Teaching to the Test

Published August 22, 2016


NOTE: This book was written in 2012, before the advent of Common Core. That decision and its repercussions, will require another book. But many of the issues discussed here are just as relevant, as they were in 2012, if not more so!

When President Obama, and others with a strong interest in education, suggest merit pay to reward good teaching as evidenced by solid standardized test results, people decry ‘teaching to the test’ as the inevitable and negative result. Many people inside and outside the educational community assume that teaching to the test is easy and that it is wrong. It seems obvious to them that teaching to the test is somehow a ‘cop out’ that any teacher can do but only inferior teachers would ever consider. The truth is far from those assumptions.
When people say teaching to the test, they really mean teaching to the state’s standards and nothing else. This could be a negative, but only if a state’s standards were somehow lacking in depth or rigor. Teaching to all of the standards usually takes great effort on the students and teachers part and requires all of a school year. Teaching to the test is a parallel but separate activity that also requires significant amounts of time spread out over the school year. Teaching to the test is difficult, requires training, much teacher prep time, classroom and homework time and more. Surprisingly, most teachers don’t know how to teach to the test and couldn’t even if they were encouraged to do so. There currently is no handbook or training! By law teachers are not allowed to even see the tests the students are taking each year.
One key assertion of this book is that teaching to the standards and simultaneously also to the test is exactly what all teachers should be doing. Most teachers would be shocked to find out that many of their students lack of reaching hoped for, even expected, standardized testing performance levels is a standardized test effect and they could have helped prevent it. If a teacher isn’t doing this test preparation, students who are fully prepared on all the state standards of any given state will more often than not under-perform on the actual test! And the above assertion assumes that the academic standards of any given state are complete and rigorous and that the state’s standardized test fully explores and tests all those standards.
Here is a good place to ask why state standards are not equal across America?
The Fordham Foundation recently had rated all 50 state math standards from ‘A’ to ‘F’ with only 3 states receiving an ‘A’ and the rest lower. Why should some students be striving for mastering standards rated as below ‘A’ level, worse yet, even in many cases as “failing’? A person could argue with the criteria that the Fordham report uses, but the wide disparity is a huge cause for alarm.
Most, if not all, teachers in America attempt to teach their students all their state’s standards for any particular subject, to the best of their ability. Conversely, almost no teachers actually teach to the test. Although these are seemingly fundamentally different endeavors, they surprisingly have actually exactly the same goals. What are those goals? Across America all stake holders want a student or class or district to be fully versed in all the state standards and then that the students can demonstrate that knowledge on the state’s standardized year ‘end’ testing. Strangely, a teacher can teach hard to the standards and yet have many of their students score below the expectations the teacher might have developed, based on classroom performance. Students, who have not been taught how to take the test and avoid its inherent, carefully designed ‘pitfalls’, will not be able to show all they know. Currently in America, whenever a teacher might be accused of teaching to the test, they are really ‘guilty’ of teaching largely or exclusively to the standards. How this is an incorrectly defined failure is the main theme of this book. A student, who knows the standards but doesn’t know how to take a standardized test, can look exactly like a student who doesn’t know all the standards.
No Child Left Behind mandates that national standardized test results will be used to assign performance grades to states, counties, districts, schools and teachers. The United States has fallen behind other industrialized nations based in these types of testing situations.
Our nation is in an education crisis. We are losing too much academic talent to “educational attrition”. There are many reasons for this, some of which won’t be addressed in this book. The issue under consideration here is should standardized testing programs be reconsidered? No one is leading a discussion of whether these test results are accurate, fair or somehow intractable. The premises on which the tests are founded are not being examined or reconsidered. Can all stakeholders be assured that state standardized tests are meaningful, fair and appropriate as the chief means for assessing academic achievement? This book addresses exactly these issues. Every parent, teacher, school administrator and politician needs to be concerned about the questions and interested in the answers.
All the stakeholders in the educational process need to redefine how they look at standardized testing. It should be seen as the one of the main goals of the year, if not the primary goal. It should be seen as a chance for all the students to demonstrate their knowledge. Rather than something to dread, wish to avoid or get through, so the students can get back to more important things, it should be seen as a crucial culminating activity. All aspects of teaching in any given year should be viewed through the following lens, ‘How will this activity or lesson help my students to perform better on the state standardized testing?’
Clearly there are some assumptions that need to be addressed in order to make these assertions and have them make sense.
1.Are the state standards for each discipline of the highest caliber for each subject area?
2. Does the particular state standard test fully align with the standards? Does the state standards test fairly and fully measure whether a student has fully learned the standards?
3. Have the teachers of that state been fully trained on how to teach the standards.
4. Does the textbook adoption list include textbooks that cover all the standards in a coherent way? And crucial to testing performance success, can the adopted textbook successfully be used in the time frame allowed before the standardized testing begins?
5. Have the teachers of the state been trained in how to present practice problems to the students, with a full understanding of how the tests are different by nature than anything they have seen all year?
6. Do the teachers of the state realize that their students test performance on the standards test should closely parallel classroom performance and grades?
7. Is teaching to the standards and to the test understood as a state and national educational priority of the highest consequence?

Besides these more particular issues are the national questions.
Why do state standards vary widely in terms of content and rigor?
Do all national stakeholders have the same understandings of what good test performance should look like and what it means?
Does our country as a whole have the commitment to education that teaching rigorously to difficult national standards would require? Who will lead this discussion in order to bring a strong national consensus?
Is the country ready to make the real commitment required to bring the best and the brightest into teaching?

‘Those that can, teach’ has to become a national slogan backed by the highest levels of thoughtful and meaningful national and state support. It must become unsupportable for anyone to continue to think that becoming a teacher is a second-class profession! The countries that we wish to compete with in the industrialized world are very selective about who they allow to teach!

Chapter 1: What is wrong with teaching to the test?

‘Teaching to the Test’ is a term heavily laden with negative connotations. A teacher that was to openly profess to be teaching to the test as their top priority would be considered many things, currently none of them positive or flattering. I feel strongly that this needs to change. If a state has carefully researched and clearly defined what it wants as state standards, which are then guidelines for student performance, why shouldn’t the teacher be actively engaged in teaching those topics and skills? If teaching to the state standards and simultaneously, teaching to the state standardized test is somehow limiting the students in that state their current standards need to improve. If classroom teachers are widely convinced that crucial things are not addressed in the standards and they have to teach them as well or shortchange their students that has to be addressed at the state level. Once the high standards are adopted, a test that accurately tests those standards has to be developed. Once these two things are in place, the necessary teacher training, textbook development and adoption can then begin.
So it must be asked again, why give a test that has so much obvious academic, demographic and visceral importance and not have teachers expressly target those standards and that test as an important if not primary goals of their teaching? If we can as a state or nation say that our standards are correct and the tests fairly evaluate the degree a student or nation has achieved those standards, why should a teacher under-value their importance. Why should a parent be ‘o.k.’ with their child’s standardized test performance not matching their classroom, graded performance? Shouldn’t strong academic performance be reflected in high, standardized test results? And if not, why not? Each component of student success on the standardized tests needs to be reconsidered and changed where appropriate.
Competent teachers all over America are teaching their students their state’s’ standards as best they know how. There is no question that some teachers need help in this regard, but the typical teacher has no lack of resolve to do their very best. But the assertion that teachers are largely not teaching their state standards is not true. But the test scores of most students do not show a demonstrated proficiency on the respective state standardized tests. If teachers are teaching the standards what else can be going on that is causing this deficit?
I would argue that a student can be very well schooled in the standards and still under-perform on their standardized test because the nature of the test often confounds the students best effort to score well. These tests are an experience unto themselves. Lack of familiarity with any situation or testing format will necessarily produce decreased performance. If we are going to use No Child Left Behind regulations as a basis to change schools and teachers, which is the statutory consequence of under-performing students, shouldn’t we do everything possible to help those students to be successful? To ask a student to take a test unlike anything they’ve seen before that year is not fair and logically leads to depressed and depressing test results. Granted some students enjoy this testing format and seemingly flourish, yet even though they do relatively better than many of their peers in percentile terms, wouldn’t many of their raw scores be even better if they didn’t also miss questions that ‘fooled’ them into answering the ‘wrong’ question or making some other test effect type of error. This idea of a test effect is crucial to understanding why a prepared student can under-perform and how their teacher can help remove the impediments to this students scoring to their true potential. It is axiomatic that not all students do as well on standardized tests as their classroom performance and grades would lead one to predict. Should there be this type and level of disconnect? Should colleges be routinely weighing significant differences between G.P.A. and standardized test results, and be wondering, which is more real, and thus the better predictor of student performance in college?
Doesn’t teaching a student how to more successfully answer any questions given in the standardized evaluation become an important part of deeper long-term understanding? If the questions are fair and the topics important, how can we consider all efforts to improve performance as not somehow integral to the teaching enterprise? I assert that all teachers should be aware of the standards and the testing environment and both arenas should be targeted as essential to successful student performance. No one is going to say that those students that score below expectations on their state’s standardized tests, but had good grades, somehow did still learn, they are going to say that the test scores show that that they didn’t!

Teachers all over America, wrestle with how to approach standardized testing. On one hand, if you teach all the state standards required by your respective state Department of Education and more specifically, your grade level, shouldn’t your students be prepared and do well? On the other hand, if you only teach what you expect the students to be tested upon, aren’t you short-changing the students of a true full education? Strangely the answer to both questions is an emphatic NO!
If your state has strong state standards in all curricular areas then teaching only that material will cover all the topics a student needs to know. Yet, those same students often will not perform as well on a given state’s testing as their teacher would expect and hope. In plain words, many will not be ready for the standardized test you worked so hard to prepare them for. If you didn’t teach them how to take the test, they will often look exactly as if they weren’t fully prepared academically. And if you are teaching a student all the information on which they will be tested, a true definition of teaching to the test, you are preparing them academically. But again, if those same students are not taught the nature of these tests and how to excel at taking them, they will as a group significantly under-perform. Teaching to the standards and teaching how to take the test are different enterprises with essentially the same goals. Teaching how to take the test has the goal of allowing the student to use all of their previously learned knowledge, to be fully reflected in their test score. The teacher, who ignores this reality, could easily find himself or herself wondering how and why their students ’failed’ to work to their potential. And this could also translate into tough questions being asked by administrators and parents about why that teacher’s teaching methods ‘apparently’ failed.
There is nothing wrong with using standardized state or national tests to evaluate our students, if they are fully prepared for these tests. This test preparation must include teaching students how to take these tests.
So how should a teacher look at a school year? The experienced teacher knows that state standards take a full year to cover especially if they are rigorous. However, the classroom teacher also has to look at the year through another lens. That lens is how do I fully prepare my students for that years state standardized testing? How soon do I start to introduce standardized test practice problems? How and when do I start to introduce test-taking strategies? If important topics are introduced late in the book how do I bring them forward in the teaching year or otherwise help the students to gain sufficient familiarity so they can answer questions about those topics. Topics not presented to students early enough will not be fully assimilated and won’t be very useful under the stress of testing. How soon is early enough? And if the state test in question is typical of many, the later topics in the textbook and in the standards they cover, are also the toughest, so harder for students to grasp. And then, because they are deemed more difficult and thus more important, they are often given a more featured role in the test. It is common for the test writers to emphasize the more difficult topics to a degree that is arguably out of balance with less tested ‘easier’ topics. If the students aren’t just as facile with the later and more difficult topics, as they are with the easier ones presented earlier in the year and earlier in the book, they are in real trouble. These are issues that are not part of teacher training. A teacher left on their own to make sense of these issues will have a difficult time of it. Why? Because the classroom teacher is typically be unaware that these questions should have been asked and need to be answered.
So let’s take these issues. It is axiomatic that test writers focus on the more difficult topics as crucial. The typical textbook builds from simple to more complex. “The hard stuff at the end” is often not approached until the end of the year. When it comes late in the year it is unfortunately often rushed and students don’t have sufficient time to digest it. If the student didn’t get ‘it’ into the long-term memory, where they can easily access the information and skills, recall under testing conditions will be difficult.
So what to do? Teachers have to formulate an understanding of what standards are those crucial ones that come late and are given increased emphasis? The textbook should be helpful not as is usually the case impediment (more on this in Ch. 5) but even a really good textbook isn’t enough by itself. The every state should have released test questions from previous years on the state Department of Education website. These questions are a powerful resource. The teacher has to look these over carefully. A careful analysis of those test questions will show which topics that usually appear late in your textbook as those final, difficult chapters and topics, will likely be the heart of the standards test. How do you keep a coherent path through your textbook and still begin to feature these crucial topics? Clearly this is not a question that each teacher should be trying to answer individually.
One method that I use is math writing. Math writing allows a topic or several related topics to be introduced in a novel way.This skill requires the student to solve a problem and then explain in writing, using all the relevant math terminology and vocabulary, what the problem is, how they solved it and how they know they are correct.
I will use as an example topic from the California state Algebra 1 standards, the Quadratic formula. In the California standards, this appears as one of the very last standards. I learned over several years from talking to my students after the California STAR (state standards) test that there were many questions about this formula and they didn’t feel they did very well.* I knew that they couldn’t have really understood it because it was covered so late in the year, right before the standards test. So I invented a math writing problem where a linear equation and a quadratic equation had to both be graphed on the same X/Y axis (note: I became familiar with math writing problems when they used to be part of a California State program called the Golden State Examinations. This program was recently cancelled due to budget constraints.) Students had to find the roots of the parabola by graphing and using the quadratic formula. Of course, this required I teach several topics out of sequence to the book. The first writing problem was fairly simple, given late in second quarter. (Early December) I reprised it with a more complex pair of equations in February. This began the process of imbedding these concepts, terminology and formulas.
I also use as practice, multiple choice ‘released’ state test questions. These are given as a test, graded on a curve and used as part of their grade. I never tell my students the raw scores. In the beginning especially, it would be too disheartening. I’ve also written a study guide of practice problems, crucial formulas and test taking tips and reminders. I have them go over this intensely for several days before the state testing begins. These measures on my part, have helped improve my overall student performances from one year, with no such interventions, to the next. I continue to look for new ways to help them succeed on these tests.
*Note: This year, 2009, after the test I questioned them about the topics tested and the number of questions approximately on each topic. The quadratic formula was involved in 9-10 questions. Other difficult, ‘later’ topics i.e. later in text and last in standards, also received heavy emphasis.

