Simplifying radicals? Who needs it?

There was a time I thought radicals should be simplified. A factor of a radicand should never be a perfect square. To do otherwise was just sloppy math—so I thought. Now, I think differently. You should consider it too.

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pic fromthemathlab.com

Making the square root of 40 look like 2 on the square root of 10 serves no real purpose in the mathematics of real life. You can readily estimate the square root of 45, but trying to do that with three square root 5 is a much more complicated task, and for what? If I am going to the fabric store and I am asked how much ribbon I want, I better not say, “4 root 2” and expect to get the correct amount. At the hardware store, I am far better reasoning that the square root of 32 feet would be a bit less than 6 feet, and a bit more than 5 and one-half feet and just say 2 yards. The lumber department does not want to hear this nonsense about radicals and square roots.  They want to cut my lumber and send me to the register to check out so they can help the next person also get a reasonable amount of lumber.

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Photograph by question_ev3rything on reddit

Now, I know, one needs to simplify radicals to combine radicals via addition, such as the square root of 27 plus the square root of 12, but seriously. This is not reality. This is a contrived problem that I have never seen come up in real life. Ever. And I sew and measure and do real life things with math—at home. It does not come up.  It’s clever, like a party trick, but not terribly useful.

I do make certain my Math 2 students can “simplify radials,” but just for the “man.” Not for real life. I used to “ding” my students (take off 1 point just to be mildly irritating to get them to conform to convention) for not simplifying radicals. I am totally rethinking that.

Reality says leave radicals as they are so they are easy to estimate to be useful and to check for reasonableness. Done.

Skirt and shirt made out of fabric that has root vegetables that are in the shapes of rectangular prisms.

Last spring’s project…square roots…get it?

 

year two–take two

Happy Saturday all! I’m supposed to be getting ready for a couple PD sessions that I am presenting over the next two weekends, but I can’t get a comment from @SAMSDrSapsara at @mrsstipemath’s Thursday night 7th grade zoom (time stamp 00:27:14) collaboration out of my mind. Dr. Jessica Sapsara said, “I feel like when I’m making my anchor charts for the things, it’s really coming…way…after the unit has passed and I’m like, alright, let’s get something up so when we’re referencing it later as units come through…let’s look at that one again…so we have better working language…visuals…” And I thought to myself, “self, that’s where you were last year, if you made them at all!” That got me thinking about even more exciting differences between years 1 and 2 of implementing Open Up Resources 6-8 Math Curriculum. I know I reflected on this earlier,  but there’s more. Here’s my summary so far:

