A simple mistake or something more?

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I’m grading tests this weekend.  My third grade group just finished up an assessment on fractions and multiplication.  It’s been about a 1-2 month journey full of investigations on this particular topic.  Many formative checkpoints were planned along the way and the unit assessment was scheduled for last week.

While reviewing the student work, I’ll sometimes write questions or direct students to a part of the question that would’ve made their answer more complete.  There are moments of pride and moments where simple mistakes drive me a bit crazy.  You see, there are only eight units with my new resource, so the assessments influence grade reporting quite significantly.

After grading the assessments, I have student analyze their results.  They comb through the test and look at how each question aligns with certain skills.  They also determine if a missed question was a fixable mistake.  I want students to be able to recognize when this occurs and fix them when they can.

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In my experience, 9/10 times students believe that the reason they missed a question was because it was a fixable mistake.  That’s not always the case.  There’s a certain amount of self-reflection and humility that’s involved in this process. Being able to be a bit more honest and communicating what a simple mistake is and what it isn’t might be in order before the next assessment.  So, what steps do students take if a missed questions isn’t a fixable mistake?  It’s one step in the right direction to admit that it isn’t fixable, but then what happens next?  Do students and teachers have plan for this, or do we move on to the next unit?

So, was it a simple mistake or something more?  This question comes up more often than not while I’m grading student work or reflecting back on a class conversation.  Some of the answers are more positive than others.  A simple calculation error can vastly impact an answer, but it may be a simple mistake and the student has a solid conceptual understanding of that skill.  But, a number model that doesn’t match the problem tells me that the student might not be certain about the operation that needs to be completed. Was the simple mistake putting the wrong operation sign in the number model?  I guess you could go down many different paths here.

The question type can influence how well and thorough a student responds. Some questions are quite poor in giving educators quality feedback to help inform instruction.  Right off the top of my head, multiple-choice and true/false questions fit that bill.  They sure are easy to grade by human or a machine.  Hooray!  But, they don’t give me quality feedback that I can use immediately.

This also has me wondering about the quality of the assessments that are given.  Measuring how proficient a student is on a particular concept doesn’t always have to come from standardized or unit assessment results. Classroom observations and formative checkpoints are beneficial and give teachers insights to what students are thinking. I want to make students’ math thinking visible.  Whether that’s using technology or not, making that thinking visible puts the teacher in a  better position. From what I observe, some of the best math task questions are open-ended and tend to have a written component where students are asked to explain their thinking.  The quality of the question and openness of the answer helps educators dig deeper into how and what students are thinking.  I think that’s why teachers are always looking out for better math tasks that help students demonstrate their understanding more accurately.

How do you help students determine if a mistake was simple or something more?


Google Forms and Presentations

Google Forms

My 3-5th grade classes are finishing up their math genius hour projects this week.  Fittingly, it’s the last the week of school so we made it just in time!  I have two days to fit in 15 project presentations.  This last round of projects lasted around two months and the final projects will soon be revealed.

Students created questions, found a math connection, researched and are presenting this week.  During the last two years students present their projects and the audience asks questions about the topic. This technique seemed to work but I tended to have the same students ask the presenter questions.  Around five or so of the same students asked the presenters questions.  In a class of 25 that’s not ideal.  It was great that the students were asking questions, but five or fewer was disappointing.  Bottom line – the audience wasn’t as engaged as they should be.  So this year I decided to give the audience more of a voice in the process during genius hour presentations.  This actually stemmed from a class that I took this spring about using Google tools in the classroom.

One of the assignments required students to create a Google Form that could be used in class.  My first thought was to create a student rubric for presentations.  I decided to create my own after dabbling around the Internet for a few examples. Initially the form was going to be used by the teacher to evaluate presentations.  After starting the form I changed my thinking.  I thought about possibly having all of my students use the same form to evaluate the presenter.  The genius hour feedback form was built from that idea. Click the image below for the form.

Feedback Form

This week my students have been using the form to evaluate their peers.  Students are asked to present their projects while the audience listens.  At the very end (not at the beginning as some students want to get a head start) students take an iPad and scan a QR code to access the Google Form.  Individual students evaluate the speaker and submit their response.  It’s not confidential as students have to pick themselves (the evaluator) and the presenter.  I tell the students that this information will not be revealed to the presenter.  So far it’s been working well.  The last presentation took less than 2 minute to collect 21 feedback submissions.  Another bonus is that you can have a class conversation about the overall quality of the presentations.