CH. 2
Why teach to the test? Or to ask the same question another way, why aren’t you teaching to the test? And more to the point, why aren’t you teaching about the nature of the test.
Each state in the United States has made a determination about what skills and information a student should learn at each grade level. These are published and then hopefully addressed by the textbook manufacturers. Teachers and districts adopt a text and then attempt to teach all the standards. The students hopefully learn all the material.
This is my working definition of teaching to the test. Any teacher, who ignores the state standards of their respective state, risks doing their students a major disservice, as well as potential loss of employment. So if most teachers are doing their jobs, which I don’t doubt at all, why don’t our students perform up to those academic standards? Of course, I am not ignoring the difficulties many schools face with absenteeism, second language issues, equal access, etc. But tens of millions of our students are being asked to take tests that they “should” be prepared for, but based on state and national test scores seemingly aren’t fully ready.
Why? Most teachers do not realize that standardized tests have ‘their’ own agenda. Although testing the state standards, they, the test authors, are embedding the test topics in problems intentionally formatted to make them confusing, ambiguous, and likely to cause students who understand the issue to still possibly get it wrong. In math circles these are often called non-routine problems. However that isn’t even totally accurate. The problems often don’t look like something the student has seen before. On standardized tests it is usually not supposed to look familiar.(More on this issue later) This can cause the student to be uncertain of how to proceed and prone to make errors or not even actively try. And in those cases where the test looks like a textbook problem, the answers are carefully considered and crafted to present to the student the most common errors made by students as alternative answers, as well as the correct answer. A student who has to do a problem with 3 steps and partway through the problem solves for a number, that number will usually be an answer. Even though the student isn’t done, he or she will see the number they just solved for as a choice. Students will often choose this partially correct ‘wrong’ answer, because the partial solution was presented as a choice. And as an alternative parallel issue, the most common student errors are provided as other choices. The test authors are obviously very aware of the common errors students make and the answers that then result and will consistently present those as choices. So if I make a sign error on a math problem and should have found positive 4 as the answer, but actually I got –4, that will be provided as a choice. The answer column will seemingly validate my unintended error. EVERY ANSWER IS THERE FOR A REASON! The answer column is used as a filter, to try to lure the unsuspecting student into a wrong choice. And the student, who knew how to do the problem and makes a silly error, looks no different than the student who is completely confused. They both got it wrong. If I was grading the students work, it would be clear who knew how to do the problem, who made a foolish accuracy error and who doesn’t understand the concept at all. Teachers of mathematics want to know if their students understand the concepts and can perform the calculations accurately. Minor errors and major errors don’t make a difference on standardized multiple choice tests. They both appear as the same gap in the student’s knowledge. Even though one student can have a solid foundation in the subject and one can be in beyond their understanding, they both will get the same score on that problem. If a student knows the material, it is also the teacher’s responsibility to help them navigate around ’embellishments’ to the basic topic or mathematical issue. “Choose the counterexample” is not a term or skill most text books teach. “Choose the answer that doesn’t fit” is not a usual homework problem in most math texts. Which number isn’t divisible by 2 or 3, will often have the student choosing the answer that is divisible by both. And there will be a number that is divisible by both provided as an answer. That is part of ‘crafting’ a typical standardized test. If a student who understands the concept involved can still get it wrong, then the teacher has succeeded and failed at the same time. Succeeded in teaching the concept and having the student learn it. And yet sadly, have ‘failed’, by not knowing they needed to teach the student how to avoid the wrong choices that can be and often are cleverly presented and too often chosen. In other words they needed to teach to the standards and also, how to effectively take the test.
There is a whole industry devoted to helping college bound students to improve their standardized test scores. When the student has the will and money, they can usually make a significant improvement in their standardized SAT test score by learning how! There are strategies on how to use the test as a resource. There are strategies on how to prepare students for problematic wording. There are strategies for helping them to become more accurate. These are artificial constructs to ‘test’ the student’s math skills. If they don’t know all the rules of the game, they will under-perform.
Teachers and their administrators need to be aware of these parallel issues. If students have not had significant guided practice in taking these types of tests and been debriefed on the common errors and how to avoid them, they will not score to their full potential. So even as the teacher is teaching the state standards, at a point in the year well before the standardized test is scheduled to be given, the standardized test taking practice must begin.
I have used practice problems for many years to try to help my students. At first, I just showed them some problems our district purchased or that were released by the state. I didn’t really do a study of the process. I assumed that if they had a little familiarity with the multiple choice format, they would do well. I was often disappointed by the results over-all. Although some students excelled, many that also should have done very well did not. I was using my classroom experience with those students to set my own mental criteria for how well I expected them to do. After a few tries, I started to give the practice problems as a test and then looked at the problems they missed and went over those topics again. This helped a little but I was still largely disappointed.
In 2008, I decided to use the commonly missed problems as a basis to make a study guide for my students. They did one practice problem on each commonly missed topic. They were also given a short written explanation of the process. Although I was still somewhat guessing the probable topics, I saw a marked overall improvement.
This year I had an epiphany. I had been logging the problems my students missed, but I never really asked myself why they were missing these problems. I just assumed they needed more practice and explanation. This year it hit me that I never had logged the most commonly chosen wrong answers. I did the data collection and began looking at the data. I began to ask the question, ”Why did they choose that wrong answer?” I began to notice that there were ‘good reasons ‘ to have made those errors. As I began to work the problems and then look at my students’ error patterns I discovered why certain wrong answers were being consistently chosen. I divided them into two categories.

Partially Correct -Totally Wrong
The first large category of error is based on the way the test writers have analyzed student response patterns. If a problem takes several steps, crucial partway solutions, ‘answers’ required to continue to the next step would be provided as a final answer. For example, if a student solved for some “X” value, that was needed to continue the problem, that value would be provided in the answer choices. Even though that was not the goal of the problem, the student would see their partial solution appearing in the answer column as a potentially correct answer. Under the pressure of the test, coupled with lack of familiarity with the intentional presentation of these partially correct choices, students were being lured into choosing ‘partially correct yet totally wrong’ answers. This is a state of affairs that students need to be trained to expect and watch for in choosing their solution. They need to make it a practice of re-reading the problem to be sure they are answering the question the problem actually was asking. They need to look at all the offered answers to be sure that they don’t jump to a wrong conclusion.
Accuracy Errors
None of the answer choices provided are random. They are not wasted. Each incorrect answer is selected and presented because it serves a purpose in offering students the most popular wrong or partially correct answers. It is a repository for the answers a student would get if they made a sign error or other common types of errors, I call accuracy errors. These are the errors where the student knew how to do the problem and still got it wrong because of an error caused by careless work or trying to work to fast. There is no accommodation in standardized multiple choice testing for students who know how to do the problem and make a minor error that leads them to a wrong answer. A classroom teacher would usually offer partial credit in these cases. The state standardized tests are not forgiving in the least. A student who knows nothing and guesses incorrectly or leaves it blank is awarded the same score as the student who knew what to do but made a minor error. NOTHING!
Teachers need to strive all year in emphasizing that accuracy is just as important as knowledge.
Clearly the teacher is striving to help their students to become knowledgeable. But the state testing does not directly honor knowledge. It only honors accuracy and that is the challenge the teacher faces in preparing their students. A student could know how to do every problem and still get a sizeable number incorrect. The nature of the testing becomes an issue in the results. This is test effect and can only be limited by careful preparation.

.Chapter 3/ The House of Math
The House of Math is built on two pillars, knowledge and accuracy. It is the teacher’s role to fully explain each lesson and concept. It is the students’ job to focus and learn the material. Strangely and unfortunately, this doesn’t always lead to math success. A test score is not based on what a student knows but rather what they show. It is accuracy that is the final determinant of testing success. Good teachers teach, good students learn, so why doesn’t that lead to 100% test scores? This chapter will be largely about the difference between knowledge based errors and accuracy errors.
Many students struggle to focus in the classroom. When the teacher is teaching, their mind and often their eyes are elsewhere much of the time. Students often need to be trained to stay present. If a student only sees pieces of an explanation of a new topic they will struggle to understand it on the homework and tests. I force my students to put down their pencils unless they are taking notes or doing practice problems. The pencil seems to have a mind and life of its own. I require students to look at the board all the time I am teaching. Students are visual learners, they have to see it as well as hear it. I jump right on the first student who wants to tell me they are listening and that should be good enough. I talk to them about how much more information in our brains is gained visually than all the other senses combined. Students need to know why they need to listen and look, and when they do, they are then usually more willing to fully focus. For the rest there is the ‘red dot’. I tell them that if they don’t want to watch me then they have to stare at the red dot on the top of my white board in the center of the room. I tell them that at least now I can pretend they are watching. Of course, the red dot is even more boring than me, so they end up seeing the lesson. I don’t allow tapping noises, zipping back packs, getting out paper, while I am teaching and discussing a lesson. Students often live in a bubble where they don’t hear their noises as other people do. I train my students to honor what I have to say as important and requiring their attention and silence.
Another key thing for teachers to understand is the difference between I.Q and effective I.Q. If a student struggles to focus and bring their best attention and effort, they perform as if they really have a lower I.Q. than they actually do. This difference between raw talent and harnessed capability is the reason a fully engaged student with reasonable intelligence will often outperform the more talented but less focused classmate. This issue of effective I.Q. goes a long way to help explain the gap between what a student is theoretically capable of doing and what they can actually present as test performance.
Teaching Focus
Students who struggle to focus also are reluctant to show all their work. They are prone to want to do it in their head. Many of the most talented always are looking for short cuts and reasons to not show the work. This is a major issue. The teacher must demand showing the work and put in place the reasons and have penalties for failure to do so. I require the student to write down the problem and show all the steps. If they don’t show all the work on homework the lesson is graded down as ‘homework lite’. On tests if they don’t show all the work, an error is given a larger number of points off if the work is not properly presented. Teaching the essential nature of discipline and rigor is required of the teacher. The teacher must be relentless in demanding this. The work must be constantly modeled fully and completely.
Partial Credit
The issue of not showing work leads right into the major issue of partial credit. Most of us didn’t encounter partial credit on a math problem or physics problem until college or possibly an enlightened high school teacher. It was used at my university to allow so many of us who didn’t fully ‘get it’ to survive. I think partial credit should be used as early as 1st or 2nd grade! As soon as there are algorithms, (i.e. rules and methods you are teaching to solve a math problem) as a teacher you should be looking at their work to see if they are learning the method and using it. A student doing two column subtraction with borrowing who finishes by subtracting 3 from 9 and gets 5, should not get the same score on that problem as the student who doesn’t understand or use borrowing at all. Knowledge based errors are far more troublesome than accuracy errors. A student can miss 5 problems in my class and get a score between 75% and 95%, out of 20 problems, depending on the nature and severity of the errors. Not all errors are created equal. If students learn to show all the work because you demand it and then reward them for doing it by giving them a higher score, they will largely buy in to what you are ‘selling’.
Another huge benefit for the teacher and the student is that if you give partial credit and a student gets a high ‘B’ or a low ‘A’ they will feel good about them selves in the immediate moment. Then you can begin to raise the bar for them. A student who gets a strong grade but sees that they made several small and avoidable errors will be open to the crucial concept of striving harder to close the gap between knowledge and accuracy.

