Making Math Mistakes

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This summer I’ve been reading a few different books.  One of them is Becoming the Math Teacher You Wish You’d Had.  It’s part of a book study that started a few weeks ago.  Kudos to Anthony for helping start the study.  I’m slowly making my way through the book, following the tag and listening to people’s comments on Voxer.  My highlighter has been busy.  I appreciate all the different teachers that Tracy showcases.  I’m currently in chapter four, which is related to making mistakes in the math classroom.

I believe making mistakes is part of the math learning process.  I don’t think I’ve always communicated that enough.  Some students that I see come into the classroom with an understanding that mistakes are evil.  They’re not only evil, but I’ve seen them used to humiliate and discourage students and peers.  I believe these types of behaviors tend to crop up when the culture of a classroom isn’t solid.  Of course there are  many other variables at play, but a classroom culture that doesn’t promote risk-taking isn’t reaching its potential.

Tracy showcases different teachers in chapter four.  All the educators highlighted seem to be able to communicate why it’s important to look at mistakes as part of the math journey.  This chapter is full of gems.  A couple takeaways that I found are found below.

  • The math teachers that are highlighted seem to understand that mistakes are opportunities.  When they happen, teachers have a choice to make.  Modeling and showing students different ways to react to mistakes is important.  Students need to be able to understand and be accustomed to making mistakes in stride.  This can be a challenge since some students stall or immediately stop when they run into a mistake. Mistakes shouldn’t be perceived as failure. If a student makes a mistake they need be able to have tools and strategies to move forward.  They need to also find the underlying reason to why the mistake or misconception happened.  Having a misconception investigation procedure in place for these instances is helpful.
  • Using classroom language that creates safety is key.  Teachers need to be able to have phrases in the bank that empower students to participate and take risks.  I found that the teachers highlighted in the book often ask questions related to students explaining their reasoning.  They also set up the classroom conversation so that students build upon each others’ responses.   Students speak their mind about math in these classrooms.  They’re not afraid to respectfully agree or disagree with their peers and explain their mathematically thinking.
  • I noticed that the teachers played multiple roles during the observation.  Teachers often gave students time to work with partners/groups to discuss their mathematical thinking.  This time of group thinking and reporting happened throughout the lessons.  Teachers often anticipated possible misconceptions and guided the classroom discussion through students’ thinking.  The teachers asked probing questions that required students to give answers that displayed their mathematical thinking.  Teachers didn’t indicate whether an answer was correct or incorrect.  Instead, educators asked students to build upon each others’ answers and referred to them as the lesson progressed.

I can take a number of the strategies identified in the observations and apply them to my own setting.  I see benefits in having a classroom conversations where students explain their math thinking.  That productive dialogue isn’t possible unless the culture of the classroom is continually supported so that students feel willing to speak about their thinking.  Students aren’t willing to take risks and explain their thinking to the class unless a positive culture exists.  That type of classroom needs to have a strong foundation.  That doesn’t take a day, or a week.  Instead, this is something that is continually supported throughout the year.  Next year I’m planning to have students use the NY/M tool again.  I’d like to add additional pieces to this tool.  I’m also planning on using more math dialogue in the classroom.  I believe students, especially those at the elementary level, need practice in verbally explaining their mathematical thinking to others.  That verbal explanation gives educators a glimpse into a student’s current understanding.  I also believe that giving students more opportunities to speak with one another about their math thinking will help them develop better explanations when they’re asked to write down their math thinking.

I’m looking forward to starting chapter five on Monday.


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!

Castle Area Project

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My fifth grade students just finished up a unit on geometry relationships.  They took the unit assessment with mixed results. I noticed that some students struggled a bit to complete problems involving geometry and formulas.  Specifically, students inconsistently used formulas correctly to find volume, area and surface area. Keeping in mind that there’s only three days left of school, I balanced the idea of having students complete a brief geometry project or dive right into a review packet.  I decided on the project.  The project is similar to another one, but has a Hogwarts type of feel and the focus is more on surface area.  I paired up students and told them that they have literally three days to complete this project as that’s how many days of school we have left.  Students were given an introduction letter that basically explains that their team’s job is to create a model of a new museum.  The museum must take the shape of a castle.

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After briefly explaining the difference between surface area and area the class was off to creating their kingdoms.  Students used nets and a map sheet to create a base for the castle.

