New School Year - Connections are the Crucial Element in a Successful Classroom!

Connections...being connected...being a part of the community of classroom...its a bold statement to say that this is the most crucial element but I truly believe it is.  If a student is not connected, not a part of the community of the classroom, then they simply become a seat filler, another paper to grade, another voice.  Think about how often you have walked into a room, a wedding, a party where most of the people know each other, have a shared connection and you are the "odd man out".  How do you feel?  While there are a few people who could jump right in and walk up to people, most people wait for someone to approach them, to bring them into a circle.  As teachers, that is our job - to bring our students into a circle, a community, a safe place where they can learn, grow, explore.  So how do we do it?  How do we make that connection, build the community that our students need?

1.  Be tuned in.  When you are in your classroom, be there mentally as well as physically.  So often we are in our classes physically, but our minds are consumed with the forms we need to complete, meetings we need to have, our families at home, what are we doing for the weekend and 1,000 other things.  As hard as it is, we need to shut those things off or at least try to and focus on being present mentally when we are in the classroom.  How this occurs is different for everyone.  It may be featuring artwork from your kids or pictures of your family so they are "with you" but not your primary focus.  It could also be making a "To-Do" list of everything you need to accomplish so that you don't forget but can focus your mind on other things.  Whatever it is that works for you and allows you to be tuned in, do it!  Your students will thank you!

2.  Create an environment that is safe.  Students, whether they are in Kindergarten, their senior year or anywhere in-between are still kids.  (If you don't believe me, just break out the crayons and see what happens! :)  They still need to feel safe and secure in order to be truly successful.  They need to know that there will be supportive words to encourage them, a safety net if they fail and accolades when they succeed.  They need to know that they won't be ridiculed by their peers if they have questions and that it is a safe place to express their thoughts.  In order to create this safe environment I suggest starting by with clear expectations (I go into more depth about this here) about what is and is not acceptable in they classroom, establish clear norms about how discussion will occur and more importantly be tuned in to what is happening in the classroom.  Do not let behavior slide for some students but come down hard others for the same behavior.  Be fair and consistent.

3.  Partner with parents.  As both a parent and a teacher I know first hand how important it is to be in a partnership with parents and not a competition.  Communicate with parents about the spectacular things that their students do, not just the negative.  I had an experience with past year where I would get notes home from my son's teacher about how his day went.  They were a checklist of six areas of behavior and rated with a smiley face, straight face or sad face.  He got quite a few sad faces, so many at one point that I started dreading getting the note at the end of the day.  I would start to have that anxious feeling in the pit of my stomach when I would see his backpack.  This teacher never sent home a note when he had a great day, just when he had a bad day.  BTW - my son is 5.  After a few weeks of this, my husband and I requested a meeting with the teacher where we expressed our concerns.  We found a way to partner with the teacher.  The communication got better, our son got happier and the behavior improved.  You can partner with parents with written communication (notes, e-mails, etc.), verbal communication (phone calls or meetings) or through practice (become the teacher that impacts their student(s) in a positive way to make a positive impact).

4.  Make a personal connection.  Humans are social.  We love to talk, to interact, to communicate and to share of ourselves.  Being a teacher allows a million ways to make a personal connection.  Notice the things that change about your students - a new haircut, cool new shoes, etc. and comment on them in a positive way.  Remember the things that a student tells you - an upcoming vacation, a new puppy, a job interview and follow up to see how it went/is going.  Attend after school activities when possible - a club, band concert, sporting event and then mention it the next day.  Share of yourself and common interests and experiences.  The more human you are, the more you can connect with your students.

5.  Nurture skills that will help them to be part of a bigger community.  We have our students for a short fraction of their lives - a semester, a year, maybe two and it is our job to help them to move on to the next part, chapter or adventure that their lives will offer.  At the high school level this means teaching them job skills, social skills and other skills that will help them to be successful in getting their first job.  We get the opportunity to help them figure if the job market, college or the military is where their strengths lie after high school.  We have the responsibility to help mold them into the (young) adults who will lead future generations.  We can give them these skills through conversation, prodding and modeling them ourselves.