100% Club

It was this exact scenario that led me to invent a very powerful tool that is used school-wide at my junior high. It is called the 100% Club. One day 20+ years ago, I over-heard one of my students make the comment “at least I got my ‘A-‘”I drove home thinking about this and why it bothered me. There is certainly nothing wrong with an A- on the face of it. What I realized was that the students had set a bar for themselves, which they had reached, and they were satisfied. I knew I wanted to address that and the 100% Club was born. For a student in any math class to come to the 100% Club pizza party at the end of each semester, they have to earn at least one 100% score on a math test. This has proven to be amazingly powerful. I can say to a student that you did really well, almost 100%. When the 100% Club is in full effect, the ‘worst’ grade for a student is the 99%! They have been known to wad up their test in disgust and toss it in the recycling. When someone asks them how bad they did, they will groan and say the horrible truth,” I got a 99%!” When this is happening I know my class is on the way to truly valuing accuracy. The giving back of tests in my classes is largely a day of celebration. If you get an ‘A’ you get a chocolate kiss or some other small edible reward. (Whether you agree or not in these types of rewards is up to you, but kids will work for chocolate and nothing tastes better than that chocolate kiss when you earned it! Form your own opinion) But my students know that they can and should be doing well. They also know and acknowledge that if they had studied harder and been better prepared they would have earned their ‘A’ as well. Although I teach junior high, I have seen my students go on to high school and earn 100% on Algebra 2, Trigonometry and even Calculus tests. I have 5 students so far who earned 100% on college calculus tests at major universities. This result is astonishing to anyone who has ever taken college calculus. What I have learned from watching my students over the years is that setting the level of accuracy they are shooting for at 100% has a galvanizing effect on the student and their study habits. The level of focus on test day, the intensity of preparation, the quality of classroom attention is all shaped by this goal of reaching perfection. This is a huge issue, as we shall see on the standardized tests, because there is no margin for error their either. On a standardized multiple choice test, a student with no clue about a problem gets the same zero points as those that knew how to do it but made a computational or accuracy error. Getting each problem correct on any test, especially a standardized test, if the student knows how to do the problem, is the goal of the student and the teacher.

How are they wrong?

If a student doesn’t know how to do a problem and you know you taught it and most students are doing fine, you know something more about that student. They are probably either having trouble focusing in class or they are not doing careful homework as practice. As the teacher works with a student on their test result and what can make it improve, the teacher has to formulate a clear idea what the problems are they are having. If you find a student is using the correct methods and making many accuracy errors you need to do an error analysis and show them where and why the errors are occurring. Most students never study a scored test as a learning opportunity. When I score a test, I go through the work looking for the errors and indicate what it is and what it should have looked like. The usual student practice is to look only at the score and then ‘file’ it. It is crucial to show the class as a group and each individual on their test, the types of errors they are making and to talk about how to avoid them. I feel that it is crucial to test your students often. Many students have error issues, which leads to the well-documented problem of test anxiety. Partial credit, as a way to reward a good although unsuccessful effort, is a good way to raise self-esteem. Testing often, while also requiring careful homework, allows them to slowly grow into becoming a better test taker and thus a better student. No matter how smart the student is, if they don’t see the results in their test scores, they will not believe they can be good at math. Them learning to look at their own work and seeing that they knew what to do, they just made an accuracy error, raises the level of cognition of the student. They can see for themselves that they know the math; they just need to be more accurate. Learning to be accurate is much easier when the student believes they know the material. They have to see for themselves that they are doing solid careful work and with a bit better focus on removing accuracy errors they too can compete for the 100% Club.

How Good is Good?
One of the amazing and humbling things about teaching and starting the 100% Club has been the long-term results. I have had students take their achievement levels to a status I never came close to in my academic career.
My first realization that the 100% Club was energizing some of my students to do something extremely special was a student I will call Jill. Although the 100% standard had produced obvious results, I never saw anyone embrace it as she did. Over the course of the year we had 25 tests of textbook material. She earned 21 100% test scores. And it was all about an intensity of focus that was amazing to watch. As other students stood outside my classroom talking before class, she already had her game face on. I had told my class, as I always did, that concentration is not something you turn on like a light switch, rather it was something you gathered. It takes time to get into the ‘zone’ of intense concentration. It isn’t a switch you turn on like a light switch. Most students heard me but only had varying degrees of understanding. Jill embraced this idea. She was able to block out idle chatter before class and get herself ready. She was able to take the test and then do the entire test again in one test period to check it! Her work had the fluid, ‘clipped’ style of someone moving their pencil with extreme confidence. One day, as she turned her test in, she told me she was having trouble concentrating. That day she earned a 98%. She could tell her focus was not perfect. I was constantly amazed to watch her take her test, in a personal world of mathematical focus that was stunning. She went on to UC San Diego. She told me one day at the local theatre where she was working that she had earned a 100% grade on her college Calculus final. I got goose bumps. People at my university (UC Irvine) prayed that no one would score above 70% on any calculus test, so there would be a big curve. Jill is what we called a ‘curve killer’! This theme would repeat.
I had another student who went to UC Davis to major in Civil Engineering. She was the only female in the class of 20+ students. They were all taking a brutal class called ”reinforced concrete”. These are the calculations required to hold up bridges, levees, etc. The professor walked into the room with their final exams. They were all stressed of course and the inevitable question was asked, ”What’s the curve?” The professor looked grim. He looked at each one and then said, “There is no curve!” The class erupted in shouts of protest! “You always have a curve, You’ve had a curve for years, this is unfair!” The professor waited for the clamor to die down. He calmly said, “ Oh, there is a curve, there is just no curve.” It took time for all of them to divine their way through this logical conundrum. Then suddenly one of them blurted out, “You mean…..?” “Yes”, The professor said, “For the first time in the history of this program someone earned 100% on the final.” Suspicious eyes roamed the room to see who had created this doomsday. The answer was given by the professor, as he gave the perfect test to my former student, the only woman in the room. She had learned years before that if you know everything on the test, you should be able to earn 100%. She walked out of that room universally disliked and respected.
Another of my former students is currently going to my alma mater, UC Irvine.
She came back to tell me that she had earned 100% on a test in advanced calculus. She even brought me a Xeroxed copy. She had another test that she apologized for, because she had made a sign error on one problem and earned ‘only’ 99%. Her work was and is amazing. A single problem filled an entire page of closely written computations. She shared with me that there were students in the class that were so smart she was in awe of them. But I asked her who has the best grade? She shyly admitted that she did! Her effective I.Q. is exactly equal to her raw I.Q. score. She is able to outperform others she identifies as brighter than herself by out working them, in test preparation and then out focusing them and having superior test accuracy.(More on this topic in Chapter 8, David vs Goliath) Learning to be this accurate is a trainable skill. Teachers can and must emphasize this. This student as a graduation gift to me made bumper stickers that said “Homework is your Friend”. She had heard that phrase from me so often and it had made a lasting impression on her. One is in the front of my classroom to this day.
A few years after Jill set the bar for 100% scores in a year,a result I thought no other student would ever reach again, another remarkable young lady began my Algebra X class. She started the year with a string of 13 straight 100% scores. It occurred to me that I should have a math award to honor these students. She finished the year by breaking Jill’s old record by receiving 22 scores of 100% out of 25 chances. I named the award after her. It is called ironically the ‘Haight Math Award’.
Another student, a young man I will call Cooper was in my Algebra X and then honors Geometry class. We butted heads at times because he was so bright that he couldn’t see the need for showing all the work that flowed so effortlessly from his brain. Fortunately by the time he graduated from high school his work ethic was rock solid. He also gave me two Xeroxed college calculus tests both over 100% because his test was perfect and then he could also solve the challenge problem. He subsequently transferred to UC Berkeley to major in Mathematics. I show these Xeroxed tests to my current students because the tendency to not want to show work is all too common.
I had another young lady get a scholarship to Santa Clara U. to major in mathematics. She was near graduation and the only female math professor was helping her choose which schools to apply to for their PHD programs. Her advisor put black lines through several schools Liz was considering and told her they were ” Good Old Boys” Schools and if you checked the “wrong” gender box you wouldn’t be seriously considered. Liz applied to and was accepted into the Illinois University Doctoral program in mathematics. She still credits the idea of perfect work, the 100% Club and my telling her often that she could be exceptional in math, as pivotal in her accomplishments.

Refuse To Be Outworked

What all the above students had in common is that they have realized that they control how hard they are willing to work. They walk into each classroom with the clear idea that no one will out work them. If a student refuses to be out worked they will always be competing for the highest grades and the best test results. Effective IQ is built upon this principal. If you can get a student to embrace it, they will excel. If you can get an entire class to embrace it or even most of the class, the teacher will find himself or herself blessed with many teachable moments. Getting the students in your class to ‘agree’ to allow you to lead them further on the odyssey of learning is a privilege that must be earned by relentlessly training them on what it takes to find academic excellence.
Many other students have come back to tell me that they sadly have no 100% tests but they have the highest grade in their high school and college class. The purpose of the 100% Club and teaching the fundamental importance of accuracy is that in the final analysis it is what you show not what you know that is the score on your test.
I should mention that although my junior high is relatively small as a middle school. I have had several former students earn 800 on the math SAT test, which is a perfect score as in highest possible grade. Many others have earned in the high 700’s which are also extremely good scores indicating math/science/engineering level talent and preparation. I am convinced that identifying accuracy, as a crucial goal, has been a major influence in all the cases mentioned above. I should also mention that the number of my former students who have earned major scholarship money and admission to top universities is now in the 100’s.
I just had a student who is a high school senior earn a $20,000 four year Coca-Cola scholarship. She was the only one in California. This young lad has been accepted to Harvard, UC Berkeley and Dartmouth among others. She told me in her 8th grade year that she would be a high school valedictorian and never wavered. Teachers have to stand as relentless encouragers and sellers of the work ethic. Many students don’t know what they can achieve until a teacher they respect convinces them that they have greatness within.
Although I do consider myself a passionate and dedicated teacher, I am far from alone in this. What I have been able to do is see into the mathematical learning environment in innovative ways that not all my colleagues have and create learning devices that have been very powerful. This isn’t something a teacher should necessarily have to do to be effective. These are ideas, skills and techniques that can be taught and all teachers can use.
I want to mention another idea here. I have seen that if a student has a solid year in math, where they learn to believe in their ability and the power of hard work, where they have seen mathematical success repeated on tests, those students will get a ‘bounce effect’. They will be somewhat impervious to one or two or even three years of indifferent teaching. Once the student has internalized that they are good in math and have come to expect success, they will continue to achieve at a high level without being dependent on the teacher to continue to give them support and validation. One excellent teacher ever other year is all a student actually needs to continually grow. Of course, the ideal is for a memorable teacher to be there every year for every student in America.

