In the year 2000, Arizona voters said that education was important the sustainability of Arizona’s economy and society. They voted in Prop 301 which promised to keep teacher salaries competitive by providing cost of living increases and performance pay, among other things. The state legislature has failed to exact the will of the voters and has instead acted on “their will.”
A lawsuit was filed to restore the missing funding for education and the state’s response was to propose Prop 123, which would borrow money from the land trust. This was spun as a way to “pump money into education,” but in fact would settle the bill for $0.07 on the dollar owed to the state’s voters, in order to fulfill Prop 301. The ruse worked and the proposition passed … but was ultimately determined illegal by a federal judge.
Now we find ourselves with a teacher shortage, one that threatens to be a true crisis. The short version of the story is that teachers are making less take-home today than they were 5 years ago. Adjusting for inflation, teachers made substantially more a decade ago, and more than that a decade before that! Below is a short video that lays out the situation today:
To learn what the #REDforED movement is all about, here’s a short video, less than a minute long:
To get involved, here are a few links.
The first link is a nonprofit that I have started, which is why content here on The Bearded Math Man has slowed. (We are pretty well up and running, and I have a big project ahead for BMM). Arizona’s Working Poor
There has been a push for computer-based learning in public education for about a decade or so now. The thinking is that students can go at their own pace, have optimally focused and differentiated remediation and instruction, and thus, students will perform better. That’s the sales pitch, anyway.
I teach remedial math courses part time at a community college (the observations made here pertain to all of education not just math), the shift was made so that 100% of these remedial math courses were taught on such computer programs. Students take placement tests where their strengths and weaknesses are accurately identified and they then work their way through lessons and assignments, with help along that way that addresses their specific short-comings. If students grasp something easily they can move quickly through the curriculum. Students that need more time can go at their own pace. At the end of the section (or chapter), students take a test and must show a predetermined level of accuracy before they’re allowed to move forward.
It sounds great, but it doesn’t work. Even if it did work and students could pass these classes in a way that prepared them for higher level classes, it would be a failure. The purpose of education is not future education.
The ugly truth here is that we’ve lost sight of the purpose of education. Education has become a numbers game where schools receive funding based on graduation rates and percentages of students passing multiple choice tests that have mysterious grading schemes behind them (70 multiple choice questions will be graded on a scale of 450 points, for example). We lull ourselves into believing we are servicing our students if they graduate or our school surpasses the state average on these tests.
The truth is that the quality of education is rapidly decreasing, seemingly in direct response to the remedies that seek to reverse this trend.
The question often asked by students, in minor rebellion to the tasks at hand in class, “When am I going to use this in my real life,” needs to be carefully considered, with honesty, by the public and by educators.
The particular skills and facts being tested are of little to no importance. What is important is the ability to be teachable, the ability to learn, which requires a lot of maturation, determination, focus and effort. The purpose of education is to create an adaptable person that can readily latch onto pertinent information and apply previous learning in new ways. An educated person should have the skills to adapt to an unknown future, a future where they are empowered to make decisions about the direction of their own lives.
Absolutely none of that happens in a computer course. The problems are static, scripted and the programs are full of basic “If-Then” commands. If a student misses this question, send them here. There’s no interpretation of why a student missed. There’s no consideration of the student as a sentient being, but instead they are reduced to a right or a wrong response.
What do students gain from computer courses? They gain those specific skills, the exact skills and knowledge that will serve little to no purpose at all in their lives after school. But, they’ll gain those skills in a setting with a higher student-teacher ratio (fewer teachers, less students), and where the teachers need not know the subject or how to teach. That’s right, it’s cheaper!
But the cost is enormous. Students will be trained how to pass tests on the computer, but will not be receiving an education. They will not develop the interpersonal skills required to be successful in college or in the work place. They will not develop as people. They will miss the experiences that separate education from training. They will be raised by computers that try to distill education down to right and wrong answers, where reward is offered for reciting facts and information without analysis, without learning to consider opposing points of view, without learning how to be challenged on what it is they think and believe.
Our youth deserve better. They deserve more.
Not only that, our young teachers (and we have an increasingly inexperienced work force in education), deserve better support from within education. Here in Arizona the attitude from the government is that the act of teaching has little to no value, certainly little to no skill, and that anybody can step in and perform the duties of teaching in a way that services the needs of young people.
