and reading and conversing with other math teachers, I have incorporated some different approaches to instructing my students and I think that has made me a better teacher. However, as I look at my classes, I am frustrated because they are so dependent. It's almost as if they don't know how to think.

In my Advanced Algebra 2 class, where we're working on solving systems of inequalities by graphing, we had this discussion about the test point today. It started first by clarifying how to determine which half to shade based on testing (0,0), which is the point I use unless it's on the line. Then the conversation shifted - can I test (1,1)? What about (2,2)? After answering the same question but with a different point for the third time, I got a bit frustrated with them. It was as if they couldn't take the concept of checking a point to represent the region and shift it to a different point. And these are supposed to be my "better" and/or "brighter" students. Granted, I didn't use the same start as I did with my Algebra 2 classes today, and that may have made a difference, but it was very frustrating to me that they couldn't transfer the idea to different points. As I said earlier, it's almost as if they don't know how to think.

Last night I was trying to get caught up on my Google Reader, this post by crstn85 caught my attention about studying for math. Since I was at 130 or so when I started trying to clear it out, I really only skimmed the post and starred it to go back and read later. However, as I continue to reflect, I think I may go back and read it much more thoroughly to see if I can use what she did with my classes. Maybe it will be helpful to them.

So here I am at the beginning of the grading period of disruptions second grading period and I have not really done anything as far as my second and third goals on my list for this school year. I have not made any progress on incorporating Reasoning and Sense Making activities into any of my classes, and although I try to ask my students questions as I help them, for the most part, I pretty much help them through their questions instead of asking them questions. I'm not sure why I am struggling with that one. I think part of it is that my Algebra 2 students are lower in ability/prior knowledge compared to previous groups and I am afraid of losing them. I suppose if I really thought about it with my Advanced Algebra 2 students I could work at it more with them, but again, there are some students in there that I could lose. I'm at a point that I'm not sure what to do, so I continue to do what I have been doing and trying to bring in some new things as I catch them in the twitter-blogosphere. Right now, I'm not feeling that's good enough. So now what?

Friday, November 04, 2011

Student Comments Part 2

Earlier in the school year, I had done a gallery walk activity with my Algebra 2 students (see this post). Our HS football team has made the playoffs for the first time in 17 years and we had a pep rally to celebrate and encourage them. This meant that I wasn't meeting with my last of my 3 Algebra 2 classes. So, I thought I might try this again.

Since we have been working on solving systems of linear equations, I set up three sets of problems - 6 solve by graphing, 6 solve by substitution, and 6 solve by elimination (only by adding or subtracting - we haven't talked about multiplying yet). I brought up the random word chooser and let each person as their name came up choose a partner. Each partner set came up to the SMART Board and rolled a die twice. First time was to determine what type of problem they got - 1 or 2 got a graphing problem, 3 or 4 got a substitution problem, and 5 or 6 got an elimination problem. Second roll determined which problem they got (I had 6 of each type). Each group got a 12" x 18" piece of construction paper with their question taped to it, markers to write their work on it, and the graphing groups got a piece of graph paper (I had 1 cm square graph paper in my room) and a ruler.

It took between 10 and 15 minutes to get them to work through their problem and get it on the paper. After all groups had their work on the paper (or mostly did), I began passing out post-it notes. Before I sent them around, I talked about how just writing "good job" or something like that was similar to clicking "Like" on Facebook - which Christopher Danielson had mentioned in this comment. I stressed the importance of being specific and really looking at the work people had done. I mentioned that just writing positive comments when something was wrong doesn't help the person do better and that it was important to constructively point out errors. I thought I had done a pretty good job of getting across what I wanted them to do.

I guess I didn't do as well as I thought because by the time I got to my third one to comment on, there was already a sizeable stack of post-its and most of them said "good job" or "nice work" or the like. I read 2 lines of the problem and found a very obvious (well, I thought it was) error - the student had put a 2 in front of the y when there wasn't one to begin with - and once again, not one student had found it. So I stopped them and reminded them to really read the work because I had found a very obvious mistake and not one student had found it. I also told them that if they were looking at a paper for 30 seconds or less, it wasn't anywhere near enough.

