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Interesting Ideas on Improving Mathematics Instruction on Select Topics

In June of 2016, I took two graduate level mathematics classes (Number Theory and Discrete) at Bemidji State University (BSU).  BSU’s focus is on improving mathematics instruction in the K-12 classrooms vs learning high level mathematics that would never apply to teachers as they work with K-12 students.  I learned some really good ways to help students understand some mathematical concepts either intuitively or with manipulatives.  Some of the topics that stuck out to me are:

•  adding, subtracting, and multiplying integers
• dividing rational numbers
• greatest common factor
• least common multiple
• permutations and combinations
• proofs of divisibility rules (you may think this topic would be boarding but it was quite interesting)

The two classes I took were back to back with class officially starting at 8:00 and going to 1:30 with a 30 minute lunch (provided) but most people were their 15 to 30 minutes early.  You may think that 5 hours of class is a lot and it would be if it were just straight lecture but BSU’s approach of asking questions, giving you time to think individually, discuss in small groups, share your ideas at the board, with only a little lecture here and there made class go pretty quickly.  The approach that BSU used in their classes had some characteristics that reminded me of Number/Math Talks.   Many of topics they shared with us to help improve the instruction of mathematics at the K-12 level arise from either Nation Science Foundation curriculums or their decades of teaching experience.

Below is a brief outline of the approach or approaches taken and a link to a video to help you understand it better.  Some of the topics apply only to upper elementary and middle school while other topics have aspects that can be applied in algebra and other high school mathematics classrooms.

The approach that BSU shared came from IMP (Interactive Mathematics Program).  The idea of adding and subtracting integers is rooted in the story of The Chef’s Hot and Cold Cubes”.  Part of the story is shown below.

In a far-off place, there was once a team of amazing chefs who cooked up the most marvelous food ever imagined.

They prepared their meals over a huge cauldron, and their work was very delicate and complex.  During the cooking process, they frequently had to change the temperature of the cauldron in order to bring out the flavors and cook the food to perfection.

They adjusted the temperature of the cooking either by adding special hot cubes or cold cubes to the cauldron or by removing some of the hot or cold cubes that were already in the cauldron.

The cold cubes were similar to ice cubes except that they didn’t melt, and the hot cubes were similar to charcoal briquettes, except they didn’t lose their heat.

If the number of cold cubes in the cauldron was the same as the number of hot cubes, the temperature of the cauldron was 0 degrees on their temperature scale.

For each hot cube that was put in the cauldron, the temperature went up one degree; for each hot cube removed, the temperature went down one degree.  Similarly, each cold cube put in lowered the temperature one degree and each cold cube removed raised it one degree.

This story sets the stage for helps students add and subtract intuitively and with manipulatives that represent a hot cube and a cold cube (the “cubes” could easily be red and blue counters or could be represented as just an “H” and “C” but giving the students something physically to touch and move is preferred). Here are three links to some videos I made to help others use this approach:  Hot Cubes and Cold Cubes (https://www.youtube.com/watch?v=SGT0aElpzzI ), Hot and Cold Cubes Practice Problems (https://www.youtube.com/watch?v=QNQYTks3wfs ), and Summarizing the adding and subtracting integers (https://www.youtube.com/watch?v=ywQHnosH_ec ).

Do not start out by showing the students how to model all eight types of problems but only show them how to model one or maybe two problem then let them try to figure out how to model the other problems.  It is good to let students be perplexed but be watchful so you can prevent the perplexity from switching to frustration.  After some time of the students working on one problem ask some groups that were able to model the problem to go to the board and demonstrate it to the class to help other groups model it.  One major mistake that most of us teachers do is to go to quickly to the summarizing of the adding and subtracting rules or letting one of the top students share the patterns (rules) they noticed for adding and subtracting integers too early.  It may take a week and a half of having the students playing with the hot and cold cubes modeling the adding and subtracting of integers before you start to summarize the rules/patterns for adding and subtracting integers.

Multiplying Integers

The multiplying of integers uses the idea of hot and cold cubes or could be done with two colored counters.  The approach used in the video link below starts with multiplication being repeated addition.  Start by modeling one problem and letting the students see if they can model the rest.  Here is a video I made on this concept Hot Cubes and Cold Cubes Multiplication (https://www.youtube.com/watch?v=urYeTDhuRt8 )

Dividing Rational Numbers

I have only seen one way in which students can use manipulatives to model dividing of rational numbers that works wells and that is the approach that I am sharing with you.  This approach uses unifix cubes to model the dividing of rational numbers.  Most people like a story and there is a short story that leads the students into modeling the problems.  Start with an easy problem then continue to asking more and more complex questions.  Here is a video I made on this concept Dividing Rational Numbers with Bricks (https://www.youtube.com/watch?v=0pMBRo46JZA )

Some of the above topics do not apply a lot to high school students but the next few topics do have some application algebra and other high school.

