Monday, February 29, 2016

Mastery (Part 2)

Details of Practical Mastery
·         Mastery Quizzes (short quizzes)
o   Mastery quizzes can be done electronically like on Moodle or can be done on paper and pencil.  The key to the individual lesson mastery quiz is to keep them short.  A typical lesson mastery quiz will have at most 4 to 5 questions and be focused on a limited number of concepts.  At Byron our mastery quizzes and tests are paper/pencil since students do not have access to the internet when they take our state assessments every spring, the ACT, or their college placement tests, and we want to prepare our students for those assessments. 
o   One of the challenges of mastery quizzes is making similar but different versions of a quiz.  You do not want to just change the numbers because students will memorize the process versus understanding the concepts.
o   After a student takes a mastery quiz, you want to provide feedback on that quiz ASAP, ideally within a minute or two since they are still emotionally invested in the quiz. 
o   Grade mastery quizzes on a 1, 2, 3, or 4 scale.  4 is perfect, 3 is proficient, 2 is some understanding of the material with room to grow, 1 is lots of opportunity to improve.
o   If a student does not do well on a quiz, for example a 2 out of 4, you still need to celebrate with the students on what they know and help focus their energies toward what they don’t know.  This explicit identification of students know and do not know is empowering as it helps them see success and focus efforts appropriately.
o   Encourage students to strive for a 4 on all mastery quizzes.  When students are pushed to get 4’s on all mastery quizzes, they do a lot better on the unit tests assuming your mastery quizzes are aligned to your unit tests
o   Students are not allowed to retake a mastery quiz on the same day they just went over their mastery quiz.  Students need a sleep cycle between going over their mastery quiz and retaking it.  We also recommended that students learn and practice the material one day then wait until the next day to take a mastery quiz.  This encourages a deeper learning and longer retention of the material. 
o   Whether you are quizzing online or paper pencil mastery quizzes, be sure to avoid the line of students waiting for you.  Students may be in line to get a mastery check, have you grade a mastery check, go over the mastery check with you, or ask a question on an assignment.  But if students are standing in line, they are wasting their class-time.  You will need to figure out a process that works for you to avoid the line that can easily occur.
·         You will need to change the physical setup of your room and have one area specifically for the mastery quizzes with another area for group work or individual work.  We have one row of desks set up for students who want to take mastery quizzes then tables or groups of desks setup for students to work on learning and practicing concepts.
·         Provide a pacing guide but encourage students work ahead since they will come across harder sections that take more time and/or may be gone someday.
·         In class encourage students to focus and make good use of every minute.  For many students that means if they focus and make good use of their time then they will not need to watch any videos or complete any math problems outside of class while others will need to do some homework for math to stay on pace.
·         Encourage all students to retake mastery quizzes on test review days, including students who got all fours on their mastery quizzes.  Supporting and encouraging long-term retention of concepts is a key goal.
·         On final review days, students are allowed to retake unit tests to improve their score, again demonstrating retention and mastery of concepts.
Benefits of Mastery
·         The culture of a mastery classroom is different.  Students have more of a growth mindset or the “I can do it if I work at it”, “not yet”, or “Try Fail, Try Fail, Try Again then Success” mindset.  In a non-mastery classroom if a student failed an assessment, it reinforces the student’s idea that “I am not good at math.” But in a mastery classroom if  students fail an assessment, they have the attitude that they are going to go back and relearn the material so that I can master the concepts.
·         There is a big change in the conversations within the mastery classroom.  The student-to-student conversations or the student-to-teacher conversations are more focused on learning the material rather than just getting the assignment done like is often true in a non-mastery classroom.
·         Students view the homework problems as a tool for learning the material versus something they have to get done even if they do not learn anything from it.
·         Students in a mastery classroom take more pride and ownership of their own learning.
·         When a student does fail your course, you have an exact record of what they know and what they do not know.  So it is relatively easy to get their grade up to a C if they want to.  This can easily be done by having them attend your “summer school” for only a few days and focus specifically on the concepts they do not understand versus needing to retake the whole course next school year or in full summer school session.