They did get slightly better and we spent the last couple of minutes talking about what good comments looked like. In particular, I was looking at a graphing problem and the people working it out had only graphed one of the two lines. There were two comments on the paper that said something to the effect of "there should be two lines here" or "you are missing a line." and I mentioned those by name. Normally I don't ask the students to reveal who wrote the comments and this time I did ask because I wanted to commend them.

For the second class, I made a couple of changes. First, I put a timer up on the SMART Board set for a minute. I told them they needed to be looking at the problem and commenting for at least a minute. Secondly, I specifically mentioned the comments from the first class that were good solid constructive comments on what was done wrong. Although I saw some students doing a better job of really looking at the problems and checking through, there were still an awful lot of "good job" type comments. I also came across at least one problem with errors that students had not caught until I got to them and stopped and reminded them to really read through the problems. I also saw some negative comments (along the "you suck" type idea) but not until after students had left.

I had hoped that maybe with a better setup students would do better on commenting on each others' papers. I was disappointed. At least with the timer, the students seemed to slow down and actually try to read through the work a little better, but it still wasn't what I had hoped. Maybe I need to go back and do an activity like Bill T suggested here where students see 2-4 versions of the same problem and they need to find the error. I'm not totally sure here.

I suppose I really need to think about what my goal was here. I think I had 2 goals: for students to practice solving systems of equations using the three methods we had talked about (which did happen - they did only work on one problem and therefore one problem type, but they did do this) and for students to critically critique each others' work. The second one didn't happen. I need to re-think how much I want to emphasize that second goal, because if it's a big deal to me, they need to be better prepared. Meanwhile, I also need to decide if this is worth doing again in its current form. I left both periods feeling a little frustrated that the comments didn't go as well as I would like and that they still need practice solving all three methods of solving systems. That's not how I hoped I'd feel after they were done today.

Wednesday, November 02, 2011


Last year with my Algebra 2's and SBG didn't seem this frustrating to me. We are in our last week of the grading period right now and of course kids seem to now care because they are in that lovely point chasing mentality. We had a quiz yesterday - same format I have been doing all year - grade 3 learning targets and feedback on 3 (although I only did two because if I waited for the third one it would have been so long since we did the earlier 3 that I don't think they would have done well). The 3 graded were graphing linear equations (slope-intercept form), write equations given 2 points or 1 point and slope of the line, and write equations given a point and a line either parallel or perpendicular to it. The 2 feedback were solve a system of equations by graphing and solve a system of equations by substitution (both 2 equations 2 variables systems). As I have been doing since the second quiz, I did give them a review page with answers over the three to-be-graded learning targets before the quiz.

The students did okay on the graded learning targets. They didn't knock it out of the park, but my real sense about this group is they are not going to be knock-it-out-of-the-park students unless they do more preparation on their own. When I got to grading the feedback portion, most them had no clue on how to graph the two equations together. Same kids who did pretty well on the graphing one equation on a coordinate plane. Same kids who were able to solve for y to get the equation in slope-intercept form couldn't do it on the back. Directions said to graph and students were trying to do substitution (most unsuccessfully). Or in several cases (way too many to count and way too many for my comfort), students didn't even attempt the feedback only problems.

It's like they don't even see the connections between material we have done previously to what we are doing now. And they certainly aren't retaining things well. Today we worked on more substitution problems, which we did Thursday and Friday (and I gave them a notes page to try to guide them to either take notes or fill in what was important) and I would say from what I saw about half still had no clue what they were doing and most of the other half was struggling to remember where to start.

I am frustrated at this point on many fronts. I am frustrated that I am teaching Algebra 1 stuff to my Algebra 2 students. I am frustrated at how long it takes them to get concepts. I know I can't change that one, but I need to find a way to deal with this better and quick or I am going to be one really frustrated teacher all year. I am frustrated at how dependent my students are. I am frustrated that I haven't done or felt like I can do my goals of incorporating reasoning and sense making materials or being less helpful (see this post for details). There is a part of me that thinks that if I tried to bring in reasoning/sense making activities that it would help my students. Then there's the other part of me who looks back at days like today where I see my students so dependent and not willing to think or try something on their own and I wonder if it's really worth trying.