Whatever you do, make connections with your students.  They really will translate to greater success for your students and for yourself as a teacher.  How do you make connections?

Task Cards - A Fantastic Resource at the Secondary Level

The first time I heard the term "Task Cards" I thought they were index cards that gave students directions for a task.  I wondered why teachers couldn't just tell students the directions verbally...  As time went on, I heard the term more and more but still never saw a concrete example of what they were.  So, I did what all curious people do - I Googled it!  While that gave me some clue, it was still not a definitive reason as to why I should pursue their use in my high school classroom.  I mean, let me be honest here - high school is overrun (like all grade levels are) with standardized testing, CCSS, and forty-seven things you must accomplish every day.  I had no time to add something new. 

Then I looked at my students and saw that something just wasn't working in my day-to-day structure.  So, I started looking for ways to better address student needs and learning levels WITHOUT adding extra work to myself.  While there were multiple options, I kept coming back to Task Cards as a viable solution.  After I tried them a couple of times, I saw some amazing results.  Students were picking up on concepts that they had struggled with, discussion in class had students participating that I had rarely heard from and most importantly, grades were rising.

So, let's talk about Task Cards!

1)  What is a Task Card?  
A task card is a card, piece of paper, piece of card stock (basically whatever works best for you) that features one problem, question or work task. (I prefer printed on cardstock and then laminated to save on costs).

Beginning Concepts Task Cards

Why only one problem/question?  I questioned this at first too, but found that it helps the students focus on just one problem, idea or skill at a time without getting overwhelmed.  It is easier for students to see on problem at a time instead of facing "I have 15 problems to complete and I can't even get number one done".  

(Most task cards can be created to be four to a page that you then cut apart into individual problems like the ones at the left).




2)  How do you use a task card?  
There are so many ways to use a task card.  A few are:

Congruent Triangles

      a)  Pair students with a partner and have them pass the cards from pair to pair every 2 minutes.
       b)  Separate the cards by concepts/difficulty and have students work in small groups depending on
             their abilities.
       c)  Use the cards as part of a larger station activity.
       d)  Have students complete one (or two) as an exit card to test for understanding
       e)  As a warm - up or bellwork assignment
       f)  Laminate a set and post them around the room for students to rotate around.  This is great for students who need/want to be up and moving around.
  
3)  How do task cards allow you to differentiate?  
Since task cards feature only one problem you can write the problems to be at individual levels.  For example, when I teach Segment Addition I can use a variety of cards.  I can use some that are just simple adding and subtracting, some that feature algebra in a one or two step equation or some that are a challenge where the problems are multi-step with variables on both sides of the equation. 


This is an example of four different levels of task cards that I use when doing the Segment Addition Postulate. It allows me to differentiate by giving students cards that are at their individual level and still assess all student understanding of the same concept!  It has been a real lifesaver!

Coordinate Distance and Midpoint

4)  How do I create a Task Card?  

The first step is to decide how many cards you want per page.  I find that four works well since it gives room to create the problem and is neither too big as to waste paper nor too small as to make it hard to read the problem/question.  You want to have a clear idea of the different levels of the concept as well as to how you want to phrase the questions being asked.  I find that laying it out first makes it easier to create them.  Once you have decided on a layout and the different levels of questions, you can start creating.  I will admit, creating task cards is a time-consuming process but once they are created, they are done!  (I find using PowerPoint the best program if your task cards require layering like mine do!)

5)  How I best utilize my resources and save myself time?  

I recommend printing your task cards on card stock, laminating and then cutting them out.  Unless you really find it necessary for students to be able to write on the cards, there is no reason to recreate the wheel every hour and print them on paper.  Additionally, if you print them on card stock and laminate them, then students can write on the cards using dry erase markers and then "erase" when they are done without damaging the card !  

