Arithmetic and Algebra, are They Different?

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.

Model	Arithmetic	Algebra
Algorithm	12  X  14     4  8 1  2  0 1  6  8
Algorithm Expanded	10  +  2    X   10  +  4          40     8 100  20         .  100 + 60 + 8	a + 4       X  2a + 3           3a + 12 2a2 +  8a         . 2a2 + 11a + 12
Expanded “foil”	(10 + 2)(10 + 4) 100 + 40 + 20 + 8     100 + 60 + 8	(a + 4)(2a + 3) 2a2 + 3a + 8a + 12   2a2 + 11a + 12
Area	100 + 40 + 20 + 8	2a2 + 3a + 8a + 12
Summary	Pair each digit of one number with digit of the other number	Pair each term of one binomial with each term of the other binomial
Extending It	123 X 456     100 + 20 + 3 X 400 + 50 + 6 (100 + 20 + 3)(400 + 50 + 6) Pair each digit of one number with digit of the other number	a + 2c + 3 X 4a + 5c + 6 (a + 2c + 3)(4a + 2c + 6) Pair each term of one trinomial with each term of the other trinomial

“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

My blog below was originally posted on Pearson Educations Research and Innovation site on July 6th, 2015.

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.

Visit http://troyfaulkner.com/  or my blog at http://troyfaulkner.blogspot.com/ for more information.

Reflection

             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

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.

Advantages for teachers?

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.

Advantages for students?

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.