Getting back to the SBG bit - with getting back their quizzes today and the end of the grading period being Friday, I had several students wanting to reassess. Most of the ones that came up knew what they had done wrong. Those who didn't, we worked through their problems and I think they better understand it. But part of my concern at the moment is that my students are working to just learn the three learning targets they are getting graded on and not working on the big picture. I don't know if that's because of the system I'm using (grade 3 - give feedback on 3). I'm not sure if I'm helping by giving them review problems on just the three learning targets being graded. Well, I guess I'm helping some because they are preparing, but I cannot continue to spend lots of time reviewing if I am quizzing every 5-8 class days. I'm not sure what the answer is to this one.

At this point I just have a bunch of questions and no answers. My thoughts are still pretty jumbled on all of this. I'm hoping that just getting some of it out here will help straighten my thoughts eventually. @cheesemonkeysf probably said it best - my subconscious has to be working through this somehow even though I don't feel like I have any answers. Hopefully it comes to me soon.

An attempt at an open question

I gave my Advanced Algebra 2 students the following question today to open class:

Write a system of equations with two equations and two variables with a solution of (2,3). Be prepared to justify your answer in two ways.

We have done solving systems of equations by graphing and substitution. We started elimination yesterday. I had hoped that they would come up with equations by working backwards - for example, 2 + 3 = 5, so x + y = 5 is one equation they could use in the system. I had hoped that once they got the equations, they would use either substitution or elimination to check the solution was (2,3).

What I got was blank stares. They had no idea where to start. I ask them to give me one equation and we'll put together a system. Then the student I have from the ED (emotionally disturbed) unit - who is bright mathematically - raises his hand and gives me an equation. Then he gives me a second one. And, right or wrong, I take the time to show them that (2,3) is a solution to it, first by substituting the values in to check and secondly by using substitution since it made the most sense for the equations he had given me.

I asked them to give me two more equations, and after a few moments, I had two more equations, one each from two different students. Demonstrated that (2,3) was the solution again. I asked them for two more equations and got two more equations, this time a little quicker from my students.

After I was done checking the third system of equations, I did talk with them a little about how I wanted them to understand what the solution to the system meant.

I'm still behind on More Good Questions and I guess I need to read some more to have a better idea of how to handle open questions with my classes. I decided to try this with my Advanced Algebra 2 class since they are a little more willing to think and work at stuff than my regular kids. I'm not totally sure what I wanted to get out this, but I had hoped for better responses from them. Will have to read more and think more before bringing in the next open question.

A win - maybe

One of my Algebra 2 students gave me absolutely no work on his quiz yesterday writing equations given two points on the line, a point and a slope of the line, or given a point and a line that is parallel or perpendicular to it. As I was walking around checking how students were doing with the problems they were to be working on, he had his head down and was visibly frustrated.

I asked him what was going on and he said that he had the right answers but got no credit for them.

I responded that I had no idea where his answers came from - if he figured them out in his head, or guessed, or looked off someone else's paper.

He replied that he's not a cheater.

"I'm pretty sure of that. So how did you come up with the answer?" was my response.

He started to explain that he used the graph and figured it out from there. He didn't have any work because he kept the grid clean so he could work out the four problems. I shared with him what I was looking for work wise and why it was important to understand how to work the problem that way (especially since problems won't always work out so neat and clean).

I told him to wait a minute and went to get some graph paper. When I came back I told him to use the graph paper and show me how he got his answers on the four problems. I left him to work through it and he came back up towards the end of the period.

He began by saying he explained it to another student and she followed so he thought he had it for me. He proceeded to explain very articulately how he arrived at the equations for the problems with 2 points and 1 point and slope given - and I should add, very accurately. He demonstrated he understood what it meant to be a point on the line and how to write the equation in slope-intercept form (which is what I had asked for). I revised his score from 0 to 5 on the spot - he demonstrated he understood it. At that point, the period was ending and we left it that he would come back tomorrow with explanations for the other two problems.

I am hopeful the student will come back with the explanations tomorrow. I am also hopeful that he will make a better effort in the future to demonstrate how he is arriving at his answers so that I can see he truly understands the concepts.

Winning here with one student, at least, I hope....

Twittereen 2011 - for Sam

I had a heckuva time trying to get the actual tweets posted and in order, so here is Twittereen 2011 in Word. Sorry for the not so great quality - it was what I did quickly so Sam could see all the fun he missed.