I recently did a lesson on the basics of reading and writing in Geometry. You know, dry, dull stuff...what's a point, line, ray, segment, how do you write an angle, what types of angles are there, and so on.
While preparing all of this information I was thinking:
How can I expose misconceptions about such material so they learn it?
Remember, just seeing the facts is comfortable for students, but not only do they not learn, they somehow find confirmation that their held misconceptions are in fact correct. It's not as wild as you think, and it's not limited to kids. I took a psychology class in college and was unknowingly part of an experiment. I was asked a question, a seemingly throw-away type. But it's trickier than it looks and nearly everybody answers wrong. But it was of such little consequence that I did not remember my answer (you weren't supposed to). Then, I was shown the correct answer and asked if that's what I had said.
Turns out the vast majority of people mis-remember that they answered correctly. That is, they answered it wrong, but it's hard for us to imagine we're wrong, and they latch on the to the idea they were right...even when it's quite obvious they weren't.
This is so powerful that to be wrong and be aware of it, being confronted with things we don't understand and such, is very uncomfortable and unpleasant. Yet, that's what is needed for learning to occur. (And I'm talking the type of knowledge where understanding is paramount to success.)
My assertions are that what Derek Muller has unconverted here goes beyond science and film.
Students are not void of knowledge in your content. They have ideas. Teaching them is more like part repair work on the frame of a house before roofing. Presenting students with correct information will not shore up their misunderstandings.
Also, students need to experience some level of cognitive discord. In education, nearly all of the "best practices" work hard to do the opposite of this. There are things like Content/Language Objectives, or SWBAT, word walls and graphic organizers. I'm not saying those things don't have their place, but that's it, they have a place when balanced with quality instruction that explores misconceptions and such.
When you can deliver a lesson that explores the misconceptions the students will be confused. But if it is student lead, they won't be lost. The amount of mental effort required is much higher than a typical delivery of information and note-taking style. However, they'll learn!
So, how to create this tension and expose misconception over some pretty dull information?
Start by asking questions and exploring answers. Do not use your authority in the subject to state if an answer is right or wrong, initially. Instead, have students share their thinking on what other students are saying.
For example, a particularly nasty question that dealt with the boring definition-based lesson I just gave was, "What is an angle?" To someone versed in geometry, this isn't a big deal. But to a kid who hasn't taken geometry, this is monumentally difficult to describe. The best response I received was, "Measuring the space between two lines." So, of course, I drew to parallel lines and asked for explanation.
Now, this is just something I wonder, but is it possible that on these boring, just the facts, type lessons that exposing misconception is more important than ever?
Regardless of how that fleshes out, challenge yourself to challenge the thinking of students by exposing misconception through dialogue. Be brave enough to explore misconception and encourage students to seek understanding by challenging the think of themselves and others. If students understand the purpose of your methods, they'll play along.
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.
Grading papers is one of the most time-consuming responsibilities of teaching. Hours upon hours can be, I argue, wasted, pouring over daily homework assignments. This article will discuss how to integrate awarding credit for daily assignments in a way that saves hours of time while increasing your awareness of student progress, increases student completion rates and better informs students regarding their progress in the subject.
This routine described here is a daily variety, not how I grade quizzes, tests or projects. However, there are some tips that apply to recording those grades later in this article.
Let’s begin with a question: What is the purpose of homework? For me, it’s practice needed for students to gain proficiency. Homework is about trying things, working out how to struggle through difficult problems and making, and learning from, mistakes. Without effective homework, students will not integrate their learning into a body of knowledge that they can draw upon for application or just recall.
The breadth of the purpose of homework and how that purpose is best served is beyond the scope of this article, but I would like to suggest that homework is something done in their notes, whenever possible. The reason being is that notes are a receipt of their learning, to be reviewed in the future to help remember observations and important facts.
Overview of How It Works:
At the beginning of class, often before the bell rings, I begin walking around the classroom stamping homework that deserves full credit. (What merits full credit is up to your discretion, but it should be a clear and consistent expectation, known to students.) As I circle the room, I look for common mistakes, ask kids if they have questions or difficulties and make small talk.
Students that didn’t do, or complete, the homework have to answer for it on the spot!
Then, I simply mark those that did not receive credit for the homework on a student roster I keep on a clip board. (For a video of how this works, visit the link here: )
Quick Notes: This method has students ready for class because they have their notes. They’ve also asked me questions if they had any, so I can begin with meaningful review. I also have forced students that are remiss to account for their actions and done so in a way that applies positive peer pressure. The scores are recorded by leaving blanks for completion and only marking those that do not get credit (which will be very few).
Credit: I award full credit or zero credit when checking homework. If a student attempted all problems, with evidence of attempt demonstrated by work shown and questions written, they get full credit. Those that fail to receive full credit have the opportunity to reclaim 80% (the percentage is arbitrary but again needs to be consistent, clear and known by all), the students must see me during tutoring time by the Friday of the week of the assignment to show that they’ve fulfilled the expectation. Students that did not attempt the homework can also see me during tutoring time (before or after school, not between class times or lunch), and receive partial credit.
