mathematical literacy

Mathematical Literacy

Integrating Literacy Development
Opportunities in Your Instruction

 

A few years ago I attended a professional development seminar designed to help American math teachers integrate best practices and strategies required for their students to be successful with the Cambridge IGCSE program.  I was a guest “reference-source,” in the seminar because of the success my students experienced in the program over the prior six years.

The IGCSE program is, in short, a college preparatory program.  By passing the end of course examinations students can demonstrate college readiness.  In my school they’re even given a high school diploma at the end of their 10th grade year, upon successful completion of the program of course.  Some have even exited the school to attend college during what should’ve been their 11th grade year.

At the end of the seminar participants were invited to ask questions.  A teacher, quite frustrated, asked, “How am I supposed to get my freshmen prepared for calculus by their senior year?  There are too many things to teach and not enough time.”  (What she was getting at is that the Cambridge curriculum is appears sparse compared to typical American curricula.  In 9th and 10th grade there are a total of 10 topics for math.)

The presenter asked me to handle the question.  I knew the answer, but could not articulate my thinking in a concise fashion.  She and I were speaking different languages.  I tried to explain that she didn’t have to teach everything.  It is better to have a solid foundation that can be applied to all of the tangential and “one-off” topics in math, than it is to have brief experience with all of those various topics.  We do not have time for both.  We cannot develop deep understanding of the fundamentals of Algebra and have students exposed to all of the iterations and applications.

All she heard was a know-it-all teacher bloviate about some theoretical ideal.  She needed practical advice.  While I tried to provide that advice, I failed, miserably, to do so.  I realized after writing this article that this information, in this article, is what I should have shared with that teacher.

Her question had a specific context, but I believe it hit the heart of one of the biggest issues faced by mathematics teachers, world-wide.

 

How do I get my students to acquire and retain mathematical thinking?

 

I’m going to offer a two-word solution:  Mathematical Literacy.

If we want our students to really learn mathematics and be flexible enough to apply that knowledge in their futures, they have to be mathematically literate.   Mathematical literacy, for our purposes here, is (1) the ability to decode information from mathematical text and (2) the ability to encode contextually relevant information in mathematical text.

A mathematically literate person can understand mathematics as it is written, but also realizes countless associations, contextual meanings, and tangential ideas.  When a mathematically literate person sees mathematical text, they don’t wonder what should be done.  They read it as information, which is decoded and analyzed.  They are able to articulate appropriate, contextually relevant mathematical responses to information provided.  A student that has developed this literacy is prepared for whatever type of math their futures may hold.  They’re not bound by our efforts, they’re not reliant on what we have directly shown them.  Instead, they’re empowered with the ability to think and communicate mathematically.

The prior two paragraphs are entirely insufficient for defining mathematical literacy.  This article is about developing mathematical literacy, not defining it.  If you’re interested in learning more about what is meant by mathematical literacy, consider listening to the On Teaching Math podcast on mathematical literacy.   You can access the podcast with this link.

The development of mathematically literate students involves two components.  First, students must make sense of problems they’ve never seen, and problems that often expose a misconception created by a person overly reliant on procedural proficiency.   Then, students must apply something they know that is contextually relevant to the problem at hand.  The key component here is they must identify the relevant concept and understanding.  They must make the association.  They cannot be following a mapped-out procedure or following instructions. 

In order for this to happen, students must have a certain degree of conceptual understanding and procedural proficiency.  However, marginal proficiency with both is sufficient.  Through developing literacy, they will improve their conceptual understand and their procedural proficiency.

Warning:  Carefully acclimate students to answering questions designed to improve literacy.  If you do the thinking, instead of teasing it out from them, you can destroy the possibility of developing literacy in students.  As we dive into a few examples we will discuss, in detail, how this works.  But, for now, understand that if you demonstrate how to solve the problem, or answer the question, students will not develop literacy.  In order to develop literacy, students need to bring in relevant conceptual understanding (may or may not be directly related to the topic being taught), and then devise a plan and monitor the appropriateness of their approach as they work through it.

If we, the teachers, make all of the connections and do all of the decision making, we’re the ones exercising our own literacy.  Literacy will not be developed through imitation.

Let’s get into how we can set up experiences for our students that will promote the development of mathematical literacy.  We will use solving simple polynomial equations in Algebra 1 as our initial  testing ground.  View these examples in their spirit, not specific application so that you can begin to craft your own questions and design their implementation.

