Best Practices

Philosophy and Best Practices

Cart Before the Horse

The easy part of teaching any curriculum is the curriculum.  The hard part of learning to teach a new curriculum is figuring out how to teach it most effectively.  For the sake of clarity, effective teaching will develop student conceptual understanding and problem solving.  Earmarks of quality teaching include retention, connection, engagement, and thoughtful discussion among students, as well as high test scores of course.

One of the problems with modern education and education research is the need to quantify everything.  The adage, If it can’t be measured it cannot be improved, has bound us with conclusions based on extremely soft data, nonsensical results, and has engaged us in an insidious pattern of behavior.  That same pattern is easily seen in students, and it drives teachers crazy.

If a student is in a class, then that class should cover new material for them.  If a student gets a good grade, they will have learned the material.  Grades measure learning and gained ability.  Students that focus on the grade in lieu of learning struggle to get good grades.  They have confused the marker of success with success.

Is there a more frustrating question to a teacher, from a student, than, “What can I do to bring my grade up?”  Isn’t the answer almost always, “… just about anything will work, so long as it involves learning.”

That is exactly what is happening today in education.  In order to conduct “scientific” research we quantify markers of success and measure them.  We lose sight of what is really important and chase what has been measured.  Teachers jump through hoops to create word walls, daily and in-depth content and language objectives, the good old SWBAT, and perform remediation that teaches students how to get quiz questions correct.

All of these actions are designed to show an increase in the markers of success but potentially do so in direct conflict of the true goal of education.  Our job as educators is to train the minds of young people.  Many best practices ease the cognitive strain to promote short-term positive results.

The end result is that between 40% to 60% of first year college students need to take remedial courses in English or Mathematics, or both.  Let us explore how this happens.

Use retention as an example.  Suppose a teacher is being coached that good teaching requires that students remember the material in say, four months.  That teacher will take steps to make sure that students remember.  There will be tricks, rhymes, and rewards for getting good test results on retention.

The problem is, this will not be the type of retention we are trying to develop.  The retention our students need is a consequence of conceptual understanding, not brute-force memorization.  With conceptual understanding comes the ability to reconstruct lost memories.  That is a great tool!  No matter what, the likelihood of losing information over time is astronomically likely.  Besides that, if a student only remembers the fact or procedure, their ability to use that fact or procedure will be bound by how they’ve been taught to use it.

Retention is just an earmark of conceptual understanding.  There’s more that comes along with conceptual understanding, like the increasing pace of integration of new material due to connections made, and improved engagement and problem solving skills.  And, perhaps most importantly, when students conceptually understand, they’re not bound by what we have taught them, they can take what they know and go further than we did!

Not everything in education has artificial restrictions placed upon it by inappropriate data seeking.  Climate and culture, for example, are two areas of focus for administrators.  Yet, climate and culture would be incredibly difficult to measure … but you can feel it the moment you walk onto a campus!  The same goes for relationship building between a teacher and the students.

Take a minute and imagine how artificial markers of determining the existence of a quality relationship between a teacher and student could be made.  Now if the efficacy of your teaching was being evaluated on this false metric, and if that metric had been engrained in the culture of education, it is likely you would be engaged in the best practices of the day that would increase the frequency of that marker.  Yet, would you really be building relationships?  What if you believed you were and you were not, and the system told you that you were doing well?  How would you know you were wrong if the system told you that you were right?

Baby and Bath Water

Now it is likely that quality outcomes are realized by chasing faulty markers of success.  For example, if you learn how many siblings each of your students has, you’ll likely open the door to conversation and find connection.  If a student attends tutoring to bring up their grades and they participate, they’ll likely learn.  By learning their grades will improve, even if they believe the act of attending tutoring is what brought about the improvement.

There is nothing inherently wrong with a Content and Language Object, or Learning Target, or SWBAT, or whatever it may be called at your school.  In fact, objectives are really needed when designing and implementing a lesson.  Without the target the lesson will meander.

Before deciding that a particular best practice is a false metric and deciding to abandon it or treating it as a matter of compliance, first see if that false metric can be employed to foster the desired outcome.  What should this best practice really build and why?  Can you make it serve that end?

A great example is allowing students to re-take a test.  If a student takes the same exactly test, especially after teacher-led remediation, of course the student will improve their grade.  But, did they really learn the content, did they develop a conceptual understanding?  Did they improve their problem solving skills?

Most likely, the conceptual understanding remained unchanged but the problem solving skills would have damaged.  It is a bad outcome when students learn that the way they solve problems is by seeking someone else to solve the problem (unless they’re bound for politics).