Greatest Common Factor and Least Common Multiple

The greatest common factor and least common multiple videos have approaches that will help students in algebra.  Start by sharing why you would want learning these concepts (LCM is used for adding and subtracting fractions while GCF is used for simplifying fractions).  With the approaches shared with us on  GCF and LCM both start with using the words of Greatest Common Factor and Least Common Multiple.   For GCF, list all the factors then identify all the common factors followed by finding the greatest common factor and the LCM does a similar approach.  But what I really like is the second approach of each that uses the prime factorization approach.  Most math teachers in middle school and high school will write the prime factorization of 48 as 24*3 but that approach leads to many misunderstandings or questions for students like “Why for Greatest Common Factor do we use the smallest exponent when we have a common factor show up for both numbers?” or “Why for Least Common Multiply do we use the largest exponent  when we have a common factor show up for both numbers?”  But the second approach of prime factorization of writing out 48 as 2*2*2*2*3 is so intuitive to students.  Here is a couple of videos I made on these concepts Greatest Common Factor (https://www.youtube.com/watch?v=ZmFefdmZgwc ) and Least Common Multiple (https://www.youtube.com/watch?v=Ty87XNcveps )

Permutations and Combinations

This approach starts out asking students how many ways you can arrange a simple word followed by slightly harder words then gets into words with repeated letters which the way it is done is fairly intuitive.  The permutations of words with repeated letters leads naturally into combinations.  The second video extends the idea of combinations by asking a question of “How many ways can parents have three girls in a family of five?”  Most of us who teach math know this is a basic combination problem but most students will think of it this way if you ask them this question the day after you introduced combinations.  This basic combination idea can be extended even more and tied into Pascal’s Triangle.  Here is a couple of videos I made on this concept Permutations and Combinations (https://www.youtube.com/watch?v=9DdLZtnk3tw ) and Ways of having kids in a family and Pascal's Triangle (https://www.youtube.com/watch?v=LaRK8bEpJlA ).

To really challenge some of your top students, ask them “How many ways can you form groups of four out of a class of 24 students?”  If you think the answer is a simple combination of twenty four choose four then you are thinking incorrectly like I was initially.  You may want to let the students struggle with it for a while then emphasize the “s” in groups, “How many ways can you form groups of four out of a class of 24 students?”  If needed follow that clarification with, “How many ways can you form one group of four out of a class of twenty four students?”

Proofs of Divisibility Rules

I would not do these proofs with elementary or middle school students since they are algebraically based but I think they would be good for high school students.  High school students have been using divisibility rules for years but they know have the algebra background to be able to prove the divisibility rules.  I would start by modeling one or two proves to my students then I would let them try one that is a little harder as they work in groups.  They will likely pick up on some patterns for doing these proofs.  These proofs are a good extension for Algebra 2 or Pre-Calculus students.  Here is a couple of videos I made on this concept Proofs of Divisibility Rules 2, 3, 4, 5, 7, 9 (https://www.youtube.com/watch?v=xmaXdamlz1Y ) and Proofs of Divisibility Rules for 11 and 13 (https://www.youtube.com/watch?v=VH0sxRkivhM )

Most of the approaches I experienced at BSU was the professor(s) asking simple questions, giving students time to think and work in small groups, followed by some more key questions and think time.  As a student this past summer at BSU, I appreciated that approach vs lots of information being thrown at me in a lecture.  I hope you are able to learn a few new ways of helping students more effectively learn some of these ideas.

Mastery

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 homogenous 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 spencer.bean@eu.dodea.edu , Michael Brust michael.brust@eu.dodea.edu , Timothy Kelly tim.kelly@eu.dodea.edu , and Corey Sullivan corey.sullivan@eu.dodea.edu , who all teach in different US Department of Defense schools in Germany) for the most part use this approach.  Their web site is http://www.flippedmath.com/.  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.

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.

Does Mastery work?  Our mastery is averaging a 28.5% increase in unit proficiency compared to our non-Mastery results in Geometry.  Proficiency is defined as the percent of students who are at or above 80% on a unit assessment.

Solving Absolute Value Equations and Inequalities from a Graphical and Algebraic Point of View.   February 10th

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    https://youtu.be/oSmeu3cwsfY

Below is an overview of the approach I took with more details in the video. https://youtu.be/oSmeu3cwsfY

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.

### The Power of "C" for Correct.                                      February 6th

I have my students do “Check for Understanding” worksheets at the beginning of my flipped class.  These worksheets are typically 3 to 5 questions covering the basics of the lesson they watched before class.  Having students complete these problems provides many interactions between the students and me to make sure they do understand the material.  If they do not understand then I can work with them and help them learn the material.   For students who are shy or quiet these worksheets provide opportunities to really know what they understand.