Suppose your students have been taught how to combine like terms, and then solve simple equations, like 3x + 3 + 4x – 5  = 23.  You can run them through countless pages of practice where they’d see every possible iteration of this type of problem.  But, you’d not be increasing their literacy or developing a deeper conceptual understanding.  That would only promote procedural proficiency, which is of course not well retained over time.

Instead, you could give students a problem like Problem A. 

The problem on its own will not promote literacy.  How you introduce the problem and your expectations of students will promote mathematical literacy.  If you work a similar problem, by changing the numbers, the students will latch onto the procedure.  They will not be pulling in various mathematical understandings they possess that are contextually relevant. 

However, without support, at least initially, students will likely be unable to even approach this type of problem.   The level of performance and thinking required of your students is likely brand new, and foreign to your students.  They will wait for you to show them how it goes, and then try to recreate what they’ve witnessed.  That is exactly what we do not want. 

If this was the first opportunity for my students to develop mathematical literacy, I’d explain my expectation and goal to them first.  The purpose of the problem is not to find an answer, but to develop the ability to understand what is written and draw in previously held understanding.  Once the understanding and associations are complete, students are practicing articulating their thinking mathematically. 

The purpose of this problem is to provide students with experiences that will prepare them for unknown futures.  This is practice that will help make them adaptable by teaching them how to think mathematically. 

A good way to start is to show students the diagram and the information, but not the question.  Ask students to brain storm about what they see, what they know, what comes to mind.  They’ll often be hesitant to state the obvious things, but those obvious things are sometimes the most difficult to see and are sometimes the most important things to notice! 

Once students have collaborated, through whole-class discussion collect and list ideas and observations on the board next to the diagram.  Many kids will have forgotten how perimeter works.  This will be a great time to shore-up that issue.

Then, after all of the observations have been recorded and discussed, show students the question.  Remind them that the steps to be followed are not what is important here.  Creating the steps to be followed is what’s important.   We want students to write mathematically, in appropriate contextual response to information provided.

Unfortunately, once this introduction has been completed, the opportunities to develop literacy with this style of problem are long gone.  The road is familiar.   Students will be remembering the process instead of making mathematical connections.  In response to this, teachers need to have two things at the ready. 

  1. Students coached to fully engage with the problems.  They cannot sit back and wait for the path to be clear.  Finding the path amidst uncertainty is the pursuit.  Once a problem has been explored, the path is found and the goal is no longer attainable. 
  2. You need a bank of problems at the ready!

Here is another, similar, but fundamentally different problem that could be used to follow Problem A.

Of course helping students develop the habits of thinking that will lead to literacy takes time.  You could easily teach students to “do” this problem in a handful of minutes.  Then, you could try to back-fill some meaning.   But then, students are learning how to “do,” the problem.  They’re not getting practice learning how to thinking mathematically.

The pay-off, however, is worth the time spent!  By learning how to make sense of mathematical information, and how to identify contextually important prior knowledge, then articulating their thinking mathematically, students will, over time, learn much more quickly.  They will also strengthen the prior knowledge through these experiences because these experiences provide opportunity to create connections between topics.  All of these benefits together result in greater retention of the new, and old, mathematical concepts.

Let’s see an example that would be appropriate for students at this level that does not involve Geometry.  Again, we are considering a group of students who can distribute and combine like terms, and solve equations in one variable. 

 

There are two boys, John and Bob.  Both boys like to collect colorful rocks.  Bob puts his rocks in his left pocket, which has a hole in it.  John finds half of the rocks that Bob drops.

If Bob found 36 total rocks, and one third fell out of his pocket, how many of Bob’s rocks did John find?

 

There is nothing special about this problem, or the previous two.  What is different is how you introduce the problems and how you coach students to approach the problems.  Encourage brainstorming, making sense of the problems.  Set the expectation that students will need to develop mathematical literacy in your class to be successful.  If it is a true expectation, and you are unwavering, but encouraging, students will develop literacy over time.

Questions that are not directly related to the topic at hand can also be used.  In my podcast, On Teaching Math, I start each episode off with a question like this.  They’re typically easily understood and involve solutions that are within reach of most people, regardless of mathematical prowess.  Also, it is often the case that the answer or discovery made by exploring the question is of little consequence.  But, what is important is that students must create a hypothesis and test it, either through independent exploration or collaboration.  As they test their hypothesis, through reflection they must decide to adjust their or approach, or through validation, continue on. 

A typical question will be:  How many times in a 24-hour period will the hands of a clock create a 90-degree angle?

