As we have been working through the Common Core State Standards this year, I have really been putting a heavier emphasis on my students being able to explain why they are getting the answer they are getting or what something represents. At times, it has felt like I am pulling teeth to get them to do it. But slowly, surely, we have been getting better. I am starting to see more students being willing to contribute an answer. Discussion is improving a little bit at a time.

I am not sure if I can pinpoint exactly why today’s discussion went well. I felt good about the questions I asked. Once I reflected on the day, I felt good about the responses I received from my students. But could I pinpoint exactly why they had a better idea today? Not really. But what I think I can feel confident about is that my students are coming around to applying the Standards of Mathematical Practice. With perseverance on my part in teaching them, they have begun to come around.

So, if you are reading this, I want to offer you some hope. Keep fighting the good fight. Push the students to use the Standards of Mathematical Practice. Post them in your room, whether the actual ones or the student-friendly ones. Make the students recite them, a la Justin Aion (sorry – I can’t find the actual post where he shares that they do this, but he does!) Whatever it is you need to do to get them to understand and, more importantly, use the Standards of Mathematical Practice, DO IT! Will it be easy? No, not at first. But it will get easier and they will improve. As far me, I will keep fighting the good fight. I’m going to enjoy today for a few minutes first.

]]>I teach in Ohio, which is a PARCC Consortia state. When Common Core was first released in 2010, grades K through 8 had their own set of mathematics standards. There are 5 domains of standards at the high school level, but they are not arranged by course. In addition, as you read the standards, there is some overlap. An Appendix (A) was added with a suggested list of which standards should go with which course (Algebra 1, Geometry, Algebra 2 or Math 1, 2, or 3), but that’s as much guidance that has been given.

So Ohio was involved with both consortia at the beginning. Eventually, they decided to go with PARCC. My students will be taking the PARCC exams this winter and spring. As I was looking at my curriculum map this fall, I was trying to figure out the order I was going to put my units in, knowing that my students were going to take the PBA in late February. Once again, I was hunting all over the PARCC website trying to find the information for each test and look through the End of Year test, and getting incredibly frustrated. As I was getting frustrated, I kind of thought about putting together something that had everything in one place – the actual Common Core Standard and all of the table information from PARCC from every document they had on Algebra 1. As I began assembling it, it made sense to me to add in the End of Year Items and Sample Tasks that were already released by PARCC in with the correct standard. While I was waiting on PARCC to release the PBAs, I also did Algebra 2. Since I teach Algebra 1, it made sense to me to do the same for Algebra 2 so I could see where my students were going. The 8th grade standards are next on my list, so that I can clearly see where my students should come from. I will add the 8th grade once I have it finished.

Once I completed these, I shared them with a few people I know who could use them. I wasn’t totally sure what I wanted to do with them. There were some varying opinions, but the bottom line opinion was that they would be useful to other teachers and it was a good idea to share them.

I have copied and pasted the information provided by the Common Core State Standards and PARCC online. At the very end of the document, you will find the links of where I pulled the information from. All I have done is formatted and arranged the information in a way that made sense to me.

So, here they are. I hope they can help you out. Please share the links.

All the best, Lisa

]]>We are starting our gear up for TMC15, which will be at Harvey Mudd College in Claremont, CA (outside of LA – map is here) from July 23-26, 2015. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc (http://bit.ly/TMC15-1). It’s an open GDoc for people to list their interests and someone who might be good to present that topic. If multiple people were interested in a session idea, he/she added a “+1” after it. The doc is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, so we really, honestly and truly **need** you to submit/present! What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing.

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 19, 2015 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Kemlage, Jami Packer, Max Ray, Glenn Waddell, and Darryl Yong

]]>After recent talk about notice&wonder, sense-making, T said, “Sounds great, but I have to finish Ch4 by end of Nov or else…(1/2)

— Annie Fetter (@MFAnnie) November 24, 2014

…I don’t have time for that ‘habits of mind’ stuff because I have to cover too much content.” What say you? (2/2)

— Annie Fetter (@MFAnnie) November 24, 2014

I didn’t think I could reply in 140 characters (or even 280), so here goes:

I used to think the same way. I have to cover x, y, and z by the end of the year and I wouldn’t have enough time to do the Math Forum Problems of the Week or problem solving or (fill in the blank of your favorite thing we don’t get to). This year I decided that I was going to do the Algebra Problems of the Week with my Algebra 1 classes. Every 2 weeks, the new AlgPoW is released and on the first day of the week, I project the scenario for my students. Our current routine is that I read them the scenario, students list their noticings and wonderings as I read and for a moment or so afterwards. They then get about 2-3 minutes to share their noticings and wonderings with a partner. I then ask each group to share one notice, which I compile. Even if all of their noticings are up there, they have to tell me which notice they had and I add a * to indicate that more than one group had it. After each group has had a chance to contribute, I ask for any additional noticings, which I add to the list. We repeat this for their wonderings. This whole process takes about 15 minutes total.

