Episode 6: Mastery

In this episode we discuss how to develop mastery, differentiate for your students, and what it is about teaching that forces us to improve our own skill with concepts and procedures.   By forcing yourself to tighten up loose thinking, fill in conceptual gaps, uncover and repair misconceptions in order to teach a topic, we become masters.  That is exactly what is required of our students if we wish them to master their content.   This episode is all about how to make that happen while pulling up those reluctant learns!

A lesson was mention in the episode.  The PowerPoint for that lesson can be downloaded by clicking on the icon below.  A printable copy of the homework can be downloaded by clicking provided in the PowerPoint.

On Teaching Math Podcast
Show Notes: Episode 6

The Problem with Mastery


Let me ask you a question.  When you stepped into the classroom for your first day of teaching fulltime, why weren’t you a mastery teacher?


Let me ask you another question.  Has there been a topic in math that you thought you had mastered, only to learn, through teaching, that there was a lot more to learn about it?




Today’s episode is a little different.  Instead of talking about a specific topic in math, we are going to talk about a philosophical misstep, if you will, in our beliefs about teaching.  We are going to see how this misconception sets us on an insidious cycle of failing to bring up our low students while also failing to truly challenge our top students.  But we’re not just going to be poking holes in education, we’re going to see why those holes exist and articulate an approach responds appropriately … to make you a better teacher.


Here’s the faulty, pillar, of education.  I have heard this echoed in education since my first days teaching.  “We teach to mastery.” 


Did you get A’s in your education courses?  I bet so.  The letter grade A should signify mastery.  Yet, it doesn’t.  Why not?  Because we can’t be taught something to mastery.  Mastery happens over time, with diligence, personal accountability, introspection and response to failure.  That can happen in an academic setting, but it is a result of a personal challenge taken up by a student.  As teachers we can provide structure for mastery to be developed, we can encourage and nurture it, but we can’t teach to it.


Maybe you think I’m splitting hairs here … but let me ask you another question.  Have you ever gone into a lesson (or unit), thinking you knew it, that you were a master of that content?  Yet, when you got into teaching it you found out you didn’t know everything about it there was to know about it.  If you had, the teaching might have gone much smoother.


Yet, after some time teaching that content you get better at teaching it, but also get better with the content itself.


Why is that?  What is it about teaching that shows the truth?  Why is trying to teach something a more accurate assessment of the level of mastery than a test on that topic?


I believe the answer is related to the ability to articulate.  When we cannot articulate our thoughts, they’re loose, sloppy, and likely incomplete or have faulty foundations.  In forcing ourselves to articulate an understanding we come to find gaps or faults in our understanding. 


It is that very reason that I define to my students that a grade of an A is mastery, and that mastery’s key earmark is the ability to articulate understanding.


Now, for the first question:  Why is it that when you first stepped into the classroom you were not a master teacher, despite grades earned in college? 


Is it possible that even in the most stringent grading system that a test, or any sort, cannot truly determine mastery?  Mastery is too nuanced and multi-faceted to be ascertained in a classroom setting.


So then, how can we teach to mastery?


I say we cannot.  But, we can teach to proficiency.  And proficiency is a wonderful thing.  I mean, being proficient at things in life is wonderful.  If you possess proficiency with a handful of key skills, you’ll be extremely successful in life!


So how does all of this apply to what’s happening our classes, right? 


By suggesting we teach to mastery, the bar of success is inappropriate.  As a result, our responses to failing to achieve our markers of success are further inappropriate.  We double-down on the source of the problem because we think that is the solution. 


What happens, typically, is we get a group of students that get it, right away.  They need to be challenged further. So we either give them harder problems, or give them access to more advanced materials.  They move at a faster pace.


The kids that don’t get it… we back up, slow down, and tell them again. 


The kids that move on, they’re not learning what they kind of got any deeper, or better.  In fact, they likely had a surface level understanding.  By moving on, whatever understanding they had will likely be lost, at least partially, and will not be strengthened.


The kids that fall behind did not do so because they didn’t have access to the information.  Telling them again, in new words, won’t help.  They need help integrating the information.


To summarize the problem.  We are not serving our advanced students by moving them more quickly through curriculum.  Going faster is not better.  Sure, they’re continually challenged, but their surface level understanding is not challenged.  They don’t know more deeply, they just no more broadly.


Our lower achieving students fall further and further behind because the reason they struggle to make progress is not appropriately addressed.  They’re not integrating the new information correctly.  They have access to the information, just don’t know what to do with it.


