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.



Which One Doesn’t Belong – Fraction Edition

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My classes have been using a WODB board this year.  The board has been a permanent fixture in my room and it has been up since August.  I came across the idea last year after reading Christopher’s idea and Joel’s example.  I’m finding that it has been a great routine for my 3-5th grade math students.  My goal was to change my WODB bulletin board every  week, but it’s really being changed around 2-3 weeks or so.  My boards started out as mainly shapes, but has moved to numbers and equations recently. Changing it less gives kids time to see other options and add more notes.

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My third grade students are in the middle of a unit on fractions.  They used number lines to multiply fractions by whole numbers earlier in the week.  The students are becoming better at multiplying fractions using visual models, although some are more wanting to multiply the numerators and denominators.

Today, the students completed an individual Which One Doesn’t Belong task.  I’ve heard of other classes doing something similar, so I thought it might work well with my kids. Students were given a criteria for success page and then asked a bunch of questions.

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Students were asked to create four different fraction multiplication models.  Students then created two different solutions for the WODB prompt.  After a brief amount of modeling, students started to create their own WODB boards.  Many students had questions about what could count for solutions?  I put it back on the students to figure out if their solutions were appropriate or not. For the most part, students did a fine job finding two different solutions.

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Students then wrote down their solutions and folded the paper to hide them.

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The students took pictures and put them in their SeeSaw accounts.  Next week, the kids will look at each others’ responses and see if their solutions match. I’m looking forward to what they observe.

Fractions and Fruit Salad

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My third graders have been studying fractions for the past few weeks.  Last week, students represented fractions as multiples of a unit fraction.  In one of the lessons they  broke apart 4/5 into four 1/5 pieces.  They used number lines to show all the different unit fraction pieces.

Afterwards, students represented fraction as multiples of a unit fraction.  They showed how addition equations and multiplication equations are related.  Students reviewed how repeated addition is similar to multplication. Students are becoming better at understanding how to multiply a whole number by a fraction and show the progression on a number line.  I had some students want to jump to multiplying the numerators and denominators , but these students had trouble when explaining why they used that process.

The next task really seemed to stretch their thinking.  It also showed me how well students grasped the idea of combining fractions with similar denominators.  The students were asked to create a three-fruit salad.  They were given fruit and the typical weight for each item.  Students needed to combine three of the fruits to total exactly five pounds.

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Students needed to create two different recipes for this task. Students needed to also show how they combined the fruits withe  a visual model and display a number model.  For the most part, students were able to combine the fruit accordingly.  An interesting tidbit was that many of them overemphasized one fruit over another.  For example, I had a few students that took 16 cups of grapes to get a total of four pounds.  Then they just found two items that totaled one pound.  I asked the students what those types of fruit salads would taste like.  They didn’t have a response, so I’m assuming we’ll have to look at the context a bit more next time.  This could also be used for a ratio/proportion lesson somewhere down the road.

Later on in the week, I gave a similar task related to a four-fruit salad with a weight of exactly six pounds.  This time the weights weren’t unit fractions.

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Students first noticed the strawberries 7/8 weight.  A few students were pretty sure that the 7/8 might not be compatible with the rest of the fractions.  Others disagreed with this idea and said that the 1/3 didn’t fit.  Students worked out this task and needed some help along the way.  Students moved towards creating just number models, as some decided to not go the visual model route.  Overall, I’m impressed with how they tackled this problem.  We’ll be discussing it tomorrow afternoon.

Polygon Hierarchy

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My fourth grade class has been exploring geometry and polygons.  They’ve been comparing polygons and looking closely at how to classify them.  Students are familiar with these shapes and can classify them by their looks. Students were confident early in the week as they were able to label polygons based on side length, angles, and sides.  Things started to change when the word hierarchy was introduced.  The word is new to most of my students so the class first reviewed the term.

Students were then given a paper full of shapes.  They cut out the shapes and used their desk to classify them.  Students classified the shapes according to their attributes.  Some students sorted the polygons by angle size, while others used symmetry or side length.

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Students were finished in about ten minutes.  The class took a gallery walk and reviewed all the other ideas.  Students discussed which polygons met or didn’t meet the category title.  Students then went back to their seats and reviewed the term hierarchy again.

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The class took the cards and taped/glued them to make their own hierarchy.  Students are starting to see the characteristics of polygons in a different light.  This is good news as later on in the year students will use polygon characteristics to find area measurements.  They’ll also be transforming the polygons on a grid in about a month.

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I’m looking forward to the polygon discussion tomorrow.

Side note:  This is my 300th post.  I had no idea that I’d be writing so much over the years, but it’s been an amazing journey.

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.


Reading Menu Projects

This year I’ve had the opportunity to work with a fifth grade reading group. My day consists of almost all math instruction, so having a reading enrichment group is something different. I appreciate the different subject matter as I tend to look at most content through a math lens. The group meets every day for about 30 minutes. This is my third year teaching this  group and I’ve become more familiar with the resources every year.