That does not mean that allowing students to re-test is a bad practice.  But it is a bad practice when done as described.  What if, in order for a student to re-test, the student had to demonstrate their conceptual understanding and problem solving before they re-tested, and then the second test was different than the first? (Changing the numbers on a test does not make it different.)  This would serve the student’s needs and encourage them to perform well.  It would provide you with data that could be used to more accurately assign a grade.  It is all around a great outcome!

How and Why

It is how and why we do what we do that makes it worth-while.  That adds an additional layer of complexity to the profession of teaching the United States today.  A teacher can follow all of the directives and engage in all the best practices as assigned by their leadership team.  They can have wonderful reteach and differentiation strategies, the best posted objectives, standards referenced lesson plans informed by classroom data, and formative assessments published by respected companies.  In their classroom you may see varied questioning techniques, students getting up and performing tasks, and the teacher “owning the room.”

I was that high school math teacher.  I was energetic, my students loved my class and proclaimed I made math fun for them.

I was also teaching remedial courses at the local community college.  When I began seeing some of those same students in my classes, I realized I was not effective.  The reason I wasn’t effective is I was putting on a show.  Sure, I followed the best practices of the day, but I failed to see what was important.

In the next few blogs, I will break down what I have discovered to be best practices that promote student conceptual understanding and improve problem solving skills.  I will try to explain why I believe they work so that you can adapt them to your needs.

Student Skills and Tools

The biggest hurdle in transitioning from Middle School to High School is the lacking set of student skills possessed by incoming Freshmen.  Students come in failing to appreciate the importance of homework, struggle to think independently, cannot communicate mathematical thinking, and are easily frustrated to the point of quitting.

This observation is not a knock on the students’ experiences in Middle School.  It is entirely likely that the brain of a 12 to 14 year old cannot develop these skills.

In the upcoming 2019/20 school year I will be running an experimental program where I use SMART Goals focused on student skills to hasten the development of those lacking student skills.  The pay-off could be huge…the development of quality student skills would transcend the classroom, even school.  Ultimately, student skills are goal-oriented problem solving and personal management skills.

Here’s how it is going to work.  During the first week of school I will teach students what SMART Goals are (read about them here if you don’t know: https://www.yourcoach.be/en/coaching-tools/smart-goal-setting.php).  We will practice setting small SMART Goals in order to learn what is required, and how to foster them.

During the first week I will also teach students what quality student skills are.  I’ve made a reference sheet of what they are, what they look like in action, and how they’re beneficial.

Perhaps the most important thing taught in the first week will be how motivation drives engagement.  If a student is deeply engaged in their studies, they’ll persevere and be successful.  The two types of motivation, intrinsic and extrinsic, are directly related to the quality of engagement.  A student motivated by reward, or fear, from grades is extrinsically motivated.  They’ll easily give up and will engage in their work at a shallow level.  Their mindset is to complete required work.  A student that is intrinsically motivated is motivated to learn.  They engage deeply and seek learning.  They persevere and find learning opportunities in their work.

At the end of the first week of school students will draft individual SMART Goals that focus on student skills.  I have created a four-week long form where students will be guided through the reflection, monitoring and fostering required to have those goals come to fruition.

If you’d like to see the documents I’ve created, they are here.  Here is the Student Skill Sheet:  https://drive.google.com/file/d/1TCDKiwWrU-Ycoc1JILbOVd_y0f5_VkxR/view?usp=sharing

Here is the Smart Goal Planner:  https://drive.google.com/file/d/1mrZtEM3sAUcYkU5pxLdatnyn9DjVqT4k/view?usp=sharing

If you’d like to follow along with how this goes, you can read my blog:  http://thebeardedmathman.com/home/blog/

 

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.

#REDforED – Arizona’s Working Poor – Arizona Educators United

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

Arizona Educators United

Save Our Schools – Arizona

AZED101

Our Youth Deserve Better – Computer Based Learning

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.

We owe our youth more.  They deserve better.

It is time to unplug.

Math is Hard

Math is Hard

A typical conversation with a failing math student, with a failing math student’s parents, or with a counselor or administrator about a failing math student either directly sites this, or is pulled in a direction like driftwood in a tide by the fact that math is hard.

A common conference would go something like:

Parents:  Why is my child failing math?

Me:  Well, let’s ask your child.  Why are you failing math?

Child:  Because math is hard.