CH. 4

Why aren’t all state standards created equal?
All over America, parents are told that their students are being taught all the state standards. But they aren’t being told that the standards their state is using as a template for math excellence and teacher guidance, might not be as rigorous as they should be. In fact, depending on the state a student lives in, the standards could vary from very daunting and excellent to failing.
The Fordham Foundation used very specific and publicly available guidelines to evaluate all 50 state standards. They came to the conclusion that only 3 states deserved an ‘A’ ranking. On the other hand, eleven states received failing grades and 19 more received grades of ‘D’. The national average was a ‘D’. The wideness of this disparity is obviously cause for alarm on many levels.
Now some will want to argue that their state is being harshly judged. One can obviously argue that Fordham criteria should be altered. This is possibly true. The question of why there is this huge disparity remains. Based on what I read in their report, there seems to be no set of criteria that would have all the standards of all the states receiving similar scores or ratings and that were acceptably high.
It is amazing that states all over America are allowing themselves to be labeled as having failing standards or ‘D’ level or even average. The standards a state adopts drive the ‘school bus’. If the standards are not rigorous enough or if there are conceptual flaws, small wonder that many students are underperforming on international standards tests. If the bar they are trying to reach is too low, what have those that reached it really achieved.
I have to ask what is going on here. Teachers all over this country are being held to standards that are widely divergent, contradictory in emphasis, ranging in depth or rigor by large amounts. It is just not defensible that the crucial concepts that need to be taught in an Algebra 1 class, for example, should vary significantly. I happen to teach in California where the Algebra 1 standards are ferocious. California received one of the 3 ‘A’s for it’s standards. To complete my coverage of all the standards before the state testing dates requires weekend homework, homework over breaks and a unrelenting pace to be ready. Homework is my students friend, even though many find this idea distasteful at first.
Why does this matter so much which state you live in and which standards you are striving to master? I had a young student who was forced by nature of a family break-up to move to a Southern state that has a failing math standard according to the Fordham report. I received a phone call at the end of the summer after this young lady had moved to the south. The math department chair wanted to know why I had recommended this girl for Geometry as a freshman as none of their students had ever done that. I said she was ready and to give her a chance. She wrote me four years later to say that not only had she been the top math student in her school, she won every award in every discipline! She received a full scholarship to Tulane University. It must be noted, that although she was a solid student she was not my most talented. But she had embraced the idea of working as hard as she could. I firmly believe that if the bar is set high and the student has come to believe that they can reach it if they work really hard, anything is possible.
I do have a few other thoughts on this issue of standards. The Fordham Group clearly states that calculators are not appropriate at early grades and shouldn’t appear in general use until late pre-algebra, at the earliest. Many state standards are in fundamental disagreement with this premise. I personally concur with the Fordham Foundation assumption. My experience is that a student can manipulate a calculator and not really understand the algorithm involved or even have any idea if their answer makes sense. And if they are inputting the problem incorrectly, they are forced to assume that the calculator doesn’t make mistakes, so their answer must be right. Garbage in, Garbage out, as the computer technology people are fond of saying. The rule at our school is that a student can’t use a calculator until they can earn test scores of 90% or better in pre-Algebra, on 3 tests in a row, where they are tested on all the basic arithmetic algorithms including whole numbers, fractions and decimals. To put it simply, they can begin to routinely use the calculator once they have proven that they don’t actually need it. They are still required to show all work on fraction problems and all Algebra problems. The calculator can only be used to check those.
It must be asked here if the nations of Europe and Asia that routinely outperform American students allow early use of calculators and if so on what terms or limitations? We are constantly comparing our students to the rest of the world on the TIMSS and PISA tests, so we must be looking at how they are preparing their students.
Another aspect of the calculator issue is the development of a work ethic in our students. I find a very strong correlation between students who are willing to learn to fully show all the work to a problem and test performance or accuracy. Many students, in my lower math classes especially, would prefer to never write anything down except answers. This unwillingness to do full and complete homework, of course, translates into increasing numbers of test errors, especially as the mathematics increases in difficulty and complexity. I think it is important to ask whether the calculator issue be only looked at as a way to ‘streamline’ math. Not only is there the question of does the student have true understand,with the attendant risks that they don’t, but calculators also must be looked at as a way to allow students to be less responsible and thus not as engaged in developing solid work habits.
But let us set aside the calculator question. The students in the Asian countries that perform so well on the TIMMS are going to school 220 to 240 days per year! If you compare 180 days to 240 days that is an extra 1/3 of a year. In a transit from K-12 that is basically 4 EXTRA YEARS OF SCHOOLING! On top of that, there is data that shows that it is the 3 month summer vacation where many students drop behind their American classmates because the parents in some families can and do offer their children a much richer and challenging summer. The data suggests that the students of all socioeconomic groups learn about as much during the school year. It is the non-school summer period when the deficits largely occur.
This opens up the question of how much school do we, as a nation, feel is appropriate and if it isn’t going to match the competition in other countries, why do we continue to draw unfair comparisons?
In 9 countries surveyed, students spent 200 or more days in school. This doesn’t take into account Juku in Japan, for example, which for the 47% of the students taking these classes, adding another 5 hours per week. When survey staff from OTL(Opportunity to Learn) asked students how many of the questions they were tested on in math they had been taught the relevant information, in Japan the number was 92%, in the United states it was 54%. Again, I must state I am not necessarily arguing for a longer year, but rather that we recognize the differences in educational environments when we make international comparisons.


Chapter 5
Why are textbooks wagging the dog?

Math teachers all over America depend on their textbook as a navigational map through the math standards of their state. They assume that if they teach from the text their students will be exposed to the proper topics in the proper sequence. So why is it that all too often math teachers are forced to teach quickly, skim or skip various lessons, chapters and topics? Since they are not often experts in mathematics, why are they are forced to make major decisions on how to proceed through their adopted textbook, which is usually too long to be completed in the school year? The question is magnified by the fact that the standards test is usually given about 85% of the way through the school year, often-late April or Early May. Even if the teacher could finish the text by June, the standardized test will be testing all the standards. Any standards taught after the testing dates will be of no value to the student in terms of their state test scores.
If the teacher is aware that all the standards will be tested and if they don’t teach all the standards before the testing begins their students won’t be fully ready, what DO THEY DO? This real problem leads to the wrong people making curricular decisions based on insufficient data.
Each state standards board should have strict guidelines on how and when to teach all the standards so all their students are completely ready. The textbook adoption process should be guided by the premise that the teacher will follow the textbook wherever it leads. If the textbook can’t be completed before the testing dates, the teachers shouldn’t be deciding how to proceed. All textbooks offered for adoption in America should be written based on one simple guiding principal, if followed and used correctly, the students and teacher will be fully supported in the march toward full fluency in all the standards, before the standards are to be tested.
So what should a math textbook ‘look like’? It should be constructed to have no more standards based lessons than can be completed before the testing dates. If this means the testing must be moved to a later date so be it. But it is unfair and worse, counter-productive to the very goal of standards based teaching, to have teachers deciding which standards need to be taught and which can be glossed over, in order to be ‘ready’ by the testing date. This places unfair pressure with potentially uneven and disastrous results upon the teachers. And the unsuspecting students and parents are assuming that all is well. Teachers shouldn’t ever be forced into deciding what is necessary to be taught and when and how fully.
Another problem teachers face is that the textbooks are usually up for adoption, which inevitably means change, every 4-5 years. Just when a teacher is becoming fluent in moving through a book, fully versed in how to teach each lesson in depth, the books are replaced. The reasons for changing the textbooks are not always based on the proper guiding principals. As ethnicity, language, socio-economic and other issues impact schools, textbooks are changed to respond to those factors. Coupling this with books that were never written to a scripted year, you get a ‘cookbook’ full of recipes to solve problems. The result is that the book becomes in simplest terms un-teachable. If such textbooks are dumped on teachers across America to wade through, that are un-wieldy and not designed to be finished, coherently before the testing dates, the test scores in that state will be depressed from what they could have been.
So what should a useful textbook have as its components?
First, mathematicians must write it. They are the true experts in these matters. They should be aligning it to the highest possible state standards. (The important issue of why all states don’t have equivalent or even possibly the same standards is covered in another chapter.) And they must be forced to write them so a teacher at any grade level can reasonably be expected to finish all the grade level standards before the testing dates.
The mathematical content should be organized to fluidly build from basic to more difficult topics. It must have the ‘due date’ of the beginning of testing squarely in focus in designing how many lessons need to be included. What should happen after testing(usually 4-6 weeks) should be a factor in the writing of the textbook, but it can’t include topics needed to prepare the students for their grade level standardized tests, for obvious reasons.
Mathematicians must also write the homework problems. They should expertly take the student from simple to more difficult practice problems. I feel that the homework problems should be limited. There should not be more than 30 problems per homework assignment. The teacher should not be choosing what practice problems to assign out of a huge, overwhelming bank of problems. I strongly feel that 2-4 practice problems, the first night on the newly presented topic, is enough. The bulk of the problems should revisit previously taught concepts. From my experience it takes students many repeated exposures to various types of problems and algorithms to fully embed them in their memory. This revisiting or spiraling allows for no re-teaching of concepts later in the year.

Looking at the textbook

Not all textbooks were written to make the school year a clearly focused march toward standardized testing dates. It is not now and never has been defined as the stated goal of textbook writing that the teacher can be assured if they use the book properly they will have covered all the standards in sufficient depth before the testing dates arrive. All too often the text is not in careful alignment to a thoughtful tour of the standards. Further, the book is usually so long that the teacher is forced, without support, to pick what to teach, what to skim and what to skip. This is not acceptable. Teachers should not be forced into this no win situation, especially when they are not allowed to know what standards are emphasized and which are lightly tested. (More on this elsewhere, but it should be noted that the standards are often tested in an inverse correlation to difficulty. In other words, the toughest topics at the end of the book that ‘YOU MIGHT GET TO’, are given additional weight by being tested with more questions than easier topics earlier in the text.) The textbook has to be the teacher support system. Teachers should not have to be wondering are all the standards emphasized properly, should I augment some areas, can I skim or skip this? Until the textbook industry stops wagging the dog and is given strict performance guidelines, the teacher is forced to try to assess what the standards test will be asking of their students. If all standards are not equally important, why shouldn’t the teachers be allowed to know that? If some things are more crucial than others why are the teachers the last to know? If all the standards are equally important why aren’t they all given an equal number of test responses? These are questions the teachers can’t answer, but if they don’t become aware that the test could be weighted toward later, harder standards, they could leave their students partially unprepared.
Teachers who have significant time in the trenches know that if you don’t give students sufficient time to assimilate topics, they will not be able to demonstrate reasonable knowledge.
If the difficult standards that often occur near the end of the book are not taught well before the standardized testing dates, the students will have difficulty demonstrating mastery of those standards. Any difficult standard that is not first taught several weeks before the standards testing will not show as an area of strength on the testing records. In a class report of test results, that standard will appear as an area of weakness. So if the textbook is of daunting length, with no guideline for how to streamline progress, what is the teacher to do?
I think that all textbooks should be 120-130 scripted lessons, focused on the entire set of state standards. There should be a spiraled set of test questions in each lesson, designed to reinforce the new topic and constantly revisiting previously taught topics. (Clearly here I am talking about math, although the issue of teach-ability is the same in all academic disciplines). The teacher who completes the book will be done before the state standardized testing begins. This is the level of support all teachers require and deserve in order to fully prepare their students. The constant changing of textbooks every few years has to end. Teachers have to develop a working familiarity with the text. They have to know that by a certain date, they should be to a certain lesson. They need to become extremely confident with the topics presented, so they can teach each lesson in depth and with a variety of examples and different approaches. The topics in the state standards in most states are not changing, so why are the textbooks constantly, radically different. I would submit that it is expressly to sell new textbook series. This does not necessarily improve the educational environment in any way for the student or the teacher. States and the federal government have a responsibility to control the textbook adoption process so that the people who matter the most in the process, the students and the teachers, are the focus.
So what must be done? The lobbying by vested interests must be derailed. The pressure to choose experts, to sit on the adoption committees, who happen to be affiliated with a particular company, has to be fully exposed and stopped. This is a scandal and can’t be tolerated when the education of our children is at stake. With hundreds of millions of dollars up for grabs, often in a single state, every 5 or so years, the pressure to get your text on the adoption lists is enormous. Representatives of the textbook industry know exactly what to do to influence the process.
If the guidelines for textbooks all revolve around teachability as I have defined it above, the need to constantly readopt can end. Becoming really facile with a mathematics textbook takes several years. Particularly in the self-contained classrooms, many teachers will admit that math is the most difficult and often least enjoyable for them to teach. It usually wasn’t a strength in high school or college for most elementary and many middle school teachers. Constantly changing the book in 5 year cycles is a huge problem, especially for them. “Look how different our book is”, should no longer be seen as a big selling point. Administrators and politicians should wonder, does all this constant changing of textbooks improve or lower test performance? What are we fixing by constantly readopting? Is this pressuChapter 6