This was the list of my students’ noticings and wonderings the first week of school:

This was the list of my students’ noticings and wonderings today:

My students have made incredible strides just in their noticings and wonderings over the course of the first 12 weeks or so of school. The quality of their noticings and wonderings is far superior to where they were at the beginning of the year. Now they are looking more for the mathematics and their wonderings are more mathematical than they were at the beginning of the year.

Do we solve every problem in class? No. What will happen next is I will give them the problem of the week that has the question. This week, they will get the PoW tomorrow. They are expected to work on it outside of class. Right now, I am working with them to focus on attempting and revising a solution. When we have work time in class, they can be working on the PoW. I have some students who diligently use their class time to work on the PoW so that they can bounce ideas off of other students or ask me for some direction. Other students will work on submitting it online. Out of my approximately 70 Algebra 1 students, about 20 of them consistently submit at least one draft to a PoW. This is probably the biggest area I am struggling with right now. I would like this number to be higher. Since it is my first year doing this, I am just kind of going with the flow right now.

What is my goal with giving my students the PoW? I want them to be exposed to mathematical situations. I want them to be able to find the pertinent mathematics in a problem situation and be able to use it to solve a problem. My hope is that by the end of the year, my students are more confident in attempting these types of problems because from what I have seen on the PARCC exam, these are skills they will need to be successful on them. But most importantly, they are skills that they will need to be successful in solving any type of problem in life. Will they encounter quadratic equations in their daily lives? Probably not. But will they encounter problems? Yes. Being a good problem solver is an important life skill. If I don’t cover all of the material in my course, yes, they’ll be lacking a little bit when heading to the next course. But if they are good problem solvers, they’ll be able to figure it out and apply what mathematics they do know to the situation. The mathematics will come. Meanwhile, I will keep plugging away at helping my students be better problem solvers. I know it is time well-spent. I can see the improvement in my own students in just over 12 weeks (we are doing our 7th Noticing and Wondering / Problem of the Week).

So, Teacher, wherever you are, give it a try. And I don’t just mean give it one try. One is not enough. This is where my students were after 2 PoWs. This is where my students were after 4 PoWs. And now you see where my students are today. 15 minutes of class time to do Noticing and Wondering every 2 weeks plus another 10 or so minutes of class time to pass out the problem and give some further direction (such as sharing all of the noticings and wonderings or having students look back at their own lists) isn’t a lot of time. You don’t need to give up a whole day of instruction. A little bit at a time will help your students out. If you have a PoW membership, you can find PoWs that will work with what you are teaching right now, which will let you work with two things at once – reinforcing your content and teaching problem solving skills. It took me three years to finally get there and I don’t regret doing it at all with my students. Just try it. I don’t think you’ll regret it.

]]>Most of my students could determine if a solution worked in an equation. I did goof and the second one did not work out as I had thought it would. What is most interesting to me is that 64% of my students could find the correct value for x, but would not use inverse operations to solve the problem. For example, on #3, rather than showing:

x + 572 = 893

– 572 – 572

x = 321

they showed:

x + 572 = 893

893 – 572 = 321

x = 321

In other words, they knew what operation to do, but they did not recognize that they needed to do the same operation to each side of the equation.

Any suggestions on how to approach instructing the students? Although what they did was correct (as far as getting the answer), I am concerned that without understanding the ideas of what an equation is and using inverse operations to solve them that they will struggle as we get to more difficult equations. Thanks in advance for any suggestions.

]]>- I can use and interpret units when solving formulas.
- I can perform unit conversions.

I found some resources online I was happy with for actually doing the unit conversions, however, I did not find anything I was real happy with for using and interpreting units when solving formulas. So I developed this:

My intent was to introduce the idea, put the students into groups of four and have each student work through 2 problems, comparing with other students in his or her group who would have done the problems to see if they got reasonable answers. When they were satisfied they had good answers and units, they would get another half sheet. We did number 1 together on the SMART Board so they would see what I wanted to see as far as work went, showing all units throughout the process. I then directed them to work through another problem on that page and then compare notes, asking for the next sheet when they were ready.