Similar Triangles

I always make sure to include an answer sheet with mine for students to hand-in when they are done.  This allows me to hold them accountable and again saves on the need to continually reprint the task cards. 

All in all, I have yet to find any drawback to using task cards!  I highly recommend trying them out.  If you are a Geometry teacher, like myself, and want to try them before dedicating large pockets of time to creating them, I recommend shopping on TeachersPayTeachers.  I have the multiple sets illustrated above individually and in a money saving bundle in my store (with many more on the way!) but a search will turn up task cards on many, many topics! 

Mini-Quizzes: Why I Will Rarely Ever Give A "Full Size" Quiz Again!

Most teachers, myself included, give quizzes about 2 times a unit.  Sometimes, I would give three if a unit was particularly long.  And I HATED it!  They always felt like mini-tests because they would cover three sections, took the majority of a class period and were historically not that good  Quizzes are supposed to be indicators of mastery of knowledge and performance of the future end of unit test.  When a quiz, however, spanned multiple topics, I was not getting that feedback.  Different students would struggle on different areas and I could not see consistent areas that needed re-teaching.   

Triangles and Congruency Quizzes

 I knew that something needed to change but I wasn't really sure how to go about it.  Until I happened to look at one of the exit tickets I was giving at the end of an hour.  I was using those as an indicator of mastery of what I had taught that hour and realized that it was the perfect length and structure for a quiz!  And so, mini-quizzes were born! 

In designing them I gave myself a few rules that have really served myself and my students well.  These rules are also my rationale as to why they are so successful! 

Proof and Logic Quizzes

1)  A mini-quiz cannot be on more than one topic.    Since one of my problems with full-sized quizzes was that I struggled to understand whether they got the topic or not, I wanted to make sure that my mini-quizzes only covered one topic.  This has been so beneficial in that I get immediate feedback on mastery of a topic before I move on to the next topic or whether I need to re-teach something.   Which leads to rule number  two.

Quadrilaterals Quizzes

2)  A mini-quiz should be given within a couple of days of a topic being taught.    Given that my purpose of the mini-quizzes is determine mastery or the need to re-teach I wanted to make sure that they were given in a timely fashion.  Most of the sections that I teach last two days.  The first day is notes and homework, with the second day being small group practice or an activity.  Usually at the end of the second day, I will take the last 15 minutes to give my mini-quiz.  This serves the dual purpose of giving it while the information is fresh and giving me time to check it and determine whether we need another day with enough time to copy whatever I will need.    It was this need for expediency

Circles Quizzes

and to save paper, that my third rule came about. 

3)  A mini-quiz cannot be more than half of a page.  I only want to assess one topic at a time and two to three questions will usually give me all of the assessment that I need.  For example, when I did my quadrilateral unit, each mini-quiz featured three topics - angles, sides and diagonals - with one question apiece.  This was more than enough for me to see if understood the properties without overloading the students.  Also, since my goal was to have these done in less than 15 minutes, I didn't want to make them so long that they could not be finished in that amount of time.  Additionally, with each quiz being half of a page, I could get to quizzes per sheet of paper and cut my copies in half!    This length restraint lead naturally to my last rule!

Right Triangle Quizzes

4)  A mini-quiz cannot be more than ten points.  The majority of the mini-quizzes I give at six to eight points, with a random few being nine or ten.  I never go over ten points.  On most mini-quizzes that happens naturally  with one point for each correct answer and one to two points for the work depending on how in-depth they had to go.  I did not want these mini-quizzes to destroy a student's grade but I also did not want them to be throwaway grades either.   My students definitely appreciate this and have a better attitude about them!

Now the big question was, would this work?  Would switching to four to six mini-quizzes per unit instead of one to two full size quizzes have an impact on grades?  The short answer - YES!!!  My class average on tests has improved by multiple percentage points!

