Category Archives: student performance

Showing Work for Mathematics

Earlier this year, my sister-in-law asked if I could help her son with his 2nd grade math as his grades had gone down as of late. I agreed and at the next opportune gathering looked things over. He had received an 85 and a 75 on multiple choice tests, but I could not discover his mistakes. In fact, I was certain that he had actually not missed any of the problems that were marked wrong. I consulted with my nephew and discovered that although he was getting correct answers, he was having those answers marked wrong because he was not showing work. I was infuriated and decided to do a little research, have a good conversation with my sister-in-law as to how this should be approached, and, of course, write this blog.

My initial problem was with the fact that the parents and child did not really have a firm grasp on why he was not doing as well on these tests, which meant poor communication from the teacher. The next issue I had concerned the expectation to show work for a multiple choice test, which seems bizarre in my view. Lastly, I was puzzled by the necessity to show work in order to earn credit. Although the first two are disconcerting and creep into my thinking, I chose the third to consider more deeply.

My initial review of internet information revealed that there were some debatable issues about showing work in general and, specifically, the requirement to show work. Some points in favor of showing work were: the process and thinking in mathematics is a vital concern for true mathematical learning and understanding; with work, teachers can find mistakes and suggest corrections; and showing work may help students avoid mistakes as it slows down the solving process for those who rush through their work. Some points of contention were: showing work discounts the value of intuition in math; the “process” for some students may be that they “did it in their heads;” prescribed solving processes may be detrimental to some students, in particular those who are advanced; and slowing down the process can lead to confusion. There really is a great deal going on with these issues which is worthy of contemplation.

The first issue that I have with the showing work requirement is that everyone learns and demonstrates they have acquired math knowledge in different ways, at different paces, and to varying degrees of depth. When one mandates a single process for learning or showing evidence of understanding, this idea of variety is devalued. I can understand the need to know that a student is not cheating or guessing when new skills are introduced, but I do not believe this is a satisfactory motivation in requiring work to be shown, especially since students can copy work as well as answers.

In my nephew’s second-grade work, he was being asked to show work for addition of two-digit numbers. Certainly, one could employ a myriad of methods to calculate these relatively simple arithmetic computations, such as: using the standard algorithm, counting on one’s fingers, using manipulatives, calculating in ones head, drawing pictures, using a calculator, using a computer, or asking a friend. Who is to say which of these is the best, most appropriate, or only acceptable method? Clearly, some of these methods would be unacceptable in an educational setting, but adults use many of these ways to add two-digit numbers regularly. As there are, in reality, various methods of arriving at an answer, why should a student be punished for doing the work in his head, or for example, not drawing number blocks representing the problem and answer? Furthermore, how would a teacher determine which method is the correct method for all students to use in order to display their understanding of the solving process? Because there are many ways to arrive at answers, many justifications ought to be acceptable, providing the best learning opportunity for each student. Additionally, there should be no punishment for choosing whichever legitimate method works best for the individual student. Lastly, if doing the work without showing anything is the best method for a student, that method should be honored not disparaged, rewarded rather than punished.

I fear that the purpose of this controversy for some teachers may be that they are unable to comprehend various methods of solving or that they possibly cannot do the work in their own heads; worse, some teachers may not comprehend how one might do the work in one’s head. Simply because the math teacher needs to show work to do certain problems does not necessarily mean the students do also. I wonder if teachers who require work for all students understand that showing work  in whichever fashion they deem the “right way” could actually be part of the roadblock to students understanding math well. Additionally, I would be concerned if they truly know their twenty students well enough to know which methods help each to be most successful. Assuming the best rather than dwelling on the worst, however, leads me to a continued exploration of the issue.