Originally, my thought was to start them off on different half sheets (so they wouldn’t go to another group to get answers possibly) and try to rotate which sheets they got. Then I had the thought that students would not understand what I wanted them to show work-wise, so I opted to start all students on the same sheet. In my first class, I rotated who got what sheet next. After watching that in action, I decided that was not my best option and I revised for the next two classes what order I wanted to go in. The present order is in my document above.

We only got through 2 half sheets, which also surprised me. I guess I expected them to work quicker on their own. Part of our hold up today was getting the Unit 2 Learning Target page and divider page set up in our Interactive Notebooks and that took more time than I realized. I was surprised how many of them could not substitute into the formula even when given the variable and its value (such as D = 400 miles). It was not a huge amount, but it was enough that it bothered me. I had hoped to spend one day on this learning target, but after the first group of students, it was obvious to me that we needed to continue tomorrow. We didn’t get anywhere near enough practice. Many of the students who had completed problems on their own did not follow directions, not using units throughout the entire problem. This is one of my larger concerns. A couple of students commented that it was “harder” to do the problems this way – why couldn’t they just do it the way they were used to? I shared with them the idea of them understanding why the units for the final answer ended up whatever they are, but I don’t think they’re buying that.

My intent was to have students practicing the skill and not be lecturing at the SMART Board, walking them through step by step on example problems. I can tell that my students are not accustomed to this kind of work, both in groups as well as having to be a little more self-reliant. When they are looking at something they have seen before that they know well or are working with something they have just been taught, they are mostly good to go. However, put something in front of them that is different and enough don’t have an idea of what to do and the questions pile in.

As I am writing this, I am still pondering how I want to change things tomorrow. My goal is to have them get through 4-5 more of the half sheets. (I’m not sure if I want to put the BMI sheet in front of them or not.) I also would like for them to work through more of the problems on their own, including the units as I directed them to. Hopefully something will click in my head between now and class time tomorrow.

]]>

Here are the noticings and wonderings from this week:

I am very pleased with their growth over the last 6 weeks. The last notice and wonder we did (two weeks ago), I was not very pleased with. I did not feel they got the idea very well. After talking to Max, I brought back up their notices and wonders a couple of days later and we discussed which wonders were questions we could perhaps solve using mathematics. Then we looked back at their notices and talked about which notices would help them answer the questions that came up from the wonders. Even though I did not have quite as many students submit solutions to the problem, we had some better starts than the previous two weeks.

What I did not anticipate was how that exercise would change how my students worked through the Notice and Wonder exercise the next time. Even though it has been almost two weeks since we reviewed their noticings and wonderings, something from it has stuck with them I think. I am hoping that they will be able to take their more solid noticings and wonderings and move forward with solving the problem.

]]>My Algebra 1 students today had a class-wide reassessment over three learning targets: solving linear equations, solving linear inequalities, and solving formulas for a specified variable. I had rearranged my units this year so that the unit with solving equations and inequalities was first since on the initial benchmark I gave, students could not remember how to solve equations and many commented that they knew they had seen it before but could not remember how to solve. Given that some of the other units I do in the beginning of the year assume that students remember some basics of solving equations and that many students could not even give me that on their benchmark, it made sense to me to rearrange the unit order so this came first.

I know from their 8th grade teacher that last year they spent “a lot of” time solving linear equations. We spent more time than I would have liked to have on these three learning targets. I was disappointed that there were few students who mastered the skill the first time around and that students were still making mistakes that I felt they shouldn’t be. Things like combining 24-15a into 9a, or subtracting 3 from each side rather than subtracting 3x from each side (or writing that they were subtracting 3 from each side and meaning that they were subtracting 3x and treating it like they were subtracting 3x from the side of the equation with 3x and like they were subtracting 3 on the other side of the equation and not subtracting 3x). Don’t get me started on the arithmetic errors. They were allowed to use calculators, so they could double check their arithmetic. Although I am encouraged that students were trying to do the mental math, I still saw many computation errors (3 times 1 is 4 – did you know that?). I had students work through a test reflection and error analysis form that Tina Cardone so graciously shared with me. I was dismayed at how many of my students did not study. We did corrections, additional practice and I got around to as many students as I could to help them. We reassessed today. Although some did better, not as many moved forward as I hoped. At this point, I am going to continue to review the concepts as we move forward and will probably continue to include this learning target on future assessments until I am better satisfied on their understanding.

