Intro to K-6 Portion of Traditional Math

The first part of the book will be devoted to grades K-6 and will be written by J. R. Wilson who taught those grades over a 30 year period. This is the introduction to the first part. The routines he describes for his classes (such as the I-We-You method) also apply to what I have done in seventh and eighth grades.

I started out to be a primary grade teacher and ended up teaching high school math as a long term substitute.  After teaching primary grades for a few years, I went to graduate school and earned an M.Ed. and Ed.S. in administration.  I worked for a few years as a curriculum specialist at a state department of education and then as staff development coordinator for a regional educational service agency serving eleven school districts.  I had a goal of becoming an elementary principal.  I achieved that goal for five years in two different positions—one as principal of two elementary schools concurrently and then as a teaching principal.  I then gravitated back to the classroom where my heart was.  At some point, I taught all subjects at every grade: kindergarten through sixth in elementary schools.  My elementary teaching was capped off with eight years teaching sixth grade.  While teaching sixth grade, I realized I enjoyed teaching math more than any other subject so I stepped out of the elementary classroom.  I obtained a middle school math endorsement and began substitute teaching in middle and high school math classes.  Several times I was called on for long term high school sub gigs teaching first and second year algebra, geometry, and remedial algebra.  Some of those gigs lasted a semester or more. 

Having taught every grade from kindergarten through sixth, I have a fair sense of what should and can successfully be taught at each grade level.  I draw heavily on my elementary teaching experience in what I share here.  It is important to keep in mind that what I share here is what I found worked for me and helped my students meet with success.  My approach and strategies may work for some but not for everyone.  Each teacher needs to find what works for them to help their students be successful in math…   and have fun.

My sixth grade experience was in an elementary school where I was responsible for all subjects.  This may have provided me with greater flexibility, at least timewise, than middle school teachers who may only teach six grade math. 

Success and Fun

Each year, in my sixth grade classes, about 25% of the students would be proficient with their multiplication facts.  By early December, 75% would be proficient.  In my observation, this roughly corresponded to student attitudes towards math.  At the beginning of the year, most of my students hated math and would have been happy, cheering loudly, if we skipped math class for the day.  As we progressed in the school year a gradual shift took place to the point where most students cheered when I would announce it was time for math.  What contributed to this shift?  I doubt there is any one factor, but I think how math class was conducted had many contributing factors.  As students began being more successful with math it became more fun for them.  Having fun and being successful are important elements.  It also helped that students could tell I enjoyed math and helping them become successful with it. 

It’s possible my attitude and enjoyment would rub off onto some of my students.  I remember running into a former student on a visit I made to one of the junior highs my students would move on to.  The former student excitedly told me that I made math come alive and be fun for her by having numbers talk to each other.  A little numerical personification reached at least one student. 

I feel I helped most of my sixth grade students make a shift to where they liked and enjoyed math.  Many would even express that it was their favorite subject.  Not for lack of trying, there were always students I could never reach.  While there may be any number of reasons they weren’t successful, they were students that put forth little to no effort to pay attention or perform during math class or with assigned work.  For many, it is difficult to be successful without effort.  For most students, things are more fun when they are successful.

Finding My Way by Using I-We-You

Each teacher needs to find what works for them.  Early in my career, I found something that worked well for me.  It ended up not just being something in my so-called bag of tricks but it was something I used most every day in class for the duration of my career.  Over the years I would tweak it here and there to help more students be successful. 

What is this something and how did I find it?  I, along with many others, call it I do, We do, You do, or I-We-You for short.  In more recent years, I have seen offerings for staff develop called Gradual Release of Responsibility that incorporates or is built upon this method.  I really can’t say I found, discovered, or developed this.  It was a gift from one of the best principals I had the pleasure and opportunity to work with.  At the end of a post observation conference, he briefly explained this method to me and suggested I give it a try.  It made sense to me and once I tried it, I continued to use this method.  In what follows, I will try to explain how I use this method.  If you are a teacher and this is new to you, I suggest you search for more information online and give it a try, tweaking the method to suit yourself.  Try different things to find what works for you. 