Education is the way out

Most people fall into the trap of thinking they know everything. It isn’t necessarily intellectual arrogance. But rather, the person who isn’t constantly challenging them self and having their knowledge base challenged will entropically settle into a mental space of “I know it all.” There is no way around this process, unless the person in question is constantly assaulted with new ideas and differing opinions that have the ongoing effect of keeping them persuaded that they must keep their eyes and mind open. This is the great value of education. I tell my students that getting rich off your education is fine, but it isn’t the greatest value or point of gaining an education. There are thoughts a person will never think, books they will never read, ideas they will never understand without a deep, wide education. Lifelong learning is the natural consequence of a proper education. Too often the national news and discourse is dominated by ideas that don’t stand up to empirical scrutiny. But the undereducated person, thinking he or she knows it all, is easily persuaded to take on as fact, ideas that don’t have real merit.
I believe teachers have to challenge their students constantly with the ‘WOW’ factor. These are ideas that just can’t be fit into neat little boxes and which force the hearer to reconsider the notion that they know all there is to know of any importance. The teacher must constantly prove the student wrong in their assumption that they ’have it all down’. Students routinely tune out their parents because they think they know it all and their parents are actually, truly wrong. The teacher can play a pivotal role in forcing back open the closed minds of their students. The natural tendency to close one’s mind must be constantly challenged. Teachers need a repository of ideas, quotes, thoughts, facts and classroom photographs that have the student enthralled, doubtful or reflective. I can present ideas from Brian Greene’s wonderful book “The Elegant Universe” to 6th, 7th, and 8th graders. Even though the underpinnings of math and science are too advanced, the basic issues are not. That time slows down when the observed object is placed into motion is a confounding idea. When the student is shown that there is no way out of the logical consequences of this notion, they are perplexed, amused, thoughtful and/or annoyed. They often go home telling their parents about this crazy idea they heard today in school. “Physics for Future Presidents”, Neil McAleer’s “The Cosmic Mind Boggling Book”, any thing by Stephen Hawking, are all sources of facts that can’t be ignored, discounted or easily put into a little convenient box. There are unlimited other books and Internet sources that fit this description. If a teacher sets this task as a necessary component of their teaching, finding cool ideas to share becomes easy, enjoyable and personally valuable.
The book “The Tipping Point” reinforced the importance of opening, and keeping student’s minds open, for me. Mr. Gladwell makes the persuasive case that peers are the single most powerful influence in the lives of all too many young people. Teachers need to point this out to their students and challenge this working paradigm. Asking a student how often a ‘friend’ tells them that they can think bigger thoughts, work harder in school, have a brighter future if they try harder, is a chance to pry open the mind that can only hear their friends self-reinforcing notions. I am constantly amazed to find students in my below grade level math classes who have real math capabilities, but the matrix of issues they carry keeps them from operationalizing that talent. And their peer group is made up almost exclusively of students who share those same ‘issues’. To slowly convince a student that they don’t belong in a below grade level class, because they have too much talent, can have the unintended consequence of getting them to change their peer groups. It is interesting and powerful to watch a class that is as a group largely dysfunctional, reach it’s ‘Tipping Point’ and begin to actually act like students. When a teacher sees this happen, it is amazing, very rewarding and empowering beyond words.
I have learned to see a school year as a 9 month gestation period. I am assisting in giving ‘educational birth’ to a new or improved student. It takes months of ‘labor’ on my part to sometimes begin to see a glacial shift within some of my more challenged students. It is so important to keep relentlessly pressing students to achieve. The teacher can’t let 6 months of seeming failure dissuade them from that 7th month of labor on behalf of each student. America’s schools are heavily impacted with students that have issues and needs. These needs often shape the availability of students to the educational environment. A teacher never knows how much of what they are offering to a student is actually reaching them. I have had students come back to see me, who I felt I hadn’t reached, only to have them tell me what an impact I had had on them over later years.
In junior high school, the adult to be is often already emerging. They can think big ideas. Einstein was already thinking profound things at this age. I have had the opportunity of watching students set their life’s course at the age of 13. I have seen students set goals in junior high, that they never deviated from over the course of 8-12 more years of education and on into their adult lives. How much is too much to expect from students this young remains an open question for me. I would say that I still tend to under-estimate what I can ask of them with an expectation that they can fully respond. The bar I set for my math student has been raised by me several times. As I continue to realize that I can and must ask more of my students for them to be ready for the educational and life challenges they will face, I am amazed to see so many take the challenge and respond. Every classroom, every school and most importantly, every person has undiscovered capabilities. I truly feel all people have greatness within them. Surely not in all areas, but it is there in some venue. Someone just has to believe in them and help them find it.
Once a teacher gets a student’s or a class’s attention. Once a student knows that the teacher clearly sees them and knows who they are, The chance to help them make a shift in their self-image presents itself. This can take a long time with many students because they have learned to be ignored or not truly ‘seen’. They will not trust a teacher until that teacher has demonstrated a resolve they have come to finally believe. Earning that opening is the power point of education. All teachers are striving for that teachable moment. The subject matter being taught is truly secondary to this process of trust building.
I want to introduce an idea here that is the single most powerful thing I have found to motivate my students and to gain their trust. I write a personal note on every math test I grade. I write to the student by name. I tell them what I see in them. The talent that is there, often unknown and unavailable to them. If they drift in class I tell them. If they are showing all their work fully and completely, I note that. I have had parents tell me that I know and see things about their child they never knew. These notes are my personal, one-on-one way to let them know I care, I know them and I won’t settle for less than the best I see them capable of achieving.
Does this take a lot of time? Yes it does!! Do I write really fast and often look like a doctor’s prescription, yes again. I tell students to figure it out. And in time they really want to figure it out. I was never told by someone I respected, besides my parents, that I had talent and should set the bar much higher. All people have a need and craving for recognition and guidance. Of course you must earn that person’s trust before they will believe what you have to say. But the extra time it takes to write these notes is some of the best time I can spend. I find the notes a challenge and a reward to write. I have to constantly monitor classes with as many as 43 Algebra X students. They are all complex and shifting daily. But when I hit a student with what they need to hear and they are open to hear it, I see in their eyes as they are reading their test note, that they have some one in their life that really gets who they are.
A final thought about writing to your students. I send all my 8th grade graduates a postcard right before they start 9th grade. I choose very cool rock-n-roll postcards. I have a chance over the summer to distill what I want to tell each student. I have had literally hundreds of students and parents thank me later. I have had 40 – 50 students that I know about, put this post card up in front of where they study, IN COLLEGE! I had a student who was striving to be a valedictorian in high school tell me her postcard kept prodding her to do her best and not give up. Teaching is truly an affirmation of life. We all have to find the ways that work for us to reach our students and help them discover their best.
re for new adoptions really about selling more books?

Chapter 7
Girls are better math students than Boys!

If I needed a test done perfectly, all work presented rigorously and precisely with the highest possible probability of 100%, I would choose a female student. This doesn’t mean I don’t have any excellent male math students. On the contrary, I’ve had many wonderful male students. But if I put all my male and female students into Bell curves for test performance, the female curve would be shifted slightly but significantly to the right.
So what am I saying here? Our nation has traditionally believed that men were superior in math. This needs to be looked at and explored. I have watched young math minds at work for 24 years. I think, as a group, males probably tend to pick up concepts more readily. It appears the process of data acquisition is slightly different between the two genders. So why does performance shift the opposite way? My observation is that females as a group are more focused, drivenand willing to out work their male counterparts. They are also more comfortable with being precise and showing all work completely. I don’t have to work as hard to convince my female math students that they need to be rigorous in showing all steps. It seems that the very fact that males will say, “I get it”, quicker as a group, than the females, leads them to then be less willing to see the need to show all the steps carefully to arrive at a conclusion or answer. Given an opportunity to ask questions and get more help, females are more willing to ask for the help. This combination of factors leads to one group being more prepared and precise on tests, which translates into superior test performance.
One can argue about whether males have more math ability than females but the more relevant question for the real world is which group performs better and why. If America’s teachers set the bar exactly as high in math for both groups and never lower their expectations for the females, they will more than meet the challenge. How people learn the concept is important, but more important is that both groups can master the concepts and excel.
This leads us to standardized test results. Males have tended to score better on standardized tests when looking at large groups. I think that the typical standardized test appeals to a certain mental matrix. The variations of problems away from the usual or familiar to the non-routine or intentionally confusing definitely seems to favor a certain mind-set. This is why I think that the practice that comes with examining typical test questions, with a careful explanation and analysis of how to proceed and how to avoid embedded pitfalls, is crucial to improving test scores across the board. If students know the standards but can be confounded by problems that are out side their ‘comfort’ or experiential range, the teacher has to know this happens and step in to help all their students.
I think it is important to mention here that 99.9% ( my unashamed guess!) of the significant mathematics being done in the world is applied math. Inventing new math is a very different enterprise, reserved for an extremely tiny portion of the math users in the world. Why does this applied math issue matter in this discussion? I have heard high school math teachers of both genders say things like, “ Males are better math students” and “Our job is to weed out those who think they are good from those that ‘really are’”. This can lead to teaching assumptions and behaviors that are not in the best interests of individuals, one complete gender group and the world in general. If I am assuming because one person gets it faster or with less effort they have a superior mind, I can then send out messages that lead the others to feel that they are not as good and should find something else to be interested in as a career. But if applied math is where the work largely is to be found and with support women can be as good if not better in utilizing applied mathematics, the assumptions need to be re-examined and addressed.
I have the famous World War 2 Riveter Rosie posters in my room. I heavily emphasize the power of hard work and that women can achieve anything they set their minds to achieving. This leads me to never assume less of my feminine gender students. On the contrary, I challenge all my students to out work the competition. This exhortation appeals to the female groups tendencies and strengths. Female engineers, analysts, scientists, etc. should be present in the workplace in fully equal numbers to males.
Based on my observations, if I was a boss of an engineering firm and I needed a report done perfectly and on time and I had to choose by gender who should be in charge, there is no question how I would choose.
A footnote to this analysis: When I did my student teaching, I had two Algebra 2 classes at a high school. I was always intrigued by data, so I collected the following information. I asked the classes:
What were your grades in Algebra 1 and Geometry?
Which they liked more?
How did they feel about themselves going into Algebra 2?

The results were fascinating. In Algebra 1, the girls had a 3.35 g.p.a. and the boys had a 2.98 In Geometry, the girls dropped to 3.0 and the boys remained pretty steady at 2.96. So overall as math students, the nod clearly went to the women. When I asked them which they liked more, it was 2-1, both groups, they preferred Algebra. But the confounding anecdotal answer to the last question was this. The boys as a group said, “Math became more difficult, I did just as well”, while the girls, as a group tended to say, ”Math got harder, I did worse!” This is huge because math is a journey (Read the Mt. Calculus story) and if confidence is damaged, the road forward becomes much more difficult. If the talent nationwide is this strong in girls everywhere, this little data analysis could help explain why and where we lose a lot of math talent. Math teachers should be constantly intervening in this process, reminding the feminine half of the math classes, as well as the male, from kindergarten on that they can achieve, they should expect it to be hard and to expect that their hard work will continue to allow them success, at any level of math.

Ch. 8

David versus Goliath
Even though this theme is covered somewhat elsewhere, I would like to revisit it in some depth here. An article by Malcolm Gladwell, in New Yorker Magazine, made me do more thinking about how to expand the possibilities for success in math. His operating premise is that if you fight by the ‘established rules’ in any endeavor or battle and you are short where ‘tall is good’, small where physical strength is dominant or somehow simply less talented, you will lose. I want to expand this idea to include the world of academia. The operating rule in education is usually that all things being equal, the smarter student will be the best student. If everyone studies the same amount, game over. But what if a student decides to shift the strategy to a relentless full court press. Studying, taking notes, forming a study group and preparing as if it was a ‘war effort’ and they refused to lose. Gladwell, Schoenfeld at U.C.Berkeley and many others who paid attention noticed that lack of talent, within a reasonable construct, can be overcome by superior preparation and effort. Relentlessly outworking the opponent cannot be underestimated. The problem with this plan is that it takes a special type of person, a toughness of mind to be able to operationalize this strategy. Just as many players and coaches can’t mentally face pressing full court, all the time, with the attendant level of physical and mental conditioning required, many students would rather submit to the ‘obvious’. Many students also ‘buy’ the notion that another student is smarter so of course they do better on tests.”
It takes a period of time, relentless reinforcement, seeing some success and then assessing and deciding whether one has the internal drive to keep this level of relentless pressure on the opponent. The person, who attempts to use effort to outperform superior raw ability, must also be willing to face the relentless pressure that their decision has placed upon them self. I have seen enough students who brought this mind set into my classroom, that I know it works. Putting the idea out there to our students and asking them to decide if they are up for this level of challenge, is a departure from typical classroom topics. Giving them enough data and allegory to help them to begin to formulate this, as a possible academic and life strategy requires some introspection on the teacher’s part as well. Placing this idea in the arena of student discourse and discovery is to create a meta-level of classroom assessment. To look at class work as a struggle where to the most mentally tough go the spoils, is to place academia on a more neutral and fair basis. If a student with reasonable ability and tremendous drive decides to operationalize this paradigm, they are committing to an academic lifetime of relentless effort, the academic equivalent of Rick Pitino’s trademark Kentucky Wildcat full-court press. You believe you can out hustle and out work your opponent until they break under the pressure. Not a typical description of a classroom setting, but to win under the old rules you have to have been born smarter, which is something no person can control.
“We sometimes think of being good at mathematics as an innate ability. You either have it or you don’t. But to Schoenfeld, it’s not so much ability as attitude. You master mathematics if you are willing to try….Success is a function of persistence and doggedness and the willingness to work hard.” From Outlier’s by Malcolm Gladwell

Ch. 9 Why my district excels in math
My school has a group of teachers that share the agreement to demand excellence. This is an expectation that comes with a high price. We are an academic public junior high school. Every class, everyday students are expected to be prepared, focused and to behave within strict behavioral guidelines. Over the course of a three years journey through our school, students go from an often shaky view of themselves and what it takes to be a student, to a clear concept of what a work ethic is for and what it can be expected to produce. This requires the teachers to relentlessly demand effort and accountability. It often means training the parents that homework is a necessary, powerful part of a child’s educational process. This means some often, quite painful adjustments for the students and the parents. For a student and their parent to learn to trust that we actually are professionals, do know what we are doing and know their child very well and even in crucial ways, better than they do, takes time. The payoff is that, although there are 8 feeder junior highs, our students go on to the local high school to earn the lion’s share of highest grades, scholarships and entrance to the finest universities, as they graduate from high school. Although this can and should be true of all schools in America, it sadly isn’t.
We are a middle school with many stable families, generally strong socio-economic indicators, generally solid parent support, excellent teachers and supportive administrators. Most importantly, the majority of our students come to school ready and willing to learn. We teach to our state standards. Our students and school score very well on the California state standardized tests. The Thomas Fordham Foundation rates the mathematics standards that our students are being tested against, the highest in the country.

