Making Mistakes in Mathematics

Action Research Paper

Summer 2008

Lindsay DeVeny

Action Research

Phase I-Problem Identification

I teach Algebra and Geometry at the high school level; therefore, I have a mixture of freshmen, sophomores, and juniors in every class.  While their needs and the way I structure my class depends slightly on what the majority grade level is in each period, for the most part teaching mathematics, in my opinion, involves three major things: knowing your areas of strength and weakness, lots of spiraled practice, and a willingness to try at all times.  It is that last one, a willingness to try at all times, that most students struggle with because they seem to come into high school with the idea that it is better not to try at all, then to try  and fail.  Therefore, my research question stems form that idea: How can I teach my students to realize that making mistakes is a part of the learning process?  In other words, I want students to internalize and then apply this statement to their work: “I should not be afraid to make mistakes because I cannot learn without them”.  Once a student is able to recognize a mistake, and then start to recognize a pattern in the mistakes they are making they can begin to internalize their areas of strength and weakness and then begin to build confidence in their strengths, and start to work on their weaknesses, and thus start to see improvement in their work.

Phase II-Plan of Action

While the idea of first making mistakes and then learning from them is an on going process throughout the entire year there are certain strategies that will need to be put into place from day one.  Below is a sample timeline of events that need to take place within the first Unit and then continue to take place during each subsequent unit.

• Day 1: Give each student a difficult maze to complete, or a game with no instructions, and allow them to play with it for awhile.  Discuss as a class the idea that no one got it “right” the first time and yet they were willing to try again and again to figure it out.  Relate this to mathematics in that making a mistake is needed so long as we learn from it.  For example, once you hit a dead end in the maze you wouldn’t go that way again, you would try a new path, thus showing you learned from your mistake.
• After the first exit ticket is returned ask students to match their answers with the answer key provided and highlight anything they seem to be missing.  Push those who seem ready to verbalize what their mistake was but writing it next to where they highlighted.  Remind students that making mistakes is okay, we want to see them so we can learn from them; therefore, not trying (a blank paper) is not helpful because there is nothing to learn from it.
• During the unit have students practice identifying mistakes in others work: either teacher given answers or other student’s.  Get them used to circling the mistake in a different color pen and writing in words what that mistake was.
• After the first quiz, have them do quiz corrections in which they will circle their mistakes and identify in words what their mistake was.  Then ask them to identify types of problems they are good at and types of problems they need to work on.  Also push them to look for patterns in their mistakes; for example, if they made five mistakes all involving adding integers incorrectly then we know this is something they need to come in and work on.
• Continue to encourage mistakes, and learning from them, the entire year.
• Continue to use test/quiz corrections and identification of areas of strength/weakness.
• Encourage students to work on their weaknesses and praise them when they start to show improvement, thus connecting practice with improvement.

Phase III-Data Collection

On the first day of class students will be placed into pairs and given various statements to place into either TRUE or FLASE statements based on their previous beliefs about mathematics.  Two of these statements are “Making a mistake in math is a good thing” and “Knowing my weaknesses will help me improve”.  As the teacher, I will observe how many student pairs place these statements into the FALSE pile.  Later in class I will place an overhead up in which I have placed the statements into their correct piles according to the way my classroom is ran and we will discuss those that seem out of place, assuming that this way of learning will be new to most students who have not had me before.  It will also give me an opportunity to see from those students who had me last year if they place those statements in the TRUE pile.  I started this “Mistakes is a part of learning” half way through the year least year, so it will be interesting to see if the idea got through to those students whom I taught last year, assuming I have some of them again.  Other types of data collection will include their exit tickets, test/quiz corrections, and their ability to accurately determine which areas they are good in and which they need more practice.  I will be keeping student portfolios, something I did not do last year, and use them as a tool to help students try and find patterns in their areas of strength and weaknesses.  I will also be able to observe the atmosphere of the classroom: do they seem to be okay with making mistakes or would they still rather not try?  Are they getting better at identifying others mistakes when they are working together or are they still depending on me to find it for them?  Are they getting better at verbalizing what their mistakes were?

Phase IV-Analysis of Data

As I look for evidence from the previously mentioned questions I can begin to determine if the strategy is working for all, most, or only some of the students.  However, a huge part of this strategy is the students being able to identify their mistakes themselves.  Therefore, their portfolios will be a major part of my data analysis and I will need to continue to look for tends throughout the entire school year.  I will be looking for improvement in leaving answers blank to actually trying something.  I will be looking for improvement in identifying mistakes by circling multiple lines to hopefully being able to circle just the mistake.  I will also be looking for a difference in their vocabulary when asked to write about their mistake; I want to push them to change from “I got 4 + -6 wrong” to “I did not add my integers correctly”.  I will also be looking to see if this process does or does not seem to show improvement in the types of mistakes certain students are making.  If a student in September is making the same mistakes all year long without any improvement then this did not work.  What I am hoping to see is improvement of mistakes over time, either less of them, or different types of them as the units change.  I will also be looking at a students ability to correctly identify their strengths and weaknesses, and along with that, a willingness to admit and then work on their weaknesses (during after school tutoring or practicing on their own at home) so that they can then see improvement over time.  I also want to find a way in which the students might be able to analyze their own data and make connections between “I practiced this and I improved” and “I was good at this, did not look at it for awhile, and now I forgot it”.  These would be powerful connections for the students to begin to make on their own especially when it came to studying for tests because then they would be able to identify what they actually need to study rather then wasting their time studying what they already know and then feeling like studying does not help them to pass a test.