Students are expected to complete these problems and have me check their work before starting their normal assignment.  I am surprised at the power of me putting a “C” for correct on their problems even for my juniors and seniors in high school.  Many students won’t start their daily assignments until they have a “C” on each problem of their “Check for Understanding” worksheet even if I tell them that their problems are all correct.  They want that “C” on every problem. (see image)

Last spring I had some students whose goal was to get the “Big C” which was one big “C” on the whole page indicating that all the problems were correct.  (see image)

I surprised one of my students who often get the “Big C”.  One day I did not give him his normal “Big C”, I supersized it by taping eight more pieces of paper around his page making it a 3 by 3 big page then I put one “Mega C” on it.  (see image)  He loved it and took it home.  The “Mega C” has been hanging on his parents refrigerator for months.

Some of my students from this last fall heard about that students “Mega C”.  They decided that they wanted to beat that students “Mega C”.  So everyday they would do their “Check for Understanding” worksheets and they wanted me to check their work and answers but they did not allow me to put any “C’s” on their pages.  They had a plan to make a supersized “Mega C”.  At the end of the semester they taped all their worksheets together into a 12 by 12 page and had me paint one huge “C” on it.

I am surprised that something as simple as putting a “C” for correct on a problem for a student will motivate them to work hard.  Even a this simple “C” motivates my juniors and seniors to go above and beyond what is expected.  Look for ways you can motivate your students to succeed as you check for understanding or learning of your content.

### Arithmetic and Algebra, are They Different?       October 26th, 2015

I have not had a lot of time to reflect this year since I have an overload (classes all day and no prep period).  But last Thursday I was at a meeting were they were discussing number/math talks (Making Number Talks Matter).  One of our tasks was to solve a problem then try to think about all the different ways a student might solve a problem, like 55 – 17 of which I thought of four ways to solve it however doing this problem with students the presenter stated that there are eight common ways to solve it.

All this got me thinking about arithmetic and algebra.  I know that algebra is very powerful and more powerful than arithmetic but are there some similarities.  Are arithmetic and algebra really that different?  Consider the following parallel.

“Algebra” is really doing the exact same processes with numbers that we do with variables, which shouldn’t surprise you considering the variables are representing numbers.  Drawing connections between our algebraic concepts to numerical calculations is important to our students understanding of mathematics.  Many times we as educators will need to redraw these connections as students get farther and farther into the algebra courses.

Student Teacher Relationships in Learning July 7th, 2015 (blog post on

Relationships in learning are important but it is only recently that I have realized how important these relationships are.  When I was lecturing – first 20 years of my teaching career – I thought I had a relationship with all students, but in reality I did not.  Yes, I interacted with students, but it was often me with the whole class. There was a whole class relationship but not many individual, personal relationships.  I normally only got to develop a real relationship with a student if they regularly came in before or after school to get help.

I started flipping my classroom in the fall of 2010, and I noticed a couple of things.  I loved not being the dispenser of information, and being out and about with the students, helping individuals or small groups of students.  Flipping allows me to have many individual conversations with the students on a daily basis since I am helping students one on one the whole class period.   I get to talk with each individual student.  I get to know them, their learning style, and their interests; I can talk about how mathematics applies to their interest.  Each student gets to know me on a personal level.  It was after I started flipping that I felt like I made many more personal connections with students and truly got to know individual students.

Evidence of Relationships Mattering.  A former student teacher shared this with me a couple of months ago, “You taught me that at the heart of teaching is healthy relationships.  You made it a priority to get to know each and every one of our students personally and make them feel welcomed, respected, cherished, and challenged

Just before Christmas, a student gave me a card that stated, “Merry Christmas!  I also want to say thank you for everything you do.  Even though I feel like I put a lot of time into Calculus, I know you put even more!  Also, just like you said that we impacted you, you definitely impact us too.  It’s nice to be able to know that there are teachers that genuinely care.   So thank you!  Anyway, I hope you and your family have a very Merry Christmas, and I will see you next year.  Thanks for dealing with me when I get frustrated.”

A couple of my students wrote a blog post “Student Perspective of Flipped Learning”.  Here is part of the post that discusses the student teacher relationship.  “The relationship that we’ve all developed with our math teacher is something that none of us would ever trade.  Not only has Flipped Classroom helped us to know him in a professional setting, but it has also allowed us to know him on a more personal level.  Even during the parts of the school year when we’re not involved in one of his classes, he still finds the time to check in with all of us.  Be it through email, during passing time in the halls, or even around the lunch table, he is always there to make sure we are doing well.”

I received a couple of notes at the end of school from students and I have shared parts of the notes that deal with  the student teacher relationship.