Monday, February 22, 2016

Mastery (Part 1)

The mastery approach allows students to work through material at their own pace.
Students will learn a concept/topic then when they are ready, they take an assessment on that concept.  If they earn 80% or above on that concept, they move on to the next concept and repeat.  If they scored less than 80% on the assessment, they need to go back and relearn the material and try the assessment again.  Homework is not part of the grade, and students view it as a tool to learn the concepts.  There are a couple of types of mastery.  The first is true mastery, and the second is practical mastery.

True Mastery
True mastery is the same as what is described above.  Students are given as much time as they need to get through the material.   A few students may only need half a year to get through a full year long course then they can either move on to the next math class or take an elective.  Most students will finish the year long course in a year while a few students may need to 1.5 years to mastery all the concepts in a course.  With true mastery it is easy for a math teacher to have several math “classes” throughout the day, but within each math “class” they have students in Algebra 1, Geometry, Algebra 2, and maybe even Pre-Calculus.   There is no need to have separate homogeneous classes since students are all working at their own pace.  Students just need a place to work on mathematics, and a teacher who can help them when they have questions, need an assessment, or need an assessment graded.  The Algebros (led by Spencer Bean , Michael Brust , Timothy Kelly , and Corey Sullivan , who all teach in different US Department of Defense schools in Germany) for the most part use this approach.  Their web site is  One difference from true mastery that the Algebros implement is that students who do not complete a yearlong course by the end of the year are required to attend summer school to get caught up.
Practical Mastery
Practical mastery is what the Byron High School mathematics department use and what a lot of schools use.  The true mastery does not work for a lot of schools because of either required seat time and/or limitations of the individual school’s master schedule (See note below for more details on why true mastery does not work in most schools).  
With practical mastery there is a set test date by which all students must take the unit test.  This is beneficial for most students since they are more motivated to get concepts completed when there is a deadline.  We give our students a recommended schedule they should follow for when they should be taking each mastery quiz for the unit.  Students who get behind sometimes need extra motivation while others need extra help from the teacher or just more time on particular topics.

Flipped Learning and Mastery (True Mastery or Practical Mastery) work well together.  Flipped Mastery starts with students watching the video and taking notes on a concept followed by practicing problems.  Once they feel confident they know the material from doing a self-check they take a mastery quiz.  Depending on how the students do on the mastery quiz they will either move onto the next section or go back and relearn the material.

Visti  and select Mastery Learning for for more information.

Tuesday, February 9, 2016

Solving Absolute Value Equations and Inequalities from a Graphical and Algebraic Point of View.

Solving Absolute Value Equations and Inequalities from a Graphical and Algebraic Point of View.
I have struggled with helping my students understand how to solve absolute value equations and inequalities with only one variable like |2x – 8| = 4, |2x – 8| > 4, or |2x – 8| < 4.  Some of my students in the past have understood it while others have not.  Most of the time they forget or do not understand to solve half the problem.  The will solve the 2x – 8 = 4 or    2x – 8 > 4 but not the other halves of those two problems or they do not understand when to use a compound inequalities versus two separate inequalities separated by an “or” statement. 
I started my lesson from scratch and approached the material from a graphical and algebraic point of view.  You can access my video lesson at
Below is an overview of the approach I took with more details in the video.
Solve |2x – 8| = 4 by graphing

 Use the above picture and solutions to help students see the solutions for |2x – 8| > 4, and |2x – 8| < 4. 
 I follow this with the details of how the algebraic method ties into the graphical and how to get the solutions algebraically.
 Tying in more than one perspective in our lessons will help students better understand and make more connections with the mathematical concepts we are trying to help them learn.  It is not easy to make many of these connections.  It takes lot of time to think and reflection plus time to design an effective lesson but it is well worth it since it helps our students learn.