I then export the file to Excel, hide the evaluator column and then print out the sheet for the student.  The student is then able to reflect on the data at a later time.

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The form needs some work as I’m thinking of making some of the questions more clear.  I’d also like to add a section on the form where students can ask the presenter questions.

Student Reflections and Assessments


This past week my third grade class took their third unit assessment.  This particular unit focused on computation of single-digit numbers, data analysis and order of operation procedures.  While grading the assessments I started to identify a few patterns in the student responses.  Specific problems were missed more often than others.  This isn’t an anomaly on assessments, but these particular problems stood out.  One skill area that seemed to jump out to me dealt with the skills of being able to identify the median, mode and range of a set of data.  These skills were introduced during the first few weeks of school and the class hasn’t revisited them in some time.  Also, I found that students were having trouble identifying the differences between factors and multiples. Some of the student responses mixed up the terms while others seemed like guesses. Both of these skill are necessary moving forward as the third grade class explores prime and composite numbers next.  A colleague and I and came up with a limited list of reasons why we thought the problems were missed.

1.) Students aren’t yet able to apply their understanding of the skill

2.) The question on the test was confusing

3.) Students made a simple mistake

Optimistically, I’d like to say that most of the mistakes fall into category two or three. I don’t think this was the case with this particular assessment.  After looking over the class results I concluded that most students that missed skill-associated problems fell into category one.  In addition to not grasping a full understanding, I felt like students were not given enough time to practice the newly learned concepts.

I believe students should be given additional opportunities to show understanding.  Coming from that thought line, I decided to have students reflect on their assessment results in their math journal.  I’ve done this in the past but I wanted to also include an addition to the process.

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After completing the page above students reflected on their performance in relation to the expectation. Students were then given a list of four problems.  The problems were similar to the most missed skills on the assessment.  Students were asked to pick three of the problems to complete.  Students were encouraged to pick skills that were missed or topics that they felt needed strengthening.

After both sheets were completed, students brought their math journals up to me and we had a brief 1:1 conference. This time is so valuable. The student and I identified skill areas that showcased strengths and areas that needed strengthening.  We then reviewed the responses to the questions on the reflection sheet.  I spent around 2-3 minutes with each student.

Students were then asked to work independently on another assignment that I planned for the day.  Overall, I thought this reflection process has helped students become self-assessors.  Students have a better understanding of their own skill level in relation to the expectation.  I plan on using this strategy a bit more as the year progresses.

Beginning the Year with Math Journals

Math JournalingLast Monday marked the first day of my 2015-16 school year. It’s been an amazing week but I’m glad the weekend is here. Yesterday my school had its annual PTO picnic and I now have some time to write. I feel like I need time to process and reflect on the whirlwind of events that’ve happened over the past week.

Last Tuesday I planned on giving student their journals. Before passing them out I decided to cover a few ground expectations. I had a discussion with the students about what the journal will be used for. In the past, students used the journal to reflect on different assessment results.  This year I want to add more goals related to journal use. Primarily, the journal will be used by and for the student to reflect on math experiences and answer prompts. I told the students that I’d be writing feedback or ask questions related to their journal responses throughout the year. Most students were relieved when I said that the journal wouldn’t be graded. I indicated that it may be used during parent/teacher conferences or to show growth over the year. The students didn’t seem to have a problem with that. I ended the brief conversation reiterating that the journal was for them.

One of my priorities this year is to have students write more in math class. I read a book this summer that exposed me to research related to connections between the brain and math.  In particular, I found that students need adequate time to process and rehearse mathematical information for it to be retained.  I feel as though newly acquired math concepts can be processed through reflections and the writing process.  I intend to have students explain their thinking and view of certain mathematical concepts. I’d also like to give students time to process what they’ve been experiencing and document those events. I feel like students, just like adults, need time to process and reflect.I feel like the math journaling process can lead to the rehearsal of math concepts. In a sense, students are practicing what they’ve been learning and personalizing it through their journal writing.

So, this week all the grade levels that I teach had an opportunity to start their math journals. All classes participated in the Marshmallow Challenge and that event related to their first journal entry

Tell me about your marshmallow challenge experience.