·         One student wrote “Mr. Faulkner, I don’t know where to begin, other than thank you.  You have not only touched my life academically, but in every other aspect as well.  You’ve taught me so much that I can’t even begin to explain how much you’ve impacted my life.  It seems so minuscule to send an email that says “thanks a ton”, but unfortunately I don’t know what else to give.  There are no words to describe how much you’ve changed my life.  Starting high school was a tough time for me, but you were always there to help.  Every conversation with you is a blessing.  You’ve given me opportunities that I never thought I would have. … You care about your students and you truly listen.  THAT is what makes you a great teacher and even better person.  You give so much and ask nothing in return.  You’ve changed my life in so many ways, and that’s all for the better.  I look forward to continuing to communicate with you in the future for advice, math help, or merely for a friend.  Thank you for all that you do and God bless.”

·         Another student, “Mr. Faulkner, You are someone that I am unbelievably thankful for.  You really opened me up and helped me realize my full potential.  Not only in math but also in life.  You work so hard and I respect that and hope to be just like that.  Lastly thank you for taking me in and being such a great mentor and friend to me.”

·         Three students wrote, “Words cannot express how grateful we are for having you as our teacher.  From Algebra 2 to Calculus, you’ve helped us grow not just as students, but ultimately has people.  You’ve shown us that without hard work and dedication, our goals are much harder to reach.  You’ve show us that sometimes you have to put your nose to the grindstone and get the job done (with laughs and espresso beans along the way).  You’ve shown us that there are no such things as stupid questions and that questions are how we become lifelong learners.  We’re proud to call you our teacher, but even more proud to call you our friend.  Thanks for all those life lessons and some math along the way.  THANKS AGAIN FOR EVERYTHING and for helping those three quiet kids in the back of Calculus class!!  Sincerely your lifelong friends the ‘Back Table Crew’.”

I tried flipping my classroom hoping that it would improve learning but I think it is the combination of flipping and an increase personal one on one connections with students that improved the student learning in my classroom.  (See https://goo.gl/hnWX0y for data on how student learning has increased in my classroom and “Positive Impact of Peer Instruction Flipped Learning” for more information about student teacher relationships in learning.)

So whether you flip or not develop those personal relationships with students, their lives and yours will be richer because of it; plus students will know that you really care about them and learning will likely increase as a result.

Reflection                                     May 13, 2015

Reflection is necessary and very beneficial, but hard to do.

Why is reflection hard to do?  Time and Screens

Many times as a teacher, husband, and father I either do not take the time to reflect or simply do not have the time.  As a teacher, I am often busy grading papers, entering grades, preparing lessons, working with students, teaching with an overload, or working on preparing a presentation.  This means I do not always have the time to ponder on what is going on.  I have no time to ask myself “How is each class going?” or “How can I improve the student learning in my classes?” Once in a while I have time to reflect at school but I often choose to talk with a colleague or go hang out with students instead of reflecting.  Both of these are important.  Positive relationships with fellow teachers are helpful and encouraging, while relationships with students can often be enjoyable, as well as educational.  Outside of school I have family responsibilities with kids’ activities, household duties, and the never ending “to do” list.

So why would I say “screens” keep me from reflecting?  When I am at school and have time I may check my email, the weather, or Twitter for interesting educational ideas.  All of these activities and apps take place within a screen.  When I am at home and have time to reflect, I may hang out with my family or just want some down time, so I turn to a screen (TV with Netflix) to relax.  All this can be good but it keeps my mind occupied so I can’t reflect.  When I am sitting waiting for my son to get done with soccer practice, I pull out my phone to check email instead of just sitting there and reflecting on my teaching or personal life.

Why is reflection important?

When you reflect you are able to see what is going well, what is going okay but could be improved upon, and what is going poorly, definitely needing to be changed and improved.  I recently had the pleasure of having some time to reflect.  I was at a conference and took the advantage of spare time to reflect with colleagues.  Conferences are normally great times to network and learn new things, but at this conference I had the joy of reflecting with two of my co-workers.  I had the pleasure of driving three and a half hours to this conference with one colleague which was a great time to talk.  For us, that meant discussing our classrooms and our schools.  Talking about what is going well, what we are doing, why, what we would like to change in our classrooms, as well as many other topics.  After three and a half hours we arrived and joined a third colleague.  The three of us continued to reflect on how things are going for a couple more hours.  We normally meet and talk one hour a week during our PLC (Professional Learning Community) but having multiple hours to just talk and reflect was wonderfully energizing.   The next night we had more time to talk and reflect over food after attending multiple breakout sessions.

Reflection is hard to do but I would encourage you to find the time or make the time, to do so.  This can be done by removing screens at least once a week for the purpose of reflection.  This time spent in thought will give you a new, fresh perspective.   It will help you see how blessed you are, and help you see the important instead of just the urgent. Having the chance to reflect for multiple hours with a colleague or friend is of great benefit as well.

Unexpected Advantages of Flipped Classroom Videos  February 22, 2015

In a flipped classroom the teacher records their lesson for students to view at another time.  There are multiple advantages to these flipped videos for the students but there are advantages to the teacher too.