  • Anchor charts—Like I said, year 1 I was lucky to make them at all. In fact, I did not understand their usefulness until it was too late. Year two, I am deliberate in their creation as well as in pointing to them during lessons and even as students present, intentionally connecting student ideas to the charts.
  • Lesson preparation—I was so tired last year that I would fall asleep as I was reading the lessons. I’d get up in the morning and get about half-way through the teacher guide and then my room was filled with kids that needed support for the other course I teach, so there went my preparation time for Math 8. Year two, I am seeing so much exciting mathematics and even more brilliance in the authoring of this curriculum. I am excited about reading it and doing it and energized by it, so much so, that I read it before I even go home the day or even two days before the lesson. I am copying and cutting my cool downs and black line masters for an entire unit at one time rather than daily. I am importing my slides for an entire unit into one Google slides presentation. I just edit daily to remove what was covered that day in preparation for the next. (Note to self…hide the slides rather than deleting them, thereby making the file useful for next year! Yep, here I am, learning through reflection!)
  • Student copies/student workbooks—Year 1 saw me rushing around daily or all day Sunday editing and copying student pages for the week ahead. Year 2, my district purchased student workbooks. That in and of itself has given me back part of my life. I am so fortunate that my district cared enough about its teachers and time and copying costs to purchase student workbooks.IMG_3565
  • Attention on student learning—this was a luxury rarely afforded in year 1 because I was so intent on my own learning. In year 2, student learning is what fuels me. Seeing the lesson by lesson progress and retention is reassuring that I am actually helping students become mathematicians.
  • Student engagement—what an improvement!. In year one, I had three students per class I felt were really with the program. In year two, I have all but three students demonstrating understanding and actively owning their learning. I need to remember that when I am beating myself up about those three students, but still, that’s 3 students too many.
  • Student results—What a difference! They are truly remarkable. When I review cool-downs I can see student thinking and reasoning and catch misconceptions so they can be addressed timely. Year one, I was lucky to get a cool-down in the same day as the lesson. Now when I pass out the cool-downs, I hear students say, “we’re done already?” Students self-assess daily as they turn their cool-down into the basket based on their level of understanding, thanks to @mrsstipemath.understanding
  • Supplemental activities—Year 1 had virtually zero supplemental activities for students. Year 2, they are actually part of the game plan. They include Desmos activities to further student learning and assess student understanding; Quizziz games so students can self-assess and develop fluency with concepts; Desmos graphing calculator sessions to quickly and easily make math visible; practice problems from the curriculum; review sessions using practice problems for unit assessment preparation. I am still trying to get to Khan Academy  exercises, but haven’t managed to get them worked in—yet.
  • Pacing—This is a non-issue in year 2. I attribute this to not over-teaching as well as to keeping moving even without 100% buy-in from students. I know the material and concepts are coming around again and both the students and I will get another crack at nailing down the standards. I also understand the learning goals more clearly and know that keeping them bite-sized is essential to student success. That my students were successful at all last year was truly a miracle.sample matrix for blog
  • Community support—In year 1, I felt like my blog was my only companion as I learned this new curriculum and relearned how to teach—or perhaps, finally learned how to teach. This year, there is so much community support. There are the organized supports such as the face book groups and Monday night twitter chats (#OpenUpMath) as well as monthly zoom sessions by the Gurus of Open Up Resources 6-8 Math. My district is also providing monthly professional development specifically for users of the Open Up Resources 6-8 Math curriculum. I also have a network of users across the country as my personal Professional Learning Community. I find it hard to believe I made it through last year without these committed educators.Chat.png
  • School-life balance—This did not exist in year one. I worked very hard, but not very smart and it took a physical toll on me. This year, I am more rested even though I am doing more each day. I manage to eat healthier, sleep more, exercise regularly, read for pleasure, find time to support my learning community and even spend time with my husband. These activities have all improved my mood and attitude and help me recover from slumps and meltdowns more quickly.Selfcareisnotselfish

Year two just keeps getting better too. I actually feel valued and appreciated by my colleagues across the country. I feel confident in my classroom and I am excited about the future. My community members experiencing year one now who are taking advantage of the support of the Open Up Resources 6-8 Math community are doing so many wonderful things for their students. I am grateful for them and want to support them as we move forward together, as a community of learners.

Here I Stand

I’ve heard it. I’ve said it. I’ve lived it. The equations section of Unit 4 of 8thgrade Open Up Resources 6-8 Math curriculum is a beast. It ramps up so quickly with little to no practice and students are lost. They are frustrated and giving up. So are teachers. So was I, until I got my head around it. Sheer conjecture, but this is my take on the whole thing.

This curriculum is designed for 8thgraders. All 8thgraders. We have three distinct levels of math classes in 8thgrade at my school. The Open Up curriculum is only being used for students who are currently at, barely at or below grade level. There is a narrow group of learners using this wide-ranging 8thgrade curriculum. Most of these learners have never truly been asked to perform work that is on-grade-level. This is the first time. They are lost and struggling and giving up.

We are taking a curriculum intended for acceleration, remediation and everything in between and using it exclusively for corrective and remedial instruction with enough access for on grade-level students to make progress. We are working hard to deliver the curriculum with fidelity. Our students are being challenged with grade-level material for, perhaps, the first time. They, in all likelihood, will not get it all. That’s ok. For many, this is their first exposure to grade-level material. Maybe they’ll get it the next time. We need to focus on the fact that students finally have access to grade-level material. We, as teachers, need to be careful not to let our well-intentioned actions take that away from them. When we take the opportunity for students to solve equations containing distribution and fractions and negative numbers and variables on both side and exchange it for 6thgrade-level equations, we are cheating our students.

