Math Error Analysis

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My third grade class finished up a cumulative assessment last week.  This particular assignment was completed independently and covered skills from January – March. The assignment spanned the last two units of study and reviewed topic of factors, multiples, composite/prime numbers, area, fractions, decimals, measurement conversions, using standard algorithms, and angles.  There was a hefty amount of content found in fairly large assignment.  It took around two classes to complete the task.

It’s my personal belief that an assessment should be worthwhile to the student and the teacher.  Why take the time to give the assessment in the first place??  Well …. don’t answer that – especially when state standardized testing is right around the corner.  : ) There are some assessments that teachers are required to give and others that are more optional.

My assessment for learning belief stems from past experiences that weren’t so thrilling.  I remember being given a graded test and then immediately moving on to the next topic of study.  There wasn’t a review of the test or even feedback.  A large letter grade (usually in a big red marker) was on the front and that was that. This left me salty.  All teachers were students at some point and this memory has stuck with me.

I like to have students review their results and take a deeper look into what they understand.  In reality the assessment should be formative and the experience is one stop along their math journey.  It should be a worthwhile event. It’s either a wasted opportunity or a time slot where students can analyze their results, use feedback, and make it more of a meaningful experience.

So back on track … These third graders took the cumulative assessment last week.  I graded them around mid-week and started to notice a few trends.  Certain problems were generally correct, while others were very troublesome for students.  Take a look at my chicken-scratch below.

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As you can tell, problems 2, 4, 8, 11 and 22 didn’t fare well.  It seemed that problems 3, 17, 18, and 21 didn’t have too many issues.  My first thought was that I might not have reviewed those concepts as much as I should have.  There are so many variables at play here that I can’t cut the poor performance on a particular question down to one reason. That doesn’t mean I can’t play detective though. My second thought revolved around the idea that directions might have been skimmed over or students weren’t quite sure what was being asked.  So, I took a closer look at the questions that were more problematic.  I looked in my highlighter stash and took out a yellow and pink.  I highlighted the problems that were more problematic pink.  Yellow was given to the problems that were more correct.

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The next day I was able to review the assessment results with the class.  I gave back the test to the students and reviewed my teacher copy with the pink and yellow with the class.  I used the document camera and made a pitstop each pink and yellow highlight and asked students what types of misconceptions could possibly exist when answering that particular question.  I was then able to offer feedback to the class.  For example, one of the directions asked students to record to multiplicative comparison statements. Many students created number models, but didn’t use statements.

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Students also mixed up factors and multiples

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Many students forgot to include 81 in the factor pair and thought they didn’t have to include it since it was in the directions.  Hmmmm…. not sure about that one.

Some of the problems required reteaching.  I thought that was  great opportunity to readdress a specific skill, but I could tell that it was more than just a silly mistake.  I think the default for students is to say that 1.) they were rushing or 2.) it was a silly mistake.  Sometimes it’s neither.  I had a mini lesson on measurement conversions.

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I also reviewed how to use the standard algorithm to add and subtract larger numbers.  Some students had trouble lining up the numbers or forgot to regroup as needed.

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I offered up some graph paper to students that needed to keep their work organized.

After the review, which took about 10-15 minutes, I gave students a second opportunity to retake the problems that were incorrect the first time around.  I ended up grading the second attempts and was excited as students made a decent amount of progress.  The majority of pink highlighted problems from earlier were correct on the second attempt.  #Eduwin! The feedback and error analysis time seemed to help clarify the directions and ended up being a valuable use of time.  I’m considering using sometime similar for the next cumulative assessment, which will most likely occur around May.

Now, I don’t use this method for all of assessments.  My third grade class has eight unit assessments a year.  After each assessment I tend to have students analyze their test performance in relation to the math standard that’s expected.  Students reflect and observe which particular math skills need bolstering and set goals based on those results.  There’s a progress monitoring piece involved as students refer back to these goals during there next unit.



Side note: I had trouble finding a title for this post.  I was debating between misconception analysis and assessment analysis.  Both seemed decent, but didn’t really reflect the post.  So I tried something different – I wrote the post and then created the title.  I feel like error analysis fits a bit more as the errors that were made weren’t necessarily misconceptions.  Also, this post has me thinking of problematic test questions.  That could be an entirely different post.