Teachers are often very busy and do not have time to regularly observe their fellow teachers.  We can often learn a lot from observing other teachers but rarely have the time.  The teachers in my department have recorded their own lessons for our common curriculum (same notes and same assessments).  We can easily “observe” each other by watching each others’ video lessons on our own time.  In our video lessons we have the same examples but we each have a different way of explaining the concepts so by watching each others’ video lessons we can gain new insights in to teaching the concepts and learn from each other.  While watching a colleague’s lesson we are exposed to a new way of explaining the material or make new connections of how this concept relates to a different area of mathematics.  It is common for me to step into a colleague’s room during his prep time and see him watching another teacher’s video lesson.   Using flipped classroom videos to improve our professional practice is a great use of our videos even though the videos were initially created to help our students.

A second advantage of flipped videos was shown to me last spring.  I had a student teacher for one of my classes.  He would watch three different teachers’ lessons to see how the experienced (old) teachers taught the material.  He would then develop his own lesson and record his video lesson using the best practices he saw in the experienced teachers’ lessons or improvements to the lesson that he was able to bring in.  As a new teacher, this was a great way for him to learn how to present the material to students in an effective manner.  As someone who started teaching in the early 1990’s, I would have found this technique very valuable.  Using flipped classroom videos as a way to help young teachers develop into a more effective teacher by “observing” multiple experienced teachers was an unexpected but great use of a flipped video.

A third advantage for teachers I have noticed is that since I have been flipping my classes I do not have nearly as many students in before or after school asking questions.  Recording video lessons requires significant time up front but in the long run the teacher gets that time back with not having as many students in before and after school needing help.

A fourth advantage of videos in a flipped learning classroom is that if a student is absent, they can watch the video before they come back to school.  Students can come back to school caught up on what they missed.  It is not uncommon for me to have students gone and they come back having watched the lessons and completed the assignments.   This means the teacher either has more time since they are not working with previously absent students before or after school to get them caught up, or the teacher can continue to work with all the students in the classroom versus ignoring most of the class while reteaching the previously absent student one on one during class time.

Each teacher that teaches one of our courses records their own videos at the request of our students.  We expect our students to watch the lesson before class but since we have multiple teacher videos for each lesson students can choose to watch which ever teacher.  Most students will watch the video lesson of their classroom teacher but some choose to watch a different teacher because that teacher’s style matches with the student’s learning style.  Some students will watch their teacher’s video lessons on a daily basis but will watch a different teacher’s lessons as a review for an assessment.  The key thing is that students are learning the material, we do not care who they are learning it from.

Students are able to rewatch lessons either from their teacher or another teacher as part of their review for an assessment.  My Augsburg College class meets once a week and students watch the lesson before class then have an assessment on that material the next week.  Almost all my college students rewatch the lesson as part of their review for an assessment while others will rewatch the lesson multiple times.

Students are able to pause and rewind the teacher during the lesson, even rewinding the teacher multiple times to understand that part of the lesson.  In a lecture a student may ask a question once, it is not likely but possible they would ask the same question a second time if they still do not understand.  If they still do not understand, they will likely give the teacher the impression that they do understand because they do not want to look stupid in front of their peers.  But with a video lesson a student can rewind and listen to part of the video lesson as many times as needed to understand the concept.

Most teachers that start out flipping their classroom anticipate the student advantages like the student can pause and rewind the video as needed, watch the lesson when they have time and that students can rewatch the lessons for an assessment.  But as you have read there are multiple unexpected advantages for the teacher like being able to “observe” their colleagues and learn from them, to helping younger teachers develop their skills and become an effective teacher, or having more time before and after school.  As I continue flipping and reflecting, I am sure more unexpected advantages will come to light.  I would encourage you to consider flipping your class, so that you can experience these advantages too.

Teach Students How to Watch Videos for Instruction.  October 15th, 2014

Today’s students are consumers of videos for entertainment.  Is watching a video for learning/instruction different than watching a video for entertainment?  Yes, how you watch a video for entertainment is totally different than watching a video for learning or instruction.  This may seem clear to adults, but isn’t necessarily for students.   Most students do not realize that there is a difference, so we as teachers need to explicitly teach our students about the difference.  Giving students tips on how to watch a video for learning/instruction, then modeling in front of the students how to watch your videos for learning/instruction is important for flipped learning success.

Below is an acronym that I use to help my students remember what to do:  LIFT.  The first thing I want my students to do when watching a video for learning/instruction is to focus on Learning  versus just starting the video and letting it play.  Having the video play before your eyes does not mean you are learning the material.  Watching a video for learning means that the students should be Involved  in the video by pausing the video at times to either write notes or work on a problem before they watch me do the problem on the video.  They may even need to rewind part of the video if they either missed something or did not understand something.  Being Involved also means taking notes.  For me that means writing down, at a minimum, all the material that I write during the video.