Parents and other interested parties accept this as sufficient reason and place the onus back on me as though I can alleviate the very nature of the subject.

I am completely fed up with the observation that math is hard.  And while refraining from profanity in response to this excuse should award me man of the year, I get it.  Math is hard.  No kidding!

It doesn’t matter what innate abilities someone has in math, eventually it will become difficult, confusing and … well, hard!  It is something everybody that learns math must face.  They must learn how to learn something that is hard, demanding and elusive.  That whole experience of, “Oh, I got it,” and then ten seconds later, “Wait a minute, I don’t get it anymore,” is something we all suffer.

When I was taking math courses in college I was certainly challenged.  At one point a formal proofs/topology class was really destroying me.  It was designed to be a bit of a gate-keeper of a course.  If you failed to posses the ethic and fortitude required to be successful in mathematics, this class would ferret out such things.

While taking this class my birthday rolled around.  I am the oldest grandchild on my father’s side and share my birthdate with my grandmother.  I am the oldest grandson.  So birthday parties are kind of a special thing for the two of us!

At the party I showed up with a small dry-erase board, a marker, rag for erasing and my book.  I didn’t have an assignment, no test coming up soon, but was well aware that I “didn’t get it.”  While friends and family hang out enjoying themselves I sat in a room with the door closed and practiced.

To be clear, I wasn’t struggling for mastery, I wasn’t fighting to get an A.  I was struggling just to get by, just to get a C in the class.

So yeah, math is hard.  Education changes you, or it should.  I’d argue if it was easy and didn’t change you, what is the purpose?  Sometimes you have to fight to get things done.

Think math is hard, try beating addiction.  How about facing cancer?  Raise children.

The difference between those that get math and those that don’t is a simple one…some are fighters while others site difficulty as sufficient reason to surrender and quit.  While that may sound harsh, there’s a little more to it than just that.

Fighters have faith and patience.  They have faith that through perseverance they will overcome.  They have the patience to persevere through hard times, knowing that it will pass and the result will be worth the endeavor.

By facing the struggles presented in math that perspective can be gained.  If math is hard for you it offers you an opportunity to learn that if you persevere, keep faith and have patience with yourself you will overcome.

Accessing Prior Knowledge in a Way That Uncovers Misconception – A Lesson

Accessing Prior Knowledge in a Way That Uncovers Misconception - A Lesson

If what we are doing in class today does not promote our understanding tomorrow, we have wasted our time.

Click to download

The order of operations and arithmetic with signed numbers combine to be the downfall of many high school math students.  As with many things, remediation does not work.  While exposing mistakes with these topics in applied contexts can be powerful, that also sometimes leaves students with the wrong impression that their new conceptual understanding is flawed.

For example, say a student is graphing a polynomial, factorable, function and they make sign errors.  Their roots might be off and they’ll likely consider their new understanding to be flawed when in fact they’re just adding incorrectly.

With careful discussion and in an environment that encourages exploring mistakes, this is a fantastic way to shore up weaknesses that are prerequisite in nature.  However, with a shy student or even a new student, this often just leads to frustration.

I’ve attached a link to a free PowerPoint activity/lesson that is designed to get students to explore combinations of arithmetic operations and the order of operations to arrive at an answer.  It challenges their understanding of both integer operations and the order of operations in a way that does not just leave them wrong, but empowers them to change what they’re doing and make use of their wrong answers.

The activity uses the scary clown from the Saw movies.  He wants to play a game.  If you’ve never seen those movies they’re likely not your taste, but teens love them!  The, playing a game, with this reference is powerful to them.

The game is that there are four-fours with spaces between, and they equal a number, like 1.  Students can add in whatever operation signs and or parentheses they want in order to make the total equal 1.

An example is:      4  4  4  4  = 3

Students may add:  + - × ÷ [  ]

So an attempt might be:  -[4÷4]+4÷4

This is of course wrong, but there will be more gained from discussing why it is wrong than simply sharing what is right.

There are often many proper solutions, and if a student gets an answer quickly, have them see how many they can find.  Another thing to do with a student that gets it too quickly is have them help another student, but not in a way where they share the answers but instead their thinking, helping the struggling student to arrive at their own answers.

Depending on the ability and enthusiasm of the class you can take one of the problems in the lesson and write four or five of that problem on the board and have people come up to the board and write in their arithmetic notation and parentheses.  Then when all of the slots are filled, you can discuss what’s right and wrong, and why, as a class.