Teachers all over America work hard to teach their students the material required by their state’s academic standards. This process is culminated in one key sense by the standardized testing programs used nation wide. These test results are then assigned to students, teachers, schools, districts and states. Everyone wants to know how they, as an individual how they did and how their school compares, against comparable schools. Depending on the constituent group, the interpretation and reaction to the results can very from satisfaction to outrage, concern to complacency. America as a nation isn’t competing well on these tests standardized tests. The assumption is that our schools are failing our students in some way. I feel strongly that given the proper educational climate and truly rigorous standards, in any given American public school, our students can perform or compete with the best. The TIMSS test is one such international test on which America’s students have under-performed compared to other industrialized nations. I once gave the half of the TIMSS that was released on the inter-net to all the students in my middle school. We as a school, with all students taking the test, performed at a very high level. There are many schools and areas in America where our students can compete with any students anywhere. The issue is that there are large swaths of our educational landscape where teachers can’t make the same assumptions that my school can and does make, about student readiness to learn, parental support of education and an administration that backs our relentlessly tough standards of accountability and performance. This national discourse on the need for education, and the underpinnings that must be in place for schools to perform, must begin.
So what have we done to improve our math scores in our district? When I started teaching in my district 20 years ago, we didn’t offer a truly rigorous Algebra 1 class. Our students were perceived to be only ready for a warm-up in Algebra 1 in 8th grade. Having had 3 years of math teaching experience previously, in another district and bringing that school up to a full Algebra 1 expectation, I knew it was possible.
With the support of the administration, I took a class of our top 6th grade math students and raised the expectation for them one full grade level. Two years later they were our first true standards based Algebra 1 junior high graduates. The high school was concerned that they wouldn’t have been prepared fully. The high school started testing entering freshman on their Algebra 1 skills. My students did extremely well. From then on we offered ‘true’ Algebra 1, based on tough standards.
I also began dialoging with our feeder elementary to get them to start tracking and allowing all students to proceed as fast as they were capable. This idea met resistance at first, but was put into place. I began to get 7th graders that were Algebra 1 ready. They took Algebra 1 in my formerly 8th grade only classes. They then went to our high school for Geometry as 8th graders and then came back for the rest of their day. The high school was again concerned that they wouldn’t be ready for a high school level math class, but based on their entry test scores on the Algebra 1 test, they were allowed into Geometry. They were generally the best Geometry students in the class. This placed them in Algebra 2 as Freshman. Again the high school had concerns.(Notice a theme?) But again they were the top performers year after year. In time the number of 7th graders doing Algebra 1 grew until we could start offering a full fledged Geometry class at our junior high. The curriculum was based upon the high school’s Geometry X class’s curriculum. This practice has continued to this present day. Last year every student in my geometry class scored Proficient or Advanced on the state standardized test of Geometry. Did they do this well because my students are smarter than other places? No, but I did spend the entire year teaching to the state’s standards and the last 2 months before the state testing dates teaching how to take the test. Remember, that teaching to the test means teaching my students all the ways that these test questions could confound them, even when they knew the relevant theorem or concept involved in the question.
Our school had the highest math test scores in our county. Our county was second in the state behind highly affluent Marin County. As I said before, the Fordham Foundation ranked the California mathematics standards the most challenging in the country, one of only 3 states receiving an ‘A’ rating. As a county, teachers here can assume that they will get to teach and that the majority of the students want to learn. Unfortunately, that basic and profoundly important assumption can’t be made in many schools in America.

Chapter 10
Are your students really ‘there’ on testing day?

Clearly this is a trick question, right? If the students are all in their desks then certainly they are ‘there’. Unfortunately, for testing purposes this could be far from the truth.
The first question, a teacher must ask himself or herself, is whether the students have ‘bought into’ the idea that this year’s standards test has some meaning or purpose for them. Being told to just do it, isn’t necessarily the motivational set that many students respond to well. The pencil could be moving and the bubbles are getting filled in but how do you know that the effort they are putting into the test will reflect their actual learning for the year? The teacher has to start early to convince their students that their best is not a negotiable quantity. There will be differing results with this campaign. Many students who under perform on a daily basis are struggling exactly because they don’t see a need or value for themselves in school. Consequently they aren’t often willing to give their best. Standards taught are not always standards learned. The teacher has to work mightily to begin to get a polar shift among their more reluctant learners. They have to work hard to help them see success. If there is no success before the state standardized testing arrives, the student will be hard pressed to even care. They could have the ability and knowledge to score well, but the lack of effort or interest will be the deciding factor in how well they do. Having a math program where the students have developed an expectation of some success is essential for any real hope of the state testing really showing what they can achieve. This is why the textbook coherence issue and the idea of a yearlong approach are so essential. If the student has seen that they can be effective test takers during the regular school year, then there is some reason to hope for a truly accurate result on the state testing. In one of the previous chapters, “The House of Math” addresses the issue of improving test results. But it can be said here that if a student is not convinced that they can do well on tests, the first signs of difficulty on the state test will quickly lead them to a mental place of resignation.
For a student to give a consistent effort over the time it takes to do a full standards test, usually 75 minutes or more, they can’t encounter many problems where they perceive ‘they have no clue’. If a students initial buy-in starts to waiver because they start encountering problems they can’t recognize or seemingly solve, the teacher will have lost their students best longitudinal or continuous efforts. Once the student checks out mentally, they usually don’t return. In other words, the students must know the standards and know how to effectively take the test, two very different propositions.
But this chapter isn’t really about whether students are prepared or even care. It is really about the state of their psyche as they enter the classroom on test day.
The Geneva Convention forbids sleep deprivation or manipulating a prisoner’s food. How is this relevant to a classroom? Well, students define the usual weekend as their time. And America’s parents largely buy into this notion. So the usual in bed by 9-9:30 during the school week is abandoned on the weekend. The meals go from those chosen by the parents for their healthfulness on week nights and are replaced by student chosen weekend foods that are largely sugar based or junk foods of all descriptions. This produces the ‘zombie effect’ that teachers all across America witness every Monday. And if those walking dead are going to be taking a standards based test on that Monday, their scores will be significantly depressed.
Having worked in a sleep laboratory as a college under-graduate, I know a bit more than the average teacher about sleep patterns and the danger of disrupting them. Even if a student sleeps for 9 or 10 hours each day of the weekend, if it is from midnight until around noon, they will be a mess on Monday. And if they have been carbohydrate loading in the form of sugar and other empty calories, it will take 2-3 days for them to fully recover. This has huge implications for which day a teacher’s academic discipline is being tested. I fought years ago to have my math tests on Thursday, always, because I noticed over many testing trials that my students expected performance was down 10-15% if I tested on Monday. It was not as bad on Friday or Tuesday but there was still a test effect. Thursday gives my students the best possible chance for test success. They have had 3 nights of normal food and sleep. And thus they have had 3 days of increasing ability to focus in class. If a teacher gives a test on Monday, only students who are willing to discipline themselves over the weekend can perform at their most efficient level.
Which day of the week your students will be tested will have a measurable impact on their performance.
Another interesting aspect of the matrix of improving student test performance is the observation that almost all underperforming students have ADD-like symptoms or behaviors. If a student is drifting in class, they are only getting bits of what you are teaching. I enforce that my students must look at me, and the board, as I am teaching a new lesson. It is amazing to watch and see how hard many students will struggle to stay with you for even 5 minutes, before their eyes start to wander. “I’m listening” doesn’t go well with me. I know that math/science are very visually driven enterprises. Seeing the words but not associating them with the ‘pictures’ I am presenting on the board, becomes highly incomplete and largely useless information. The standardized testing format assumes that all students can bring a consistent and high level of concentration to the testing environment. Every teacher knows that this is a highly variable skill. I have observed that the ability to focus and concentrate can be improved. It requires constantly reinforcing the requirement and need to pay attention auditorally and visually. Many of our students have never kept a high level of concentration for an hour in their entire life. (With the interesting exception playing video games.)
As noted elsewhere, a student with an I.Q. of 130 operating at 100% concentration, will out perform a student with an I.Q. of 150 using only 80% of their ability to focus. An interesting discussion topic for all teachers across America to engage in with their students is based around the question, “Are you willing to be outworked?” The student who can commit early in their life to use all their abilities on a consistent basis will maximize their educational and life opportunities. Many students think the smartest students get the best grades and are thus the most successful in school and life.
Spending the entire year developing an expectation of hard work and quality focus is a crucial component to solid standardized test results. Students who can barely concentrate through a 15 minute classroom presentation, are hard pressed to really concentrate for the protracted periods standardized testing requires. And if the student is mentally ‘drifting’, lowered test scores is the unfortunate consequence. Many variables play into whether a student will be able to perform up to their true academic potential.
“In a completely rational society, the best of us would aspire to be teachers and the rest of us would have to settle for something less, because passing civilization along from one generation to the next ought to be the highest honor and the highest responsibility anyone can have.” Quote from Lee Iacocca

Chapter 11
Should you be a teacher?
If you are not saying yes to this question what is wrong with you? Granted not everyone can be an effective teacher, but why are you already sure you shouldn’t chose this life path?
Every person who attends college should consider being a teacher. Everyone knows the negatives. Only teachers know the positives.
There is no feeling or reward that can touch making a true difference in someone’s life. The power and joy of motherhood is largely about being a child’s first teacher. Fathers can get there also if they choose, but mothers know this feeling at their core.
Teaching at its finest is about making a life-long difference. It is about helping a person find their true self, what they are capable of achieving and helping them operationalize those goals and dreams. The future of our country is going to be largely created in classrooms across this land. The opportunity to participate in this future eclipses virtually any other vocation one can choose.
To find the strength to continue, day in and day out, to give of yourself to every child, even when they are being difficult or seemingly unresponsive is a tough task. Parents often say to me that they can’t understand how I can be in a room with 30-40 7th and 8th graders. They will honestly say they have difficulty sometimes being with their teenager one-on-one. They are really taken aback when I say that ‘I like it’. They are often astonished to find that I can write a note on a math test and capture qualities in their child that they had never seen or noticed. By the end of a school year I will routinely know more about their child in certain crucial ways than they do. The reason is that as a teacher you have to pay careful attention to survive. We are all psychologists masquerading as teachers of academic disciplines. A true teacher is about operationalizing talent, the academic curriculum should always secondary
A child will not learn until they are comfortable in school. Finding out whom each child is to help create that climate for learning is the most fundamental aspect of our job description. Children are relentless in their drive to have what they need. If you can structure you class and you behavior to help them find these things, they will respond. This requires that you come with your battery fully charged everyday.
As I am writing this book, a former student is presenting her doctoral thesis in math to her committee and undergoing the grilling that comes with this amazing achievement. This student feels that I was instrumental in setting her feet on the path to this day. I am humbled and amazed by this young lady. This is just one of many success stories I can share. Every teacher who comes with passion, commitment and determination has their own stories. It is largely what fuels our redoubled efforts every year. I am a better teacher this year than last. I continue to look for the better way to do all the aspects of this profession. Seeing how powerful I can be in the lives of young people constantly reinforces to me how important being a teacher is in our society.
Our society doesn’t yet fully appreciate as a whole what teachers do. The parent that sees a dramatic change in attitude and performance will see this effect and usually thank the teacher. But this hasn’t yet reached a critical mass. Our society does not yet attract enough of the people who would be the finest teachers, because as a nation teaching only receives lip service. People say they want to support education but they don’t see schools as the essential vehicle for their child’s success. And as such, they are not willing to support the reforms that would make education the pre-eminent career choice it should be. Mr. Iacocca is not wrong. There needs to be a strong on-going effort at all levels of society and government to place education in the proper perspective. People ask me out loud why I teach, when there are so many other more rewarding careers. Until I can tell people I am a teacher and have them really get how wonderful that statement is, I know there is more work to be done.
We should be choosing the best and brightest to enter teacher training. This requires a fundamental shift in the definition, scholarship support and other significant forms of re-enforcement. Until school districts can set up their recruiting table next to IBM, Sony, Google, etc. and be able to offer similar career opportunities, the field is tilted away from many top candidates choosing teaching. Becoming a teacher should have no economic penalties attached. The other professions and career choices should have to wait for the selection of teacher training candidates to be completed, before they get to recruit from the remaining pool of graduates.
The dinner guests  were sitting around the table discussing life.  One man, a CEO,  decided to explain the problem with education.
He argued, ‘What’s a kid going to learn  from someone who decided his best option in life was to become a teacher?’

