Tag Archives: high school

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.

Judge Rules California Tenure Laws Unconstitutional

Oakland_Court_House_California_USA2I was dismayed to learn of this decision this afternoon and decided that it was worthy of my attention and writing. I will immediately admit that I do not have all the facts and information and have only read three or four pieces on this topic. Let me also preface my comments with the disclosure that I have taught underserved students for the majority of my 18-year teaching career.

According to a CNN article (http://www.cnn.com/2014/06/10/justice/california-teacher-tenure-lawsuit/), “a California judge ruled as unconstitutional Tuesday the state’s teacher tenure, dismissal and layoff laws, saying they keep bad teachers in the classroom and force out promising good ones.” Having taught for a while, I realized I needed a full brush up on tenure.

What is Tenure?

Commonly, the purpose of tenure is to ensure that teachers will not be dismissed for reasons unrelated to their academic performance, such as personal issues with an administrator or school board member, the fact that they make more money than a first year teacher does, the teaching of classic literature that some moms decided is inappropriate for their precious daughters, or pursuit of academic strategies that do not support a rich benefactor of the district. Additionally, a 2008 Time Magazine article (http://content.time.com/time/nation/article/0,8599,1859505,00.html) noted, “In the 1920s, female teachers could be fired for getting married or getting pregnant or (gasp) wearing pants.” Many in education also believe tenure to be a factor in the choice to become a teacher, providing safety and consistency in employment.

Earning tenure.

In the districts that I have worked in, teachers were given two years to prove that they are worthy of being taken off of the provisional (1-year) contract and upgraded to a tenure-type, continuous contract. If they have not shown this level of proficiency in two years, there is a third year in which they can prove it; I believe they are automatically switched to a tenure contract if there is no reason to dismiss them after three provisional years. During the two/three year provisional period, the teacher’s contract can be nonrenewed without extensive cost or proof of ineffectiveness as probationary teachers do not have the same due process as tenured teachers.

The ruling.

The case was developed by an organization that seems to be anti-public education/pro-reform, Students Matter, and is backed by a billionaire reformer. They have also been linked with Michelle Rhee’s Students First organization, which seems to be an anti-teacher and anti-union/pro-reform organization. Students Matter supported nine students suing the state of California because teachers earn tenure too quickly and once they have tenure are nearly impossible to fire. Poor and minority students suffer the most, according to the judge, as “grossly ineffective teachers” work in their schools more often. The Time magazine article highlighted bad teachers saved by tenure: “A Connecticut teacher received a mere 30-day suspension for helping students cheat on a standardized test; one California school board spent $8,000 to fire an instructor who preferred using R-rated movies instead of books; a Florida teacher remained in the classroom for a year despite incidents in which she threw books at her students and demanded they referred to her as ‘Ms. God.’” {Incidentally, I find it interesting that Rhee taped her 8-year-old students’ mouths shut in her first year of teaching.} Arne Duncan supported the judges decision: “At a minimum, …the court decision, if upheld, will bring to California ‘a new framework for the teaching profession that protects students’ rights to equal educational opportunities while providing teachers the support, respect and rewarding careers they deserve’” (CNN). “An expert called by the defendants estimated there are as many as
8,250 ‘grossly ineffective’ teachers in the state — or up to 3% statewide, the judge said” (CNN).

My opinions.

I cannot stand “grossly ineffective teachers” and have worked with my fair share. I wish there was a fair, easier way to dismiss these teachers. Kids and fellow teachers can easily point to the worst teacher in the building; it baffles my mind to see the administration allow them to continue. For every $100,000, ten year dismissal out there, there are a multitude of options that could have been used to dismiss the teacher more easily. Ultimately, I agree that the worst of the worst should not be able to continue teaching. I have solutions for this dilemma.

I believe it takes about three years in one teaching position to get it figured out. Year one is scary, being an on-your-own (mostly) on-the-job training environment. Year two is developmental. Year three should go swimmingly. Therefore, I would agree to extending California’s tenure beginning to three years. The laws being unconstitutional, however, seems to be a stretch to me. If those laws are unconstitutional, then all tenure and firing protection laws could easily be argued as unconstitutional. This may actually be where this lawsuit is headed in the big picture.

And there my agreement ends. I challenge the motives for the lawsuit based on who is paying for it and the big ideas it seems to support. I do not think charter schools are the answer to all the ills in education; their track record is not that great. That one great charter school you know of down the street is a drop in the bucket of all charter schools. Additionally, removing unions from the American landscape seems to be a flawed path. They came into being for a reason, we should not allow history to repeat itself here. Again, unions are not the issue, but the massive power they have accumulated may be a problem in the other direction. Throwing out babies with bath water is disconcerting.

As cited in the court case, 8,250 teachers (3%) created a need in the judge’s mind to erase the entire system. That seems mathematically illogical. Those teachers ought to indicate a need to change some things around, as I have indicated, not eradicate everything ever created in the world of educational employment. This point actually seems to argue against itself, admitting that 97% of the teachers in the state are at least competent enough to not be included in the statistic. With so many effective teachers performing so well under the umbrella of this system, why would you consider removing the umbrella and getting everyone wet? That type of thinking gives the feel of a knee-jerk reaction.

Every job has grossly ineffective employees: police, politicians, dog trainers, hotel and restaurant employees, etc. They are not all protected by firing policies, and yet the bad ones exist. They do not all earn tenure, and yet the bad ones exist. This is perplexing, to be sure. How would changing the tenure and firing policies lead to better teachers? Training, mentoring, attitude, and experience lead to better teachers. Losing tenure and making it easy to fire teachers will lead to only one thing for certain: less teachers.

Replacing 8,250 teachers would be an incredible challenge, let alone 50,000 (random estimate) when administrators have little challenge to their firings. I am certain that California has a teacher shortage already, which is the main reason bad teachers are allowed to remain. Without enough teachers for the kids, what happens? Overcrowded classrooms and the extensive use of substitutes (less qualified teachers) will become the norm. These poor conditions make all teachers less effective and create a worse job environment. With that having been done, you can look into the eyes of your potential recruits at universities and sob as they laugh in your face because they would rather do ANYTHING but become a teacher in such a repressive environment. Tenure for teachers is one of the few things you can rely on as a 20-year-old deciding on a career. Honestly, school board policies, state boards of education, and the federal government already infringe enough on educational ideas such as homework policies, state testing accountability for teachers, and the teaching of controversial topics in science and history. Speaking mathematically, teachers, regardless of research, personal philosophy, or personal effectiveness, are already heavily influenced concerning calculators, memorization of facts, and algorithmic vs. conceptual approaches to math instruction. The more reasons we have in this country for young men and women not  to go into education, the worse this country will become.

And let me lastly address the concern for the poor and minority students. In reality, this is a governmental/social issue and should truly be addressed. Tenure or not, old firing policies or new, poor and minority kids living in worse neighborhoods with less of the benefits of living in this great nation will have worse schools and worse teachers. They also have worse medical care, worse police and fire protection, and worse McDonald’s. This decision will do nothing to improve California education, in my opinion, but it will work to further destroy American public education as a whole. This is most likely the intent, anyway, and is sad.


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.

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.