But I am frustrated tonight. I am seeing that my students are not retaining what they are learning, whether it is something they learned recently or not as recently. That concerns me greatly. We are being told that as we are setting up our curriculum map and deconstructing the standards that although we should be aware of prior skills that should have been taught at previous grade levels, we are not to reteach it if it was supposed to be previously taught. As far as our map goes, we are to assume it was taught and not include it. I should still check to see what their understanding is, do a brief reteach if needed, and move on to the learning target I need to teach for my course. So, since solving linear equations is an 8th grade skill according to Common Core, I should do a brief reteach and move on. A brief reteach was not enough and I am not sure how much good continuing to review will do. So I will continue to incorporate elements of the learning target as we move forward, reviewing as I can and moving forward.

I look at my students and I wonder if they really get it. Do they understand that what we are doing now will be present in all sorts of concepts as we move forward? I tell them that and we talk about the connections back to previous material. But do they really “get it?” Do they understand that if they don’t learn it now beyond just trying to pass the test that they are hurting themselves? That if they cannot recall prior knowledge that it is harder to do well in the course? I honestly wonder. Do they really realize that now that they are in high school they don’t move on to the next course unless they pass the one they are in right now? These freshmen will have to earn points on the PARCC exams which will help determine if they graduate from high school. If they cannot earn enough points on the PARCC exam in Algebra 1 towards graduation, they will have to make them up in later courses. How are they going to do better in a more difficult course if they cannot get through the basic algebra? I can share these concerns with my students but it is not real to them. Graduation, college, and jobs are so far off to them. Heck, for some, the Homecoming Dance is so far off to them and it’s 11 days from now. For some of my students, they are focused on what it will take to get them through the day (my district has greater than 50% of its students on Free and Reduced Lunch).

When students have come from an environment where they feel they haven’t had to retain information (whether it is true or not), how do you change that mindset? How do you impress upon them the importance of retaining prior knowledge so they can build upon it at a later point? I feel like I tie back to prior knowledge often so students can see the connections, but I wonder now if I am really doing as good of a job of that as I could be. When the students’ mind seems to be focused on the here and now (and maybe the near future), how do you get them to see the importance of learning something for longer than the time it takes to take the test? And, as always, how do you help students learn that lesson without sacrificing too much time since we have much content to teach as well? I don’t have an answer that I am happy with right now. I’m certainly looking for your thoughts, especially if you have been able to be successful where I don’t feel I have been.

]]>Today we were discussing dividing fractions. I began with Andrew Stadel’s “Give me more sandwiches.” As we were going over how to divide fractions, one of my students asked if we could divide fractions like we multiply fractions, where we would divide the numerators and divide the denominators. We talked about why that would not make sense. In the particular problem, we would of had 9 divided by 4, which would not have worked out evenly. When I asked my class for a first step, another student suggested to get common denominators. At this point, I remembered what someone (I cannot remember who! – UPDATE: Dave Coffey had blogged this 2 years ago) had shared a few years ago. I told my students to hold that thought and then we proceeded to solve the problem as we had done the first one.

Once we finished the problem, I worked through the same problem by first getting common denominators, and then dividing across.

The one thing I wish I would have done is written the fraction over 1 and then the final answer of the fraction. We talked through that but I didn’t write it down. We did the second problem both ways also. The students who had originated the thoughts said by the end of the second problem that they thought “my way” was easier. I responded that it probably is, but if that is the way it made sense to them, they were welcome to work it out that way.

I was really pleased with my students’ willingness to ask questions and to ask about other approaches today. It was a good class period.

]]>I will be honest, I have pretty much just put the problems in front of my students with very little instruction to date. We begin on Monday (or Tuesday) by doing Noticing and Wondering. First, students list what they notice and wonder on a piece of paper. Then, they share with their neighbor. Finally, we share out to the class and I record them on the white board. Here are my 3 classes’ Noticings and Wonderings for Kristina’s Code:

I haven’t done anything else with them at this point. After a couple of days, I then give the students the problem. I cannot remember why I didn’t go back to noticings and wonderings with the first problem we did. I did not go back to noticings and wonderings on Kristina’s Code because they received the paper copy of the problem on a day that I was not there. After some reflection and seeing first draft responses, I think it is important to go back to the noticings in particular to help students see where to start or what to look for in the problem. I give students a couple of days to submit a first draft response to me. Right now, I am giving them the option to turn in a written response or to submit online through the Math Forum’s interface. I did have more students submit online for the second problem compared to the first problem. I do not have easy access to a computer lab to get all students onto a computer. I do have four computers in my classroom, however, it usually takes about 5 minutes for students to log in and get to the internet, so sometimes using the classroom computers is not practical for me.