This would also make a fantastic white-board activity, although recording thoughts and realizations in their notes is very important, because:

If what we are doing in class today does not promote our understanding tomorrow, we have wasted our time.

 

Teaching, Learning and Habits

How Habits
and
Education Collide

 

The best definition I have come across for a habit is, “action without thought.”  A quick search on the internet says that a habit is, a settled or regular tendency or practice, especially on that is hard to give up …

We certainly need habits, especially in education.  Students, in order to be successful, need to be in the habit of being on time, having their homework done properly, whatever the classroom norms and expectations are need to habitual.  In other words, the day to day activities of school should be done automatically, without thought or the student needing to be reminded.

And certainly we want students to be in habits when it comes to performance.  For example, putting their name on their paper, showing appropriate work, employing effective questioning strategies and the like all end in higher levels of academic performance.

But what about what they’re learning.  Are we teaching them habits, that is, the action without the thought?  I say that we definitely are, and that is in direct conflict with the purpose of education.  That purpose is to give people the opportunity to learn how to think in a safe environment where the messiness that comes from the process of learning to think does not have major consequences.

As with most thinks related to teaching, this is highly nuanced and subjective, and there are certainly times where teaching a kid a habit that leads to a right answer or desired outcome is best.  That’s part of what makes education so powerful is that you can learn from what others before have done and take the next step, right?

What makes this double tricky is that we grade the results of habits.  Can a student see a prompt and spit out an appropriate output?  If so, they’ve obviously learned, right?

If you’re an expert in the field you’re teaching, you most likely approach problems at the level you’re teaching habitually.  Little reference to the ideas at play is required for you to arrive at a solution.

If you’re not an expert but have enough background to teach the topic, you’ve probably brushed up with some Khan Academy videos or the like, where you were shown those efficient methods and techniques that are the ways the expert acting habitually would do.

If a student is able to pass a standardized test they must also possess these habits.  However, if they’re taught the actions without thought, the process alone, they have no way to connect what they’re doing to other things.

Let’s consider how thinking and problem solving really works.  After all, learning to think is the purpose of education, right?  It’s highly unlikely that any student will have a practical use for 90% of the materials learned in your class.  But the learning that takes place, that is entirely useful and practical!

In thinking and problem solving the issue at hand must have a level of novelty.  If not, a habitual approach will be successful and little thinking will take place.  The problem must first be grappled with and understood and then the person dealing with this task can generate some ideas.  These ideas are the conceptual understanding of the task at hand.  From these ideas come the actions, the steps taken.  Upon completion review of the entire undertaking is performed and if the outcome was desirable, success can be claimed.

Often it is the case that not only is success claimed, but all similar problems now have a heuristic background.  Upon further review and generalization and actual procedure can be articulated.

Since the procedure is the measured and share-able portion of this entire development, that is what is written in books and what is measured on tests.

Yet, it all came from a conceptual understanding, an idea.  The idea initiates the procedure.

To not allow students access to the time and level of involvement required to explore ideas and develop heuristic approaches to problems is to rob them of the very purpose of education.  They do not learn how to learn when they are trained to follow steps given a particular input.  That’s training.  Sit Ubu, sit. Good dog!

It is certainly a challenge and uncomfortable for all parties involved to have students develop this level of understanding and explore without explicit direction.  However, it is the absence of such things that has education in the United States in such a terrible predicament.

My challenge to you, the reader, is to pick an overarching, big idea in your topic, something that is coming up next, and develop an activity/problem that will require a lot of thinking and little direction from you.  Make it something where the student result can be assessed as correct or incorrect based on the concepts at play, or by reverting back to the original question itself.

What you’ll find is that the students uncover connections that you have forgotten or taken for granted, or maybe never realized at all.  Over time, with regular activities/lessons like this they will begin to adjust to what is expected of them and they’ll increasingly enjoy actual learning!

Let me know what you think by leaving me a comment.

 

Thank you once again for reading.

Philip Brown

Teaching Conceptual Understanding Flow Chart for Educators

Focus on Conceptual Understanding
Flow Chart
for
Educators

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.

  1. 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.

  2. 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.

Philip Brown

 

Why Remediation Fails

Why Remediation Fails

Students that struggle unwittingly do two things that ensure they continue to struggle with concepts and procedures.  Students can go to tutoring over and again, and sometimes it works, but it’s a long and frustrating journey.

I’ve fallen victim to these two habits myself, we all have.  How students learn in school is not any different than how adults learn outside of school.  Learning is identifying something that’s wrong and replacing it with something that is right, or at least more efficient.