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Students were required to subtract the circle of one of the towers from the square base.  That was interesting, as some students decided to count the squares, while others jumped right in and used a formula.  I gave students a formula reference sheet but also had them access the OpenMathRef page.  That was a fantastic resource as kids could manipulative the object to see how the surface area changes.  Students also noticed that they needed to measure the slant height of the pyramids and cones.  This was a new experience for them.

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Students cut out the nets and glued/taped them to the map.

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They had to make sure to cut out the nets correctly or they wouldn’t “close” and finish the shape.  Students filled out their formula sheets after or during the construction process.  One sheet was designated for the surface area and the other for floor area.

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Students have made progress during the last two days.

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Next week they have one day to finish.  I’m planning on having those that finish complete  a stop-motion video of their city with action figures.  I’m looking forward to seeing the student creations.

Side note:  On Friday students were given were given the opportunity to use the Crafty Cut app.  Kudus to Trever for finding this gem.  Basically, students are given a shape and they have to make one or two cuts to create an additional shape.  We only used the free version and it worked just fine for my fifth grade class.  I thought it helped kids see geometric shapes in a different light.

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.


Camping and Rates Project

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My third grade class is nearing the end of a unit on rates. We’ve been discussing tables and how to use them to record rate information. The students have been given a number of opportunities to fill in missing sections of tables when not all the information is present. They’ve been required to find unit prices of items to comparison shop.  We spent a couple days just on that topic. Whenever money is involved I think the kids are just a tad more interested in the problem. For the most part, students were able to find the unit price of items. It was a bit of a challenge to round items to the nearest cent. I used Fawn’s activity to help student explore this concept.  For example, 32.4 cents per ounce is different than 32 cents per ounce. When to round was also an issue, but I believe some review helped ease this concern. Near mid-week, students were picking up steam in having a better understanding of rates and how to find unit prices to shop for a “better buy” when given two options.

I introduced a camping rates activity on Wednesday. This was the first time that I’ve tried this activity as I have time to use it before the end of the year approaches. Here’s a brief overview: Students are going to be going on a camping trip. The student is responsible for shopping for the food (adding the unit prices), sleeping bags (for the entire family) and tent. They can spend up to $50 for the food and their complete total has to be less than $500.

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Students used Amazon to find the items. The most challenging part in this assignment was finding tents and sleeping bags that were the appropriate sizes. The tent had to be at least 100 square feet and each sleeping bag at least 15 square feet. That’s where some problems starting to bloom. Converting the measurements for the tents and sleeping bag took some time. Most sleeping bags were small enough that their measurements were in inches and students needed to record their answers in feet.

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The tent dimensions were in feet so that didn’t cause much of an issue. Students had to figure out which dimension indicated the height and not include that in the square feet. Although, some students thought that was an important piece. Maybe I’ll change this assignment up next year to add a height component. So this took a bit of explaining and guidance, but we worked out the kinks. Students used tables to convert square inches to square feet.

Next week, students will be creating a video of their camping activity. They’ll be taking some screen shots, explaining why they picked each item, (I chose food/this particular tent/sleeping bag because …) explain the unit prices of the food items, describe the process used to find the square feet of the sleeping bag and tent, and how they were able to keep the total cost below $500. I believe they’ll be using Adobe Spark to create this presentation. I need to remember to cap the videos at around three minutes, as some go a bit too long as they might talk more than they needed. I’m looking forward to seeing how this turns out next week.

Thinking About Math Misconceptions

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My students have about one month of school left.  It’s hard to believe that the 2016-17 school year will soon be over.  This year I’ve been attempting to have my kids think more about their mathematical understanding.  Putting aside time to do this hasn’t been easy and there’s been a struggle, but I believe we’re making progress.  One of the most impactful pieces to this has been the inclusion of a more standards-based approach when it comes to student work.

One way in which I’ve had students think more about their thinking is to give students opportunities to redo assignments.  Students are given a second attempt to complete an assignment after they complete a reflection sheet.  The sheet is below.

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The goal is to improve and move from the NY – not yet to a M- Met.  Students are required to analyze their assignment and staple on the NY–>M sheet before turning it back in.  I’ve changed this sheet over the past few months as I started to notice that some students were a lot more successful at redoing their assignments and receiving full credit than others.