Another question that is simpler is: Why is 5 the only prime number that is the sum of the previous two prime numbers?

One more example is:  What number less than 100 has the greatest amount of unique prime factors?

These types of problems are a great way to give students experiences that develop mathematical literacy.  The way a person must engage with those problems is the same way a mathematically literate person can engage with our last example.

One last positive outcome from these questions is that a lot of meaning will be exposed. Students will likely discover things you never thought of.  That is a great outcome and a great way to include activities that promote academic discussion into your classroom.

This final example is a favorite question that can be used to develop literacy.  An ancillary benefit is realized for students who failed to obtain the solution.  In review, students will have a deeper understanding of exactly what the concept at hand with this topic really means.

Suppose you’ve taught your students the mechanics of functions.  They can read and perform operations from examining the notation, they can perform function arithmetic, maybe composition of functions, and they can find inverse functions.  I selected the words, can find, here because they indicate procedure, not concept!

Here’s the question:  Given that f(x) = 2x, what is the value of x when f -1(x) = 4?

When I first saw this question on a Cambridge IGCSE examination I thought the question was entirely unfair!  In fact, I was asked by a person outside of my district how kids could solve this.  The students taking the test had no experience with how to find the inverse of the function!  

When the test results were released I was shocked to see that the majority of my students answered the question correctly.  I could not believe it.  Upon questioning, students explained that the question was easy because the input and output for a function and its inverse are reversed.  For example,  if g(2) = 3, then g-1(3) = 2.  So, if the output of the inverse of function f is four, then the input for the function f is four.  Then, f (4) = 24, which is 16. 

Because the students understood the concept and had practice applying concepts in new ways, they were successfully able to answer a difficult question correctly!  To make it even better, they answered a question that I had never dreamt of before.  This is a great example of the power of mathematical literacy.

Let’s pull it all together here.  To develop mathematical literacy students must apply conceptual understanding in non-routine applications.  This will likely be a shift in engagement for students and teachers.  As such, we, the teachers, must orchestrate a series of experiences that will help students make this shift.  We start students off with simple to understand questions that are non-algorithmic in nature, and gradually move to more complicated application of the concepts at hand.  All the while, we increasingly move students to more independent thinking, where they collaborate AFTER they've have created and executed a plan. The pay-off is well worth the time and effort required!  This is absolutely a case where going slow early can speed things up over time! 

Your devotion and consistent application are required to help students develop mathematical literacy.  You will need to incorporate these style of problems and the appropriate pedagogy into your lessons.  Students will need opportunities to practice their literacy on homework, quizzes and tests.  Many of the students will require continual encouragement and reiteration of the relevance of their efforts (why it is important for them, that they develop literacy). 

If your students develop mathematical literacy under your tutelage, then you will have served the future needs of that student well.  They will be prepared for an unknown future because they will be empowered with the ability to think, and communicate, mathematically.

If you are looking for questions that can be used to promote mathematical literacy within the application of a specific topic in math, please leave me a comment below.  I have a large collection of these types of questions built over the years.  

 

 

This

Let’s talk about THE question in a teacher’s life … the baleful, “When am I going to use this in my real life?”

Yeah, that one.

The honest answer is probably, never… and they know it.  Why else ask, if not to subvert and diminish your role and purpose?  They don’t really want an answer.  What they want ... is just to watch you squirm, or to hear what B.S. you might spit out.  Either way, your class is now off the tracks!

However, it is a valid question, in its essence.  What is our purpose here?  It is a fair question, and one that needs to be answered.  And don’t be one of those people that thinks all content is applicable to daily life … it is NOT, nor should it be.  The purpose of education is not to train kids for every possible situation in life, but to equip them with an education so they can adapt to any possible situation.  The purpose of education is the development of the mind.  Sometimes we learn tricky things for the same reason some accountants lift heavy weights at the gym.

So let’s talk about how to change the answer to THE question.  Let’s turn it on its head.  Let’s answer the question: When am I going to use this in my real life, it in a way that swings the pendulum back in your direction.  Let’s answer the question in a way that stops the division and animosity that fosters the question in the first place.

See, the reality is, when a kid asks that question, they’re doing you a favor.  They’re providing you with insight that perhaps you’re serving the wrong this.  And if you’re not, whatever purpose you serve is not apparent to that student!