And while those in education throw their hands up in disgust, they follow suit by finding quick, easy and cheap solutions to the ever-expanding problem of lack of quality education, especially here in Arizona. Instead of providing meaningful professional development and support for teachers, teachers are blamed for their short comings. Instead of being coached and developed, they are being replaced by something cheaper and quicker, something that is fully compliant.
I fully believe that a teacher that can be replaced by a computer should be. I also believe that a computer cannot provide the inspiration, motivation, the example, mentorsing and support that young people need.
The objection to my point of view is that teachers aren’t being replaced, they are still in contact with students. This is true, the contact exists, but in a different capacity. Just like iPads haven’t replaced parents, the quality of parenting has suffered. The appeal of having a child engaged, and not misbehaving, because they are on a computer, or iPad, is undeniable. But the purpose of parenting is not to find ways for children to leave them alone. Similar, the role of education is to to find ways to get kids to sit down and pass multiple tests. Children are difficult to deal with. Limiting that difficulty does not mean you are better fulfilling your duty to the young!
The role of a teacher in a computer-based course is far removed from the role of a teacher in a traditional classroom. While students are “learning” from a computer, the role of the “teacher” is to monitor for cheating and to make sure students stay off of social media sites. Sometimes policies are in place where teachers quantitatively evaluate the amount of notes a student has taken to help it seem like a student is performing student-like tasks.
Students learning on computer are policed by teachers. The relationship becomes one of subjects being compliant with authority.
The most powerful tool a teacher has is the human connection with students. That connection can help a student that sees no value in studying History appreciate the meaning behind those list of events in the textbook. A teacher can contextualize and make relevant information inaccessible to young learners, opening up a new world of thinking and appreciation for them. None of that is tested of course.
A teacher inspired me to become a math teacher, not because of her passion for math, but because of how she conducted her business as a teacher. Before that I wished to work in the Game and Fish Department, perhaps as a game warden. That would have been a wonderful career. Consider though, over the last decade, I have had countless students express their appreciation of how I changed their thinking about math, how I made it something dynamic and fluid, something human. Math went from a barrier, in the way of dreams, to a platform, upon which successful can be realized. Those things happened because of human connection.
Focus on Conceptual Understanding
Teaching by concept alone will lead to inefficiencies in students. They will, in effect, be reinventing a large part of the wheel at every turn. (See what I did there?) We have all witness what focus on procedure alone does. It leaves students will a bunch of isolated skills that they do not recognize out of context. Out of context here literally means changing the font or using a different set of variables.
An example is the topic/skill of finding the lowest common multiple of greatest common factor. Students are well versed in many procedures, yet of course, mix the two up. That is, they’ll claim a GCF (greatest common factor) is a LCM (lowest common multiple). This is NOT their fault. They don’t understand the difference between a multiple and a factor. They don’t see how those two are applied in other mathematical calculations, even though in order to perform the majority of operations with fractions, those are required.
The focus in education has shifted, and like large bodies do, they swing too far. More than likely the focus has been too great on concept and avoidance of procedure and rote memory of basic math facts. That’s a discussion for another time.
I’d like to help you, the teacher, strike a good balance. Unlike big publishers or professional development companies, I am in the classroom, trying these methods with all of my topics and a wide variety of students. It is highly successful.
One key component of the success is removing yourself from the role of, “The Human Wikipedia,” in the room. Think of yourself more as a coach than a teacher. The knowledge you possess cannot be possessed by the students simply by you telling or explaining what you know to them. They must experience it themselves and grapple with the misconceptions to make sense of things. You’re a facilitator of discussions and explorations, and quite importantly, you’re a guide. No need to chase too many rabbit holes. When a level of understanding is achieved it is up to you to help bring closure, probably through a discussion and writing activity where students write down their explanations of what they’ve learned. Then, that’s when homework changes from uncovering misconceptions to solidifying understanding and making efficient processes that are repeatable.
I’ve harped on many of those things in the past. If you have questions about any of those ways in which homework is used to help learning, please feel free to leave a comment or send an email.