It is the act of identifying something that is wrong that is the hitch here, the hold up.  The first of the things students do when presented with remediation, that is review materials or a review of what went wrong before, is they morph what they’re seeing to fit what they know.  Of course if they did that the other direction, things would be great.  But that’s not how we learn.

It is imperative to recognize that we develop new learning by relating it to old knowledge.  We don’t just replace all that we’ve developed over time with this new thing.  Instead, we create connections between what’s already in our noggins and what is new.  The more connections we have, the stronger the new learning is and the more quickly it happens.

Consider someone learning to cook.  Say, they learned that Worcestershire sauce is yummy and delicious on steak.  Some spills over into potatoes and that’s not too bad either.  It’s not even unpleasant when it mixes with green beans or broccoli. With some experimentation we can learn that it’s good with chicken, rice and mushrooms.

What’s the thing we know?  Worcestershire sauce makes things taste good.  Not wrong, but not a very deep understanding, right?

Now let’s say this person want to make some desserts.  Someone hands them some cream and tells them to whip it up, so it can top a pie.  Why, they might ask.  Well, to make the pie better, of course.

This whipped cream is new information, it’s something different than what they know.  It’s fundamentally different than Worcestershire sauce.  Yet, whipped cream is supposed to make food better, just like Worcestershire sauce does.  So what students do, in effect, is say, oh, whipped cream is the same as Worcestershire sauce, and I’m used to Worcestershire, so let’s just use that instead.  Same thing after all, right?

A similar thing happens when trying to train someone to use the computer.  They know how to do a set of things and try to use those processes to manipulate this new software.

That is, instead of seeing the new protocol for interfacing with the software as completely new, they instead relate it to what they had done in the past.  They fail to replace old knowledge with new.  Instead, they see the new information as the same thing as what they already have at hand.

How do we, as teachers, combat that phenomenon?  Well, we have to expose what they believe as fundamentally different than what’s right.  We have to expose their misconception as being, well, a misconception that is not aligned with reality.

That’s a tricky thing to do, especially in math, for two reasons.  The first reason is that often in math we are dealing with abstractions.  We can’t have them taste Worcestershire topped cherry pie.  The second reason, especially for math, is that when students see a procedure performed, they feel they understand if they believe they’re able to follow that procedure. (That is not that they are able to perform the procedure themselves.)

That second reason that it is tricky to expose misconception is the second thing that students do, they latch onto procedure.  It makes them feel grounded, even if they are obviously off-base!

How many times has this happened?  You, as the teacher, review a quiz question with students.  They sit there, take notes as you work through a problem.  They all exclaim they can’t believe how dumb they are, how could they have missed that?  They get it now, right?

No.  They don’t.  They followed what you did, you doing all of the thinking along the way.  A large percentage of students will be no better off than before the review.  In some ways, some will be worse because they’ll now think they understand.  Before the review, they just knew they were wrong, probably had no idea why.

What can we do?

This is a tricky thing to answer, dependent on too many variables to articulate a clean protocol.  However, I think I have some ideas that will help in general.

First, when developing a review lesson, test or quiz review, or remediation lesson, you need to have students confront some mistakes.  Maybe they need to try a problem and get it wrong.

Once the misconception is exposed, address why it’s wrong, what’s wrong with it.  Don’t discuss what is right immediately, they’ll translate that to fit what they believe (and that is wrong).  Expose why the misconception is in fact wrong, on a fundamental level.

Next, if possible, arrive at the right conclusion without process or procedure.  Is there a way to think through the conception at play and arrive at what is right?  If so, that’s beautiful.

The last thing is that this new learning will be soft in their heads, a fragile thing.  They need to make a record of what they’ve learned, in their own writing, preferably on the old quiz or next to the thing they used to believe was true.  It’ll be a reminder, because they’ll go for that Worcestershire sauce again when they shouldn’t!  Old habits, they die hard!

I tried something along these lines in a video I prepared for a remedial math class at a community college.  The topic is fractions.  I tried to show how common denominators work without treating them like they were stupid, because they’re not, they just never had to learn fractions, and tried to do so without use of a process.

As I explored the inner workings, and why various things were wrong, I began describing what needed to be done, but the focus was conceptual.  The video is posted here at the end of this article.

This is a topic I hope to explore more in detail, how to help promote the efficacy of remediation and tutoring.  I am working on some experiments I’d like to try to determine more closely the behind the scenes workings here.  Until that time, thank you for reading, thank you for your time.

 

Philip Brown