 

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Reflection and Math Goals

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Two of my classes took assessments this week.  These are considered unit assessments and are related to math skills that the class has been working on over the past 1-2 months.  My fourth grade class just finished up a fraction unit, while fifth graders ended a unit on equations. I tend to grade the tests and then pass them back in the next day or two.  Seeing that it takes so much class time to give these tests (and the grading) I want students to be able to use these assessments.  By using them, I mean that students should be able to look at them with formative lens and purposefully reflect on the results.  Usually the assessment process looks like this:

Stage 1

  • Assessments are passed back to students
  • Students review their score and are excited or disappointed
  • Students try to figure out how everyone else did

Stage 2

  • Teacher reviews the assessment solutions with the class
  • Students ask questions about why or maybe how they can get additional credit
  • Students see where fixable mistakes exist

Stage 3

  • Students receive their math journals
  • Students fill out a reflection sheet looking at skill strengths and areas to improve
  • Students indicate the most memorable activity and why
  • The teacher and student meet and sign-off on the test analysis and reflection portion

 

Okay, so stages 1-3 have been happening in my classroom for the past seven or so years.  It’s become part of my classroom’s math routine.  I see benefits in having students reflect on their progress on assessments, but I also want students to look at an assessment beyond the grade itself.  I’ve blogged about this evolution before. I stopped putting actual letter grades on assessments because of this.  I also considered taking off the point totals as well, but ended up keeping them since it was on the grade report anyway.

I see value in the student reflection component.  I believe students feel empowered when they’re given more control, choice, and access in the classroom.  This year I’ve added my own stage 4.  I’ve added this for a couple different reasons.  One, I’ve noticed that students that don’t necessarily meet their own expectations are really hard on themselves.  They often react negatively on the reflection component and I don’t want students to feel worse after reflecting on their performance.  I want this to be a valuable experience and growth opportunity.  Two, my students have kept their math journal for multiple years.  Some of them are jam packed with notes, reflections, and foldables.  You’d be surprised at how much is in some of these journals.  One thing that students continually tell me is that they love going back in their journal and looking at what they completed over the past few years.  They see that their mathematical writing has changed as well as the concepts that they’ve encountered.  It’s similar to a math yearbook to many of my students.  My third reason is that I’ve always been interested in how students perceive themselves as math students.  Over the years, I’ve emphasized that creating an individual math identity is important. I emphasize this at my school’s back to school session. This math identity shouldn’t come from a parent, but instilled within.  Being able to see students for multiple years allows me more of an opportunity to do this.  Also,  I’m excited to share this at NCTM and learn with other educators about the goal setting and monitoring process. This has been an area of growth for me as I’m continually refining the student math reflection process.

So, here’s stage four:

Stage 4

  • Students review and rate their perceived effort level and attention to detail
  • Students provide an example of where their effort level increased
  • Students create a math goal that will be achieved by the end of the year
  • Student indicate how they know that the goal will be met
  • The teacher and student sign-off on the reflection sheet

 


 

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Don’t get me wrong, this type of reflection is time consuming.  Whenever I discuss this process with other teachers I get quite a few questions about how to find the time.   Meeting 1:1 with kids to discuss their goal takes time and usually the other students are in stations or working on something independently. I can usually finish up meeting with the kids over 1-2 classes.  Instruction still occurs during this time, it’s just not a whole-group model.

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I’ve attempted many strategies to move kids away from comparing their score with others.  One strategy that seemed to work well was to have students go to stations and then I passed out the assessments.  I realized later that they just compared the results when they left the classroom.  I want to shift the paradigm to more of an individual growth model.  It’s a challenge.  Through the years, I believe progress has been made in this, but more needs to be done.