He reminded  the other dinner guests what they say about  teachers: ‘Those who can, do. Those who can’t, teach.’

To emphasize his point he said to another guest; ‘You’re a teacher, Bonnie.  Be  honest. What do you make?’

Bonnie, who  had a reputation for honesty and frankness  replied, ‘You want to know what I make? (She paused for a  second, then began….)

‘Well, I make  kids work harder than they ever thought they  could.

I make  kids sit through 40 minutes of class time when their parents can’t make them sit for 5 without  an I Pod, Game Cube or movie rental.

“You want to know what I make.” (She  paused again and looked at each and every person  at the table.)

I make kids wonder.

I make them question.

I make them apologize and mean it.

I make them  have respect and take responsibility for their actions.

I teach them to write and then I make them write..  Keyboarding isn’t everything.

I make them read, read, read.

I make them show all their work in math. They use their brain, not the man-made calculator.

I make my students from other countries learn everything they need to  know in English while preserving their unique cultural identity.

I make my classroom a place where all my students feel safe.

I make them understand that if they use the gifts they were given, work hard, and follow their hearts, they can succeed in life.’

(Bonnie paused one last time, then continued.)

‘Then, when people try to judge me by what I make, with me knowing money isn’t everything, I can hold my head up high and  pay no attention because they are ignorant…  You want to know what I make?

I MAKE A DIFFERENCE. What do you make Mr. CEO?’    His jaw dropped, he went silent.
THOSE THAT CAN, TEACH! Everyone else just goes to work.

CH. 12

Why should our students care?
As teachers we assign work and hope, implore, demand, encourage, request, and require that it get done. The degree to which any given student is willing to participate in our process is highly variable student to student. When the state standards’ testing arrives, we are operating with the same set of assumptions. They, our students, will do their best and show what they know. This is clearly a highly suspect set of assumptions. The student who truly cares about their school work, where there are grades attached and consequences for poor performance, will usually try close to their best on the state testing. But, what about our students who are less than committed to school, what are reasonable expectations? What hold do we have on them? What is their buy-in to this process? What does a test score ‘mean’ if the student didn’t give their legitimate best effort?
I spend lots of time before the testing each year trying to create an attitudinal set that the students should care. But this year, as in others, students ask me why should they care and what possible impact does this have on their future.
I talk about self-esteem, always doing your best, that the test scores go into the cum folder and ‘follow’ you. I talk about teachers in the future could look to see if they were proficient on earlier standards. But the bottom line is that students who have a sense of integrity about everything they do try the hardest and it is then spread out all the way to students not even reading the questions and having a ‘bubble-in’ party.
There is no doubt in my mind that if there were extrinsic benefits more students would give a better and thus truer sample of what they actually know. As a nation we are asking students to report to us through this testing process what they have learned. But without an agreement shared by all or most all of the students, this will be a highly suspect arrangement.
I think there should be rewards for reaching test benchmarks. There should be rewards for improved performance year to year. These could be college tuition credits, guarantees of jobs upon graduation or entrance and support in trade and tech schools as is common in Europe. I don’t claim to have the answer here only the question. I don’t know many adults that would continue to do their best at a difficult task year after year if they did not see a tangible purpose and/or reward. We can say all we want, as teachers and parents, but the students decide what they think things are worth or not worth and the effort they feel that is justified. Our culture has changed since I was a child in school. When I was young there was not seemingly as much necessity to justify a students performance to them. We were simply asked to do things and we did them. Now students expect clear and reasonable answers to why they should do things or they will often choose to opt out of our process. The assumption that our students aren’t learning all they should because their test scores are not outstanding assumes that the score reflects a true picture of their knowledge level. I sincerely think this assumption needs to be examined. Isn’t it worth the cost of providing tangible rewards and justifications to our students to get them to attempt to maximize their educational opportunities? As a nation, every dollar spent in the process of improving the education of our citizenry will be more than returned in tax revenues and improved worker capability in the marketplace. Every lawyer taking the Bar exam knows exactly why they are taking it and what the rewards can be to them. Every college bound senior knows the value of s strong performance on the SAT or ACT. Every person looking to work for the post office, military, or the business sector knows why he or she want to do their very best on those respective evaluation tests.
Why should our students try their best on these state standards tests?
It is a question that they want answered, whether they ask or not.

Chapter 14

Thoughts for teachers to share with students

1. Homework is your friend
2. Are you willing to be out worked?
3. You can do more than you think
4. A strong education will change your life
5. Without a strong education there are thoughts you will never think, ideas you will never understand
6. It is your attitude more than anything else that controls your future
7. If you continue to do the same thing then it is only reasonable to expect the same result. If you want ‘A’s then you must do what that takes.
8. Hard work can neutralize and surpass greater talent. Talk often, with your students, about effective I.Q. versus potential I.Q.

Thoughts for teachers to think about

1. You have basically 85% of the school year to teach 100% of the academic standards of your state
2. You should focus on the end of the year state standards test as a crucial culminating event
3. If you don’t teach them how to take the standards test they won’t be ready
4. A student can know all the standards and still under-perform on the state test
5. Every student deserves your best every day
6. Be relentless in demanding excellence
7. Every student needs to have their confidence in themselves reinforced, often, to build a resilient sense of self-esteem.
8. Students need to see you energetic, confident, on task, willing to listen, prepared, able to laugh, everyday!
9. Dress for success. I wear a tie everyday to school and students do notice. If they see that you see teaching as important, it will nudge some into a deeper appreciation of what school represents.
10. Find something to share with your students several times a week, that makes them stop and think. They tend to think they know everything! Prove to them they don’t, in such a way as they go home wondering or thinking about what you shared. When parents start to hear these things at the dinner table, then you know you are doing something special.
Possible CH 14
Every Student Deserves A Great Teacher

Carnegie Report
How to get the best to teach
Radical change in 20 years
Journey to Mt. Calculus
The 100% Club and why it matters
Why should we not have a great teacher every year?
Running the math gauntlet
Becoming a teacher shouldn’t cost extra
What is the benefit to cost ratio of not getting the best candidates?
What every parent should know and be willing to pay for.
Do you really want the best for your child?
Gender issues
Applied Calculus versus theoretical
Make high schools springboards to success in colleges , not a weeding out process
The Impulse to public service
Find great quotes i.e. Iacocca
Parents must demand great teachers and be willing to support what it takes to ‘get’ them
Teachers have a chance to impart the most valuable gift a person can give, a vision of that person that gives them impetus to achieve.
How to recruit the best and brightest
What does a great recruit look like?
Every new teacher needs a mentor for support
A fully educated generation could fundamentally change the future of America (Carnegie report of 1980’s)
The failings of a society are a failure of education. You have great difficulty taking advantage of a truly educated person.
Teaching junior high, why I like it best
Public schools are not and will not fail(ing) if the basic requirements are present
Timss test we gave
What does education mean? What does it allow someone to do? Necessity for an educated populace
If you want to give a gift, it should be a complete education
Why other cultures revere teachers
What is ‘behind’ the screen or activity of teaching? (Finding out who that person is and what they are capable of and then encouraging/demanding thy reach for those goals. Keep reaching deeper , renew your quest
The teachable moment
All people are capable of important things
A life of service
Were you ever told that you could do something great, by someone besides your parents/
Once a teacher has won the respect of the students, anything is possible
Chapter 15

Test Taking Strategies
1. If you have covered all your state standards, then your students need to believe strongly that they have the skills in place to be able to do all the problems.
2. There is no penalty for guessing on multiple choice tests in California as a wrong answer and no answer are given the same score, zero.
3. A student has to know that multiple choice problems don’t usually require extensive computations. There is usually a concept, like ‘division by zero’, that if recognized, allows a correct choice quickly. Since the tests are meant to be completed in usually a block of time of set length, a minute to 2 minutes is all a student should expect to spend on a particular problem.
4. Remember that the ability to concentrate at a high level is not an ability that students can maintain indefinitely, encourage them to make a best guess when stumped and move on. Have them put a mark by those problems they weren’t sure about and hopefully, they can come back to them.
5. Have them mark out in text booklet answers they can definitely eliminate. Actually line it out, so they stop looking at them. Then if they must guess, they are only looking a 2 or 3 to choose from.
6. Look at all answers before choosing, even if they see one that agrees with their answer. Sometimes they will recognize that they have only partially solved the problem. The answer column is a source for information.
7. Some problems will give clues to solving others. Have them keep a broader perspective, especially if they guessed on a similar problem earlier in the test.
8. Always read the test questions at least twice. Underline what you are to solve for and all the relevant information. Surprisingly often, students correctly solve the ‘wrong question’, having misread what they were to have done.
9. Use a study guide right up to the start of the test. (See my example) They are to then write into the test booklet important and difficult formulae, so they don’t have to actually remember them. Utilizing the short term memory in this way can help immensely.
10. Use all the available time! Go back over test, look especially hard at those you starred as difficult or where you guessed.
11. Work backwards from the given answers in the multiple choice test! This is a “Guess and Check” type strategy except we know that one of the four answers is correct. Often by just trying one answer, we can get a sense of what the correct answer must be.
12. Don’t rush but don’t get ‘hung up’ on a particular problem, either. Spending 10 minutes to solve a problem and as a consequence not finishing or getting too tired to really concentrate is not a good use of time or energy.
13. Before you bubble in your choice of an answer, check again that the answer you want to choose actually answers the question being asked. If a student solves for ‘X’ , for example, and doesn’t substitute that value back into the problem to find the length of a side to a rectangle, as they were instructed to do in the problem, they will find the ‘X’ value as a choice, but it will be an incorrect choice!
14. Answer every question, even if you must guess!
15. Have students be on the lookout for words like “NOT” or “Counter-example.” These are confounding words to make the problem more difficult than it would otherwise be. Many of my students choose the correct answer, when the problem said, “Which answer is not true!” In homework, students never intentionally solve to find what answer is not true. We call that a mistake on a test but standards tests have their own ‘rules’.
16. Get lots of sleep and eat a highly nutritious breakfast. Give them an idea what nutritious means. Most students eat to make their mouth ‘happy’ not their body! Explain that sleep deprivation or poor nutrition will make a negative difference in their test score.
Chapter 16- Example problems as resource for teachers

These problems are taken from the Released Test questions from the State of California Dept. of Education, available on their website. Grades from early elementary through Algebra 2. These are all multiple choice. An answer is either right or wrong. Complete accuracy is essential! This exercise is for you as the instructor to begin to realize why wrong answers can look correct and how to help your students see the pitfalls. A year of emphasizing 100% accuracy pays off on reducing accuracy errors.
1. Which equation is equivalent to
5X-2(7X+1)= 14X ?

A -9X -2 = 14X
B -9X +1 = 14X
C -9X +2 =14X
D 12X-1 =14X
Commentary: Answer ‘A’ is correct. But Both ‘B’ and ‘C’ were popular wrong answers. The ‘B’ answer is ‘possible’ if the student makes a grievous sign error finding the difference between 2 and 1 and then forgetting the negative sign. ‘C’ was the most popular wrong answer, where the student knew to distribute the 2 across both terms in the parenthesis but failed to include the negative sign the second time they multiplied. Test writers are aware of the most common student errors and provide those as answers!

2. A 120-foot-long rope is cut into three pieces. The first piece is twice as long as the second piece of rope. The third piece is three times as long as the second piece of rope. What is the length of the longest piece of rope.

A 20 foot
B 40 feet
C 60 feet
D 80 feet
Commentary: Clearly the ability to read and decode is crucial in this problem. I recommend in the study guide chapter to have students underline all key facts and what they are solving for. This will turn out to be crucial in this problem. Answer ‘C’ is correct. But all three other choices are partially correct and that is the nature of many of these problems. PARTIALLY CORRECT, TOTALLY WRONG. If the student called the shortest rope ‘X’, and then the others 2X and 3X, then they understand the key Algebraic concept being tested. But if they divided 120 feet by 6X they would get the ‘A’ answer of 20 feet! So they solved correctly for ‘X’ and then chose answer ‘A’ because it agreed with them. They forgot to read again, what where they to solve for. The first piece of rope was twice as long. Some students went for the doubling of ‘X’, the first sentence in the problem, and got 40 feet, also wrong. And the second piece of rope subtracted from the total was 80 feet the incorrect answer ‘D’ . A student can know a lot of Algebra on this problem and still get the wrong answer and thus no credit at all!

