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.

Saturday, July 16, 2016

Why do we Continue to Teach...Part 2

Well, it took a bit longer to get to part 2 than I had planned. Things like finishing a school year and planning for next year took some precedence. Regardless, if you need a refresher on part 1, here it is.  The premise of the problem is "Why do we continue to teach math that has little application to a “normal” person?"  The first post went the route of justifying why people are asking the question and the problems with the foundation of the question.  This post now leads into what other teachers are saying.  To clarify, I polled my staff and some of other math teachers around Wisconsin.  Their thoughts ranged from dumbfounded, to overly traditional, to outright angry.  Most of the teachers had very similar comments.  Finally, my closing thoughts on this topic.  I must say, it ranks up there as one of my least favorite questions to answer.

The overly traditional teacher:
I must admit that this teacher is the one I would not want discussing this prompt with others.  Why must we teach post Geometry mathematics when there are programs that can do it for us?  "Because students need to know this information.  It is important for being a mathematician.  It is what pure mathematics is about."
Personally, I feel these teachers are so far off base it is scary.  Why does a student need to know this information?  Where can we show students they will actually apply these concepts.  In my life, as a math teacher, outside of school I have not had the pleasure of factoring a degree 3 polynomial.  Nor do I ever feel like I will. The final two comments made from what I am calling the overly traditional teacher bother me the most.  It is rare that we have "mathematicians" come out of school.  It is one of the highest need majors in careers yet one of the least populated in post-secondary education.  Being a mathematician isn't even about pure mathematics.  A general mathematician doesn't sit around solving proofs on his coffee table.  A mathematician applies mathematical theories and techniques to solve practical problems in business, engineering, the sciences, and other fields (  Overly traditional teachers tend to feel they are producing life-long math people.  They miss the point of mathematics.  It is an area of study that can enhance other fields by making them more efficient and effective.  Since they miss the point, they also miss many of the best ways to meet the needs of their students.  Their flaw is not with a lack of instructional ability but with a lack of knowledge of their students and because of that, instruction that can engage students.

The Politically Correct teacher:
It was a struggle to put a name to these responses.  Partly because I agree with their answer to the question but don't feel they have hit it all.   These teachers believe we need to teach post Geometry mathematics because we are working on their "Critical thinking skills."  When they get in the real world (no matter what their job is) they are going to be required to think critically and solve problems on the fly.  The better their critical thinking skills are the better chance they have of advancing in their career.

On the whole it is hard to disagree.  However, there are many, far more engaging ways to teach critical thinking skills than higher level mathematics.  Why would we teach the math and be so specific to the solutions and methodologies if it was about critical thinking?  Critical thinking skills can be taught in many ways.  Sites such as list numerous strategies that engage students without teaching mathematics.  Math in itself doesn't teach critical thinking.  Students will perform better if they have the ability to think critically.  It is much more of a shared relationship.  Does math help develop these skills.  There is no question about it.  I would argue more than any other subject in Secondary instruction.

The honest Abe with a touch of angry teacher:
The responses from these teachers were all well stated and...long.  It makes the most sense to simply put one of their responses in here since I cannot say it better.
"I was hoping not to reply to this,  as this whole line of questioning just reflects how little these people know, understand or care about math and it is very frustrating.  We have to justify ourselves...why math?  Really?  Why do we teach anything then?  
My first response would be how little they care about the development of our students reasoning and logical thinking ability as they develop their minds in adolescence.  Yes I understand a Geometry proof or multistep complex type solving equation my never be directly used in that students future, but the ability to think and reason clearly is very important.   Do they wish our students to never know how to make a political candidate decision based on logic, or decide if as a family they should spend a large amout to have surgery on a pet, or let the animal pass. We need thinking skills and math helps in that development.
Second,  I would offer to come over to their house, dismantle their TV dish or cable, ask for their cell phone, and tell them you will be happy to dispose of them since math is so unimportant.  Lets' take away some of those benefits that have been given since you don’t' care why they exist.  Okay, so not every student is going to come up with the next and best greatest math idea.  But should we not at least proceed on the premise that someone could be that person. We should at least give them a chance to improve the next generations lives, maybe figure out pollution problems, energy usage, etc...  No math, no improvements!
Finally, I would tell them how sorry I feel for them.  They truly do not see math for the beauty and wonder it offers in their lives.  Math is everywhere.  It is integral part of all our lives if you just know where to look, and be open for it.  If you have no base understanding, like Geometry and Algebra you will never see that wonder.  If they choose to close their minds to what is all around them, then just let them know how you feel sorry for their loss, but don't let your close mindfulness affect our students."