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The student math goals are interesting.  I had to have a brief mini lesson on the topic of math goal setting as many students wanted to initially make a goal of “getting everything right on the next test.”  I think many students were more interested in thinking of what their parents wanted and not necessarily a specific goal for themselves. Keep in mind these are 3-5th graders.  After a few different attempts, students started to make goals that were more skill focused.  Some students are now writing goals about “becoming better a dividing fractions”, “divide decimals accurately”, “become better at solving for x with one-step equations.”  While conferring with the kids I’m reminding them that the goals need to be measurable.

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After the assessment students review their math journals and monitor whether they’ve met their goal or not.  If not, they write down why or possibly change their goal.  I’ll then meet with the student and sign-off on the goal.  My next step is to involve parents in the goal and have a more frequent monitoring process.

Moving Towards 2018

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It’s the last day of 2017 and I’m in reflection mode.  It’s around two degrees outside right now and I’m planning on staying inside with a warm cup of coffee.  By the time I’m done with this coffee, I’m hoping to have finished up this post.  This is allowing me some time to look at what I can change and keep the same for next year.  It’s been a year of ups and downs and many different commitments along the way.  I’ll be splitting up this post into ideas that I’d like to continue and some possible additions for the new year.

Continue

  • I’d like to keep my arrival and flow chart process in place.  I’ve been using this process with K-5 this year and so far it has been working fairly well.  Over the years I’ve noticed that the first five minutes of class are golden.  Getting the students thinking about math quickly after they enter the room is an important piece and is also helps get the class moving in the right direction from the get-go.
  • Using WODB, Estimation180 and Scholastic math magazines for my bell-ringer work.  I believe these daily tasks are beneficial to students and gets them thinking about math in different ways
  • I’ve used a daily agenda this year.  It’s visible to students as they enter the classroom.  I’ve used a slide made in PowerPoint or Keynote and it includes all the daily activities for the class.  I believe it has eliminated a lot of the “what are we doing today” questions that I’ve heard before.  It doesn’t eliminate all of them, but I’ve noticed a huge reduction. Students can take a brief look and get a general idea of the tasks that are planned out.  I’d say that the class rarely makes it through everything on the agenda, but it helps keep the students (and teacher!) aware of today’s happenings.
  • I want to continue to have a balanced instructional approach in the math classroom.  I tend to use bell-ringers and have certain math routines that stay the same, but changing up the lessons and tasks has benefits.  Designing lessons throughout the week that has students working with partners, group conversations, including technology components, and having whole-class conversations tends to help students encounter math in different ways.  It also adds an unexpected element that students sometimes need at this level.
  • I’ve been using a digital planbook this year.  It’s been a great way to plan out lessons  away from school.  Also, it has helped me leave school at a decent time this year – a struggle many teachers have.  Being able to create the lessons and then copy and paste the lesson into my agenda slides have been an efficient process year.

Changes/Additions

  • I’d like to be more intentional in planning out my math questions during lessons.  Creating questions that are open-ended, yet give students time to truly think about mathematics in multi-faceted ways can be challenging.  Depositing a question in a specific place within a lesson can yield dividends later on in the lesson and throughout the year.
  • I’d like to actually use my planning time for planning purposes.  That sounds odd while I’m writing this down.  Like most teachers, I have a certain amount of time that is deemed for “planning”, but I tend to not use it for that purpose.  Generally I use it to check emails, copy, call parents, or check-in with other teachers.  Ideally, I’d like to use it for planning out or modify my upcoming lessons.  I think this is more of an effort on my part to use this time for actual planning.
  • When planning out my lessons I’d like to add more of a cyclical design.  Lessons are usually designed to meet one specific mastery objective.  This is often required at certain schools/districts and is part of the evaluation. The assessments and tasks are related to that one objective.  I’d like to include more opportunities for students to review past concepts.  This also moves students away from thinking that “fractions are done” since we finished a unit on that particular topic.  Having a revision review is such an important topic and I feel like I could write an entire post on just that topic.   I’m continuing to look for ways to make a 2-3 times a week commitment for this purpose.
  • I’d like to commit to being more aware of what is being taught to students after they move on from my classroom.  It shouldn’t be a mystery to what I’m preparing students for, although there’s sometimes a disconnect between what happens at a 5th grade level and in middle school classrooms.  I’d like to check out how the standards that I teach connect to what students will experience in 7th and 8th grade classrooms.