With all of that said, let’s get into it. The chart at below is a general idea of how concept can be established and explored, how procedures can be introduced as a way of generalizing patterns and features of the concept, and last, how that concept can be used to introduce a connecting concept, or consequence of that concept.
Here’s the idea. The rectangular shapes are lessons, or whole group discussions. Everything with an arrow is student work where your job is to encourage and direct. Typically, it is a bad idea to explain things during this time. Instead, encourage students to find other students in the room that they trust that might be able to explain what it is that’s confusing them.
Another big idea during this time is to encourage students to articulate what it is that is confusing them. When students say, “I don’t get it,” they’re helpless. They’re not even thinking about what is causing trouble. By forcing them to reflect on what’s causing the trouble, they’ll likely find their way through the confusion. For you to step in and let them off the hook will only make them have to face that point of confusion later, and it will be bigger and the nature of the confusion will be less clear to them.
A great topic to use an example of this works is exponents. All of the “rules” of exponents come from the idea that exponents are repeated multiplication, of the same number. The difficulty in exponents comes from students inability to read the notation properly, especially when groups are involved.
Let’s briefly explore how this chart can help guide your planning with something like exponents.
Concept: Introduce the notation, perhaps tying it in to how multiplication is written to describe repeated addition of the same number.
3 × 5 = 3 + 3 + 3 + 3 + 3
35 = 3 × 3 × 3 × 3 × 3
Some conceptual questions would be things like providing three different expressions written with exponents and having the students pick the two that are the same. Another way to do this is to give the students an expression and then give them a choice of five other expressions, often which may contain more than one equivalent expression, and have the students pick which match.
During such matching activities keep in mind that the students having the right answer is not necessarily a reflection of understanding. Without the proper explanation, accurate and concise, they likely do not know. Their results of being right will not be repeatable.
Also, when exploring things like this, tell the students that they should write down the examples, but students that will learn will focus most of their notes on their thoughts and questions. This is especially true since we are NOT discussing procedure.
(If you’d like to see some examples of these types of conceptual questions you can find them in the PowerPoint attached here.)
During the questioning of concepts you should chase misconceptions and show how they do not match up with what is true. Always focus on the fact that it is through mistakes that students are learning. Thank students, praise them for participating even when they’re not sure they’re right. We all hate being wrong, and students are often insecure and fear being judged harshly for being wrong.
After exploring the misconceptions and then finding patterns and developing some procedure it is a good time for them to practice what they’ve learned, AKA, homework.
When reviewing the homework the next day make sure things are determined right or wrong by referring to the concept, not finding mistakes in procedure. Of course some refinement of procedure is appropriate when reviewing homework, but that should be for the sake of efficiency, not understanding! This is likely a huge shift for teacher and student!
An in-class, open note pop-quiz is a good follow up, depending on the ability of the students and complexity of the topic. If I were to do such an activity, I would make sure the grades are not too punitive, providing credit to those that correct errors, or perhaps grade it like homework, on completion, not correctness.
If that in-class pop-quiz doesn’t work, a subsequent, more complicated homework assignment is in order. This next assignment should change the language of what’s being learned. Rephrase instructions or change some of the look of the problems so that students are not finding false clues by recognizing patterns in the problems themselves that have more to do with you, or the author of the work, than the concept at hand.
It is also a good idea to throw a few problems that tie into the next topic in, stretch problems, you could call them. Use reviewing these problems to introduce the next concept. I often do this without telling the students the new lesson has begun. It works well because students should be taking notes on their homework assignment in pen (not erasing mistakes but instead annotating them).
Two observations about these practices.
Student involvement is key. Of course, students don’t learn if they’re not involved, but their involvement is less needed for a tradition, stand up and lecture while students take notes, type of classroom setting. These methods are truly student focused and student driven.
As the teacher you must anticipate the questions and points of confusion. Do not have answers at the ready, but perhaps simple problems that students can explore so they can discover clarity. Be ready to show a consequence of their misconceptions or perhaps a problem that simplifies their misconception so they can see it.
Textbooks are woefully inadequate as a resource here. You need many books and resources in order to provide students with exposure to concepts, conceptual problems, and different levels of practice problems (the last practice problems can often come from books). The last set of problems, the stretch problems that connect what they’ve learned with what is coming next I have never seen in a textbook.
