Tuesday, January 3, 2017

The Truth Behind the Standard Algorithm

For as long as I can remember seeing in pictures, reading in books, and watching old TV shows, the standard algorithm has been the staple of mathematics during the elementary years.  These algorithms are burnt into our brain through images of the old school house, blackboards, and crummy movies.  However, they have maintained in instruction for an assortment of reasons. 

1.     They help most students calculate math.
2.     Our parents learned through them so therefore, children have also.
3.     Teachers tend to teach the way they are taught…thus the algorithms continue.
4.     The Common Core has them stated as necessary parts to instruction in grades 3-6.

Recent instructional pedagogy has produced strong data to support no longer using the standard algorithm as the main form of instruction.  The changes started in the late 1990’s and are now being pushed further by people such as Jo Boaler.  Their efforts are based off an understanding of mathematics rather than just calculation.  With all of our advancements, the United States continues to be one of the few remaining developed countries that use the standard algorithm as the main form of instruction. 

Personally, I couldn’t agree more with the changes being pushed in recent years.  Since starting as a K-12 Math Coordinator we have been discussing, developing, creating, and presenting alternatives to these algorithms that have more to do with understanding than calculating.  We have been working against traditional math trying to encourage students to do more than calculation.  I believe we can expect so much more from our children than rote mathematics.  I believe we need to focus on the “why” rather than giving students the “how.”

Lets go on a journey through some of the biggest reasons why teachers keep emphasizing these algorithms and why we as professionals need to make the decision to move on.

The methods in the algorithms are needed to learn the upper levels of mathematics
Forgive me but I started with my favorite reason most people give to keep the algorithms.  It is not enough to say that after grade 6 or 7 calculators are doing the vast majority of the dirty work in calculating math.  It is more important to understand that the methods used in the algorithms are not used in upper levels of math.  The only algorithm that reappears consistently is the division algorithm, which comes back when dividing polynomials.  Even that method for dividing polynomials is an inefficient method as compared to synthetic division or graphing solutions.  In all of these cases there are apps that can do much of the computation for us.  This doesn’t mean it isn’t important to know how to do these steps but that its’ importance is minimal as compared to the much larger picture of what the outcome of the division means.

The standard algorithm for multiplication is purely gone.  For some time area models have replaced the algorithm.  Even that is an incomplete comparison because we are comparing polynomials with multiple terms, not numbers.  Polynomial multiplication is closer to the partial products method than the standard algorithm.  Furthermore, the methods used emphasize the meaning of multiplication.  Not just calculating for a solution.

It is the methods parents know so we must teach it that way to help with home-school communication
It is ironic that statements similar to this one surface about math when they don’t surface about reading or writing.  I believe it has much more to do with the procedural drill and kill approaches taken when current teachers/parents were learning math.  In schools, we used to teach keyboarding at the high school level.  We taught students lattice multiplication in the late 1990’s and early 2000’s.  We went through phonics, to whole language, and back again. 

This goes to show that times change and we need to move with them.  With the technology of today we can communicate our methods of instruction with parents and more importantly the reasons why instructional methods are changing.  We need to emphasize instructing parents as much as instructing our students.

To be clear, we own this problem.  The problem is communication, not knowledge.

The algorithms work.  Why change what is working?
I would argue that the algorithms are not working.  In third grade, students learn to add multi-digit numbers together.  This addition should be fluent by the end of the year.  However, in fourth grade teachers are always re-assessing and arguing that the students don’t know how to add multi-digit numbers.   The same is true in fifth grade, sixth grade and so on. 

Is the problem that students don’t know how to do it or are not retaining the knowledge?  The answer, based on student performance is obvious.  Students are proficient at the skill in each grade level but when reassessed the following year no longer show the same level of proficiency.  The students don’t retain the process.  However, when using alternative methods such as partial sums they not only retain the ability to add they perform it at a fluency level doing much of the calculations in their head.  They learn that adding the hundreds, tens and then ones makes it easier to get the solution.  It also gives them a much better understanding of place value which means when the students transfer into multiplication it makes more sense.  