Beginning Concepts Quizzes

  Student retention of topics from one unit to the next has improved.  Additionally, I get far less moaning when I say clear your desks because they know why I am doing it and that even if they have a bad day, it would hurt that much!  Unless someone forces me to, I will never go back to doing full length quizzes!

Back-To-School Money Saving Tips for Teachers and Parents!

I've watched the shows about people who use coupons to the extreme and am in awe of their dedication and consistency.  I wish I could be that kind of money saver, but unfortunately, it just is never going to happen.  However, when it comes to back to school supplies I tap into my inner extreme couponer and start pulling out ads, finding coupons and trying to combine the coupons with the sales.  Teachers and parents spend so much on back to school supplies!  Seriously, I have a list of 22 items for my upcoming first grader and 24 items for my upcoming 3rd grader!  With that in mind I wanted to share a few of my and some of my friends favorite Back-To-School money saving tips!





1)  Big box office supply stores start with their back to school savings in July!  I discovered this by accident a couple of years ago when I was looking up on one of their websites and happened to glance at the weekly ad and saw things for a penny, a nickle and a quarter.  I was like a penny?  No, that can't be right...but it was!  This week alone I was able to get:

*  packs of 3 glue sticks for a penny each
*  70 page notebooks for a penny each
*  packs of 3 pink erasers for a penny each
*  handheld 2 hole sharpeners for you guessed it, a penny each.

Now of course these stores have limits - usually three each but you can go back multiple times, take a friend and some even extend the limits for teachers - just ask.  Some also have a minimum purchase (usually $5) but that can easily be met by the things that are on sale for a dollar apiece that you would be buying anyway (for me this often colored paper or dry erase markers).   I already have next week outlined because they have 2-pocket folders and packs of 100 index cards for $0.01 apiece and colored pencils for $0.50 starting tomorrow! :)





2)  Jameson from Lessons with Coffee suggests going to thrift stores! "There is so much stuff there from rugs, binders, storage options and even fabric for bulletin boards at fractions of the dollar of what you would pay at the bigger stores."  I agree completely!  I have no blinds or any other window covering in my classroom so the sun is always beating in and heating up the room.  I went to a local thrift store and picked up fabric that I fashion curtains out of!  Not only does it cool
the room, but it looks amazing too!  In addition to thrift stores, you can also check out garage/yard sales especially for storage bins and other containers for holding books and supplies.

3) Use teacher discounts and rewards programs.  Multiple stores not only have special discounts for teachers but they also have rewards programs.  Barnes and Noble Bookstores has a teacher discount program as do official teacher stores like Lakeshore Learning.  The major office supply stores like Office Max, Office Depot and Staples have a special teacher rewards program where you earn money back, coupons and other deals on purchases.  Often these discounts can also be combined with coupons so double savings!

4)  Prioritize your shopping list and buy in bulk when possible.  The reality is that as much as we want everything we don't need everything.  When sitting down to figure out what you are going to buy for your classroom, your upcoming school year or for your child(ren)'s up coming school year that sometimes you have to realize that everything is just not possible.  Money is limited and some supplies are more necessary than others.  Figure out what is the most important, most crucial and what is going to have the greatest impact and purchase those things first.  Once you have figured out the most necessary things see if you can buy them in bulk.  While the bigger package is going to be more expensive it will more often than not work out to be cheaper per individual piece than if bought smaller amounts.  Also, don't be afraid to use the store/off-brand for supplies that are "consumable".  A glue stick is a glue stick regardless of what label is on it.  Construction paper is the same paper if it is a store brand or a name brand!

5)  Jamie from Miss Math Dork takes it a step further and encourages you to "Make a budget! Or else it's too easy to keep buying "one more thing". Shop in stores that have a sale going on, and check out thrift shops and yard sales for nice storage options.  I try to buy a little bit at a time all year, and keep it stored until BTS time."  Knowing what you have to spend and setting a limit will not only make you a frugal shopper but will also help you to stick to your list (and you should always make a list)!