The first response was more geared towards helping students recognize the teamwork involved as the classroom was, and currently is, building a community of learners. The second journal entries were content specific

How do arrays help me multiply?

What’s the difference between rays and line segments?

Students gave me a variety of responses. Some were lengthy with many pictures, while others barely scratched the two sentence mark.


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ArraysRegardless of the length, I’m finding that some of the students’ math understanding is revealed in their writing and pictures. I’ve already been able to find misunderstandings through student journal entries that might have taken longer to expose using other means. Also, writing feedback to the students gives me an opportunity to extend understanding by asking questions that lead students to question their responses. This year I’m attempting to use math journals more regularly and so far (only 5 days) we’re off to a decent start.  At some point I’d like to have student complete a journal prompt related to how they use my feedback in their journals.

Feedback Opportunities


Last week I took part in a Twitter conversation about student feedback. The discussion evolved into why direct feedback is often more efficient than vague “good job” teacher responses. Most of us agreed that feedback and exploration is often the cornerstone in having students recognize misconceptions and build mathematical understanding. That feedback is so essential and is a vital ingredient in the learning process.  The conversation had me thinking of the different feedback opportunities that exist in and out of the classroom.

Everyone receives feedback. In a school setting that feedback can come in the form of a supervisor, team, students and so many others.  Beyond this, I receive feedback when I forget to shut my car door or leave the lights on. A loud beep blasts out of the speaker and I need to go back and fix the problem. When creating a document and I forget to save before exiting I receive a “do you want to save” message. Absolutely I want to save, and the feedback (or reminder) gives me an opportunity to do so. I could give other examples but the point is that feedback comes in forms that might not be associated with classroom use.

I feel as though giving students feedback is intentionally setting the stage for student improvement. What happens if the student doesn’t utilize that feedback?  What good is the feedback if it sits on the paper? I feel like this happens more often than I would like to admit.

Giving students multiple opportunities to utilize feedback can lead to action.  That action may lead students to make changes in how they approach problems and concepts.  Although the teacher is one of the main feedback systems, it shouldn’t and isn’t the only option. While thinking of feedback, I started to brainstorm some possible feedback systems that can aid in the learning process.  I can picture these systems being used to give feedback after some type of formative assessment or instruction.

Teacher – The teacher is one of the best feedback tools in the classroom. Fielding student questions, clarifying  and anticipating next steps all play a role in how a teacher responds with feedback. Teachers all around the world offer feedback, so much that it becomes part of their daily lives.  The feedback from teachers can be observed in written or verbal form.

Students – Peer editing and group work can be powerful. Of course, modeling and front loading needs to occur before this becomes an amazing tool. When students discuss answers with each other it opens up a door for feedback. Students can explain their reasoning and be critical friends in the process. Group work provides opportunities for students to become better at explaining their mathematical thinking and processes. Hearing how other students explain their thinking can lead students to an explanation that might not have been perceived before.

Math Classroom Conversations – Math class conversations can be beneficial to all involved.  This also takes modeling before becoming a positive aspect of the classroom experience.  Asking open-ended math questions and having students respond can lead students to ask additional questions.  Feedback can be provided during this entire process while students construct understandings. Classroom conversations often involve some type of whole group question, group response and feedback.

Games – Math games can provide students with a low-risk opportunity to practice skills and show their understanding. I find that when students use math games they engage socially, think strategically and practice skills in the process. Board, card, dice and app games all provide feedback in different ways. Feedback is given in how the other students react to each other, how the answer is revealed and in the scoring element. Math games open up a door of possibilities and adds some competitiveness. Apps have helped revolutionize this idea. Kahoot and Socrative have gaming elements that provide students with additional feedback that can be used to inform instructional decisions.

Adaptive Software – No, this shouldn’t be the only method of feedback. Keeping that in mind, the feedback given through adaptive software can be be helpful to a point. Regardless of the adapted score or level, this type of feedback might not be tailored to the individual student.  Although adaptive apps/software is a field that’s improving (as tech startups hire education professionals), this type of feedback isn’t as accurate as some of the other methods above.

How do you give feedback opportunities in the classroom?



Addressing Misconceptions


Students in third grade are exploring measurement this week. As students progress through the unit I feel as though they are becoming more efficient in converting Metric units. Near the end of the class today students started debating the differences between US Customary and Metric. The class than started completing an activity where they had to measure different insect lengths.  Students worked in groups to accomplish this task.