And there I am, taking work that is at grade-level and breaking it down into bits and pieces that my students can understand and taking it off grade-level. I’m reading to them rather than having them read the problems themselves. I’m giving in. I’m using a curriculum designed to meet the needs of a diverse group of learners with a group of learners who, for the most part, don’t want to be there. I have got to do better so my students have a chance to do better. I’ve started giving out Life Savers to students for getting a good start on activities. Hopefully, only I catch the connection there.

Students do not know how to put in the sustained work required to do the learning that needs to be done to get on grade-level. They do not know how to reach longer-term goals on their own.  Rather than getting frustrated with the students and the curriculum, we as teachers, need to rise to the challenge and be the bridge that finally gets these students access to grade level work. Yes. It will take multiple years, but I would rather be the start of their access to grade-level work rather than the continuation of subpar standards.

There is so much immediate gratification in the lives of students that gets in the way of the time it takes to do the work required to reach longer-term goals.  None of these students fell behind in the last year or two. Fact is they were never caught up to start with. This is just the first time they have ever even had the chance to see and do work that is on grade-level. They are 13 and 14. Yes, they are going to struggle. Yes, we are going to struggle right along with them.  We owe it to them to finally challenge them with what they deserve. All students deserve access to grade level content. Period. Taking Martin Luther out of context, “Here I stand. I can do no other.”

martin luther at luther college

Tracking is the start of all this below grade-level activity. We say we want all students to succeed, but how can they? There is no way to “jump the track” they are assigned to if they do not have a crack at the actual expectations of the grade. At-grade-level progress needs to be accessed and assessed for all learners. Watering down standards and short-changing learners who have historically struggled will never get them where they should be. Please honor our students by honoring their access to grade-level material. It is probable that many may not get it, but some will. Chances are, the ones that don’t get it weren’t going to get the watered-down version either. At grade level material gives all students a chance to meet and exceed expectations. Expect great things from yourself and your students.

A Tale of Two Years

“It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness…” Thanks to Charles Dickens for igniting my thought process.

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This is year two for me teaching 8thgrade math using Illustrative Mathematics’ curriculum published and distributed by Open Up Resources 6-8 Math. Year two is everything I had hoped it would be. I marked my Unit 3 assessments Wednesday and was pretty pleased with the understandings my learners demonstrated. I tallied the individual scores just to get a macro feeling of how the classes were doing this year on the Unit 3 formal assessment. I felt pretty good about student work, but mildly disappointed in myself. As I was marking, I wish I had some student copies from last year’s assessment to compare similarly situated learners’ progressions. (Year one people—make and save a couple student copies to look at next year for each assessment. I thought I was going to remember, but I didn’t.) I did find my teacher copy with the tally marks of last years’ results. Drum roll please…the scores are better. And more students are able to put their understandings on display on the assessment. And, dig this, I am a month ahead of last years’ pace. And the results are significant. [Timeline correction…Last year, I started Unit 4 on February 2nd. February 2nd!!!! This year, I started Unit 4 December 7th. That’s almost 2 months ahead and we have already had 7 or 8 days out of school due to weather issues. And my learners understand more and perform better.] This is all because it is year two. My class sizes are a bit larger than last year, but the mix of historical performance measures (skills, abilities, levels—all horrible words to describe children’s current placement) is about the same. This year-two teaching experience with this curriculum is a wonderful feeling. I attribute improvements to the following changes, in no particular order.