Phase V-Plan for Future Action

If this action research confirms what I believe it will, allowing students to make mistakes and showing them how to learn from those mistakes will improve student performance, I will need to, as a teacher, change my attitude towards mistakes as well.  For example, if a student makes a mistake on the board in front of the class I will still praise that effort and then have another student try, praising the “right” student no more than the “wrong” student, and then as a class we will identify the mistake and talk about ways to learn from it.  I will treat mistakes as teachable moments rather than allow them to put a student’s self esteem down.  If the action research goes well I would be writing about how to view mistakes as a positive thing, as long as students are taught how to learn from them.  I would also be writing about different techniques used to show that mistakes are positives: games, test/quiz corrections, frequency of mistakes analysis, and analysis of strengths and weaknesses.  Finally, I would be making the argument that mathematics teachers should be more concerned with teaching students to identify and learn from mistakes rather than getting the correct answer all the time.

Goals and Vision Setting

Classroom Climate Goals

• I return all exit tickets, tests, quizzes, and reflections the next day with a grade on it.
• I use a standards based grading system that encourages real learning over a long period of time.
• I only use a student’s most recent demonstration of knowledge to grade them.
• I have students track their levels of progress all throughout the year.
• I use only positive recognitions within my classroom.
• I get 100% attention when I give my cue.

Vision Statement

I believe that it takes different students a different amount of time to master new material; therefore, students are only graded on their most recent demonstration of knowledge.  This encourages students to always try because a “bad” grade at the start of a unit can be replaced by a “good” grade at the end of a unit; it also encourages them to learn from their mistakes.

Technical Management Plan

Last year when I wanted my student’s attention I typically just stood in the front of the room and said “All eyes on me”, and then waited, for as long as it took, for everyone to stop talking.  While it never took more than a minute for everyone to stop talking I know that it was not the most effective cue.  I have thought a lot about what I want to use as a cue this year, and since I will be teaching only freshman, I am going to use the clap once, clap twice strategy I saw one of my professors demonstrate at CSULB.  The reason I like this strategy is because I found that it was almost impossible to continue talking while I was trying to duplicate the teacher’s clapping pattern.

To implement this strategy I plan to have them work in pairs on the first day of school.  When I decide that they have had enough time I will say “If you can hear me clap once, (then I will clap my hands)…if you can hear me clap twice (again I will also clap my hands)…clap once (again clapping with them), clap twice (clapping with them).”  I will then inform my students that this is what I will do when I want their attention and we will practice it several times that first day.  What I know I will need to work on is expecting 100% attention after the cue and making that expectation clear.

During transitions I use a timer.  I state that they have so much time to do something: clean up, get into groups, find a partner, work some problems, etc.  I will then give a few warning times, for example saying “You have 30 seconds”, finally I count down the last 10 to five seconds.  I never give more than one minute for transitions.  This worked extremely well last year and I plan to continue to use it this year.  I, again, just want to improve on getting the students to understand that I expect 100% of them to be ready when that timer goes off.  I typically introduce this technique on the second day of class during the first transition.

Homework in my class is called “suggested practice”.  It is always assigned, but never collected.  It takes awhile, but I teach my students that in order to improve/maintain their levels of understanding they must be practicing the material, past and present.  When their levels start to go down I ask them how much of the suggested practice they have been doing.  These types of questions are always given in a written reflection form, and students begin to see the connection between practice and understanding.  I always have the solutions to the suggested practice worked out and they are welcome to come in after school or possibly during certain times in class, to compare their answers to mine and look for mistakes on the ones that they missed.  I found this policy the best for my classroom because too many students were not doing their assigned homework and I needed to come up with a way for them to see that homework would help their grade, rather than seeing it as something they either choose to do or not to do.

Motivation, Expectations, Emotional Climate

When it comes to motivating students I often feel there is only one thing I know so far: no one strategy works for every student; that being said I have tried to develop a motivational strategy in which students are only trying to better themselves and are not concerned with what others in the class are doing.  I want my students to understand that not everyone learns things at the same pace and that is okay.  If in September I have one student who seems to “get everything” and another who feels lost that is okay.  I have my students from day one think about learning in levels.  I break down what I want them to learn into learning targets.  Learning targets are statements such as “I can solve a one-step equation” or “I can translate a verbal statement into an equation and solve it”.  These learning targets are the expectations for the class; they are expected to master all the learning targets by the end of the year.

I constantly ask them to determine what level they feel they are 4 (I totally got this!), 3 (I understand what to do but maybe make a few simple mistakes), 2 (I depend on my notes/someone else to help me with this), or 1 (I don’t know how to start this).  As the year progresses I not only ask what level they feel they are at but I also start asking how they think they can get to the next level or how they think they can maintain their current level.  I constantly spiral my material so that come December they are still expected to know how to do everything we have covered since September.

