# 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.

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.

# Tear Down the Math Education Reform Wall

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 the 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 only 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 themselves.  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.

# New TEA teacher evaluation plan flawed?

Having read “TEA teacher evaluation plan flawed”  by Gary G. Godsey, I did some research to discover what the state of Texas had gotten itself into this time. Basically, local school districts will have the opportunity to evaluate teachers in a new system that includes 20% of the score for student achievement on state tests. That achievement will be determined by a Value Added Model (VAM) when appropriate, which the state of Texas noted would only apply for 25% of its teachers. The primary issues center around whether it is reasonable to rate teachers on student test scores from the high stakes STAAR exams and whether the VAM is actually an effective or fair measure of student growth. I am including some thoughts below.

1. Many teachers do not approve of the STAAR exams as effective or fair measurements for student learning, so how could you expect support for using those exam results to evaluate teacher performance?

a. As a math teacher, I understand that when students can’t read, they perform poorly on the math tests, which is not necessarily a reflection of their math ability. If the tests are not measuring students’ learning accurately, how can that be used to evaluate a teacher’s effectiveness?

2. If all teachers are not evaluated on the same system, there is already a reason to abandon the system. I can argue both sides of this idea. Why should I be evaluated with the VAM when 75% of my peers are not? Why can’t I be evaluated with a VAM?

3. The quote, “run them through complex mathematical equations,” tells me a few things. Most importantly that the designers of the model do not want those involved with the applications of that model to know what is going on.

a. Teachers will not understand how they are being evaluated. Common teachers, as well as most adults, are challenged by higher mathematics, let alone mysterious complexities in equations.

b. Administrators will not understand how they are going to be evaluating teachers, either.

c. The state appears to want mystery to be the guiding principal for revealing ambiguous information that will not be justified and can be statistically skewed to report anything they want. If they want to say teachers are doing great according to VAM, they will be able to. Likewise, they can report the opposite. They will be able to do this because “complex mathematical equations” means pretty much no one will have a clue as to the intricacies of the calculations.

d. I have seen “complex mathematical equations” that the state has used for other things, such as TAKS testing, even though they do not make these easily accessible or allow for evaluation of their equations. My understanding of many VAMs is that they refuse to allow others to see their models and indicate that there are multiple models available for various situations. It is also highly questionable whether competing VAMs would indicate the same performance reviews in the same situations.

4. On the VAM side, teachers should stop complaining about that “one kid” who will have a bad day and do poorly on the state exam. The VAMs I have read about supposedly take an average of many aspects and would not be severely skewed by one child. Thus, the argument becomes, “my entire class had a bad day.” Then, teachers will sound a little silly.

5. Also on the VAM side, if it were possible to see that Mr. X not only had horrible results for state testing with this year’s class, but every class over the last five years, this information should be valuable. On top of that, what if the other aspects of his evaluations show him to be a poor teacher, and attempts to improve him through professional development have achieved nothing. At that point, aren’t we doing more of a disservice to not only the last 5-6 years of students but future multitudes of students as well by not removing this teacher from his assignment? A better education for all students is a valid argument for teacher evaluation, and complaints about VAM in this regard would be less impactful.

6. Because VAM is generally based on previous test scores, more scores equates to a better model. Therefore, if 3rd grade is your first testing year, 3rd grade teachers should NOT be evaluated with VAM. 4th grade teachers are probably in the same boat as they only have one year to build a model from. What is a sufficient number of previous test results before VAM is actually valid? [These points emphasize the claim that only 25% of teachers will be evaluated with the VAM.]

a. A corollary to the previous point would be when a math student enters high school. Test results from previous years do not necessarily lead towards the “new” mathematical study of Algebra. Furthermore, Geometry is completely different than Algebra. Great Algebra students can perform poorly in Geometry and vice versa. These are just examples. Simply put, the VAM may not make much sense as a straightforward continuation of studies to be tested in the next grade. Additionally, since Algebra is the only tested subject in Texas high schools, the vast majority of math teachers could not be evaluated with this system.

b. Similarly, how do you create a VAM result for U.S. History when the kids have not been evaluated in social studies for at least two years and never for U.S. History content?

7. Similarly, teachers need to have accumulated data on which to compare themselves. So, first year teachers should not be evaluated with a VAM. Like the previous point, how long will a teacher need to have been teaching before the VAM is valid?

8. If the VAM system relies on randomized student placement in teachers’ classrooms, there is a problem with the model as this is rarely the case in any school system.

9. I have read that VAMs consider a massive amount of data that can vary from situation to situation. Additionally, they fill in missing gaps with best guesses. Anyone with any knowledge of statistical analysis knows that the more variables involved in an equation, the more challenging it is to determine what the results mean and which variable is causing the results in each situation. I am not saying it is impossible, but a large number of variables in an equation about raw materials in business, for example, would be more likely to produce a usable result than the variables involved with the tremendously complex realities of human beings.

Having read up on VAMs and seen the impressive controversial debates, I am concerned about teachers receiving poor evaluations or being fired based on highly variable, convoluted results. With only one-fourth of teachers even qualifying for VAM evaluation and the other 75% being evaluated with district-by-district and course-by-course varying measures, I would recommend that Texas does not implement the new teacher evaluation system. I envision lawsuit after lawsuit (already happening in Houston ISD). I know that they will implement it, and the federal government will continue to push for it, but I thought I would get my two cents worth out there.

The following are the best reads I found. mostly opposed to VAMs [incidentally, the information I read in favor of VAMs came primarily from the creators of the VAMs themselves and nowhere else.]:
VAMs and the “Dummies in Charge”
EVAAS’s SAS Inc.: “The Frackers of the Educational World”

# Judge Rules California Tenure Laws Unconstitutional

I 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?

Fran’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.

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.

# 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.

# 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?

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?