Student Self-Reflection and Common Math Errors

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My fourth grade students took their first unit assessment of the year last Wednesday.  This is the first class to take an assessment this school year.  The unit took around four weeks and explored topics such as area, volume, number sentences, and a few different pre-algebra skills.  This year I’ve been approaching student reflection and unit assessments differently.

Students were given their study guide during the first couple days of the first unit.  The study guide included questions that covered topics that would be taught throughout the unit.  At first students were confused about how to complete items that we haven’t covered yet.  Eventually students became more comfortable with the new study guide procedure as we explored topics and they completed the study guide as the unit progressed.  There were a couple of students that lost their study guides, but they were able to print it off from my school website.  I reviewed the study guide with the class the day before the test.  It took around 10-15 minutes to review, instead of around 40-50, which has been the norm in the past.

After students finished the study guide the class reviewed the skills that were going to be assessed.  Students informally rated where they were at in relation to the skill.  I decided to move in this direction as I’m finding that reflection on achievement or perceived achievement doesn’t always have to happen after the assessment.

Students took the test and I passed back the results the next day.  Like in past years, I have my students fill out a test reflection and goal setting page.  This page is placed in their math journals and I review it with each student.  I decided to use Pam’s idea on lagging homework/coding and add this to my student reflections.  Last year my students used a reflection sheet that indicated problems that were correct or incorrect and they developed goals based on what they perceived as strengths and improvement areas.  This year I’m attempting to go deeper and have students look at not only correct/incorrect, but also at error analysis.

So I handed back the tests and displayed an image on the whiteboard.

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I told the students that we’d be using coding in math today.  I reviewed the different symbols and what they represented with a test that was already graded.  Each question would be given a code of correct, label / calculation error, misconception, or math explanation. I gave multiples examples of what these might look like on an assessment.   I spent the bulk of my time introducing this tool to the misconception symbol (or as some students say the “X-Men” symbol) to the students.  After a decent amount of time discussing what that looks like, students had a good feel for why they might use the math explanation symbol.

I then passed out the sheet to the students.

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Students went through their individual test and coded each question based on the key.  At first many students wanted to use the label/ calculation error code for wrong answers, but then they stopped and really looked at why their answer didn’t meet the expectation. In some cases, yes, it was a label issue.  Other times it was an insufficient math explanation.  Most of the students were actually looking at their test through  different lens.  Some were still fixated on the grade and points, but I could see a shift in perception for others.  That’s an #eduwin in my book.

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After students filled out the top portion of the reflection sheet they moved to the rest of the sheet.

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Students filled out the remaining part of the reflection sheet.  They then brought up their test and math journal to review the entry.  At this time I discussed the students’ reflection and perception of their math journey and I made a few suggestions in preparation for the next unit.

At some point I’d like create an “If This Than That” type of process for students as they code their results.  For example, If a student is finding that their math explanations need improvement then they can ________________ .  This type of growth focus might also help students see themselves as more owners of their learning.  I’m looking forward to using this same process with my third and fifth grade classes next week.

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Third Grade Math Confidence

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My third grade students have been working on rounding and estimating this week.  It’s been a challenge as these concepts are fairly new to the entire class.  We’ve only been in school for only three weeks but I feel like we’re in stride now.  Kids and teachers both are in a routines and tests are already on the schedule.

Back to rounding and estimating.   So students have been struggling a bit with these two concepts as we head towards using the standard algorithm. With that struggle comes a shake in math confidence.  Students needed to be reminded of our class expectation of “lean into the struggle” many times during the past week.  It’s interesting how a student’s math confidence changes throughout a unit, or even throughout the year.  This third grade class in particular is working on becoming more aware of their math performance compared to what’s expected.  In order to reach that goal, I dug back into my files and found a simple, yet powerful tool that might help students on this awareness math journey.

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Basically, students first read the top row goal. They were then given a die to create an example of the goal.  During this process students circled one of the emoji symbols to indicate their confidence level.  The extremely giddy emoji indicates that they could teach another student how to complete the goal.  The OK smiley means that you’re fairly confident, but feel like you might not be able to answer a similar question in a different context.  The straight line emoji means that you’re confidence is lacking and you might need some extra help.  This paper wasn’t graded and that was communicated to the students.