Once I receive the first drafts, I take about 2 days to respond to students. What I have done so far is to write at least one “I notice” and one “I wonder” on a post-it to the student if they have submitted a written response. If the student has responded online, I try to keep my reply to the length of the student’s original response if possible. I also try to do at least one “I notice” and one “I wonder.” The “I notice” is something that caught my attention about the student’s response. Sometimes, this is really hard. I try to pick something that is at least in the right direction. The “I wonder” is something I want my student to address in their second draft. I then pass back their first drafts and let my students know that I have responded online and give them a couple of days to revise their solution. The first time I did this, I went through with them briefly in small groups what I had done on the post-it and showed them the Math Forum interface for a second time, trying to encourage them to submit online. I will pass back their first drafts tomorrow from Kristina’s Code and will mention a couple of things as a whole they should be looking to add to their responses. I will also encourage them to submit or re-submit online.

At this point, that is all I have done with the PoWs. I did go over the solution from the last PoW with the students. I put up on the SMART Board 4 different students’ solutions from the Teacher Packet. I had them do a brief notice and wonder with two of the solutions. These solutions happened to be close to the quality of work that they had submitted to me. We talked about the four levels their solution could be: Novice, Apprentice, Practitioner, or Expert. We focused primarily on two things – communication of their solution and the solution itself. I did not rate any of the students solutions the first time around, but I did share where most of them fell (which was apprentice). I also shared with them that my goal for them was to submit a better quality solution than the previous problem. I wish I could say they met my goal. What I have seen so far is that most of them are at the same quality or lower than the last problem. I had one solution (so far) that was better quality and was actually a very strong solution to the problem on the first draft. I have a couple of solutions who communicated much better their thought processes but did not have correct solutions, which I’ll take as a win. Many solutions still had no explanations but at least had some work. I did also get less submissions this time around, which was a little disappointing.

Something I have not done with both of these PoWs is read through the teacher packet before bringing the problem to the students. Honestly, this has been due to not remembering it was there and being in a little bit of a time crunch as I have been preparing for my classes. Before I do the next PoW, I will definitely take some time to read through the Teacher Packet. Something else I definitely want to do with the next PoW is to better tie in the students’ list of noticings as we begin the problem. I think if they had looked at the list, they would have had a better first attempt at the problem. This may have also helped some students to actually attempt the problem. I have not introduced the scoring rubric yet to students and I am considering introducing the full rubric to them with the next PoW. I deliberately did not introduce it during the first problem. I am trying to gradually bring in parts of the process so that I hopefully do not overwhelm them.

I will say that I have found the PoW experience valuable so far. I think in the long run, it will help my students improve as problem solvers. One of the things I am still wrestling with in my mind is how to factor this into their grade. When we talked about it at EnCoMPASS, I seem to recall that many teachers do not factor it into their grades. Some did. I have notes from some conversations that I need to revisit and decide what I want to do with this both in the short term and long term. One of my concerns right now is increasing student participation in the PoWs. I was a little disappointed to see a significant drop off in student participation from the first problem to the second problem. Not every student completed the first problem and there was a smaller number who completed the second. I really don’t want to see a further drop off in student participation.

I do have a knowledge of the resources that each PoW offers, although I am probably not using them to their full potential. The Math Forum offers courses to help you become familiar with the PoWs, how to mentor students, and more. Right now, I feel like I need the most help with commenting to my own students, so I am planning on taking the Learning from Student Work course beginning October 2nd along with some of the other EnCoMPASS Fellows. If you’re interested in joining us, registration closes on October 2nd.

Although doing the PoWs has taken some time in class, I believe that problem solving is such a valuable skill and taking the time to work with problem solving will benefit my students in the long run. Like with anything, it takes time to develop. I am continuing to learn from the experience of teaching with the PoWs and continue to tweak how I am working with them in my class. I’d love to hear from those of you who are using the PoWs in your own classrooms so that I can learn and improve how I am using them in my classroom.

]]>