You’re going to have to be creative. I am trying to publish my materials and questions as I go through this year, but even so, they relate closely to my interpretation and view of the topic, the heuristic framework I developed. Yours is likely different and so the ways in which you can stretch understanding or expose misconception will vary slightly.
I hope this has been helpful. It is something I hope to explore more fully and deeply. Whenever I have been able to employ these methods the results have been powerful. Students learn and they retain their learning. I’ve been refining these methods over the past six years or so and my students have realized great success from it.
I thank you again for reading and hope this helps. Please let me know what questions you have, just leave a comment.
The following is highly contentious. Many of the situations discussed here should ultimately be considered on an individual basis. The purpose of this is not to create a rubber-stamp solution to all problems that arise with grade assignment and student ability and or performance, but is to provide a general framework so that those individual decisions can be made in fairness and with respect to what is best for the student.
In a previous post I asked about a student in summer school that obviously knew Algebra 1 (he earned 100% on his quizzes and tests), but failed during the year because he didn’t do his classwork. The question is, Does he deserve to fail Algebra 1?
When you flip the situation around it is equally interesting. There are many kids who work hard, but do not really understand or learn the math. Do they deserve to pass based on the merits of effort?
The real issue with both of these situations is what grades mean, or what should they mean. When I worked at Cochise Community College I adopted their definition of letter grades which is described below:
A – Mastery
B – Fluency
C – Proficiency
D – Lacking Proficiency
Those are clean and inoffensive definitions of grades. A student with an A has mastered the material. To be fluent means you can navigate the materials but not without error. To be proficient means you can get the job done, but there are some gaps in ability, but the student can demonstrate a measurable level of command of all of the objectives. Students who earn a D are not able to demonstrate proficiency.
A student who struggles with the material does not deserve an A, even if they worked harder than those who earned an A. This might seem unfair, but unless the objective of the class is to teach the value of hard work, to reward the hardworking, but barely proficient, student with a label of mastery is to cheat the student and cheapen the merit of your class.
Do these definitions mean that a lazy kid that get 95% on the final exam deserves an A, but that a hard working kid that gets a 52% on the same final deserves an F? I say, with a few qualifications, yes.
Is this really fair to the student who works hard but has not yet realized an appropriate level of mastery to be awarded a passing grade? (I used the phrase, “has not yet,” instead of, “cannot,” to acknowledge the belief that students can learn, and if they are motivated and working, the only question will be the time scale of when they learn the material.)
I would say, for a math class, that the best thing that can happen is they are awarded the appropriate grade, an F. Consider if this student is given a passing grade and the class is a prerequisite course? They’re truly set up for failure in the subsequent class.
There is perhaps no worse example of bad teaching that remains within legals bounds than to inappropriately assign grades to students. If a student deserves a C based on ability, but is given an A based on effort, they will believe they are doing everything right and do not need to improve in order to achieve similar success in subsequent courses.
But to give a student who possesses mastery a failing grade in a class because of lack of work ethic is to teach the student that passing classes is a matter of compliance. Behave and you’ll be rewarded. Those kids are taught that grades are not a reflection of knowledge or ability, and that means that education is not about learning. To me, this is an injustice.
I do not believe in the efficacy of these objective lessons. That would be, failing a student based on the notion that they do not deserve to pass because they are lazy. I believe that given meaningful and challenging opportunities, most of these highly intelligent, but seemingly lazy, students will show themselves to be hard working with amazing focus and direction and incredible capacity for quality work.
What about percentages. Is it appropriate that an 80% is a B, if a B means fluency?
When I first began teaching I would have said, absolutely, a student does not deserve an A if they scored an 87% on their test. Since then I’ve changed my mind. Some topics require higher than 90% accuracy to be awarded an A, while with other topics, mastery might be far below 90%.
The level of complexity, variability of solutions and length of assessment all must be considered. This is why sometimes a grading rubric is far superior to assigning grades based on a percentage of correctness or completion.
I teach a curriculum that is designed and tested by Cambridge University, the IGCSE test is what students take. They have a very different way of assigning and defining grades than we use here in the United States. Without going into details about how they do the specifics, they assign large portions of credit based on evidence of appropriate thinking. In other words, if a student demonstrates understanding they will receive passing credit. But, to achieve a high grade, mastery is truly measured. And yet, in math at least, the percentages of correctness for mastery are usually in the mid-70’s. This is because the nature of the questions asked are often non-procedural and the method of solution is not clear, students cannot be trained on how to answer the questions they face on IGCSE exams.