Let’s back up and take a larger scope view of the situation.  Especially in today’s educational climate, teachers stay in the classroom to be of service to students.  That’s it.  Teachers are blamed for all of the woes of society, for the failings of education, called lazy, and everybody seems to know what they’d do to fix it, if they were a teacher!

That sentiment, why teachers stay in the classroom, is the gateway to changing the answer to THE question.  Teachers are only in the classroom for the benefit of the students.  And surely, a teacher wouldn’t take on the sacrifices they do to stay teaching only to waste the time of their students, right?

Well … no, but kind of yes, too.

Let’s talk about job pressures … failing percentages in your classes, average scores on “high stakes tests.”   Those are big deals!  The test results are used to evaluate schools and teachers.  Administrators can be rewarded or fired on the basis of such things!

After all, good test results must be the sign of a great school.  Bad test results, well, that is really the teacher’s fault!  Yet, if a teacher holds a standard aligned with test results, the class failing rate will be too high, unacceptable, and a sign of bad teaching.  There will be unpleasant parent meetings; counselors, and administrators asking what’s being done to help the student, as if the student is hapless, a victim of the inevitable.

What am I doing to help this student?  I don’t know … showing up to work every day with lesson plans, a warm welcome and words of encouragement?  Oh, and I tell him to pick up his pencil fourteen times an hour, that’s a start, right?

Through either threat or blarney, bean-counters and pencil-pushers outside the classroom press hard to ensure that the teacher is performing due diligence to achieve high test grades.  Parents hover, students object, and through it all teachers are led to one inevitable observation: this is not why I teach.

Do those test scores really matter for students?  Sure, maybe ACT or SAT, AP, IGCSE or IB tests matter.  But those are the culmination of years of work.  Does it matter, to the student, if they pass their local state’s Common Core exam?  Not one bit.

In my real life, everybody will be impressed that I got a 3 on my Common Core State Test in English.

The purpose of an education is not to be able to pass a test.  The purpose of education is the development of the mind.  An educated person should be adaptable, thoughtful, able to communicate and appreciate various points of view other than their own.  An educated person should have perseverance and confidence, creativity and curiosity about the world around them.

A person that is educated should have an enriched life as a byproduct of their education.

When am I going to use this in my real life?

Well, that depends on what you mean by this.

Are you, as an educator, teaching this to help kids pass a test, or get a good grade in your class?  Are you teaching this to help them to know how to do their homework?  If so, there’s no judgement or blame.  Sometimes you have to make concessions just to get through the day.  We want kids to be successful.  The problem is, what are we using as markers of success?

But what if you could make this align with why you come back every year?

If you teach students about factoring polynomials so they can complete a worksheet, and maybe pass a quiz, your this is not powerful.  That is not why you teach.  Why make it what you teach?

The trick is to devise ways to teach kids how to think, to encourage creative problem solving and develop communication … to give them an education, while they learn how to factor a polynomial.

Personally, I never want a student to say to me, “Mr. Brown, you’re the only reason I got through math class.”  That’s too low of a standard.  That is exactly the this that makes THE question so damning to our efforts.  I don’t show up every day so my students can pass a quiz or test, or get a minimal passing grade in math class.

I show up every day to provide a challenge to my students, so they can test themselves and be better tomorrow than they were today.  And by better, I don’t mean greater proficiency at rationalizing the denominator. I mean of better mind.

To me, the best compliment a student can give a teacher is, “You taught me how to learn.”  In learning to learn, all of the pieces of an education are there.  To know how to learn you must be a problem solver, a critical thinker, be reflective, confident, and resourceful. A student that knows how to learn is prepared for an unknown future.

Ask yourself:  By teaching this, what’s being learned?  Are you just rehashing the same old lessons, just giving the same information the students could get on Khan Academy?  Are you asking them questions that can easily be answered by PhotoMath?  Are you printing another worksheet off of KutaSoft?

Challenge yourself to raise the bar.  Forget the bean-counters.  They’ll be happy when they see the results because when a student that knows how to learn takes a silly test, they do well!  Not only that, they’ll stand out when compared with students who were taught the content of the course only.

Unfortunately, if you’ve been dragged to the point where you realize, this is not why I got into teaching, and it consumes your day, you cannot answer THE student’s question honestly without using the word never.

You’re unlikely to find much guidance in the education industry that will change the this in THE question.  The industry sells books and professional development designed to get students to pass the test.  Their livelihood is generated from keeping the this we don’t want in THE question.

It is on us, educators dealing with students daily, to change the this.