Side Notes:

  • I’m currently in the process of getting through module 3 of my NBCT certification. Watching yourself teach is a bit cringeworthy, but I’m making progress with the editing.  If everything goes well, I should be getting my credentials next December.  This seems like a long way off and I know that there’s a lot of work that needs to be done before then.  Also, I’m hoping to connect, learn and share with my PLN during the NCTM conference in April.

I’m looking forward to 2018.  See you next year!

Attending to Precision

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Last week I read through chapter five of Becoming the Math Teacher You Wish You’d Had.  Reading this chapter made me wish that school was still in session.  There were times when I was reading that I stopped and reflected on how I manage expectations in the classroom.  Specifically, I thought about how I emphasize the need to be precise during math lessons.  More often than not, the precision aspect is related to computation mistakes as well as issues related to missing or incorrect units.  I address this so many times during the year.  So many that I can’t count the amount of times that it’s mentioned.  I think most math teachers have been there.  In most cases I’ve observed students being able to show their understanding of a particular concept, but they don’t show it on assessment.  A label might be incorrect or a one-digit calculation completely changes an answer.  I see this all the time with adding units related to linear, square, and cubic measurements.  A student may get the answer correct, but the label doesn’t match.  I have issues when students place cm^2 when the label should be cm^3.  There’s a big difference there and it has me questioning whether the student understands the difference between area and volume.  There has to be a better way than just reminding students to check for errors or make a reasonableness check.

A couple of the examples that were showcased also emphasize using precise language.  Avoiding the word “it” and being specific are highlighted.  I find myself repeating certain phrases in class.  Not using “it” to describe a particular unit would be on my repeat list.  Instead of using that devil of a word, teachers can emphasize and have students label the ambiguous “it” into something more accurate.  Incorrect labels are a killer in my class, so this is something I continually emphasize.

Estimating can also play an important role in attending to precision.  My third grade class uses Estimation180 just about every day.  We made it all the way to day 149 last year.  We were pretty pumped about that much progress.  It was a productive struggle and heartening to see how much progress was made.  As time went on students became more accurate with their estimates.  That thought process transitioned to other aspects of math class.  I asked the students to have reasonableness checks before turning in an assignment.  The check doesn’t always happen, but when it does it’s a golden opportunity.  I’ve had some students use a checklist to record whether they’ve estimated first to see if their answer is reasonable.  Again, it’s not always used but I believe it benefits students.

Games can be great opportunities for students to be reminded to attend to precision.  Some games are great for this, others aren’t and bring an anxiety component to the table.   I was reminded of the negative impact of timed tests and elimination games.  I’m not a fan of timed fact tests in the classroom and haven’t used them for years.  More recently, I’ve used timed Kahoots or other elimination games.   Some students are more engaged when there’s a competition component.  This chapter brings awareness to how emphasizing speed can be damaging.  Most of the time these games are low-risk, but they do bring anxiety and can cause some students to withdraw.

Guided class activities like pattern creation can be helpful in reminding students to attend to precision. Using student-created patterns ( ___, ____, 56, ____, _____ ) to develop unique solutions can be utilized to show understanding of numbers.  Students can create a multitude of patterns with this.  It also challenges students to find a pattern that no one else has.  I’ll be keeping this in mind as I plan out next school year.

It seems that students will always need to be reminded to add correct units, review their work and attend to precision.  Having strategies and tools available to address this will be helpful moving forward.

End of the Year Feedback

 

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My school year ended last Wednesday and I’m now getting around to looking at student survey results.  This year I decided to change up my survey and make it more detail oriented, as I wasn’t really getting enough valuable information before.  Instead of creating my own (like in the past) I came across Pernille’s gem of a survey.  I know that she teaches at a middle school, but I thought the survey would be valuable for my kids just as well.  So I basically copied all the questions into my own Google Form, created a QR code and had students scan the code to complete the survey during the last two days of school.  Students already knew their report cards grades and they were asked to place their names on the feedback survey.  This is the first time that I’ve taken the anonymity out of the equation.  In doing so, I was hoping that students would answer the questions more honestly, which I believe actually ended up being the case.  The survey took around 15-20 minutes of time and it was pleasing to actually see students put effort into this task.  I had 54 total responses.  Of course there were absences, but I thought that size wasn’t bad, seeing that I have approximately 60 kids that I see in grades 3-5 every day.