How Do Students Earn Grades
How a student can earn a grade varies, or should, depending on subject and age, and perhaps even minor topic within the subject. I believe that separating student work into weighted categories is an appropriate method of helping make transparent to the student how their grade will be assigned. It also by-passes the tricky question of, “What is a point?” For me, a homework assignment is worth 5 points, they’re assigned daily, except Fridays, for a total of 20 points for the week. Yet, a quiz might only be worth 12 points, but will be a far more accurate representation of student’s ability on the topic.
By assigning weights to the categories, this can be easily balanced. This begs the question, how do you weight the categories?
But what about the student who works, performs all assigned tasks, but can only demonstrate a level of understanding best described as “Lacking Proficiency?” Shouldn’t hard work be rewarded?
And whatever your beliefs on these questions, would your opinion change depending on the age of the student, or perhaps the subject? Should a Chemistry student be rewarded for effort in the same way they’d be rewarded for effort in a Dance class?
At some point, nobody cares about potential or effort. If a child’s mother wants his room clean, she knows he has the potential to clean it, but if he fails to do so, the potential matters not. And if he’s really trying to get it done, but cannot master the discipline to carry through the task, does the effort really matter?
Here is how I set up my grades for high school. It is nuanced and complicated, but I’ll give the outline. Note that for college classes I use a different system.
In high school I weigh categories of grades and have changed the percentages and categories over time until I settled on what seems to work best. These work for my students because it seems to motivate the lazy-smart students and also rewards the hardworking – low aptitude student, because if they remain persistent, they will learn.
I believe extra credit should be awarded for students that perhaps help others, or for extraordinary performance. However, a student should NOT be allowed to raise their grade through extra credit. That is, at the end of the term a student is given a pile of work, that if performed, will raise their grade. This is bad teaching!
The difference between a quiz and a test is similar to the difference between a doctor’s check-up versus an autopsy. The quiz is a chance to see how things are going and adjust accordingly. The test is final. In high school I award credit for homework based on completion, but do not accept late homework.
While I wish that effort equaled success, it doesn’t always work that way…depending on how you define success. For example, I can try as hard as possible to paint a world-famous landscape, but will likely fail if my measure of success is producing a world-famous piece of art. That said, I believe there is a reward beyond measure only discovered with true effort. Our potential, our best, is not static, it changes. It changes in respect to our current level of effort. We can never fulfill our potential, you see. It is always slightly above how hard we are trying. So, if you’re not really trying, your potential decreases, but if you’re pushing your limits, the limits themselves stretch. That is the real downfall of those with an inherent talent that never learn to push themselves. Their potential decreases, dropping down to just higher than their level of effort.
I greatly reward effort, encourage it and makes positive examples of how effort promotes success. However, I do not assign grades to effort. How hard someone needs to try in a given subject to be successful varies entirely upon the student’s aptitude. And suppose you have a truly gifted student, they could be great, if they learn to work hard, right?
Well, perhaps, but there’s more than work ethic involved in greatness. What role does passion play? Take a great young musician and over-structure their training and practice, they’ll burn out. You’ll snuff their passion.
I asked the boy whose situation started this whole conversation if he felt he deserved to be in summer school. Before he answered I explained that I didn’t have an expected answer, I didn’t really know if he belonged in summer school or not. Without hesitation, he said he did deserve summer school, because, he said, he was lazy.
So maybe the kid will learn that if he’s lazy he gets punished. But he also learns that grades are arbitrary, with respect to ability.
I do not like objective lessons, do not believe them to be effective. I prefer a punishment that fits the crime, but also one that redirects the offender, allows them to correct their action.
I cannot say in this child’s case specifically, I was not there and I am not judging his teacher, but perhaps a quicker punishment that redirected him could have also taught him that being lazy was unacceptable and at the same time also allowed him to see grades as a reflection of his abilities.
All that said, this is highly contentious and varies incredibly depending on particular situations of students.