The next time a student asks THE question … ask yourself, why?  Why did they ask?  Which this am I serving, the one designed for test scores, or the one educating students?

Thoughts on Teaching

Foundation
Foundation

1. The goal: If the question, "When am I going to use this in my real life," derails your class, there's a problem with your purpose and goal. The truth is, almost nothing after taught 5th grade is knowledge used daily. The purpose of education is not to teach MLA formatting or how to factor a polynomial.

The goal is to develop a careful, thoughtful and resourceful young person that is adaptable, a problem solver and has perseverance. That's the destination. The particular subject serves as (1) the vehicle to arrive at the destination, and (2) an exploration into potential aptitude and interest, (3) as well as a foundation of reference knowledge.

2. Autonomy: When students understand they're in charge of education outcomes and find value and validation from their efforts, they'll perform.

In other words, when they do it for themselves and receive appropriate praise and feedback for progress, their potential and performance will increase.

3. Letting Go: Some kids aren't ready. I barely passed Algebra 1 as a freshman in HS...in fact, I'm sure that 60% final semester grade was rounded generously. Yet, I ended up with a BS in Math.

You, the teacher, cannot reach them all. Leave the door open, realize every misstep is a chance to teach them, but learning is done on their end, not ours.

If a kid fails, let them. Work with them to succeed, but hold firm to the standard. If you falter, and pass a student that didn't deserve it, the value of the accomplishments of other students will be discounted.

Why I'm sharing this is to color this short story:

The last three years I had 100% passing rate by all takers, not cherry picking, on IGCSE, around 10% passing rate in AZ. This year I'm pretty sure at least one student will fail. They earned the first F grade I have assigned in six years in that class.

That student just wasn't ready. At the end, the student came begging to get a passing grade. I explained to the student that while they were close to passing, to change their grade would be a grand insult...it would say that I did not believe they were capable of performing as well as their peers.

The next day the student approached me. I thought, ut oh, more grade grabbing negotiation...but to the student's credit, they just thanked me, said they're glad for the F and will do better in the future. No more crying, no hang-dog look...but instead a confidence because the student was capable and will be in the future. Perhaps now, the student is ready.

I don't want students to say, I only got through math because of you, Mr. Brown. That would make math the destination, not the vehicle. Best compliment a teacher can get is, you taught me to learn.

Confuse Them So They Learn

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.

Give it a shot, let me know how it goes.

Once again, thank you for your time.

What Do Grades Really Mean?

What Do Grades Mean

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.

Tests – 40%
Quizzes – 25%
Homework – 25%
Other – 10%

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.

Rewarding Effort?

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.

Grades

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.  

 

The Smallest Things Can Cause Huge Problems for Students




preemptive


Pre-Emptive Explanation

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
like:





f(
x
)
=3x
x
2

+1




f(
2
)



MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa
qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaaaaaa@43D9@


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:





f(
2
)
=3(
2
)

2
2

+1




f(
2
)
=6+4+1




f(
2
)
=11


MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaeyypa0JaaG4mamaabmaa
baGaaGOmaaGaayjkaiaawMcaaiabgkHiTiaaikdadaahaaWcbeqaai
aaikdaaaGccqGHRaWkcaaIXaaabaGaamOzamaabmaabaGaaGOmaaGa
ayjkaiaawMcaaiabg2da9iaaiAdacqGHRaWkcaaI0aGaey4kaSIaaG
ymaaqaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaacqGH9aqp
caaIXaGaaGymaaaaaa@4F4E@


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
they mean.

Here’s what the
students actually read:





f(
x
)
=3x
x
2

+1




f(
2
)
=3(
2
)
+
(

2

)

2

+1


MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGMb
WaaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0JaaG4maiaadIha
cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaa
qaaiaadAgadaqadaqaaiaaikdaaiaawIcacaGLPaaacqGH9aqpcaaI
ZaWaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacq
GHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa
ey4kaSIaaGymaaaaaa@4E85@


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
mathematics.

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.

 

How to Save Time Grading

 

How to Grade Efficiently

and Promote Assignment Completion

 

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).

 

Things NOT Taught in Teaching College

What College Should Teach You About Teaching

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."

Number 5:  You Only Need 2 Pairs of Pants (men)

Monday wear pair 1.  Tuesday wear pair 2.  Wednesday wear pair 1.  Thursday wear pair 2.  Friday is usually casual day, wear jeans.  DONE!

 

Math Can Not Be Taught, Only Learned

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,

The Bearded Math Man