Phase 1:  I Do.  This phase introduces what we will be working on.  I would explicitly work examples as I explain each step taken to work the problem.  I would cycle through this step working as many examples as necessary until it seems students are ready to try one on their own.  After a few examples, asking questions like, “What should I do next?” can serve as quick check on student understanding.  During this time, students shouldn’t have a pencil in their hand and should be paying attention by listening and watching. 

Phase 2:  We Do.  This phase is guided practice.  In this phase, I would put up a problem just like what I worked in Phase 1 and ask the students to try working the problem on their own.  While they are working, I would walk around the room seeing how students were doing.  I might comment or give pointers as I view student work.  This is also an opportunity to provide some individual help as needed.  I would also make a mental note of types of errors I observed students making.  When it appears most students have done what they are able, I ask them to put their pencils down and pay attention.  I then work through the problem for them and explain it step by step.  This gives students a chance to self-assess by checking their own work.  I may also point out common errors people make and suggest ways to avoid them.  When I work and explain the problem, it is another opportunity to pose questions to class like, “What do I do next?” and “Why would I do that?”  I also ask if anyone has any questions and address them if there are.  We cycle through this process until it appears everyone is getting it. Then we move on to Phase 3.   In Phase 2, some teachers may have students work as partners.  Even if students work with a partner, the teacher should make sure each student is able to be successful and that one student isn’t carrying their partner.  Some teachers may want to have students use individual slate boards or dry erase lapboards and raise them up to show when they complete the problem.  Those were usually never available for me so I grew accustomed to having students use scratch paper and would have students raise their hand as a signal to me when they completed a problem.

Phase 3:  You Do.  This phase is independent practice.  At the end of Phase 2, I would assign a set of practice problems for the students to work independently and provide in class work time sufficient for students to complete the set.  We would then check the assignment at the beginning of math the following day.  Most students would realize that if they used their time well, they would finish their work in class.  Some wouldn’t use their work time well and would then have the remainder of their assignment as homework.  Those students who didn’t use their class work time well had a tendency to struggle since it wasn’t fresh in their mind when they got home and they might not have anyone available to ask for help.  Student use, or lack of, class work time, led to two types of comments from parents.  Parents of students who used their time well and finished the assignment in class would comment their students never had any homework.  I would explain to them it was a good thing since their student uses their class work time well.  I would let parents know, they can always ask and expect their student to bring their completed assignment home to show them.  Some parents of students who didn’t use their class work time well, might comment or complain their student either had too much homework or they didn’t understand the work.  I would let parents know adequate work time was being provided in class with help available and ask them to encourage their students to make better use of the work time.

When I started using this method, I used an overhead projector.  Later, I used a document camera and a projector.  While these are nice to use, if they aren’t available, a chalkboard or white board will also work.

Scratch paper time.  I always had a bin of scratch paper in the classroom.  Students knew it was for them to use when they needed it and they could get some at any time.  I would explain the I do, We do, You do phases to my classes early in the year.  After that, I would refer to it as scratch paper time.   When I would announce, “It’s scratch paper time,” students knew to clear their desks of everything except a pencil and some scratch paper and be ready to pay attention.  It became an routine with clear expectations and an opportunity for students to comfortably strive for success.

Three things I would use the I-we-you strategy for are pre-teaching, prepping for current work, and backfilling deficiencies.  The first was to introduce something new and pre-teach.  If possible, I would try to cycle through Phases 1 and 2 with examples from one or two lessons ahead of our current work.  That way, when we got to those problems in a few days, students were already familiar with them. 

The second way was to prep students for the assignment they were going to get that day.  Prior to this prep, we might have a quick review of the previous days work.  This review often would have already taken place as we would check our work from the previous day’s assignment.  Since the current work would often build upon the previous day’s work, a quick review and making the connection to how it relates is helpful.

The third way is to backfill deficiencies.  There were always skills students should have coming into sixth grade that they didn’t all have.  We would use I-we-you to help bring everyone up to speed on those skills.  These may be skills that hadn’t been taught, were taught differently by different teachers, or students just didn’t get for whatever reason. 