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I decided to have a a brief classroom discussion to talk about how analyzing our math work can help us identify where we should target improvement efforts.  I put two slides up on the whiteboard to frame the discussion.

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Indicates what the student wrote and how it impacted the second attempt.

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The class discussed the two slides and the student responses.  I emphasized the need to critically analyze their work before redoing it a second time.  Being specific with the comments also plays a role in how well a student performs again.  I also thought it might be a decent idea to start discussing key misconceptions before the class gets back their assignments.  This already happens, but spending more time discussing prevalent misconceptions beyond “simple errors” might be helpful moving forward.  I’m sure I’ll refine the reflection sheets over the summer, but I like the progress that students are making along their mathematical journey.



Exploring the Distributive Property


My fifth grade crew has recently been exploring the distributive property.  What’s interesting is that most of the students have used the property before, but it just wasn’t labeled.  Students mentioned this as I introduced the concept earlier last week.  Although most of the students in the class had background knowledge of how to distribute numbers, the level of that understanding differs depending on the student. The majority of students have mastered many concepts related to number sense, but pre-algebra concepts are fairly new to them. That is one of the reasons that I chose to beef up a lesson related to the distributive property. I have a few specific resources related to teaching the distributive property that I thought might be helpful for this lesson.

The substantive mathematical idea of this month-long instructional unit is to have students experience algebra and use it with geometry/measurement ideas with algebra notation. Later on in the unit students explore the distributive property, apply order of operations, simplify expressions, solve equations, utilize the Pythagorean theorem and use size-change factors. 

The lesson began with an agenda.  The mastery objective for the day was “students will be able to identify and use the distributive property to simply expressions.” I briefly explained and drew an example of the distributive property on the whiteboard. At this point I wanted students to get a quick overview of the distributive property in action. This quick overview seemed to help introduce the concept to students that haven’t seen it before.

Students were then placed in groups to complete an initial distributive property activity. A scenario was given where students were asked to purchase three gifts for three different grandkids. Each grandchild would receive the same items. Students were asked to supply number sentences.  Feel free to download the sheet here.

The groups presented their findings and number sentences. During this time I was able to showcase how the distributive property can be utilized in this scenario. Based on the responses, students were still having trouble identifying and using the distributive property. Also, I was finding that students were adding each individual number instead of using a more efficient distributive property. Seeing that students needed more practice opportunities, I decided to move on to a rectangle method activity.


Students were then asked to find the area of the above rectangle using two different numbers sentences. I chose this particular assignment because of the math connection opportunities. Students were recently studying measurement concepts during the last unit and it’s still fresh in their minds. So, students were given rectangles that were split into segments and they were asked to show different number sentences find the area of the shaded portion.  I placed the page on the document camera and the class reviewed it together. Students were given time to reflect, make connections and ask questions during this time. I also gave students an opportunity to preview the next few lessons and see how understanding the distributive property will help them as they simplify expressions later in the week.

The distributive property activity contributes to the students’ developmental conceptual understanding of the mathematical idea. Students are asked to create a rectangle, divide it, and then use two different number sentences to showcase the shaded area. Students are using factoring strategies to group numbers in order to find the area. In doing this, students are acknowledging that the distributive property is evident in the combination process.

Screen Shot 2017-04-30 at 4.52.51 PMScreen Shot 2017-04-30 at 4.52.59 PMScreen Shot 2017-04-30 at 4.53.08 PMI believe there were challenges evident when I presented these mathematical ideas to the class. Students often come into class with preconceived notions that parentheses are only used during problems involving the order of operations. I believe that the students’ understanding of the distributive property was strengthened through the use of the rectangle area activity. Although their understanding seemed to improve, some students need to be guided through the activity. They were unsure of how to start the problem and some needed prompts.

I believe that the student work I collected suggests that the next step in my instruction is to expand on being able to use the distributive property and combine it with translating equations into expressions.  The next sequential step is to use equations to solve problems involving integers. Although students have used integers in the past, it may be beneficial to review how negative integers impact the distributive process. Also, as I gave students feedback, I wondered if they would’ve been able to complete the same number sentences, but distribute the numbers from both sides of the parentheses. For example, could they connect that 5(11 – 10) is the same as (11-10)5 ? They’ve only encountered the first example, so this may be something worth investigating for the next time I plan this lesson sequence.  Having practice with these types of problems will benefit students, as they need to have experiences with using signed numbers with expressions.