Math Error Analysis

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

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

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

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

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

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

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

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Students also mixed up factors and multiples

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

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

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

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

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

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



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

 

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Student Reflections and Assessments

Reflecting

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.

Student Surveys and the Reflection Process

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Yesterday was the last day of the school year for my students.  The end of the school year tends to be filled with excitement and pride as students transition from one grade to another. During this time of the year I usually give my students a feedback survey. I tell the classes that I’ll be using the information to change next year’s classes for the better.  I’ve been using this method for the past few years and find it valuable in preparing for the fall.  Most of the questions that I ask tend to stay the same while I add a few others depending on what I’m focusing in on for the year.  This year I asked a few questions related to feedback and student refections.  These particular questions stem from some of the district’s initiatives, as we’re emphasizing Hattie and Dweck’s research.  Next year we will be focusing on them even more and I believe they’ll be part of a formal walk through process.  So I gave the survey to 50 3 – 5th graders and collected the data.  The survey that I used can be accessed here.

I took the 50 students responses and had Excel calculate the averages for all of the questions. Below are few highlights from the feedback and reflection questions.  I used a 1 – 10 rating, with 1 being all the time and 10 being never.

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My takeaways:

I have to keep in mind that elementary students are taking this survey.  It’s valuable, but I feel like a third grader will perceive a question possibly different than a fifth grader.  Regardless, the data is valuable in my mind.  I looked over the numbers and shared this information with another class.  After showing the data we had a great conversation about reflecting on our learning.  Our conversation looked at the connection between allowing reflection opportunities and how they impact our learning.  We started connecting parts of the survey as a cause/effect scenario.  The conversation wasn’t too deep, but worthwhile as students made connections.  We decided that reflecting on our learning can be impactful, but not necessarily help a person understand a particular concept.  Feedback, reflection and opportunities to take action need to all be place. What seemed to be lacking this year were opportunities for students to reflect AND take action based on that reflection.  It’s important to reflect, but without any action or change in perception the act might not be reaching its full potential.  I decided to write an informal flow chart indicating the process that the classes tended to use.

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I told the students that one of my homework assignments over the summer is to provide ways to make student reflection opportunities more efficient.  This is something I’ll be revisiting in the fall with my new classes.

Reflecting on Effort

Last Monday my school started its third trimester marking period. As this new trimester begins students were given time to reflect on the past trimester. While the students brainstormed what to write I gave each one their personal file.

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For as long as I can remember teachers at my school have kept a file for every student in their class. This all-important file holds paper assessments, report cards and anecdotal notes taken throughout the year.  This file is also what’s usually laid out during parent teacher conferences.

To the students surprise, I gave each one their own file for the reflection opportunity.  Prio to handing out the files I made sure there wasn’t anything confidential in the files.  Students were then asked to analyze all of their assessments and reflect on the second trimester. Students paged through their assessments and journal entries and filled out the sheet below.

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I set aside about 30 minutes for students to look through their personal file and write their response. I wanted the students to analyze their own effort level. It’s interesting how students took on an ownership role as they took the file.  They took this role seriously.

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Some of the students took the entire amount of time while others needed more. When students finished they brought up their file and journal to discuss their views with me.  I  had a brief conversation with the students about their reflection and asked them questions related to their effort level.  The students and I discussed how the statement below applies to what they produce in class.

Effective Effor

Although there’s room for improvement, I feel like the class is making positive strides in being able to reflect on experiences without solely looking at the grade.  During one brief conference I asked a student whether they felt like effort in math class eventually leads to achievement.  The student responded, “Not completely, but effort level impacts my overall grade.”  Sometimes I find this to be a perception battle of grades/points vs learning experiences.  Providing students opportunities to reflect can help balance this perception.

Student Self-Reflections

Reflection
photo credit: karola riegler photography via photopin cc

Over the past few years my teaching practice has evolved.  Growth in the teaching profession often occurs through experience and professional development.  As continuous learners, teachers generally hone in on their craft over time.  I believe reflecting on teaching experiences plays a role in the professional growth of an educator.

  • How often are teachers able to reflect on their craft?

I’d hope that it would be more often than not at all.  Personally, reflecting on past experiences can lead to better decision making and goal setting in the future.  They’re many ways in which educators can reflect.  Off the top of my head I can think of:  after a professional development session, reading or commenting on a blog post, participating in an education twitter chat, attending workshops, and many more.

  • If educators feel that reflecting on experiences is important, why not give students opportunities to reflect on their progress?

Absolutely.  One way in which reflection has been beneficial in my classroom is actually rooted in the formative assessment process.  Local formative assessments give quality information that can be used to drive instruction in the classroom, while other data (standardized assessments) are used for district/state/nation purposes.  Formative assessment data not only serves the teacher, but it also informs students of areas of strengths and concerns.  Last year I decided to have my students use a reflection journal to analyze their own achievement levels in class.  Students reviewed their formative assessments, usually in the form of exit cards, and wrote a short paragraph regarding how they performed.  I asked the students to write a few sentences related to how close they are in understanding the concepts observed on the exit card.  Every so often, generally after a grading period, students were guided to setting individual goals for themselves. These goals were based on the journal entries and learning experiences throughout the grading period.  This process required modeling during the introduction phase, but after two grading periods the students were ready to complete this independently.

I vaguely remember using journals during my K-12 experience.  The teachers that assigned the journal entries rarely wrote any comments back to me.  This peeved me as a student and I’m over it still does as an educator.  Therefore, I make a conscious attempt to review all the student reflection journals and write short individualized comments to the students.  The comments show the students that their teacher is aware and cares about their progress.  This action is especially important to students that might not be as assertive in class or might be embarrassed to state how they truly feel.  I place an emphasis on the student created goal. Student goals are highlighted  as I will often share them with the parents to ensure that we’re all working towards the same end goal.

I also find that the student reflection journals show student growth on a personal level.  When growth is evident, students often gain confidence in setting new goals.  Reflecting on progress made can be a tremendous opportunity to set goals.  These goals can empower students to own their learning.

Side note:

 Reflections can take on many different forms.  Incorporating various prompts throughout the entire school year also communicates to the students that goals don’t have to be directly associated with scores.  In the past I’ve used field trips, current events, literature, and problem based learning activities for reflection journal prompts.  

* Feel free to visit Helen Barret’s reflection for learning site for more information on this topic.