Tag Archives: Mathematics

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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Do calculators make us smart or dumb?

Calculators help adults, speeding up tiresome divisions of real world money problems, for example.Calculators help explore high level mathematics by utilizing repetitive graphing techniques, for example.

The question is: For basic arithmetic, simple graphing, fundamental equation solving, and other low-level skills, does the use, or the use prior to mastery, of calculators make children less capable of learning higher levels of mathematics?

  1.  What do students think?
  2.  What do parents think?
  3.  What do math teachers think?
  4.  How early is too early for each concept?
  5.  How does this compare to other technological advancements, such as spell checkers or search engines?
  6.  How much of the basics in math should be required without a calculator?
  7.  Which students are most affected by this phenomenon?
  8. How do we compare internationally with our use of calculators in schools?

Any thoughts?

 

Improving elementary math education

I posted this on an old blog post (http://larrycuban.wordpress.com/2009/12/18/everything-you-need-to-know-about-education-reform-by-rona-wilensky/#comment-11864), but wanted to think about it more, so I thought I might post it here:

I am a little late here – 3 years. However…

I have long thought a reorganization of current resources may solve many of the problems with math education in the early years. Most of us will probably agree that if children do not learn math early, they are most likely not going to excel at math later; in reality, these kids will often struggle just to meet basic levels for testing these days.

Here is a possible plan:
Instead of having 5 teachers on a grade-level be the “know-it-all” for all subjects, which it may be being suggested they are not, perhaps you could have three generalists, one math specialist, and one math/science specialist. In the beginning, these teachers would simply be chosen from those available. As time goes on, however, administrators could hire specifically to fill the math specialist position for each grade level. A massive reorganization would be required, but it makes more sense to me.

This is just the beginning of the idea, but I thought I would throw it out there. This requires no additional money, training, etc., simply a reallocation of the available resources.

Feel free to respond.
Brett Bothwell

[If you respond to a blog post three years late, would anybody read it?]

High school tracking for math classes

This is my first post on this new blog. I have a number of ideas to get up on the page, but need to find the time to get it going. This post stems from my doctoral research. I would love to hear what you think about my thoughts.

The current situation for Texas high school math education, generally speaking:

  1. The most advanced students typically take Algebra 1 in 8th grade, Geometry in 9th, Algebra 2 in 10th, Pre-calculus in 11th, and Calculus in 12th.
  2. The least advanced students take Algebra 1 twice-9th grade and 10th, Geometry in 10th and/or 11th, and finish with Mathematical Models and Applications in12th, resulting in a minimum diploma.
  3. Obviously, there are a myriad of situations in between. At worst, though, the least productive high school math students are 3 years behind at graduation. Of course, the level of learning is probably well below as well.

I would like to propose a new system for my pretend high school:

  1. The highest level, in general, would be called a University STEM track, which has Calculus or AP Statistics  as the expected final course.
  2. The second level, in general, would be called a University non-STEM track, which  has Pre-calculus or AP Statistics as the expected final course.
  3. The third level, in general, would be called a Junior College track, which has Pre-calculus or Algebra 2 as the expected final course.
  4. The fourth level, in general, would be called a Vocational track, which has Algebra 2 or MMA as the expected final course.
  5. The fifth level, in general, would be called a Liberal Arts/Fine Arts track, which has MMA as the expected final course.
These five tracks can all lead to postsecondary educational opportunities, and each more closely relates to the ability, desire, and future plans for individual students.
Different tracks do not have to mean "bad."
An Algebra 2 example of how curricula would match the graduation tracks, while not being overly burdensome on teachers follows:
  1. The university tracks(1 and 2) are to be parallel Algebra 2 classes. They should have similar six weeks calendars. The STEM curriculum/course should be more rigorous and in all ways a more challenging math class. Teachers may need to modify pacing, depth, and style, but the curricula should be similar enough as to not be considered a whole different course.
  2. The junior college track would be the present on-level Algebra 2 course and is the stepping stone between the two major level differences. If a child wanted to take a step down from the university track or up from the vocational or fine/liberal arts tracks. It is unlikely that a student will be following a liberal arts track and choose to move to the university STEM track and be successful. However, track switches are possible, and this is a good compromise.
  3. The vocational and liberal arts tracks would, like the university tracks, be parallel course tracks. The  major difference, however, would be the theme of the courses pointing to non-mathematical or job cluster math. These math courses would be taught relatively close to the watered down versions that are acceptable today and are considered minimal. Depth of abstract math understanding is not the goal in this class, though learning Algebra 2 methods and thinking is still imperative.

Track choices:

  1. “Placement” tests before 6th grade and 8th grade would be implemented to determine appropriate tracking levels, along with teacher recommendation and input from parents, students, and counselors.
  2. Choice would be the norm, with input being expected from teachers and counselors to best match students with a track.
  3. Additionally, brief, non-comprehensive, parent/student reviews of placement would be conducted semi-annually. If no changes were expected, the process could be bypassed.
  4. Comprehensive reviews would be conducted annually, involving all parties. If no changes were expected, the process could be bypassed.

This is not the final development of the ideas, but a good start towards making high school math education more student friendly.