While there are varying levels of successful solving, most people skip steps when completing math problems. Upon mastery, for example, arithmetic generally becomes memorized information and does not require that steps be shown, pictures be drawn, or fingers be counted. Although showing work when acquiring new skills is most likely appropriate, as students escape the need to show work, there no longer should be that same requirement. For some students, mastery may occur after showing their work a single time. In fact, the ultimate goal for students is typically to accurately perform arithmetic without showing any work. As mathematics becomes more complicated, beyond arithmetic, students will require varying numbers of steps or representations over varying amounts of time in order to develop mastery; sometimes, no steps are required at these higher levels of complexity. Furthermore, showing work may limit creativity as students could be showing work the one and only way they were taught rather than truly exploring mathematics; showing work could be so time consuming as to limit deeper analysis of mathematical ideas. Lastly, showing work is a process of explaining how one arrived at an answer, which is similar to teaching; all students do not necessarily make adequate teachers.

My last issue concerns the grading aspect of showing work. If showing work is part of the grading process, it is critical that this requirement is clearly communicated, relevant, important, and necessary; without all of those characteristics, grades should not require the showing of work. If a teachers are requiring students to show work simply because they are concerned about cheating, some other options may be to develop multiple versions of questions, make problems that have unique solutions for each student, monitor the class better, or create an environment less conducive to cheating. Also, would paragraph or short answers not be suited better for evaluating understanding than multiple choice? In the end, my concerns with requiring students to show work for credit all boil down to whether the teacher invested significant effort in analyzing this grading requirement, intellectually determined its value for each assignment, and considered alternate methods for evaluating student understanding.

As with many issues in math education, my interest is in determining whether we are helping or hurting students and discovering where others’ beliefs fall on the spectrum, but ultimately I have little authority to make impactful changes across the board. I write this blog, I discuss the issue with my sister-in-law and others, and I hope that others will invest the time to question and seek answers along with us. I am pretty satisfied that I believe the blanket teaching practice of showing work for all students on all problems on every assignment is antieducational and, importantly, detrimental to the mathematical advancement of our brightest math students.

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Student Performance Determines Teacher Success – Fair or Not?

The primary difficulty I have with student performance determining teacher performance is that natural ability seems to play a role in this phenomenon. My experience as a teacher for sixteen years has shown me a few things.

  • I taught 8th graders in an affluent district with kids who had always been successful on our state tests. Parents ran the school with an administration that let kids get away with almost anything for fear of parent interference, the principal was fired (actually promoted to the district offices), and the students did not seem to learn as well as I would have liked. I worked hard that year trying to get 8th grade math into their brains, but did not feel as successful as I had in the past or in the future with that endeavor. In the end, almost all of my students passed the state test that year.
  • I taught Pre-AP Algebra 2 students for four years. I had virtually 100% pass the state tests year after year.
  • I have taught in a dropout recovery program for the last six years. The state test has become easier, and I have become better at working with the students over the years. While there are years when there are great successes and times when things do not go as well, we are still able to get about 80% of the students to eventually pass the state tests. These are mostly kids who would have skipped the state tests at their home campus or failed, some with a record of failing the state math test every single year since they started taking the tests (usually third grade). My feeling is that every kid that passes should be a celebration, but if I happen to run into a semester with a 50% failing rate (which will eventually become 80% or so), that could be devastating to my performance review.

The second issue I have stems from the preceding information. Why would I want to work with the most challenging students to find some success when I could simply work with the best kids in the best districts and cruise through state testing results regardless of how much education was going on? Teachers will fight for the best kids in the best districts. Teachers will fight to get rid of kids on their rosters who have shown a lack of success over the years. Teachers will lie, cheat, and steal to give the appearance of student learning through a state test, especially in lower grades where science and social studies are not tested, for example. How does the fifth grade teacher who is responsible for kids passing a science test for the first time deal with the fact that the third and fourth grade teachers did not teach science to focus on math, which was being tested? Why would I share my teaching strategies that have shown success with my peers (competitors) because my successes will improve my chances of getting the better classes and not helping the other teachers will help weed them out? Why would I help a new teacher who is essentially trying to get my job when I have a track record of success with the GT kids?

Ultimately the only fair way to assess teachers’ effectiveness through student learning is to be able to determine exactly what each student has learned in the past, determine exactly what knowledge and skills have been added purely from that teacher’s efforts, and compare each students potential to the realization of that potential each year. I am pretty sure none of that is possible, let alone through a mostly multiple choice state test given on one day of the year.