During the beginning of your course with the students, you should model what it looks like for students to be Involved in your video.  You can and should tell your students how to be Involved in the video, but you must model it for them.  To model being Involved in the video, play the video for the lesson in the classroom then pause the video at different points and tell the students why you paused the video and what they should be doing during the pause.  At times rewind the video and explain to the students why you are rewinding the video at this spot.  Depending on your students, you may have to do this for several days before they fully understand when to pause/rewind and why.  You may even want to tell your students in the initial videos when to pause/rewind the videos to help the students to get use to pausing/rewinding videos.

LIFT when Watching Videos for Learning.

·                  Learning -  be concerned about learning vs. just getting the video watched

·                  Involved - be involved in the video by pausing/rewinding the video and take notes

·                  Focus - focus your attention on the video

·                 Tech - turn off notifications, close extra tabs, put electronic devices away, close  extra apps, do not answer cell phone calls and messages, listen with Headphones

Modified from Lisa Light's FIT acronym

Technology is such an integral part of our students’ lives.  This simple acronym helps me remind students how they need to engage differently with their technology depending on whether they are using it for instruction or entertainment.

Why Some Teachers Don’t Flip                    August 13th, 2014

The following is a summary of an email exchange with Ed Mass of CrazyForEducation.com of which Ed posted on CrazyForEducation.com.

Ed asked, “Why don’t other departments in your school Flip since all five math department teachers have Flip and have had some success with Flipping?”  I reply that, “I do not know why they have not Flipped.  However, I will offer some possible ideas.”

My responses (listed in order of the most likely reasons to the least likely reasons):

·         Many teachers are resistant to change. They teach the way they were taught. Whether new, or teaching the same way for decades, they don't adapt easily to changes.

ResponseIf there is a resistance to change, I ask, "Why?" How does the resistance to change serve you? How does it serve your students?

·         Some teachers like being "in control." They feel that if their students are quiet, they are in control of the classroom. They also like being the center of attention (the power) and being the source of information. It gives them a sense of importance.

Response: As a teacher, are we here for our own ego and self-worth or are we here to help students learn and do what is best for our students? If you answer to help students learn and do what is best for them, consider Flipping.

·         It takes time to record the lessons up front and some teachers do not want to take that time.

Response: What they do not realize, even if we have told them, is that you get that time back when you are Flipping because you do not have students in your room before and after school nearly as much as prior to Flipping.

·         Some teachers are so busy just trying to keep their head above water that they do not have time to think about ways to change.

Response: If you are so overwhelmed, then you should look at ways to change and improve, that make you more efficient with your time and effective in both teaching and your students' learning.

·         If I Flip my lessons, what will I do in class?

Response: This is a great question. The possibilities are endless but some teachers are afraid to answer the question.

·         Some teachers have the myth that if I Flip then my students will not need me.

Response: I have experienced the opposite and my students have told me the same. That is, in a Flipped Classroom the teacher is even more important. It's easy to replace a lecture. However, a teacher that engages students and makes the classroom fun, while enhancing students' learning, is highly valued by students and administrators, and much more difficult to replace.

·         Some teachers view my math department as a little crazy or "just out there" and they do not want to be like us.

Response: Why not? What is better than higher student outcomes while students take more responsibility for their own learning and have more fun while doing so?

·         Some teachers are not comfortable having their lessons online where anyone could look at them including their principal or fellow teachers. In other words, they are insecure about their own teaching ability or lessons.

Response: What better way to learn, and improve our lessons and teaching, than from the critique/feedback of other professionals? Plus, you have the opportunity to view other teachers' lessons and learn from them. You can do this whether you Flip or choose not to Flip.

·         Some teachers are not comfortable with learning the technology to record and post lessons.

Response: It is fairly easy whether you use the SMART Recorder software or the Ink2Go software recommended by CrazyForEducation. It is super easy to upload them and use all the enhanced features in the CrazyForEducation system. You can post them on YouTube but then you have all those ads and uncontrolled content that don't exist anywhere in the CrazyForEducation system. They are totally ad-free.

·         Flipping is just a math department thing. Some teachers do not understand how they could flip their subject.

Response: I have ideas on how they could Flip their subject and I could help some teachers generate ideas. Here are several thoughts to get you started:

* Have students learn about Art outside of class so they can spend class time creating Art pieces.

* Have students learn about different physical activities/games (PE) outside of class so the students can use class time engaging in those physical activities/games.

* Have students learn about Spanish outside of class so they can spend class time reading Spanish, writing Spanish, speaking Spanish.

* Have students learn about science outside of class so that students can have more time to actually do science; play with stuff to get first hand experience and better understand science; conduct more science experiments.

* Have students learn about cooking outside of class so they can spend class time actually cooking.

* In social studies, have students learn the basics of different topics in history outside of class so that in class the students can be in large group discussions with the teacher about the material, and in small groups creating projects related to what they learned outside of class.