  • Focusing on not over-teaching is a giant change affecting pace issues of the past. This revelation came during summer PD sessions with the good people of Illustrative Mathematics fame.
  • I am also better at formative assessment on the spot so I am not waiting until I can look at a stack of cool-downs to know if students got the intended learning or not. I have better interventions, earlier.
  • This year I made a commitment to move along regardless of stragglers. If I have learned one thing comparing last year to this year it is that you are going to have stragglers no matter how slow and thoroughly you go. Slowing down only harms the learners who are ready to move. It’s like walking in a line with a class of students. No matter how slowly you walk, the end of the line gets further and further from the front. Continually stopping so they can catch up is necessary, but holds everyone back. I need to work at incentivizing the back of the line, aka the stragglers, so they want to keep up and be part of the learning group.
  • This year I am using the practice problems whenever I can. This may be as filler at the end of class or as a pre-class activity to review the prior day’s concepts. I also assign targeted practice problems as Unit reviews before a Unit test. I’ve also put in a Quizizz activity once in a while. I’m going to give the Kahn Academy practice problems a shot next. That retrieval/practice routine must be played out regularly.
  • The launch for each activity is better year 2 because I know where I am going. I know what the focus of the activity truly is and I know where the stumbling blocks are. I am not removing the productive struggle, but I am better with my instructions and communication to learners of the expected outcomes.
  • Both activity and lesson syntheses are better than last year in that they exist, usually. They are focused and tight. They tie to the lesson summary or they come with a note or a highlight on the activity page in the workbook to seal the deal. I am still working to improve all of this and it’s not nearly as good and tight as it reads here.
  • The professional development, I am receiving on the unit materials and elements, is better this year since I can actually attend the sessions. Last year they were not held at a time when I could make it without missing one class of each of my two courses during the day, plus I was sooooo far behind I was afraid to be gone.
  • This year I have the support of my nationwide PLC. I am also supporting other users of the curriculum so I get better and think more carefully as I respond to inquiries and participate in twitter chats. I also feel a sense of accountability to my nationwide PLC and this makes me prepare and research at a much higher level than I otherwise would.
  • I spend more quality time reflecting as I prepare Guru Zoom chats and draft the weekly #OpenUpMath Twitter chat questions. Nothing sharpens skills like leadership.
  • Taking some of the preparation off of me and putting it on students, by having student workbooks, is a positive change for which I am grateful. I am not certain learners took the copied version of the activities last year as serious as they do the official workbooks this year. I am also able to invest in better preparation because I am not making copies of the materials. Workbooks also save valuable class time not having to pass out papers.

unit 3 question

Just look at the understanding that is demonstrated here. This learner is testing algebraically as well as graphically to determine if the ordered pairs are indeed solutions to the given equation. And look! Going beyond what was given, she tests a point not provided (4, 3) that works in the equation to see that it is on the line. I cried–in the best possible way.

Not everything is rosy, however. Horizontal and vertical lines as well at the shifting of proportional lines (y=kx from 7thgrade) was horrible last year and no better this year for the most part. Why??? I know I did a better, more explicit job connecting points and coordinates and lines. I did better, but the students weren’t doing enough. I need to beef that up and I am going to go back through those lessons again and see what it is I am NOT doing. The lessons are good in the moment, but they are not sticking with learners. I have got to make them stick. There needs to be struggling retrieval and spaced practice. All those sticky things must happen more and better than I have been doing. I will also check with my Tweople and see what their experiences and remedies are This is one area that did not improve from last year, yet.

I am going to continue to tweak and ponder, reflect and revise. I pledge that to my self and to my PLC.

Look Who’s Learning

It’s no secret I am an Open Up Resources 6-8 Math junkie. I love what the curriculum does for learners. Today, I want to expand my definition of learners. I am a teacher, but I am also a learner. I have learned more about teaching, learning and math in the past twelve months than I learned in the prior 14 years. I have Open Up Resources 6-8 Math, Illustrative Mathematics and my international PLC to thank for that.

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It’s not just that I have quality material with insights spelled out for me. It’s not that my students are challenged and required to think differently than they ever have. All those things are good. And true. The fact of the matter is that this curriculum makes me a better teacher. I teach better than I ever have. I am more prepared and I have greater insights. I am a better listener. I want to hear how my students are interpreting the math. I want students to share their interpretations. As prepared as I feel each day, I seem to always see and hear and feel something new about the math my students are doing. I am learning. I am the learner and I LOVE IT. Open Up Resources 6-8 Math curriculum makes me a better teacher and learner. Every. Day.