Here’s what I’m proposing.  When a student learns a concept and procedure in class, they’re not masters.  They’re merely proficient.  To move from proficient towards mastery requires a few things. First, the individual must be eager and able to take on the challenge of digging deeper.  Second, they must have the opportunities that will expose their gaps and misunderstandings that whatever measure of success they’ve performed so far failed to do.


In other words, when a student gets it, early on in a class, they don’t need to move on.  They need to move deeper.  They need to be challenged with questions that they never considered before, much like we did when we stepped in to teach logarithms, for example. 


I was floored that students failed to understand logarithms as well as I did once I presented them with the information.  It was well paced, clear, clean, and correct.  The results from the students were anything but clear, clean and correct. 


It was in learning to appreciate what they saw, learns to appreciate their misconceptions led me to deeper understanding of the topic.  It is this experience that we need to provide for our top-level students.


By having those students that “get it,” more quickly available to provide assistance to those that struggle, we give those top kids opportunity to more deeply develop their understanding.  They will come away having uncovered their own gaps and misconceptions, much like we did when we began teaching.


But what about those low kids that don’t get it?   The dynamic between peers is far different.  When peers hold each other accountable and give feedback as to why the struggle persist, it is not coming from an authority figure, but from a peer.  It is better received.


Beyond that, when a struggling student seeks help, they’ve taken a huge step towards resolution of difficulty.  They’ve owned their role and are seeking to take control.


Here’s how I set it up. 


Say we are learning about cyclic quadrilaterals in a geometry class.  All of the facts have been presented and now it is time for students to apply the information to correctly answer questions. 


Kids that get answers right haven’t mastered the material, and depending on how easy it was, they might just as soon forget as they did gain the knowledge.  But they’re ahead of those that are struggling to make sense of it all. 


By asking students to first check with other students for correctness, and then those that are right to seek someone who needs help, and those that need help to seek someone that is right, differentiation that is appropriate has been established!


Now of course the students all need coaching on how to work together.  They need to know what good, quality help is, and what harmful help is.  And all that is not possible if the behavioral pieces are not in place.


The only reason students are in class is to learn.  Nobody gets up early to catch the bus to hear Johnny cut jokes.  It might be entertaining, but it’s not worth the price of admission.

Instead, Johnny could be working to either get better, or to help someone else get better.  The message is that, “If you can’t work with Johnny because you two get off topic too much, then choose someone else.  If I have to choose for you, I will … but it is less effective.  Then you’re doing what you’re supposed to do because I made you.  You’ll serve that command, instead of serving your purpose of being here which is to learn.”

Through that type of conversation I get kids to be wise and smart about who they work with.  It’s a tricky thing and everyone will do it differently.  I could get into that in more detail in another episode.


But once students are coached on how to pick partners, and how to behave when they’re offered time to collaborate, coaching the quality of the collaboration is required.


If a student is told what to write, and they just write it, they’re not learning.  It is when students can determine on their own what to do that they’re learning.  So it is important to coach the students on this.  If they’re a helper, the moment they tell the other student exactly what to do is the moment the learning is done.  They’re just complying at this point, not thinking.  Instead, help them to identify what they do know, and then help them figure out how to use what they do know. 


If they’re getting help, they cannot allow themselves to settle for, “I don’t get it.”  That’s a dead-end.  They must be able to articulate the source of the trouble, what it is they’re failing to understand and so on.  It is often the case that with that very articulation of confusion comes insight.


Through this collaboration, students on all levels learn to communicate mathematically.  They learn to speak and listen, and to empathize with others.  It creates a positive, team-feeling atmosphere. 


One of my classroom mottos is, “A rising tide floats all ships.”  When students begin helping each other do well, they see that it helps themselves improve.  As such, over a short period of time students develop a vested interest in the success of the class as a whole.  Those that struggle are pulled up by those that are advanced.  The advanced students are challenged to articulate their thinking, which requires formalization of thinking and filling and correction of gaps and holes in understanding.


And, perhaps best of all, your role as a teacher is changed.  You change from the person responsible for all people learning, and your primary role in addressing that charge is the dissemination of information.  What you change into is a facilitator of education.  You set up opportunities for students to learn and then guide them through it, each kid performing at their own level, but together.


Any teacher worth their salt knows that good teaching has little to do with clear explanation and answering questions in ways students understanding.  Good teaching is helping students to integrate information, to make sense of what’s been learned.