Performance Inspection

In this section, I am referring to tests used in class to test what has been taught, rather than state mandated assessments.  These would either be tests I would make, ones accompanying the textbook we were using, or district made tests for units in our math program.  Without dwelling on it and never spending an inordinate amount of time, I would orient my classes on test taking skills. 

I always made sure we would work on every kind of problem that would be on the test    Examples of every kind of problem that would be on one of our chapter tests were gone over, explained, and practiced during scratch time.  After one test, I remember hearing one my students say to another student that I showed them how to do every problem on the test.  Test what you teach.

The results of a test, for me, were as much an indicator of how well I had taught the material as it was an indicator of how well the students were doing.  Results would let me know if we needed to spend more time on any specific skill as well as which students might need some individual help.  Pop quizzes with three to five problems are a good way to get a read on how students are doing. 

Early each year, I would familiarize myself with the report card so I would know what kinds of things I needed to record grades for.  When I taught sixth grade, there was always one item that troubled me.  I had to provide a grade for each student on math reasoning.  Every year, I would search to see if there was a clear definition for what that meant as well as a way to assess it.  I never really found anything helpful so I would identify some problems that I would use as indicators.  I was always concerned and wanted to be prepared with something to show a parent to justify a reasoning grade if any parent asked what it was based on.  Fortunately, no parent ever asked about this.

Checking Work – Self-Assessment

At the beginning of math class, I would have students get out their assignment from the previous day along with a checking utensil.  I would then provide the answers to the problems, usually by saying the answers out loud as I walked around the room making note of who completed the assignment and who didn’t.  This required students to really pay attention and follow along as they checked their own work.  They were to mark any problem they didn’t get correct.  When I finished giving out the answers, I would ask if anyone wanted me to repeat any answers or if there were any problems they wanted me to work through for them. 

For most of my sixth graders, checking their own paper was a shift from pleasing the teacher to assessing themselves.  At first, many students would hurriedly try to erase a wrong answer and write the correct one on their paper without regard for why they may have missed it.  Eventually, students would begin trying to figure our why they missed a problem.  This gave students immediate feedback on their work and the opportunity to get further explanation on problems they may miss.  I didn’t collect assignment papers on a regular basis.  I might collect them on random days so students would know they needed to be prepared.

Once I started having students check their own work, I began to see more of them take responsibility for their learning.  Rather than trying to please me as the teacher, more of them were striving to be successful with their math work.  More students, if they couldn’t figure out why they missed a problem, would ask me to work the problem for them so they could figure out where they made their error.

Separating the wheat from the chaff

It’s important to know what’s important for students to learn.  Standards can be helpful as one guide for a teacher to use and may be even more helpful if the teacher is familiar with the standards for two, three, or more grades beyond the one they teach.  Knowing the math and standards beyond the grade one teaches shows where things are going and can help one better know what is important.  Even better for an elementary teacher is being familiar and comfortable with algebra.  The instructional focus should be on what is important for a student to be successful as their math instruction advances through the years.

Fibinacci numbers is something I have seen a fourth grade teacher spend several weeks on.  Fibinacci numbers are interesting and can be fun for students to work with when exploring patterns.  Is it important to spend a lot of instructional time working with this pattern?  This pattern only gets brief mention in some calculus texts and seldom is mentioned in math programs through high school.  Time would be better spent at this grade level ensuring students are fluent with multiplication facts and are able to successfully complete division problems. Fibinacci numbers:  wheat or chaff?

Early in my career while teaching second grade, we had a new math text that spent a lot of time on base 8.  The idea behind this was to help students understand place value.  Since it was in the book we had to use, I did address this but didn’t spend too much time on it.  I didn’t see this as important.  Since then, without teaching base 8, I didn’t see students having difficulty with place value.  Base 8 did confuse students (and a lot of teachers and parents).  In my life and career, I have never needed to use base 8 and have not come across it being used or needed in algebra, geometry, or pre-calculus.  Base 8:  wheat or chaff?

A good math book will be filled with wheat and have little, if any, chaff.  Unfortunately, publishers seem to have a greater interest in selling their books and programs than providing the best material for providing a solid math foundation for students.  As a result, it is important for elementary teachers to know what’s important, leave out or not spend much time on the unimportant (like base 8), and add in important things a text or program leaves out (like telling time).