3. Which number serves as a counterexample to the statement below?


A 100
B 57
C 30
D 25

Commentary: I mention elsewhere in the book that words like ‘counterexample’ are extremely confounding to even very good math students. This question ‘morphed’ into several different questions when my students did it on a practice test. They read this as “What number isn’t divisible by 2 or 3? So ‘A’ became a popular choice. And it must be constantly reinforced for students that choosing the first answer is tempting and thus the test writers often insert the most popular wrong answer here. Clearly 100 isn’t divisible by 3 so… 57 isn’t divisible by 2 so… Same logic applies here, so ‘B’ was also chosen. ‘C’ was popular because the ‘counter’ part of ‘counterexample’ slipped by them. They ended up reading it as “Which number is divisible by both”. This is clearly not a difficult problem conceptually yet it counts the same as using the quadratic formula correctly. Practice in the logic of problems like this is essential.
4. What is the conclusion of the statement below?

IF X² =4, THEN X= -2 OR X = 2

A X² = 4
B X = -2
C X = 2
D X = -2 or X = 2

Commentary: This is a simple problem if the student knows the difference between the usual meaning of conclusion and the meaning of the word conclusion when used in an IF,THEN statement. Now if they don’t understand the word at all ‘A’ could be chosen but that was not common among my students but ‘B’ and especially ‘C’ were chosen because they were in the later part of the statement. Especially ‘C’ since it was the end. However in logic the entire phrase after the word ‘then’ is the conclusion. This problem is ‘simple’ and turns on that word.

Chapter 17

The Journey to Mt. Calculus

Life seems to be many things when we are born. But a force that few really understand is going to shape our lives.
We are all born onto the continent of mathematics. Forces are at work that will determine our whole lives. And no one was going to clearly explain the situation to you….until now.
We all spend our early lives on the vast plains of Arithmetic. A happy place where our fingers and toes are out numeric friends. They help us to do sums and takeaways. Things make number sense. Money made is added, money spent is subtracted. Given a choice, few of us would ever choose to leave this world of numerical reality.
But we are not given a choice. We are forced to set for on a quest. No one tells us that is what is happening, no one explains the goals, the point or the reasons. We are just told we are going to school. You must go. It is the law.
And so we go. And for a time our toes and fingers are enough. Later, we start using blocks and coins for numbers larger than 20. But this is pretty comfortable and all seem at ease. Slowly as the years goo by, we are introduced to multiplication and division. Although it is a stretch for some, everyone is eventually Okay and we continue to be confident and content.
‘Suddenly’, we confront fractions, decimals and percentages. Things are slightly confused and confusing now. We so rarely multiply or divide pieces of things in the real world. Yet in the world of Arithmetic it is assumed. Some of our friends stumble and fall here and turn back. We don’t see them again.
As we enter the teen years, a time of inherent stress and confusion, a change occurs. Our teachers begin to speak of a land called Algebra. A highland place where we must go. We are not told why we must go. We are not told that it is not even the real goal. We are all going so we go.
Algebra is a strange world. We learn to talk of things that are not there as if they were. We call them unknowns and we spend a year trying to find these unknowns. It is a hard year for some. Drawing pictures of a two-dimensional worlds controlled by mysterious “X”s and “Y”s. Some of our friends are just mesmerized by this process. Their eyes glaze over and they seem to go away. We try to reach them, help them, but many don’t seem to hear us or understand us. After many years and much suffering, they turn back from the quest. The foothills of Algebra take an enormous toll on the human spirit. Some will never recover.
But the rest of us seem fine. Algebra is a strange world, but it is exciting and interesting. With a good teacher, we have mastered the unknowns. The pictures that were once like hieroglyphics, now make sense. We have completed the journey through the foothills of Algebra and all is well with us. This is about to change.
Oracles meet with us and tell us that our journey is not over. Algebra is not the goal! We are shocked.
It gets worse!
We are told by the oracles that we must leave our beloved continent. The place where we learned to count our blessings must be abandoned.
We must travel out from our continent to an island far off shore. The island is called Geometry. The Oracles don’t tell us why we must go there. They say it has always been so. The ancient ones, the Greeks, created this great place. We must go there to live for 15 months. From June to June and on to September. If we survive on the island and are successful we can return.
Since we are dutiful children we trust the Oracles. We go.
If Algebra is a strange world, than Geometry is beyond words.
We start with axioms that make no sense. We are told that every ‘thing’ starts from a point that has no dimension but still ‘is’. Where is it we ask? We are told to believe that such a thing can be. Then we are told that a line is a succession of these things that have no dimension. But how, we ask, can we put them side to side and they have length but no width? Distance is made up of these non-things. Area becomes even more bizarre.
We talk of circles, triangles,and squares, of angles, tangents, and arcs. We are to see them in our minds. We are to be able to turn them in our minds and still make sense of them. This is called rotation.
Many of us, often times the girls, find this place much more difficult than Algebra ever was. Those who want things explained just get more pictures. The explanations never made much sense and the pictures are worse. Those who are struggling grow frustrated, angry and confused. They had been so strong at Algebra, often better than the boys. Yet this Geometry is so strange. What does this have to do with Algebra they ask? No answer is given. “It is important, you must learn it.”
A story is told to us that ever since the Middle Ages or ancient time, a person was not considered educated unless they had learned Geometry. That once it was the only Mathematics. It is old, venerable and required. If you can not do well here we are repeatedly told, then maybe you are not ready for….
It is never said clearly, but the message is still sent. If you can not do well here, then you don’t have what it takes to go on! This is a hard message. Many among us, whom we trust and respect as bright and hardworking, are doubting themselves. We try to boost their spirits but serious damage has been done. Their confidence has been shaken. These cracks are very hard to repair.
Finally the 15 months pass and we return to the mainland of the continent of mathematics. Some among us turn back to return to our homeland. There they are received with some respect, but they have failed in their quest. Those who never were even able to go to the island of Geometry hold them in some respect, but that is little comfort.
Those of us who “did well” on the Island of Geometry meet again with the Oracles. They tell us that now the real journey begins. And now the real climbing starts. Next to the Plateau of Algebra 1 begins the low mountains of Algebra 2. We must continue our journey, our quest.
Since we are dutiful. we must go on. Some question why, but now the oracleThe world is again three dimensionals are again silent.
We quickly find that the mountains of Algebra 2 are not so bad. Much that we learned in the foothills of Algebra 1 is very useful. We wonder at times what use our Geometry is on this part of our journey. We think of our friends who turned back and realize that if we hadn’t gone out to the Island of Geometry, they would still be with us.
But there is so much work to do that we don’t dwell on this long. Besides, The Oracles must know what is best for us.
Some among us falter as we climb higher in the mountains of Algebra 2. Their confidence has not recovered from the difficulties in Geometry. They doubt that they can go on. And one by one, we lose other of our friends. And many of these were good students of Algebra 1. They were confident then, but now they are broken spirits.
We pass on without them.
The mountains of Algebra 2 turn out to be mountains in name only. For we now face even tougher climbs. Before us loom the craggy peaks of of Trigonometry.
The Oracles just point and most of us trudge on. Others return home to receive the lesser rewards of those who tried hard but failed.
Trigonometry is a powerful and wonderful world of rarefied heights. The world is again three dimensional. The language of the natives is different but we soon learn it. Much of what we learned in Algebra 1 and 2 is useful to us. By continuing to work hard and believing in ourselves, we press on.The year of climbing pass and we journey through this land. Some of us are slightly injured, all of us are weary of all the years of effort. But many feel strong with all they have accomplished.
That night we pitch camp. Out of the darkness, into the light of our campfire, the Oracles again appear. They look at us and are not smiling. Have we not done well? Have we not done all that they asked?
Yes, they tell us. You have accomplished much. But you are not done. And if you do not accomplish this final task, it will have all been for almost nothing.
The words are like burning swords thrust into our sides. The torment is beyond words.
Slowly we recover our senses. The inevitable question is asked. “What must we do next?” The Oracles tell us that sunrise all will be clear.
We have a fitful night of nightmare dreams. We think of all who have turned back. We dream of our homes and how nice it would be to go there. To be done with all this work and suffering that seems to have no point. Was this the secret message of Geometry.
We awaken to the sun rising. It would be a glorious day if we where just there for the view. But we are not enjoying THIS view! Before us stands a mountain that makes all we have done before seem like so little. It is Mt. Calculus.
The Oracles tell us that this is the goal. This is the culmination of our quest. If we can climb mount Calculus then we can return to our homeland as conquering heroes.
We speak among ourselves. “How can we continue to trust the Oracles?” “What if there is more after this?” Some can not face the task before us. Our ranks are thinned again!
Nothing can fully prepare a person for this task. The Oracles tell us that many will fail. That this is as it must be. Our homeland can not use too many conquering heroes.
We begin the climl.
The air is thin and it is hard to breathe. Even at rest, one seems to use energy just being on this mountain. We are light headed and unsure of ourselves. Derivatives and Integrations swim in our minds.
The guides that have been assigned to us by the Oracles seem to know the way but they won’t make it clear. They jeer at us, if we question this madness. They challenge us to turn back. Go home if you are not tough enough. We don’t really want you here. The women among us took an unmerciful mental beating. They were repeatedly told that they belonged somewhere else. That they weren’t tough enough. That they didn’t have the talent for such a task. Some broke under this relentless pressure. But others took to the task with a resolve that was chilling to observe.
Their eyes said, “Nothing will stop me!” And so we struggled on. Some were better equipped for this climb than others. For those lucky ones, the directions of the guides actually made sense. Many of these would later become guides themselves. Just as merciless to the later generations.
But for most of us the climb was just brutalizing work. Never sure of what was to come next or why. The mountain became our life. We helped each other as we could. Puzzling over the strange explanations the guides gave us, we made some sense out of their words. Teaching ourselves as we went, we passed the tests the guides gave us. We rarely got the answers completely correct. But in their condescending way, we learned to love the words, ” partial credit will be given.” They allowed us to continue. For they held our destinies in their hands. They told us it was a privilege to suffer like this. They had to endure this pain and so must we. Their eyes held no solace. If we could not take the pain, they were more than willing to let us fail and turn back.
For months we toiled, making what seemed to be progress. Our guides continued their never ending harangues of “explanation.” Occasionally one of us would have the temerity to ask why we must we make this climb? Why must we learn these hard lessons?
The retribution was swift! Such questions were greeted with a tirade of invective that would wither even the hardiest among us. Those who looked back at Sodom and Gomorrah faired better those who asked our guides WHY! These questions were asked less and less.
On we toiled.
Then one day, a miracle happened! The guides smiled at us. We looked at each other and assumed that our fates must be sealed. Something truly horrific must be about to happen.
Just then, the Oracles walked out of the mist. The mountain had been shrouded in this cloud cover every day. The Oracles gathered us together. Some of us could barely move. Bleeding and battered, this had been no beauty contest. Many of us had only made progress these last few weeks by crawling. We had long since ceased begging for mercy.
Now the Oracles spoke gently, almost reverently, to us. “You have finished your task!”
Just then the sun broke through the clouds. What we had assumed to be just another shoulder on this endless mountain was actually the summit.
When the sun broke upon us, it was as if the Heavens had opened. We glorified in the wonderful warmth. The Oracles now told us many wonderful things.
“You have all harbored dreams for your lives. These dreams will now come to pass.”
“Some of you have wanted to be doctors. It will be allowed. Some of you wanted to be architects, engineers, scientists,veterinarians and other professions, it will be allowed.”
“For you have proven your worth. When you return to your homeland it will be different for you few. You will be given respect like no others. You will be allowed to live in glorious homes and receive all the most wonderful things your country offers, that others will not. This is the reason that you set forth on this quest. You had to prove that you were worthy of receiving all of the societies rewards.
Although we didn’t all see how conquering Mt. Calculus should allow us all these wonderful things, we surely didn’t want to argue. And those of us who actually enjoyed the climbing were returning as guides. They already had that terrible look in their eyes. As guardians of the summit, they too would be merciless.
Only the truly worthy would achieve the peak under their leadership. For Mt. Calculus is the academic Holy Grail!

Note: Our society does use math as a filter. If you can not complete college level Calculus, the door remains closed on many of the most profound and rewarding life paths. It might not seem fair or relevant to decide who pursues medicine, optometry, veterinary and other professions based on a skill that they will rarely if ever use. But it is a fact and students need to know it and be mentally toughened to face the task ahead. Many math teachers unfortunately see themselves as ‘guardians of the summit’ and whose task is to “weed out those who are not truly worthy”, not truly math gifted!
Until the situation in our society changes, this story will remain very relevant.


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