During this time I traveled to each group and intentionally eavesdropped on the conversations. Students asked me questions and I listened and asked questions back.  I then moved on to the next group. I wanted the students to work together and persevere. Some students started to talk about the measurement of different objects around the room.  I especially paid close attention to the questions that students were asking each other. This was a great opportunity to check-in on some of the misconceptions that were flying around the room.  I jotted down some of the conversations as the students came back to the large group.

We had around five minutes left in class to review the questions that I noted. I wrote the questions that I heard on the whiteboard.  I was able to clarify some of the responses and answer other questions. This time was definitely worthwhile. The students seemed to appreciate the time as well. During our next group activity I’d like to do something similar, but not completely rely on my less-than-stellar eavesdropping skills. Instead, I’m thinking of having the students periodically use Post-it notes to ask questions. This could turn into a “wonder wall” type of activity. The students could then place the questions in a bin and we can review them throughout the unit. I think this type activity is one way to proactively address misconceptions and answer questions as students grow in their mathematical understanding.

Bridging Procedural and Conceptual Understanding

Yesterday I was putting together a few math projects when a Tweet caught my eye. The Tweet below started a short conversation that I thought was interesting.

David’s Tweet had many responses.  Most responses revealed that educators tend to side with solving one problem ten different ways rather than having students solve ten similar problems.  I started to reflect on how teachers give assignments that ask students to complete repetitive problems that often reinforce procedural mathematical thinking.  I also started to think how in an effort to provide practice, teachers may focus on procedural aspects first and then move towards practical application.  I find this happens frequently with math concepts at the elementary level.  What I don’t find often is the viewpoint that practicing procedural aspects can be embedded in solving specific problems multiple ways.  This type of thinking reminds me of number collection boxes.

Regardless of the assignment I want to be able to give specific feedback.  A larger problem that involves multiple steps can provide opportunities for teachers to pinpoint where misconceptions are and give direct feedback.  This isn’t always possible with ten similar shorter problems.  Below is an example of a few problems that you may find in a fifth grade classroom.  I don’t condone using these types of problems as they are definitely utlized, but I think we need to ask what’s being assessed when students complete this type of problem?  Students are simply asked to find the volume and show a number model.  I appreciate how the problems ask students to show their number model, but these types of problems seem to measure procedural understanding.  Do students know the formula?  Yes, well then they can answer many of these problems, even 10 in a row.



I think the above problems have a place in the classroom, but shouldn’t necessarily be the norm.  Usually these types of problems are found on homework sheets.  The problem below which was adapted from a recent fifth grade test is more challenging, but gives students opportunities to showcase their own mathematical understanding and persevere.  Some would say that these two problems are completely different.  I would agree, but similar concepts are being assessed.  They do look different and the second requires more skills to complete.  Students need to be able to use their procedural understanding and apply it to the situation.  Also, one key element that’s missing from the first problem is the student explanation.  Students are required to show their mathematical thinking in the second problem.  This is big shift and can reveal student misconceptions more clearly than the first problem.  I struggled with the decision, but eventually had students work in groups to complete the problem below.  Students were allowed to use any of the tools in the classroom to find a solution.



At first, all groups struggled with this problem.  Near the end of class all the groups presented their findings.  What’s interesting is that all the groups had different answers and ways in which they came to their conclusions.  I was able to offer opportunities for students to see and ask questions about different math strategies.  During the next class I was able to pull each group and give feedback.  This activity took a good amount of time to complete, but I feel like it was worth the commitment.

Through this experience and others I’m continuing to find that it takes a “bridge” to connect the procedural and application pieces.  At times I feel like there’s an assumption that if students are able to answer 10 similar procedural problems that they will be able to simply apply that knowledge in a multi-step problem.  This isn’t always the case and sometimes the bridge doesn’t fully form immediately.  Performance tasks, similar to the problem above can be one way in which teachers can help the transition from procedural understanding to practical application.  Being able to apply that knowledge to a math performance task can be a challenge for some students.  When teachers focus so much on the procedural, that’s the only context that students see and practice.  A blend between procedural and application needs to be established within the classroom.  I feel like activities like this help bridge this gap.

How do you bridge mechanical and conceptual understanding?