I leave this blog with a final thought.  Watch a student as they progress through Kindergarten to first and then second grade.  Students don’t naturally develop the traditional algorithms for addition or subtraction.  Instead, they focus on concepts that deal with place value.  The traditional algorithm must be learned through a teacher that directly teaches it.  That alone should tell us what we should be doing.  I believe there is a place for these algorithms.  However, only if they are taught after the sense making methods are discovered. 

As always, I don’t consider my opinion to be fact.  Because of that I have linked a few articles that support both sides of this story.  Enjoy the reads and come to your conclusion.  Please share it with me.  Hopefully we can learn together.

Thursday, December 8, 2016

What do I feel all new math teachers need to know...

I have the opportunity to talk to prospective Elementary teachers at the University of WI - Stevens Point today.  The talk is focused on problem solving and how it integrates into the classroom.  However, we are going to talk about much more than that.  I am bringing a few examples of things with me.  Such as a few articles I wrote, some Math Thinkers to demonstrate problem solving, some sample assessments, and a basic form that I use to evaluate instruction in a classroom.  I am also bringing a few books that I feel are important for every new (and existing) teacher to read.  The goal is to expose these current students to what we are going to expect from them as they become professionals leading our future students.

This is a whole lot of stuff.  However I want the future teachers to understand thinking at different grade levels.  The key to the whole conversation is not the tasks but the thinking the tasks promote and how a teacher promotes it.  Something that cannot be explained by a sheet of paper.  This year I have been lucky enough to be able to be in more classrooms than ever.  It just makes it more obvious that as teachers, we think a whole lot more than our students.  At first read that sounds like a "duh" statement but the reality is it needs to be the other way around.  In fact, we don't even let them get to the thinking because we "save" them from failure.

We, as professionals need to realize that the sage on the stage can no longer be a viable instructional method at any level.  We need to realize that the more math (please don't confuse this word with calculation - instead read it as pattern finding) students are doing in the classroom the more math they will learn.  

My hope with these conversations is the new instructional force coming into the field understands the expectations and can hit the ground running.  The reality is this will take some time.  If you are a prospective teacher just reading this blog for the first time, check out some of the links on the right.  They are resources from great teachers.  More than anything, be creative and try something new.  If you have talked for 10 minutes or more while teaching a classroom...you have talked for too long.  

For the experienced teachers reading this blog, post a comment that you feel new teachers could benefit from.  

Monday, July 18, 2016

A New Perspective...Can we look at post-Algebra 1 Math Differently?

A question was posed rather carefully on a previous blog that needs to be addressed.  

"Running the risk of inciting the ire of math teachers across the world, "What qualifies as "post-Geometric" math?" Does it have to be Pre-Calculus and Calculus? Does Statistics count? What about a class like Machinist Math, where the application of the mathematical concepts ties directly to the problem solving process directly related to operating and maintaining manufacturing machines? Is it possible to address the fundamental skills that math allows us to practice with a different kind of math curriculum than the "standard?"

Let's start with some background since this question really reminds me of a Rubik's Cube.  Depending on the way you look at it there are several answers.  There is little to no disagreement that Algebra 1 is necessary to understand for all students.  It is a gateway, not only to math but other subjects.  According to KnowRe, Algebra 1 presents students with academic challenges they have not yet had to face. Algebra is often the first course in which students deal with abstract reasoning and problem solving. This abstract reasoning helps connect the dots between historical events and what is currently happening in the world.  It opens areas of science that cannot be explored without it.  It has direct links to music and art along with a litany of applications in sports.  

What about Geometry?  Is that necessary.  Prior to 2010 it would not be difficult to say yes.  However, with the adoption of the Common Core State Standards, much of Geometry has moved into 5th - 8th grade where it is used as an application of current learning.  This should remove a significant portion of the years learning as long as students are retaining knowledge (which is a major issue in all subject areas).  The remaining portions of Geometry would be ground-level knowledge for trigonometry, applications in circles, etc...  Knowledge that the current U.S. citizen walking down the street doesn't have.  Should they have it?  That is the debate.  

