Tag Archives: math

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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What Happened to Fran?

tornadoFran’s voice rang out like a gunshot amidst a large, crowded room. With the turmoil surrounding Texas math education and the plethora of data I was collecting in this focus group interview with six Algebra 2 teachers, I was suffering from information overload. At this early stage in my data collection process for my doctoral dissertation, every new statement added to the tornado of statistics, facts, and emotions swirling in my neophyte brain.

I was fighting to maintain my professional demeanor; meanwhile, I was giddy as these teachers not only wanted to participate in my study, but they had a lot of great quotes and thoughts about my research concerns. But then, like the turning point in a great thriller, Fran responded, and I sensed the chills scaling my spine as I realized, “This just got real.”

In attempting to discover the professional opinions of Algebra 2 teachers concerning the changing math landscape, my interview questions encompassed the Algebra-for-All movement, college versus workplace preparation, tracking, the 4×4 Recommended High School Plan (RHSP), graduation rates, and the impact on individual teachers’ teaching environments. Concerning the impact on teachers of the policy positions of the state of Texas, I asked about the panel’s desire to continue teaching high school mathematics. Their answers were contemplative but measured with a determination to remain in the profession despite the enormous challenges and silent agreement in spite of the policymakers. Fran dissented, however, as she spoke honestly with piercing, young eyes stopping and starting, “Yes… The things that we’ve talked about have… me wavering on whether I do want to teach… high school math anymore.”

I had already shifted my young research mind from wanting to prove my point as the impetus for getting a doctorate to wanting to truly find out what teachers believed. Now, my purpose shifted again from wanting to discover teachers’ opinions to needing to tell their story. I realized at that point that the voices of the teachers I would come into contact with along the way were significantly unknown, and their yearning to be heard was often emotionally overpowering. Ultimately, at least I heard their voices; my belief is that their professional opinions should matter to policy makers.

 Interacting with ten teachers at two separate high schools in two focus group interviews; ninety-one respondents to a lengthy online questionnaire; and three individuals during in-depth interviews, I discovered that these Algebra 2 teachers were optimistic about the potential impact of Algebra 2 on all students but were pessimistic regarding the realities of Texas’s expectations for all students. The teachers revealed a number of interesting opinions: graduation rates would probably be negatively affected by graduation and math requirements; Algebra for all was unlikely to be successful because students were generally unprepared for Algebra 1, let alone Algebra 2, and this level of mathematics is overwhelming for many students; honest assessment reveals that all students will not be going to college; high schools ought to work harder at developing alternative paths to graduation for children; requirements involving Algebra 2 need to be reevaluated; a one-size-fits-all approach is doomed to fail; the RHSP is not having the expected positive impact on students or education; tracking is valuable and should be expanded for mathematics while being purposefully monitored to emphasize and maximize success; recent changes were not improving student learning or opportunities for postsecondary endeavors: and, lastly,  more than one-third of the participants had a lessened desire to teach math.

In the end, the doctoral study process was powerful and enlightening. I found that consensus on most issues is difficult to achieve, but the Algebra 2 teachers in my study were passionate and informed members of the educational community who felt that their input was seriously undervalued by decision makers. I am hopeful that I am able to get some of their sentiments into the ears of governmental leaders, which may lead to positive social changes. I have received a lot of great feedback so far from those I have communicated my results to, with one explosively loud exception; when I e-mailed the executive summary of my dissertation to Fran early the following school year, the e-mail was returned with a delivery failure indicating she no longer worked in her previous position. I wonder if Fran will be an example or a trend.question mark

Full dissertation (Teachers’ Perceptions of State Decision-Making Processes for Mathematics Curricula by Brett Bothwell, Ed. D.) can be found in online databases or at http://gradworks.umi.com/35/44/3544187.html.

Potential Research

So, now that I have secured my Ed.D., I may have an opportunity to reach out into the world and do a little research. I can certainly say that I have many topics of interest, however. Which of the following do you see as being more interesting or having more potential for development through research?

Classroom management.

VESTED.

Elementary math education.

High school math.

Algebra 2.

Mathematics teacher preparation.

Teacher preparation.

Classroom assessments.

State assessments.

Educational policy.

Charter schools.

Education finance.

Chess.

Mathematical games.

Curriculum development.

Math curriculum development (secondary and/or elementary).

I am sure I have more, but these popped into my head as I was writing this.

Is technology the answer to educational woes?

Who does not love a new piece of technology in their hands? This is true especially when that technology is specifically designed to improve that persons life by simplifying a dreaded task: calculators, word processors, apps, GPS, etc. The question becomes for education, does or will technology improve education and most importantly the learning of students?

I recently read Larry Cuban’s post: http://www.washingtonpost.com/blogs/answer-sheet/post/the-technology-mistake-confusing-access-to-information-with-becoming-educated/2012/06/17/gJQAt8PFkV_blog.html

This really got me thinking about technology as a teacher in a 1-1 school, where all students have a laptop computer in their hands every day in every class. As a math teacher, also, I have experienced the calculator revolution: https://rootingformatheducation.wordpress.com/2012/06/29/do-calculators-make-us-smart-or-dumb/

Technology absolutely has the potential to improve learning. However, as Cuban pointed out, technology is often poorly implemented in classrooms. My experience has shown that teachers are less willing to integrate technology into new modes of learning than as tools to do the same “learning” a different way. Schools and districts prefer to use technology as ways around learning in the classroom, with less effective strategies as credit recovery or original coursework learned through self-taught information and video explorations similar to online classes in colleges. These opportunities often are misused, provide too-easy possibilities for cheating, or do not match the learning styles of the students.

My primary question about technology concerns purpose: Are we pushing technology into the classroom because of its effectiveness in improving learning or is it being pushed as a money making opportunity? Next, I wonder why educational technology infusion has not become more standardized nationwide. Are teachers the specific roadblock? Is there so little educational technology support and software that everyone is just standing around waiting for the next big thing? Do students dislike technology as a tool for learning? Should education recruit the Halo team to create a Algebra 2 based video game, for example, featuring conic sections and three-variable systems of equations?

Ultimately, why has technology not been fully integrated into learning like cell phones in society or television into households? What are the specific challenges to merging the worlds of education and technology?