Like I do every year, I critically analyze the results.  I look at survey results as a risk, but also an opportunity to see what the kids perceive.  They don’t always communicate what they’re thinking and this is a small window-like opportunity to catch their perception.  I tend to question the results every year, but have come to peace with an understanding that I look at trends, not necessarily every number.  Like most data, I find the individual comments to be the most beneficial.  I won’t be delving into that too much here, but here are a couple key findings:

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Students averaged a 3.43 for this question.  Part of me is glad that it wasn’t below three as I don’t want students to perceive the class as being light on challenge.  I want students do be able to put in effort, work hard, set goals and see that their effort has produced results.  This doesn’t always happen.  Also, the word difficulty is subjective and what someone determines as a challenge, they might not consider it difficult.  This is becoming even more evident as my school continues to embrace growth mindset philosophies.

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Okay, the good ole homework question.  I gave homework around 2-3 times a week and it’s used for practice/reinforcement.  Students rated this as a 2.85, which means I should be giving more, right?  Haha.  I believe students analyze this question and compare the amount of homework received in their homeroom vs. my class.  Over the years I’ve given less and less homework.  Early in my teaching career I used to give homework Monday-Friday, but have reduced that amount during the last five years.  It’s interesting to see the students’ perspective on this heavily debated subject.  Maybe next year I could add a question related to whether the homework helped reinforce concepts for students?  We’ll see.

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I really like this question.  It’s risky as I don’t want the numbers to be the same, but it’s also beneficial because I truly want to see how students’ perceptions of their own growth have changed.  The first question came up with an average 7.67, which I was pretty pumped about.  Most students that I see perceive math as something positive.  Having that perception helps my purpose and it’s a also a credit to past teachers.  The second question rang up as a 9.15.  This was a helpful validation to show that students perceptions about math can change over time.  It also emphasizes the larger picture that math is more than rote memorization/processes and it surrounds our daily life.  I also wonder whether removing the anonymity portion influencd this score in some way.

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This question made me a little anxious.  I feel like knowing a student and developing a positive rapport is such an important component.  It came in as a 4.13.  While looking over the data I found that students that didn’t perform as well rated this much lower than those that did.  Spending time asking about students’ lives is important. Time is such  valuable commodity in classrooms and ensuring that you know a bit more about students can benefit all involved.

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Some students said that I could attend their sporting events or ask about what they did over the summer.  Other students said that I could’ve used a survey at the beginning of the year and not just at the end.  Ideally, it’s probably a decent idea to give a perception survey at the beginning of the year to get to know the students.  I didn’t do that this year, but will most likely put one together for next year.  It’s on the docket.

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The responses that I received on the “Anything Else” question surprised me. I’ve never used this before so I wasn’t anticipating results, but I was pleasantly surprised.  About a third of the students mentioned class activities that they enjoyed or told me about how they’ve changed over the school year.  Some students commented about certain math activities that they thought were valuable.  Making it mandatory probably also played a role in why students added more than a “No” to the comment field.  In the future I’ll be adding an “anything else” question to my survey.


Well, now that the school year is over it’s on to planning the next!

Memorable Moments

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My school year ends in about two weeks.  It’s tough to believe, but the school year is almost over.  The kids know it, the administrators do, and so do the teachers.  Classroom decorations are coming down and boxes are being packed. Summer is just around the corner and I’m in reflection mode.  Overall, it’s been a productive year.  I took a few risks and tried out a few new activities this year.  They mostly turned out well, and I’m keeping the majority of them for next year’s classes.  In a week, I’ll be surveying my kids and asking them about their favorite math activities and memorable experiences.  Through this process I’m asking students to reflect on their math experiences this year.  I’m also asking them to comment on how their perception of math has changed over the year.   In years past, some students have commented that they enjoyed certain activities, but what they remember is the activity, not necessarily the math involved.  This often comes up when my elementary students come back to see their teachers after moving on to middle or high school.