6)  Figure out what you already have.  I cannot count the number of times that I have shopped for my classroom, bought something and then when I set-up my room in the fall I find out that not only did I already have it, but in some cases I have it it in abundance.  For example, I found a great deal on index cards, then realized that I had 20 packs stuck in a cupboard in my closet - whoops!  I recommend taking the time to make a list of what you already have. Making a list of what you already have not only keeps you from spending money that you don't need to but it also encourages you to examine what is really essential.  What you do you use a lot of (glue sticks, paper, etc.) and running out of and are there things that you thought would be so great but you never used ... or forgot about?

7)  Invest in quality versions of the reusable supplies.  While spending a little more on quality will not save you money immediately, it will in the long run.  If you know that you are going to be splitting students up into groups often, invest in some high quality totes so that you have supply boxes ready and able to go.  Since scissors are used constantly, invest in higher quality brands so that they hold up to the constant wear and tear of 100+ students using them in one
day.  Of course, still look for these to be on sale (I just got very high quality 7 - inch scissors on sale for $1 apiece that were regularly $7 each) but be willing to pay a little more for the things that are non-consumable.

I would love to hear your money saving tips!  Please share them below!


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Interactive Notebooks - The Perfect Combination of Lecture and Handout Notes!

Lecture notes = stand and deliver in many minds.  Guided (fill-in-the-blank) notes = spoon feeding in other minds.  Neither one is better or worse than the other.  Instead they are structured to meet your students at the place that they are at.  The reality is, though, that in a class of 28 - 35 no one is at the same place.  So how do you meet them where they are at while retaining the flow of the class?  This is a struggle that I have had for the better part of 20 years through a 7 - period day, 6 - period day, trimesters, semesters, block and probably more that I cannot remember!  In addition, we have tried to be more "green" and "paperless" in my district as well which further adds to the struggle when you are trying to not make multi-page handouts that will just be (hopefully) recycled at the end of the semester.  So this year, I am trying  going to pilot something different in one of my classes - Interactive Notebooks - that I think will be the "perfect" marriage between the two methods!  I wanted to share with you why and how I'm going to go about it!

Cut and Paste

1)  Interactive Notebooks help students to stay organized helps them to reference back to old material.  One of my biggest frustrations is when students are trying to find something in a folder that is jam-packed with old assignments, notes, handouts and 1,000 other pieces of paper.  While it is fantastic that they are keeping all of these things, they serve no purpose if they can't find anything in them!  By using an interactive notebook students have all of their notes, IN ORDER, and can actually find them when they need them.  Some teachers also have students number the pages and keep a table of contents (which makes so much sense) so that they know exactly where the material is.  This is a fantastic thing, especially at the secondary level when material spirals and keeps building upon each other.  Although I did not use INB this year, I had a couple of my honor's students

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bring out their INB from the previous year (yes they kept them!) to look up previous ideas!

2)  Interactive Notebooks help to hold students accountable for and engage them in their own learning.  With many of guided notes, I find that a few of my students literally only fill-in-the blanks and don't write anything else down.  While this is by far a slim percentage of my students, as a teacher I want to reach them all.  I asked my students why this was and the common response was
that they don't know how to take notes and that they write slow (especially if the blanks are words that they are not familiar with).   INBs allow teachers to go more in-depth and help students to focus in on small pieces of information which leads to students writing more down.  Students also seem to feel more ownership because they can annotate easier and add things around the page.  

Angles of Elevation and Depression

3)  Interactive Notebooks should save paper, resources and time.  Since the interactive notebook pages are smaller and many less pages than traditional notes it should save on both paper and copy machine resources.  Most INB pages can be printed two to a page or are tabs that you can paste into the regular notebook and write under, both of which will save on paper.  Fewer copies means less time at the copy machine which means more time for everything else!  Additionally, since INB pages tend to be shorter, it will be easier to go over them more in-depth with absent students instead of having to wait for them to copy down multiple pages of notes from other students first.  Instead you can create an Interactive Notebook of your own that can act as a key.  This way  absent students can just borrow yours to fill in theirs and you can copy just one or two pages of your notebook if you need to send something home instead of multiple pages.