* In English, Flip the grammar rules, Flip basic writing techniques, so in class the students can practice writing with the teacher there to guide them. Flip citation formats; Flip the student writers conference, where the teacher either audio records, or video records while showing the student's paper, their thoughts and comments about the student's work.

* In music, Flip the following: how to play a particular note, background of a musical piece, sing/play a particular style.

* In our school, the foods teacher is having his students record themselves making a particular food at home.

* Our music teachers are having students record themselves playing/singing a particular piece outside of class.

·         Some teachers are not comfortable with having their voice/image online.

Response: Whether you like it or not there is already a lot of material online about us.

One last note: Typically Special Education teachers like a Flipped Classroom because the teachers can either a) watch the lessons themselves to better help their students and/or b) they can have the students watch, and replay, the video lessons to better understand the material. Teachers also watch the lessons with the students to help them learn how to engage in the lesson, pausing and replaying, and taking notes. A Flipped Classroom gives Special Education teachers more options to assist their students in learning processes.

### Friday, January 31st, 2014

After twenty plus years of teaching and trying many different things to improve student learning, I just finished my second year (4th semester) of using the Peer Instruction Flipped Learning model.  The traditional flip helped my students learn more and do better on their assessments than lecture ever did, but the peer instruction flipped learning model has outperformed all other learning models that I have tried and really helped my students excel in mathematics.  I have used common assessments for many years here in my school, so I am able to concretely compare my lecture results, to the traditional flipped learning results, to my peer instruction flipped learning results.

Background
Peer Instruction is where the students still learn the material before class which, for me, means the students are watching my video lessons before class. When the students come to class, I have two to four questions up on the board related to the video lesson they just watched.  Students sit down and answer, or try to answer, the questions without talking with anyone, but students may use their notes frp, the video lesson.  Once students have answered the questions, then they turn to their neighbor and discuss their answer with them.  If they disagree on the answer, they try to convince each other of their answer.  During this process students are discussing/arguing/debating mathematics in my room and getting at their thinking and reasons for their answer.  All the student discussions are great.  I love hearing students deeply engaged in discussions.  They are also emotionally engaged in the learning process since they trying to defend their answers or wondering if they are right.

Results
I look at proficiencies (number of students at or above 80% on tests and finals) in gauging student learning.  When I switched from lecture to the traditional flipped classroom, my Calculus proficiencies rose 3.4% and Pre-Calculus rose 6.5%.  Two years ago (four semesters) I switched to the peer instruction flipped learning model.  My four semester average of peer instruction of Calculus proficiencies rose to 84.9% (a 13.6% increase), Pre-Calculus rose to 85.5 % (an 11.3% increase), and Algebra 2 rose to 95.7% (12.8%).  These four semester averages of peer instruction are great validation for me that students are learning more.
I just finished my most successful semester yet.  I had great connections with students (see “Relationships Matter in Learning, Jan 16th 2014” blog post) and really good proficiencies.  My proficiencies just from this fall are:

• Calculus: 89.0% (a 17.7% increase compared to lecture)
• Pre-Calculus: 86.6% (a 12.4% increase compared to lecture)
• Algebra 2: 98% (a 15.1% increase compared to lecture)

Note: I am missing Algebra 2 Traditional Flip compared to lecture due to having a yearlong student teacher during that time and I did not feel I could use his results.  Interested in learning about my  former, amazing student teacher who is now on staff with me? Visit mrpethan.com and see his textbook free, open source statistics curriculum.

Conclusion
I cannot say for sure why this last semester was so good in regard to student learning, but I had  developed really good relationships with all my students.  It is possible that those good relationships made the difference.  It is possible it was just the group of students I had.
The peer instruction process is effective by itself, but if you couple that with the relationships that develop in the flipped environment, the Peer Instruction Flipped Learning model is a very powerful model.  I would encourage you to try the Peer Instruction Flipped Learning model.  To learn more about how I use peer instruction in my room visit troyfaulkner.com and click on Peer Instruction.

### One Small Change = Big Difference in Student Learning

Background:  My calculus class has been going pretty well these last few years.  Last spring I redid some videos on the lessons that students struggled with and that made a big change for that chapter in how student did on their test.  Their learning of the material greatly increased; my test proficiencies for that chapter increased from 58% to 100%.  I define proficiency as the number of students that are at 80% or above on their assessments.
But over the last four years, I have noticed students’ grades typically dropped in the second half of the class.  On average, students who had below a 94% at the half-way mark would drop on average 3.57% in their overall grade.  Students who had above a 94% at the halfway mark would drop on average 1.35% in their overall grade.  I figured it was because the last half of the course was significantly harder (volumes of revolution, integration by parts, and doing calculus on transcendental functions) and initially dismissed it as something out of my control because of the difficulty of the material.