But the rule of being due the Friday of the week assigned is big. The purpose of homework is practice. Without proper practice skills and knowledge are not developed. Homework is not about compliance and fulfilling an expectation with a grade as a reward. Students that are hustling to complete homework from two months prior are likely not promoting their understanding of current materials. Plus, by having the time requirement applied to the homework policy, students are not enabled to fall too far behind.
The added bonus is that you will not be buried with make-up work the last week before grades are due to be reported!
Work to be Turned In: If the nature of the work is not something that can be kept and must be turned in, have the students pass their work forward by row. As you collect each row’s stack, count them. If a row’s stack is incomplete, ask who in the row didn’t turn in the work.
If students can NOT fulfill the expectation and only receive a bad grade from it, and that reprimand comes well after the unwanted behavior, they will quite happily go along thinking nothing bad is going to happen. Having to answer, publically, for their lack of work, especially when the vast majority will work, is a powerful deterrent! Just as when checking the work of students and asking those who failed to complete for an explanation, this keeps them accountable and will increase the amount of students completing their work.
When collecting the papers, alternate the direction of the stacks and do not mix them up when grading. This will allow you to quick return the papers after you’ve been done. If it is a daily practice type of work turned in, I’d suggest awarding full or no credit and only recording, again on the printed class roster, those that were awarded no credit.
Recording Grades: Whether you’ve collected daily practice or are carefully grading quizzes and tests, how you record those grades can either waste your time, or greatly reduce the amount of time spent.
By recording each grade as it is calculated by hand on the student roster it is quick and easy to transfer them to the computer. This is a huge time-saving practice. You don’t need to hunt on the computer screen for each student, and do so for each assignment. When they’re recorded by hand, you can simply enter the column of numbers in the computer. When the last name lines up with the last number that you entered, you know they’re all entered correctly.
By following this method, the data entry side of grading is done in a few moments of time instead of over hours, working through those stacks of papers, again!
Final Thoughts: By looking at, and discussing, homework with students on an individual basis, very briefly, you gain insight into their progress. They get a chance to ask questions. Students that need a little bit of motivation receive it as an immediate consequence for poor behavior, rather than waiting until the end of the quarter, when a lot of pressure will be placed on you to help them bring up their grades.
This routine has proven to be a cornerstone of my classroom management. It gives me a way to set the expectation that we are here to learn and that learning is done through work and reflection. Students that need discipline receive it immediately and in a way they find uncomfortable, but it is done so with an invitation that guides them to the desired behavior (of completing their work).
As a salty veteran teacher it is almost sweet seeing the hopeful expectation in the eyes of new teachers. They've just graduated college and they are ready to fix education. Thing is, there is much to learn that's not covered in college. I'd like to share some of those things with you. Whether you're a salty veteran or wet-behind-the-ears, I think there's something here for you.
Number 1: The Most Important Skill for Teachers
There is no better skill for a teacher than the ability to get along with others. This is especially true for those teaching high school. In high school you'll be navigating around 150 students a day, all with blossoming personalities, body odor, love-interests, extravagant behavior and mood-swings. If you can't find it in yourself to be gracious for the outrageous behaviors, you'll be in for an unpleasant career.
The thing I always try to remember is that I would NOT want to be judged today for the person I was when I was 15 years old.
Number 2: Say NO to Your Boss
This is probably the most powerful for new teachers, but all can be victims of being over-worked. It's true, there's a great need for man-power at a high school. Class sponsors, club sponsors, coaches, curriculum projects, prom, after school activities and so on are all roles that need to be filled. The eager, the new, the young and energetic ... well, they're the group most likely to say yes when asked to take on these tasks, so they'll likely be asked first.
But new teachers are the last who should be taking on these additional duties.
You have a limited bandwidth and the more you try to do with that bandwidth, the lower the quality. Plus, there's a STEEP learning curve to teaching. The first year should be spent doing nothing but learning how to teach, refining your procedures and practices. Seriously, spend a lot of energy focusing on how to be efficient and effective.
Saying No to your boss isn't easy, but you can manage. You won't get fired, they need you. Just explain that you don't want to take on more than you can handle. Once you've got a strong grip on the teaching side of things you'll explore taking on other duties.
Number 3: Don't Grade Everything
Just because students did it doesn't mean you have to grade it. Sometimes participation or completion is all that needs to be noted. Think of it this way...the purpose of them working is to promote their learning. If grading doesn't inform students about their progress (are they even going to consider why they were marked wrong?), and if it doesn't provide meaningful insight for you regarding their progress, then why grade?
And often reviewing the materials completed by students as a class is far more informative to both you and the students than sitting at a desk looking through each problem, making notes for the students and recording all of the scores.
Number 4: Don't Try and Pacify Parents
If a parent is upset, let them be upset. If you have a good structure for how their child earns their grade, stick with it. "Johnny is failing because Johnny hasn't done homework. Because he hasn't done homework he hasn't learned and he fails the quizzes. Johnny fails to take advantage of the remediation offered for his quizzes and then fails the tests. At the end of each class I can get Johnny to understand what he needs to understand. But then he is responsible for performing the assignments to make his learning permanent."
Again, if parents are upset about grades, stick to your guns. Whatever your late policy is, stick with it. I personally do NOT allow late homework past the Friday of the week it was assigned. End of story, not open for discussion.
Use this line: "We can't fix the past, can only use the lessons learned from those mistakes to inform our future actions."
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,