Similar Triangles Cut and Paste

4)  Interactive Notebooks meet students where they are at.  Interactive notebooks allow students to pick their pace.  If they need more examples, they can write them on the notebook page around the INB paste in.  Or if they want to explain how different ideas tie together, they can write the page number of the related page on the INB page and have quick reference.  Additionally, for those students who have accommodations, INB pages are easily differentiated to meet their needs.  I have even found that when surveyed, my honor's level kids liked the idea.  They stated that sometimes it is nice to have the diagrams done so that they can go further and explore the ideas more in-depth instead of getting bogged down with drawing everything.

Congruent Triangles Cut and Paste

I am going to try out Interactive Notebooks in one or two of my classes for the first couple weeks of school and do traditional guided notes in my other.  After a couple of weeks, I will let my students chose which one they like better.  I suspect it is going to the Interactive Notebooks.  I'll keep you posted!   Click here to get my Interactive Notebook pages on Angles Formed by Parallel Lines for free!   I would love to hear how you use Interactive Notebooks in your classes!  Please share below!  

Activities to Encourage Collaboration #1: Surface Area and Volume of a Sand Castle!

In a time and society where students spend more time communication through text messages, Facebook, Twitter, Instagram and 14 other social media platforms that I cannot even begin to name, I find an ever increasing need to get my students talking to each other fact to face.  As a result I have been on a quest this year to implement as many collaborative activities as I can.   I have used many of them throughout this school year and have had some amazing results that include increased communication, retention of information, assessments grades and more positive attitudes (overall)!   Throughout the summer I will be sharing some of my favorites, some of my other favorite math teacher-authors and many others so that hopefully they can become your favorites too!

Today I am excited to share with you my Surface Area and Volume of a Sand Castle activity!  As we were finishing our three-dimensional figures unit in Geometry I was looking for a really good way to a) get the students talking and b) show them how the different figures can share dimensions to build the structures that we see on a daily basis.  Since I do not possess architectural skills and summer is upon us I decided to build a Sand Castle (as "Do You Want To Build A Snowman" is running through my head).  I started with a goal of including as
many of the main solids that I could and managed to include prisms, cylinders, cones, pyramids and even a hemisphere!  I worked to have the solids share bases, sides and dimensions whenever possible.  This is what I came up with!            
I also came up with a second version that has the figure divided into 11 smaller figures to help struggling students visualize a path to follow to solve it.  Additionally, this helps students to organize their work so that you and they can identify an error if they make one.  (I did not, however, hand this out to begin with as I wanted to see what they would do with it first!)


Before implementing this as partner/group collaboration piece I sat down and created a list of questions that I could ask as I walked around the room to point students in the right direction, get them thinking, communicating and solving without actually giving them the answer.  Some of the questions that I came up with:

1)  Are there any surfaces that aren't exposed?  Alternatively - are there any surfaces that shouldn't be used in surface area?

2)  Have you thought about breaking any of the larger figures into smaller ones?

3)  How are you arranging your work so that you can go back and check it later?

4)  Are there any dimensions that you don't have?  How can you find them?

5)  Do the unused surfaces from the surface area get used for volume?

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Finally the day arrived to implement this and I must say, it went AMAZINGLY!  After my students got over the expected moans and groans and sat down to start working on it, they had fun with it.  I heard great discussion, collaboration and genuinely helping each other understand instead of just giving each other the answer.  I set forth the "rule" that their final answers had to be within ten of mine (to account for rounding error) and that whoever was the closest won a prize (extra credit, candy, excusing of an assignment, ect.).  My students quickly turned it into a competition and worked hard to earn the prize.  I ended up with multiple students hitting my answer down to almost the decimal point - which is great! :)  Based on the feedback I can honestly say that they enjoyed it and felt that it really reinforced the concepts we have been learning in this unit!  