Let me know what you think, agree or disagree. Leave me a comment.
It is often the case,
for the mathematically-insecure, that the slightest point of confusion can
completely undermine their determination.
Consider a beginning Algebra student that is learning how to evaluate functions
A confident student is
likely to make the same error as the insecure student, but their reactions will
be totally different. Below would be a
typical incorrect answer that students will make:
The correct answer is
3, and the mistake is that -22 = -4, because it is really subtract
two-squared. And when students make this mistake it provides a great chance to
help them learn to read math, especially how exponents are written and what
Here’s what the
students actually read:
A confident student
will be receptive to this without much encouragement from you. However, the insecure student will completely
shut down, having found validation of their worst fears about their future in
There are times when
leaving traps for students is a great way to expose a misconception, and in
those cases, preemptively trying to prevent them from making the mistake would
actually, in the long run, be counter-productive. Students would likely be mimicking what’s
being taught, but would never uncover their misconception through correct
answer getting. Mistakes are a huge part
of learning and good math teaching is not about getting kids to avoid wrong
answers, but instead to learn from them.
But there are times
when explaining a common mistake, rooted in some prerequisite knowledge, is
worth uncovering ahead of time. This -22
squared is one of those things, in my opinion, that is appropriately explained
before the mistakes are made.
Math is something that cannot be taught, but can be learned. Yet, math is taught in a top-down style, as if access to information will make a student successful, and remediation is rehearsal of that same information. Earnest students copy down everything, exactly like the teacher has written on the board, but often still struggle and fail to comprehend what is happening. I argue that if copying things down was a worthy exercise, why not just copy the textbook, cover to cover. Of course such an activity would yield little benefit at all because math is a thing you do more than it is a thing you know. Math is only partly knowledge based and the facts are rarely the issue that causes trouble for students. I’d like to propose that you, either parent, student, administrator or teacher, considers math in a different light and perhaps with some adjustment the subject that caused such frustration will be a source of celebration.
There are many things that cannot be taught but can be learned. A few examples are riding a bike, playing an instrument, creative writing and teaching. Without question knowledge is a key component to all of these things, but it is rarely the limiting factor to success or performance. Instead, the skill involved is usually the greatest limiting factor. I argue that to learn these things a series of mistakes, incrementally increasing in complexity, must be made in order to learn. Let’s see if this will make more sense with a pair of analogies.
First, watching someone perform something that is largely skill-based is of little use. Consider driving a car. A fifteen year old child has spent their entire life observing other people drive. And yet, when they get behind the wheel for the first time, they’re hopelessly dangerous to themselves and all others on, or just near, the roads!
Second considering learning to ride a bike. Sure, the parts of the bike are explained to the child, but they have to get on and try on their own. The actually learning doesn’t really occur until the parent lets go (letting go is huge!) and the child rolls along for a few feet until they fall over. Eventually they get the hang of the balance but then crash because they don’t know how to stop. After they master braking they crash because they don’t know how to turn. And then speed, terrain, and other obstacles get thrown in the mix. Each skill must be mastered in order. Preemptively explaining the skills, or practicing them out of context does not help the child learn to ride a bike. They must make the mistakes, reflect, adjust and try again.
What a math teacher can provide is the information required, but more importantly feedback, direction and encouragement. If a student understands that making mistakes isn’t just part of learning, but that a mistake is the opportunity to learn (and without it only imitation has occurred), and a teacher helps provide guidance, encouragement and feedback, then both parties will experience far greater success. When a math teacher completes a problem for a student it is similar to an adult taking the bicycle away from the child and riding it for them. When a student gives up on a problem, it’s as if they stopped the car and got out, allowing the adult to drive them home.
The job of math teacher is perhaps a bad arrangement of words. Coach, mentor or sponsor is perhaps more appropriate. There is no magic series of words, chanted under any circumstance, that will enlighten a struggling student. The frustration making mistakes should be cast in a different light, a positive light. The responsibility of learning is entirely on the student. They cannot look to teachers, friends or tutors for much beyond explanation of facts.
In a future post I will explain how too much direction and top-down teaching of math promotes failure of retention and inability to apply skills in new applications. But for now, please consider that math cannot be taught. A teacher cannot teach it, but can help a student to learn.
Thank you for reading,