Tag Archives: math teacher

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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Tear Down the Math Education Reform Wall

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Recent news about math education in the United States shows that the students in this nation are lagging behind other countries and this crisis is getting progressively worse. It is a story as old as I can remember. In fact, perhaps you can answer this question, class: When was the last time that U.S. math education was considered to be doing well? My research on this subject travels back to 1957 with the Sputnik launch. Of course, the launch indicated that we were not doing well since we were outdone by the Russians, which means math education was lagging even then. The “New Math” ushered in a new approach to fix this problem in education. New math was a flop because it was too rigorous for most students. Thus, the nation embarked on a decades long pursuit of a math education system that would make all students successful in math and get us back to the self-determined role of worldwide leaders in math and science education. Because we are apparently still failing to achieve that goal, I began to consider why this was the case. I have many educational opinions, and several of them are wrapped into my explanation for the continued and progressing poor performance followed by an alternative approach to deliver us from this dilemma.

How did we get here?

First of all, the most recent realm in which to determine our failure is international testing. We perform worse than expected time after time. However, there have been plenty of educated responses explaining how we are being unfairly compared to countries who are not playing the same testing game we are, such as countries that only test their elite students, countries who should not be compared because of homogeneity and economic diversity disparities, etc. The analysis is available to suggest that the results are not as abysmal as we are lead to believe.

Secondly, media and politicians thrive on the need to show failures and a need to repair. Without such news, there would be less flash to their reporting. Without such a political platform, there would be no need to change the political posts already filled by successful leaders. As such, those who have the power to influence public thinking thrust the concept of poor math education upon us every chance they get. I can admit that I have never heard a politician state how awesome math education is in this country. Likewise, any news I do read that is positive about the results of math education are usually localized or temporary, such as for this year’s test for 4th graders; this reporting is usually accompanied by other areas where the math results are poor, perhaps 8th grade results.

Thirdly, I believe the majority of the push in math education since I became a secondary math teacher in 1995 has been towards dumbing down math education, removing “drill and kill,” making math accessible for all students, changing the focus from math as a “right or wrong” proposition into a purely conceptual thinking process, and steering away from the fact-based, skill-driven instruction towards a cooperative, discussion-style, discovery learning process. One reason for this change, perhaps, is that the best mathematical thinkers usually do not pursue careers in education; that is not to say that no great mathematicians become educators. Instead, many prefer to pursue more lucrative careers or opportunities that  provide a greater sphere of influence than the often distasteful educational universe. Without a significant presence of math professionals, the greater power in education tends to be held by those who are more likely to have struggled with their own math education. With a majority of non-math professionals controlling the curricula and instruction for math education, the prevalent push is for more and more approaches to math education that skew away from pure math instruction. Instead of accepting math for its position in the wider educational picture, these reformers who shy from traditional math try to make it fluffy/fuzzy or disguise the necessary rigor of math.

Fourthly, the more prevalent these non-math approaches to instruction have become, the worse the nation’s performance has become. With a poor instructional approach over decades, the teachers of our students are developing and presenting these poorer offerings, especially since they are the product of this system. The more traditional math teachers who present more traditional math instruction are attacked consistently and pointed out as mazethe problem, though the more prevalent alternative math education has been present long enough to have significantly impacted math education. At this point, the myriad of alternative approaches to math education (attempts to fix a broken system) have pervaded our culture for more than half a century and have created a maze of confusion. It follows that the alternative approaches to math education have failed to produce the changes constantly pursued.

How do we progress from here?

I am better at math than you! I have always been better at math than you and will always be. Of course, this is not targeted for all other people, but for approximately 90% of the rest of this nation, these statements are true. I am a high school math teacher who excelled in math classes from elementary through college. Although I have a tendency towards conceit, the information I am reporting here is arguably factual. Although I did not know when I was five that I would be a math teacher one day, I did know that I was very good at math and enjoyed it. With all of this being said, I am going to present a theory that will not be politically correct.

The “right” thing to do these days seems to be to tell every young child that they can be great at math. Some students have high levels of math aptitude and interest and could excel in math following a rigorous education program advancing considerably faster than is available generally. However, some students do not have a natural affinity for math nor natural talent. In today’s society, it has been determined that we must design a system of education for these students promoting the ideas that they can do math, should want to do math, and should enjoy math. If everyone would simply love math, everyone would be great at math, and we would dominate the world in the fields of cognition, education, and economics. The main complication with this philosophy is that our society values freedom of choice above education. Thusly, the dual-edged sword not A Nation at Risk - Averageonly forces those students who would prefer to do less work with abstract, rigorous mathematics to actually invest in mathematics more deeply than desired, but it also asserts to those who would be inclined to excel in mathematics and pursue advanced mathematical studies that anyone can do mathematics, thereby minimizing their special relationship with mathematics; at the same time that the curricula are being watered down for the most likely to succeed in and pursue mathematical endeavors, there is little benefit for the reformers’ “liberal arts” approach to mathematics for those students who are more likely to avoid mathematical studies as they age when they are given more choice in their coursework. The result of these efforts is mediocrity! This matches a criticism levied back in 1983 with one of the most famous calls to action, A Nation at Risk: “We talk a good fight about wanting to have excellent schools when in fact we’re content to have average ones.”

Ultimately, I believe that much of the reform in math education is catering to the least common denominator while hoping that the best of the best can still rise to the top. In the long run, as evidenced by the reformers own chastisement, the alternative approaches to mathematics education are failing to produce the desired results. I propose a different approach. I suggest that we institute a much greater level of rigor in the lowest grades with the purpose of discovering the divergent populations of students distinguished by comparative natural talent and comparative natural interest. In order to accomplish this, two major changes need to occur. Primarily, we need to place teachers in the lowest grades who are math specialists with high math aptitude and possibly some mathematical emphasis in their college work or professional development. Secondarily, we need to raise the amount of time spent with mathematics in those early grades. I have considered the disparity in time spent with English/Language Arts versus math activities, especially in the lower grades and believe that the lesser importance for math is a key challenge to successful math education throughout the K-12 system.

With these changes, we would be able to identify students with mathematical strengths and weaknesses. For those students who show little interest and/or ability, we move them along with the gentler, reform movement approach, maintaining high levels of expectation. These students may be placed on a path wherein Algebra 1 is taken in 10th grade. But, for those students who show greater interest and/or ability, we move them along with a more international, challenging approach. For these students, seventh grade ought to be the target for taking Algebra 1. Young children who enjoy mathematics will enjoy being pushed to excel, while those who prefer the myriad of options other than mathematics will enjoy a more compatible avenue. Especially because one size clearly does not fit all, this approach to mathematics education has the feel of honoring individuals rather than expecting a robotic product at the end of our assembly line school system. I feel as though these divergent paths to successful math education also addresses the psychoemotional needs of our students, which can be a significant factor in improving learning.

In the end, mathematics education reformers are consistently building walls that try to separate traditional from alternative practices and quite possibly teachers and students from the goal of greater math achievement. At the same time, students from all achievement levels are building walls of apathy and disinterest towards math instruction around SuccessStairs-400x250themselves.  Teachers, caught in the middle, help build all of these walls, attempting to appease all participants in the system, but generally satisfying no one. It is time to break down these walls and reuse the building materials to erect stairs of success for all students. This can be accomplished, ought to be considered, and should be implemented immediately.