I was starved for quality professional development ever since Common Core was rolled out. The minimal training received on Common Core was ill-conceived and not informative. I remember building fences with posts that were the supporting standards and rails that were supplemental or connecting standards. But I didn’t learn how what I was to teach was better for my students. Everything was theoretical. There was no comparison of how students were expected to learn differently or how I was expected to teach differently. I didn’t practice teaching anything. I was also not given resources with which to teach other than an outline. I learned much more about Common Core as I was teaching seeing my students make connections I had not previously considered. Because North Carolina gutted education funding on the heels of Common Core’s implementation, teachers were left to find and develop their own resources in order to teach and learn the standards. That took a while and I learned much through the school of hard-knocks. Eventually, I was seeing the connections and the wisdom of the sequencing. I was excited. Unsupported, but excited.

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Then Common Core got killed in the court of public opinion. Had Common Core had extensive, quality public relations communications, I like to think the educated public would have bought into the sequencing dictated by the standards. But they didn’t. The only public communication was parents posting homework problems without a context and calling the whole thing folly. There were even teachers that ignored the adoption of the standards and taught what and as they always had. Perfect strangers would come up to me in the grocery store line and see my teacher ID and say something like, “What do you think about Common Core?” expecting me to bash it. I wouldn’t bat an eye. I just said, “ I love it. The connections that students make all throughout the standards are solid. The insights that both students and teachers are gaining are profound. I hope the state has to courage to do the right thing for our schools and ignore the ignorant neigh-saying public. Why do you ask?” I typically didn’t get a response.

I want a do-over on the release of the Common Core Standards. I would start by rolling it out to the elementary schools for three years. Move to the middle schools for a year or two and then finally get to the high schools. Changing horses in the middle of the stream and then providing little to no professional development for teachers and no quality public relations education killed Common Core. Teachers were asked to implement without understanding. Teachers don’t do that. Teachers demand and deserve to understand why. That is a sad fact that was disregarded. We are now getting a do-over, of sorts, via Open Up Resources 6-8 Math, but better. It’s the third word…RESOURCES.

I am fortunate that my district is investing in teacher professional development with the implementation of Open Up Resources 6-8 Math. It seems the phrase Common Core is now somehow forbidden by the state. What we teach are the Common Core standards, but they are simply rebranded. Clever. But the standards are better than solid and it’s working so let’s just keep that little secret to ourselves. Open Up Resources 6-8 Math and the professional development I am receiving through Illustrative Mathematics are helping me connect my teaching to the standards with better understanding and execution than ever before.

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Now for the real purpose of this post. I am concerned there are teachers reluctant to change their ways regarding what they teach and how they teach it and some of these teachers are merely going through the motions as they roll out the Open Up Resources 6-8 Math curriculum. Teachers are learners too. Remember, you can’t get more out of something than you put into it. Just because something is good and nationally acclaimed doesn’t mean it will work without effort. Teaching is work. Teaching with Open Up Resources 6-8 Math is a lot of work. It is this work that is making me a better teacher and it’s hard. But my students are worth it. I am worth it. I am grateful that my district sees that we are all worth it.

Open letter to middle school administrators allocating fewer than 60 minutes a day and/or five days a week for mathematics classes…

Dear Middle School Leaders:

Learning takes time. That is fact without regard to subject matter or grade level. You are a professional in the field of education. You know that truth.

I hear from some users of Open Up Resources 6-8 Math from around the country about the amount of time allocated for math instruction daily. The lowest report is 39 minutes of math class daily. Another low mark is math class every-other-day for 45 to 60 minutes. Please carefully consider these questions. What is at the center of your master planning? Is it the students and their learning? Is the scheduled time currently allocated for academics in the best interest of students or can it be improved? Is your decision to allot minimal learning time grounded in research or is it merely convenient? Is it, as I have heard, the way it’s always been? That is not defensible. What is more important than your students’ learning? There appear to be institutions out there playing school rather than creating environments where students can authentically learn. Please make your business about educating learners and not about playing school.

I am being bold asking these sincere questions because your teachers cannot be. Your teachers love their students and they love what they do and they want to keep doing it. The schedule, however, is making the challenging job of teaching impossible. In the 13 years I have been in the classroom, I have had 60, 70 and 90-minute math classes. 90 minutes is way too long. 60 or 70 minutes work. That provides enough time for students to grapple with concepts and come to resolutions. Fewer than 60 minutes requires teachers to shortcut student learning experiences in favor of algorithms. Students are required to drink from the fire hose of math and they are ill equipped to do so. Surely, you, as the leader of an educational institution know better and are in a position to rectify the scheduling situation. Surely.