I find that each year brings new ideas and this year is no different. I always tend to ask question about making relevant connections to the content that I teach. This year my students are studying Hamlet. They’re not delving too deep into the original text. In fact, we’re reading this book and have been exploring Hamlet for the past month. It’s been an exciting journey. Along with reviewing the play, the class used a character map, learned about Will, and viewed clips from a contemporary portrayal of Hamlet with David Tennant.

The class is now in the final stretch of our Hamlet unit. So, for the last unit I decided to try something different. Ideally, I’d like to have students remember Hamlet when they encounter it again in a few years. I decided to use a menu board approach.  Each student picked one project below.

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The class then reviewed a criteria for success rubric. Honestly, the rubric seems quite intense at first. But in all fairness, I needed to have a rubric that actually encompassed all of the menu items.

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I made sure to have the students review the part on the left side. In that past, I’ve found that sometimes students might pick a project that is less challenging. I was hoping to be proven wrong with this project.


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Students “signed-off” on the project and were committed.  I find value in having students actual write that they agree to the criteria.  I think it adds an ownership element that isn’t always there.  It also reminds me what resources to pick up before next class.

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I was pleasantly surprised to see that all of the menu items were picked – some more than others.

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Students were then given the remainder of that Monday to work on the projects. Near the end of class I told the students the plan for the rest of the week.  They had the next four days to complete their menu item. My job was to gather materials and the technology that was needed. I had to find more technology since my school isn’t 1:1. I begged and borrowed from the other teachers in my building to get enough Chromebooks and iPads to make the projects feasible. Priority for iPad and Chromebook use was given to the stop-motion-video and board game creators. I was pleasantly surprised to find that some of my kids wanted to create a video game using Scratch.  One of my favorites was a duel between Hamlet and Laertes, where Hamlet always wins.

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Near the end of the week most students were finished, although a few voluntarily came in during their recess to finish up the project. The next Monday was designed for feedback.

Over the weekend I created a Google Form for student feedback. Students scanned the code when they entered the class.  Each student filled out the feedback form and reviewed another student’s project.

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You can view the sheet here. Currently, the class is halfway through giving feedback because we’ve had a slight interruption because of Parcc testing. Tomorrow the class will be giving additional feedback. My plan is to print out the feedback and give the responses (without the names) to each student. The authors will then have an opportunity to analyze the feedback and give responses as needed.

The student engagement for this project was top notch and I was impressed with the quality of work produced.  This reading menu has me wondering how a menu system could be applied in the math classroom.  So far, I haven’t had as much success with a menu in the math classroom.  I’ve used choice boards, but they haven’ been anything spectacular. Anyone have success with this?  This topic is something to ponder before heading off into spring break next week.

Fraction Division – Models and Strategies

My fourth grade students have been exploring fractions.  They’ve become familiar with how to add, subtract, and multiply fractions.  They just started to divide fractions earlier in the week.  Whenever I introduce fraction division I tend to have one or two kids that raise their hand quickly.   Their quickly raised hand tends to cause me to slow down and prepare.   They comment that there’s a “fast” way to divide fractions that they learned at Kumon or from someone at home.  Sadly, that trick is infamous number 1 on the NCTM’s Tricks that Expire!  These students can explain what to do, (change the numerator and denominator of the second fraction and multiply) but struggle when pushed to explain why it works.  I feel like at times these particular students inadvertently or purposefly convince others in the class that this method is the quickest.  Some agreed, but introducing this idea at the begging caused unneeded confusion.

I shifted the discussion to the meaning of the fraction bar.  One of the students mentioned that the fraction 1/2 is the same as 1 divided by 2.  Another student said that is the same as 0.5.  This conversation was productive and moved the discussion back on course.  Students started to build upon each response and were able to start thinking more about their own understanding of fractions.  I then introduced the idea of fraction as division.  This resonated well with students and I could tell that they were really thinking about how they view fractions.  I then put this problem on the board.

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Students thought for a little while and then decided to split up each fraction into three pieces.  They then counted up the pieces to find 9.

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I then introduced students to a common numerator and denominator model.  Students thought about this problem and then started making a few guesses.

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One thing that seemed to shift this thinking was to look at fraction as division.  In my years of teaching this seems to make quite a few connections   Many students know that a half of a half is a quarter, but are a confused when it comes to dividing a half.  One student mentioned that they both have common denominators and that might be useful when dividing.  Another student said that a fraction is division, so you could divide the numerators and denominators.

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The class agreed that this will work as long as the denominators are the same. They also concluded that if the denominators aren’t the same, we can find an equivalent fraction to create ones that are. This conversation lasted for about five minutes.  It was productive and not once was there mention of a “fast” method to divide fractions.  I’m hoping that students hold on to visual models and using a variety of strategies when dividing fractions in the future.  Next week, we’ll be investigating how to divide mixed numbers.  That’ll most likely happen after our week long PARCC adventure.