I have found that when students are tracking their levels they begin to make connections to their behavior and the progression of their levels.  I then work with students on understanding that what I want to see is improvement.  If a student has levels of 1’s and 2’s before a test I try to get them to understand that they should pick a few learning targets to focus on and we then together analyze how they “focused” on improving those levels and if their strategy did or did not work and why.  As the year goes on students begin to find study strategies that work for them and when they see those levels go up they are motivated to try it again.

At least once a week my students are given a choice as to which learning targets they want to practice.  As a class we talk about ways to determine which learning target(s) to practice such as; if they have a level 3 maybe they work on that learning target for 30 minutes and then move onto a learning target they have a 1 or 2 on and stay there as long as they need to.  This strategy allows for differentiation among my students.  There are always stations designed for specific learning targets as well as two others: a station for those who feel they are a level 4 on everything, and a station for those who need basic skills practice before they can master a learning target.  I make it very clear that the stations do not mean anything; they have no “smart/slow person” label to them, each station is simply designed to help individual students with what they need help on.

This type of motivational system depends on student’s intrinsic motivation/desire to want to learn/have success within a classroom.  It also works with my “mistakes are a part of learning” strategy because mistakes will happen, no one is perfect, and it is how students learn from those mistakes that matter.  For example, if a student is staying at a level 2 even though they say they are “working really hard” on that learning target an analysis of their progress needs to be made.  I direct the student to match their solutions with the solution key and circle their mistakes.  I then ask the student to try and find a pattern in these mistakes; thus, if the student is constantly making a mistake when fractions are involved then I direct them to the skills station on adding fractions.  Or if a student claims they “studied” and still stayed at a level 2 I ask them to describe in detail how they studied and for how long they studied, thus, we can begin to determine why this study method is not working for them.

Studying is a huge motivational issue when it comes to mathematics.  Students tend to say that they studied “really hard” and yet still fail.  What I have found is that one of two things is happening there: either the student is saying they studied because they know that is what they are supposed to say and did not actually do it, or the student does not know how/what to study.  Once we deal with these issues and the student begins to see their hard work paying off they do become more and more motivated to improve.

Whole Class level Goals and Strategies

In order to promote classroom bonds with my students we do various teambuilding activities through the year, always making the point that if we do not all succeed, then we have failed.  My favorite of these activities is the one in which each member of the class is given a puzzle piece and then as a class they are to piece the puzzle together.  Also, since my classroom is structured to have a lot of partner/group interactions we do fish bowl activities in which when I see/hear members of the class working well together I either have the class circle around and watch, thus the term fish bowl, or I write on the board things I hear students saying that I like (without their names), thus they learn the type of language I want to hear them using.  I also give students many chances to speak with each other through the use of think-pair-shares and I allow them as many choices as possible.  They can pick their partners, pick the learning target they want to work on, they can pick their seats, etc. but at all times my expectations for their behavior does not change; they work from bell to bell period.  When students fail to meet that expectation I have various strategies I use to deal with them.

The most common problems I have to deal with are students working on other class work while in my room, students talking/off task, and students doodling.  When I see a student working on other class work/writing a note, I simply walk over to them, making it a point to get fairly close (invading their personal space), tell them they have not made a good choice, ask them for whatever it is they are working on, and then tear it right down the middle and throw it away.  I then speak to them after class about what happened and how it can be avoided in the future.  When a student is off task/talking I again walk over to them, invade their personal space, inform them that they made a bad decision and move both of their seats, not just one.  I then talk to them after class.  When I have a student that is a doodler, I make an agreement with them; they are free to doodle as long as they have completed their work first.

When a situation with a student becomes difficult I ask them to step outside.  I continue with the class until I can give them problems to work on or get them into their stations and I then go out and talk with that student.  I usually ask why they did what they did and I give them a choice: they may reenter the room and work as expected, or they may chose to complete the work in other room (either way they are still expected to do the assignment).  If it is a truly bad situation, possibly something like a fight, I send them directly to the office and talk with them afterwards, although this rarely happens.

If a problem with a student happens on a continual basis I first try to talk with them and see if I can determine what is going on.  I then speak with their other teachers, including their advisory teacher, to see if it is only within my classroom or if similar things are happening in other classes as well.  From there, we either try to deal with it as teachers and students, but if that does not work then we have a SST (student success team) meeting that includes the parents and a follow up plan for dealing with the behavior.

However, when it comes to student behavior within my classroom I am a firm believer in handling it myself.  I feel that the second I send the student away I have sent a message that I have given up on them, and I do not want my students to feel that way.  This is why I phrase everything within the terms of choices; students who step outside choose to come back or go to another room, students choose their seats and have to deal with that choice if it goes bad, students choose to do work they know they are not do to.  I try to instill in them that school, and life, is all about the choices they make, and the consequences that follow from those choices and that the best thing to do is recognize when they have made a mistake and learn from it.