The Change:  This fall when I taught the class, I decided I would require students who were below a 97% in the class to do “daily problems” and turn them in every day.  The daily problems were three or four questions that student needed to turn in before school every day except on test days.  This is different from my normal assignments that students do where they have access to the solutions manual and can always check their work, but for these daily problems there was no solutions manual for them to use to see if they were right.  Students would turn these daily problems in before school.  If they got them all right, that was great.  If not, then they would get the problems back and have to do corrections before the end of class.  If students did not do the problems or did not get the correction problems turned in, then they had to “hang out” with me during their lunch and do calculus.
This change did a couple of things.  It was a reality check for the students to see if they really did understand the material without looking at the answers and for me to know which students were struggling with the material or which students I needed to work with more to help them understand the material.  This was a good check for understanding for the students since this class does not have quizzes and is based mainly on their tests grades and final.

The Results:  So, what are the results of this one small change?  Well, 80% of the students found the daily problems helpful in understanding the material better.  But better yet is what happened to students’ grades from halfway through the class to the end of the class.  Instead of student grades dropping by 1.35% to 3.57%, the students’ grades actually went up by 0.25% to 0.56% even though the material was significantly harder.  Another great thing that happened was the number of “A’s” rose significantly from 43% to 76%.  The number of “B’s” dropped because a lot of students went from the “B” range to the “A” range.  This change also kept a couple of my “B” students from dropping into the “C” range.
Calculus has a difficult reputation, even among math teachers. When I shared my findings with a colleague, he replied, "And Faulkner’s Calc ain't no cakewalk.  Wow.”  His reaction certainly helped validate the positive impact this small change has had.

Since I am a data-guy, here is the summary data of the big changes:

This small change impacting students also had an impact on me.  I was reminded that “change” doesn’t have to be a huge undertaking or a massive “re-do”; small things can also greatly impact student learning in more ways than I could have imagined.  I was reminded once again that the journey is all about continuous improvement, for students and teachers.

## Thursday, January 16th, 2014

Relationships Matter in Learning

When I was lecturing, I thought I had a relationship with all students, but in reality I did not.  Yes, I interacted with students, but it was often me with the whole class. There was a whole class relationship but not many individual, personal relationships.  I normally only got to develop a real relationship with a student if he/she regularly came in before or after school to get help.

In the fall of 2010, I started flipping my classroom, and I noticed a couple of things.  I loved not being the dispenser of information, being able to be out with the students, helping individuals or small groups of students.  I get to talk with each individual student.  I get to know them, their learning style, and their interests.  Each student gets to know me on a personal level.  As I am working with students and getting to know them and their interests, I can talk about how mathematics applies to their interests.

Personal relationship often leads to students putting in more effort.

My Calculus 1 class is taught for college credit through a university and is taught at a very high level.  My calculus class last spring had incoming state math scores 15 points lower (a 100 point scale) than prior classes, but they out performed every prior calculus class taught by lecture and flipped.  I developed very good relationships with each student in that class.   In my Algebra 2 class, we missed two days last week because of the extreme cold (50 degree below wind chill) in the middle of a challenging chapter on trigonometry, yet their quiz average was 92% (historical average 82%).  With these deeper personal relationships, it is like I say, “Jump!” and the students respond with “How high?” and no matter how high I set the bar, they jump to that level.  Reflecting on this, I feel like I am able to get more effort/work out of my students because of the personal relationship I have with each student.  So when students know that their teacher genuinely cares for them on a personal level, they do not mind putting in the extra effort for that teacher.

More evidence of relationships mattering.  This past November, I was surprised by my students.  On a Tuesday, I got a phone call during class letting me know that my father-in-law had passed away unexpectedly.   I ended up leaving school in the middle of the day, but right before I left, a couple of students came back to my room to express their condolences.  That evening I received two emails from separate students expressing their condolences.  The next morning I received a card from one student.  I was touched by all these expressions of sympathy.   But what really brought tears to my eyes was on Wednesday at the end of school another student left a card for me on my desk.  I looked at it later when I had time, and every single student from all my classes had signed the card expressing their condolences to me at the passing of my father-in-law.  After returning from the funeral, I shared with my students how they impacted me and shared how touched I was by their card.  In the past when I was lecturing and experienced a lost, I do not recall any student saying anything to me.   I feel that I only experienced this out-pouring of sympathy from my students because of having those personal relationships with each student.

Right before Christmas, a student gave me a card that stated, “Merry Christmas!  I also want to say thank you for everything you do.  Even though I feel like I put a lot of time into Calculus, I know you put even more!  Also, just like you said that we impacted you, you definitely impact us too.  It’s nice to be able to know that there are teachers that genuinely care, and even cooler to know that you are one who also shares my beliefs.  … So thank you!  Anyway, I hope you and your family have a very Merry Christmas, and I will see you next year.  Thanks for dealing with me when I get frustrated.”

I think some of the increase in learning that I have seen in my classroom has been because of having students doing mathematics in class as a result of using flipped learning, but the other part of the increase in learning has been because of the relationships that have occurred as a result of flipped learning.