Please recognize that learning is a process and give your students time to go through that process. Please honor your teachers by giving them time to promote and support student learning the way they know, as professionals, is best. Days and minutes matter when it comes to developing conceptual understandings. Learning takes time.

Truly yours,

Sara B. Vaughn, M.Ed., NBCT

Home Communication Logs (HCL)

What can I do instead of homework to make a difference with consistently, under-performing students? I need something that will provide learners with the opportunity to intentionally think about and talk about math outside the classroom. I also want to make sure family members, as stakeholders in children’s educations, are kept up to date with what students are learning.

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To help address all of these issues, I developed the HCL—Home Communication Log. Students are required to discuss math outside of school for five to twenty minutes each day with an adult. Students then turn in completed logs each Friday.

So far, results are mixed. I still have reluctant students not turning in the logs. I even had a couple students copy somebody else’s log the first week. But, most of the submissions are good. Parents and children are talking about math daily, and without doing homework! I suggest that students review the lesson summary, from the student workbook (Open Up Resources 6-8 Math) if they are stumped about something to talk about with their selected adult.

After the first week I asked students, “when do you talk to your parents or special adult?” A few told me that they have dinner at the table every night as a family. My heart melted. I didn’t think families did that any more. A couple kids told me that mom or dad is not around in the evening so they had trouble with the log. I told them it was fine to talk on the phone with their parent. I also offered up staff members as special adults for students who have trouble connecting at home. I also told them to come see me before school or at lunch if they wanted.

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I know this is not perfect, but it is more than I have done in the past as far as homework. True confession: What prompted this year’s HCL is a devastating failure on my part last year. I had a student performing below grade-level and, here’s the bad part, because that is what I saw in her file, I didn’t contact the parent. I saw it as normal for that student. I made a terrible assumption that I am embarrassed and sad about. The parent came in and we talked about the situation and I apologized. I then tutored the student each morning she came in. I provided additional information for afterschool tutors as well as private tutors for this student. She still performed below grade level and didn’t grow a bit according to state test results. And it was my fault. I didn’t push her enough. I didn’t let her parents know that she was sandbagging all year. I failed her–my student.

So, fast-forward one school year. I now have 38 kids in one class. 75% of them were determined by the state to be below grade level last year. These learners are like the kid I let down last year. I know a simple Home Communication Log is not going to fix that, but neither is homework. That’s why I thought I’d try this log. Students at my school keep Reading Logs so I thought this would fit in with what they were accustomed to doing. Perhaps it is again, naive on my part, but if parents see their kids making progress, they can encourage them. If parents see their kid struggling to articulate what went on in class that day, that should alert parents to an issue and again prompt encouragement and/or action. But once again, the students that don’t turn in the logs are the same students who would not have done homework had it been issued. I don’t think it’s a coincidence that these are the same students having trouble making their paper and pencil make contact during class. I do my best to check with all 38 students throughout each activity. Much of my time is spent getting reluctant learners started. They want to hide in a class of 38, but I try not to let that happen. Truth is, if a reluctant learner is also a wallflower, I may miss him or her. It pains me to admit it, but it’s true. That happened, as I said.

So, back to the 75%. On a good day I feel like half of them make progress. Am I supposed to feel good about that? Is that enough? No. Clearly the answer is no. I am doing the best I know how to do, but my boat is taking on water and I feel like I’m bailing out with a sieve. I love my kids and that helps for sure, but love is not enough. I need more. My learners deserve more. Maybe they wasted time in earlier grades. So what! All that matters now is how we fix it.

So now back to HCLs. I’m toying with the idea of adding just 1 practice problem from the workbook to the HCL per day for next quarter. At that point it could be a spiraled question. I don’t know though. If you read this, please know, this is my processing process. This is how I think things through. Don’t take advice from me like I’m some expert. I’m just figuring out this teaching thing one day at a time like every other honest educator in the world.

I am going to continue to refine and adjust the HCL as I work with it this year. If you have suggestions, please let me know in the comments.