I have put the entire activity, including a multi-page answer key that highlights each piece and how to find their surface area and volume up in my teacherspayteachers store.  You can pick it for FREE here :)  I would LOVE to hear how you use it and implement it! Please comment below!

Math Misconception: The Final Answer is The Only Part That Matters

For the longest time, I really thought that getting the right answer was the goal and really the only thing that mattered.  This misconception stuck with me all the way through elementary, middle and high school and was really not challenged until college.  Through my K-12 experience I showed my work because I could not do the math without it. However, there was no reward for it.  There was not partial credit, it was all about the correct answer.  As a struggling math student (to the point where I actually started looking into careers that did not involve math) this really frustrated me because I was often unable to find the correct answer.  (Especially in Geometry when writing proofs - ugh!)

It was not until I took College Algebra my freshman year of college that the work behind the problem became just as important, if not more so, than the correct answer.  It was also, at that point, that math "clicked".  The emphasis on showing my work helped me to the understand, make connections, and to truly appreciate the beauty of mathematics.  Moreover, it also cemented my career path - to become a high school math teacher - who knew!  Now as a teacher, here is what I explain, emphasize and illustrate to my students about why the final answer is not always the best answer to the problem.

1)  Showing your work helps you to understand the "why" the answer is correct.  My favorite question to ask my student whenever they give me an answer during a class discussion, during group work or even individual practice as I am walking around assisting them is "why".  Why did you choose that formula, why did you draw that diagram,or why did you did set-up that equation?  If my students can't answer why, I will help them out with leading questions or allow other students to do so (depending on the circumstance).  I tell them upfront on day one, if you can't tell me "why" then I am going to keep asking.  Sometimes it makes them laugh, but as I am consistent with making all students answer it, it also helps them to make it to reason two.

2)  Showing your work helps you to understand the connections between  different topics within a class and different branches of math .  We as teachers, parents and adults, all know that math is circular in its connections to each other.  When I teach parallel lines, for example, I review and apply the properties of angles that we study in our foundations chapter, setting up and solving algebraic equations that they learned in a previous class as well as the idea of determining whether or not their final answer in reasonable.  If a student is not showing their work, they are not able to see these connections in action.  They are not able to see how the math keeps "coming back".  As the year goes on, these connections get even deeper.  By the time I teacher my final unit - Three-dimensional figures - we are tying together area, linear equations, squares and square roots, triangles and at least four other topics!  So often, I hear students say - wait, didn't we do x or y before and can't we use w and z here?  The more we stress showing their work, being able to answer "why" and "where" the answer came from, the more we can move students to the most important reason I see for looking beyond just the final answer.  

3)  Showing your work helps you to retain the knowledge beyond just a quiz, test or other assessment.  As much as we don't like it, standardized testing is a reality of this world that we live in.  My students are gearing up for this in the new few weeks.  Having done multiple types of practice with them through the year, I know how much inter-connection of topics and how much prior knowledge they need to retain.  But it goes beyond that, beyond a single test (normally) during their junior year of high school.  It goes on to a better attitude about math and a better appreciation how much impact math has the world around us.  I seriously cannot count the number of times that I hear "I hate math" , "ugh, math was my worst subject" or "I never use math" when people find out that I am a math teacher.  For those that I am lucky enough to continue the conversation with, the reason often comes out that they never truly understood why they had to do it or how it all worked.   

While I definitely don't have all the answers, I do think that they more we can emphasize showing how you do the math, understanding why you need to do it and how the different ideas work together we are setting our students up for better performance on tests, in college, in their career paths and more.  Yes, the final answer matters - we need to balance our checkbooks, we need to buy the right amount of paint or carpet - but we also need to know how to consistently arrive it!  

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