Other students comment that they enjoyed more of the procedural aspects of math because they were easier to complete and understand.  Looking back at my own math experience, I don’t really remember getting excited about learning certain math skills/concepts during an activity.  My memory isn’t connected to the particular skills that I learned during these activities.  The activities were meaningful to me and I’m assuming that the skills transferred, but I mostly remember how I felt in math class.   My math teachers, specifically the ones I had after middle school impacted my perception of mathematics.  I remember math activities, how my teacher viewed math, working with other students, math manipulative and math projects.  As my students reflect on their math journey this year I need to keep in mind the influence that teachers have along the way.

On a daily basis, students will use skills and make math connections that align with posted mastery objectives.  What students remember might be completely different than the stated objective for the day.  I feel as though part of my job is to have students make meaningful math connections on a daily basis.  Activities that spur these types of opportunities are beneficial.  Creating opportunities for these memorable math activities is part of the job and it’s one of the reasons that I enjoy opening up my classroom door in the morning.

 

Improving How Students Analyze Their Work

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One of my goals this year was to have students analyze their own work, make observations and improve. These observations have improved this year by a light margin. For example, students get back their graded paper and look over how they did. Most students look at the top for their points or some type of feedback. Some look for where something was marked incorrectly, while others look for a place in their binder to place the paper. The good news is that students are looking at their graded papers with a more critical eye. That’s a win in my book. Students are starting to observe where they needed to elaborate or change a procedure. That’s good, but the time spent looking at what to change is still minimal.

This year I introduced the NY/M model. Students were a bit hesitant at first, but I’m finding some pockets of success. Those pockets are not just related to the new model, but also a whole range of opportunities that have been put in place for students to understand where a mistake might’ve occurred. Ideally, I’d like to have students identify how the mistake or error happened and to curb that action in the future. Don’t get me wrong, I’m all for making mistakes in order to learn, but some errors impact an entire answer and I’d like students to be able to identify where that’s happening. Being able to self-reflect in order to improve is a beneficial skill.

In an attempt to provide multiple opportunities for error analysis, I’ve intentionally planned for students to identify their own math misconceptions. This has taken many different forms. I believe that students that can identify math misconceptions may be better able to proceed without making them in the future. Three tools/strategies that have been helpful in this endeavor are found below.

  • Nearpod has been a useful too this year. Specifically, having students show their work using the draw tool has helped other students identify misconceptions within their own understanding. Displaying the work on the whiteboard without a name has been especially helpful, as a student might not be embarrassed, yet the class can still learn from that particular person. I’ve used this as an opportunity to look at positive elements of student work and also look for areas that need some bolstering.
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What do you notice?

 

  • Lately I’ve been giving feedback on student papers and incorporating that into my agendas. Before passing back the papers I review the misconception list and answer questions then. I then pass out the papers and students complete the NY/M process. Generally, students make very similar errors and I attempt to address this while reviewing the agenda. This has decreased the amount of questions that students ask related to why/how to improve their answer to receive a M.

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  • On the paper I’m making a renewed effort to write feedback on homework and projects. The feedback takes many different forms and isn’t necessarily in a narrative form. Sometimes I ask question and other times I might circle/underline a specific portion that needs strengthening. This method often elicits student questions as it’s not as clear-cut as other methods. Regardless, it’s another way for students to analyze their work, make changes and turn it back in a second time.

Why is this important to me? Well, I believe that students should be provided additional opportunities to showcase their understanding. At times, I feel as though there’s a gap between what math work they show and what they’re capable of showing. Giving feedback, along with another opportunity to improve, tends to help my students show a real-time understanding of a particular concept. Ideally, this would seamlessly work and all students would move from an NYàM. It’s not all roses though. I’d say at least 50% of the students improve on their second attempt, but I’d like to see more. I believe we’re making progress and have more to go, but I believe we’re on the right track. I’m encouraged to see that this model is slowly and slightly changing the review, redo and improve cycle. This has me thinking of how to expand on it for next year. Stay tuned!