When I have the opportunity to attend a math conference my thinking typically falls with the majority of others.  However, when I am in my district or visiting other districts working with secondary math teachers, my thinking tends to be the minority.  Why that is would qualify for another blog.  Regardless, there is a much bigger question here that needs to be answered.  That said, the answer is complicated. 

What does the student need to succeed in their career/life? 

Every time I am posed with student placement decisions this is my go to question.  What makes this so hard is a few conditions we cannot change.
  1. Many students have no idea what they want to do after high school.
  2. If a student does have an idea of what they want to do, too many parents steer students away from their choice.  
  3. Career choices are changing so fast most of the careers our current students will have don't exist currently.
  4. Some of the most growing fields are seen as "manual labor" when the reality is they are highly specialized (Machinist) involving lots of mathematics.

The Common Core State Standards do not require a student to pass Algebra 2.  They require some of the skills in a typical Algebra 2 course but not all of them.  Therefore, Algebra 2 is not required.  In fact, most, if not all of the skills are taught in the 1st semester of a typical Algebra 2 course.  The remainder of the skills are instructed as the next set of foundations needed for further study in mathematics.  "Taking and successfully completing an Algebra II course, which once certified high school students' mastery of advanced topics in algebra and solid preparation for college-level mathematics, no longer means what it once did," writes Tom Loveless of the Brooking Institution in a blog post. "The credentialing integrity of Algebra II has weakened."  Tom Loveless bases this statement on a correlation between the number of students passing an Algebra 2 course and the nations NAEP scores.  The Algebra 2 barrier has been a sticking point with post-secondary education.  For many institutions, it is required for entrance.  Even for students with no interest in mathematics.

So if Algebra 2 is not something all students should have what is?  The answer in my mind is nothing.  This is not to mean no math class.  It means there should not be one answer to this question.  Lets face some facts.  The important learning that happened in my life to get me where I am now did not happen in high school or college.  It happened after several years in my field when I realized, "Wow, I really need to know more about that."  I took courses, attended workshops, and read.  When I was done I read some more.  The constant for students who struggle in school is a lack of interest.  It goes back to the question listed above.  What does the student need to succeed in their career/life?  Note, the question is not "What do we currently have that we can put the student in?"

By no means do I imply that students should not take Algebra 2, Pre-Calculus, Statistics or Calculus.  I mean they should take what interests them.  If a student has no interest in math but can't wait to corner someone in order to argue politics.  They are more than likely not going into a field where Calculus will make a difference.  It is quite possible their path may need to go through a more formal statistics course (which needs more math background than Algebra 1 but not a full year of Algebra 2).  As the original questioner stated, what about a Machinist course?  Definitely!  Consider for a moment that student who doesn't do his/her homework but spends time in the wood/metal shop "playing" with materials.  Currently they don't need Algebra 2.  Although, it should be noted that much of a machinist job lies in content of a 1st semester Algebra 2 course.  In my district we did a small challenge this year.  The Machine Tool course was working on some models, determining the lengths and angles needed to make a part.  It should be noted that the math is completely integrated into the course.  The teacher took the same schematic to his Algebra 2 class and gave them a period to find the necessary measurements.  After a period, they were no farther than when they started.  The Machine Tool students, who would never pass an Algebra 2 class, had it all figured out.  

Here is my answer to a long thought process.  We don't need all students to take anything past Algebra 1.  However, we need all students to take the math they need to move forward in what they want to do.  We need instructional options for all students.  An abbreviated upper algebra course for those students who don't now need everything Algebra 2 has to offer but yet might entice them to go farther if they so desire.  We need courses such as Calculus for those students who know their path lies in mathematics.  I once read a report that only 9% of college majors require calculus.  Then why is that where the crown of mathematics is held.  We need the upper level mathematics courses to continue to move innovation because that content will possibly be helpful in the next innovation.  However, not everyone is an innovator in that way.  The key is to find where that child wants to be innovative, then tailor the plan to set the student on a path towards their goal.  

There are far too many resources available to not allow a student to tailor their own path.