Regardless of the emoji that is circled, students are required to attempt each goal.  Some students were very elaborate with explaining their thinking, while others tried to make their answer as concise as possible.  After completing this students submitted their work to an online portfolio system so parents can also observe progress that’s been made.  So far it’s been a success.  I’d like to use this simple tool for the rest of the first unit and possibly the next.  It takes time, but as usual in education, the teacher has to decide whether it’s worth that time or not.  In my case, the student reflection has meaning and it’s directly tied to the goals of the class.  I’m looking forward to seeing how these responses change over time. Feel free to click here for a copy of the sheet if you’d like one.

NCTM Reflection

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NCTM 2018 finished up yesterday.  It was a whirlwind of experiences and it has taken a while, but I feel like I’m caught up on my sleep.  Overall, it was a memorable time in DC and the weather was terrific for the most part.   In this post I’m going to put together a couple brief takeaways during each day.  Emphasis on brief, as there’s so much that happened over the past few days.

My flight arrived late Wednesday afternoon. Later that night I was fortunate enough to attend the NCTM game night.  Check out the tag for a few Tweets.  I was a bit reluctant as I was going solo, but decided to try it out anyways.  Glad I went.  I think teachers need this type of time to meet each other and build community.  I found the games intriguing and the conversations even better. Kudos to the volunteers and designers of the game night. Everyone that I encountered was welcoming and inviting. I love the idea of the Pac-Man (@ericholscher ‘s idea) tables.  This is something I’d like to bring back to my own school’s staff meetings. It was here that I met many people face-to-face that I’ve known and followed online for years.  It was great to connect and engage in conversations that extend beyond Tweets and direct messages.  There are too many to mention in this post, but it was a pleasure to meet so many inspirational people in person.

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Thursday was really the first day of the conference.  My hotel was about five blocks away from the conference and it was a great walk each morning.  Friday was the exception as it was raining.  I went to visit a few different sessions related to integrating math practices and technology tools.  Kudos to the presenters that also included short-links so that I can view the presentations later.  Annie shared her presentation on math tools and strategies.  I went out to the gorgeous city center for lunch.  Came back and learned about integers and the orb strategy from a group of three teachers.  By this time I had a decent understanding of where the rooms were located and how to navigate from one part of the conference to the other.  I dropped by the #MTBOS (I forgot to pack my #mtbos shirt from a few years back) booth multiple times throughout the day.  I was also able to meet my #msmathchat pals Casey and Bryan face to face.  Both are passionate educators and it was awesome to meet them in person.

Friday started off with a lot of rain.  I walked/ran to the conference center.  I attended a session on how to integrate mathematical practices better.  I also was fortunate enough to learn about Bongard problems – a new concept to me.

I didn’t get it as first, but was in rhythm after a couple practice tries.  I’m still looking for ways to integrate this into my math classes.  My presentation was at 11:30 and seemed to go well.  My only wish was that I had an extra 10-15 minutes of time.   I guess time management plays a role here.  Maybe I should apply for a full session in San Diego next year?  You can find the slides here.  During the afternoon I was able to learn more about math practices and attended a middle school Desmos session on equations and paper cups.  It was here that I actually learned how to use Desmos for the first time.  I’ve tinkered a bit with it this year and have used it for lessons, but found application potential at this session.  I closed out the day with a session on a partnership between the University of Delaware and a geometry lesson study.  It was interesting to hear how the university partnered with the local school districts in designing a lesson study.  Afterwards, I went out to meet some friends for dinner.

 

It was a good trip and worth the sub plans.  Excellent to meet many of my pln in person.  These people are truly changing math classrooms for the better.  There are still some people that I wanted to meet, but didn’t get an opportunity to do so.  Time was limited and so were the sessions.  Maybe next time. The people at this conference are inspiring.  Many of the presenters are still in the classroom or working with schools and I’m encouraged to see the work that they’re accomplishing and willing to share with the math community.  I’m looking forward to finishing off the school year strong.

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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.

 

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.