Although the anger clearly spews from this particular teacher the depth of the response is precious.  I can't make this better than it is.  What needs to be said is that I side closest with this teacher.  We teach post Geometric math because we need people who can make the change in the world they want to see.  We don't know who they are or when or if they will gain an interest in math.  What I can guarantee you is that if we don't teach these topics we won't have those who can make the advancements we want to see.  This is a question that needs to be put to rest.  Math has the same importance as reading in many ways.  Each subject by itself doesn't mean much.  However, tied to other content and math and reading become of such an importance that without them progress would subside.

Instead of asking why we need to teach them, we should be listing all the things that would happen if we didn't.  I don't think there is enough room on a blog for that though.

Why do we Continue to Teach...Part 1

Recently, a common question has been repeatedly posed to me, one which makes me uncomfortable on several levels.  The first being that the answer seems so simple that it begs me to wonder why bother asking it. The second being that the answer, although simple, makes me wonder if we are doing the best we can, or if we are simply giving someone verbal justice.
Why do we continue to teach math that has little application to a “normal” person?
This question begs at the systemic problem of post Geometry mathematics: “Those who do not understand mathematics, feel it is unimportant.” Dispensing with the simplistic, selfish nature of the question, the lack of vision of the person posing the question, and the refusal to answer it with “because we always have”, on its surface, the answer seems simple.

The logic associated with the late secondary mathematics allows students to engage in careers they would not be able to engage in without it. Essentially, it opens doors. The technological advancements that have prompted this question are things such as Photomath and other internet sites, along with the theory that we need to prepare students for what they will be doing in their careers, not a general path of instruction.

Let’s tackle these one at a time. The first argument typically presented is that if apps such as Photomath or sites such as Google can solve the problem, why do I need to learn about it? (see a previous blog for more introduction). This argument is fraught with problems, the least being with the instructor who is allowing students to live in simple DOK 1 styles of questioning. Those are the only questions that these apps can handle. Anything related to an actual scenario to utilize the math is well beyond the capabilities of these programs.
Furthermore, if we isolate the problem to a DOK 1 situation, as shown below, it opens up an additional area of concern.
1/3 (x+3/6)=1/3
Using apps such as Photomath produces a solution like the one shown below.
Screen Shot 2016-05-10 at 9.30.48 PM
The goal of a problem such as this is to see if students understand the conceptual nature of mathematics. We want students to understand the solution process so that they can open the door to higher levels of mathematics. However, if they understand number sense, they can quickly see that (x+1/2) must be equal to one because 1/3(1) = (1/3). Therefore, the only way (x+1/2) = 1 is if x = 1/2. What takes photomath 17 steps can be solved in less than three.
Knowing “math” is not only more efficient, it is more effective. When I teach math, I want students to understand the solving process, or what most people believe upper mathematics is about. That said, I want them to understand mathematics so much more than I even care about the process.

The second argument is more daunting and more bothersome. Adults who have struggled in math, or never truly understood mathematics, believe it is solving equations for the sake of solving equations. They believe it is completing proofs for things that have no meaning, so they can somehow be more fulfilled by the process of just doing the proof. They see math careers as teachers, professors, and engineers. The reality is so far from this perception. It is the equivalent of telling someone that they know how to make an automobile because they know how to drive one.

Kiplinger listed the top 10 college majors for 2015-16. Each of the top 10 has a major focus, if not entire focus on mathematics. In addition, CareerCast lists the top 10 professions to enter. Of those professions, seven are focused on mathematics.

This isn’t math for the sake of doing math. It is math for the purpose of what math is. It is about seeing a pattern, or a logical process, in a situation and either finding a solution or a pathway to improving that situation.

In the past few years, math has helped solve problems in heath care, computers, and safety. In the near future, math will help to solve efficient energy solutions, global warming, and many more issues of our time.

Some would argue that science will be responsible for solving these problems. But I would argue that science is a subset of mathematics. Math by itself doesn’t solve problems, which is why many school districts are implementing STEM initiatives. One of the most significant realizations among educators over the past several years is that there are really two core subjects in education — English and mathematics. Without a solid foundation in both of these, career opportunities become drastically limited.

Beyond these responses, what is the best way to respond to the question of, “why do we continue to teach courses like Algebra 2, Pre-Calculus, and so on?”

Is there a better way?

I thought about my answer to this question a lot for this blog post. I plan to split my response across three parts. In part two, I will convey how my staff feels about the question and some of their responses. And in the final post, I will cover some possible answers to the question.

Look for part two of this blog post soon!