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.

• How to Teach in a Group Centered Classroom
• Elementary Math Instruction "Look-For's"
• Task - Picnic Lunch for 3rd Grade
• Task - New Fish Tank for 3rd Grade
• Task - How Much Does it All Cost for 5th Grade
• Formative Activity - Going to the Movies for 5th Grade
• Grade 1 Assessment - Adding and Subtracting Multiples of 10
• Grade 3 Assessment - Place Value and Addition and Subtraction
• Grade 3 Performance Task - Arcade Tickets
• Grade 5 Assessment - Multiplying Whole Numbers and Decimals
• Grade 5 Assessment - Multiplying and Dividing Fractions and Mixed Numbers
• Task - Who is Correct for 1st Grade
• Task - IPod Usage for 3rd Grade
• Grade 1 Assessment - Working with Equality
• Unfinished Task - Shriners Donation from the Packers - Grade Unknown
• Book - What's Math Got to Do With It by Jo Boaler
• Book - Accessible Mathematics by Steve Leinwand
• Book - Principles to Actions by NCTM

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.  

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?"

en.wikipedia.org

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.  

www.hwporter.org

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.

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 (www.bls.gov).  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 www.criticalthinking.org 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.

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!

Give Students the Tools needed to Solve Problems...

In my previous blog, Questions that Spark Student Curiosity, I discussed ways to ask questions that would engage students. France Snyder commented, “What is the best way to evaluate our students? What is the effect on our students’ engagement, retention, and transference of skills? What is overall best for our students?” I’ve thought a lot about how we evaluate students over the past few weeks, as we have just completed first semester final exams. 

While prepping my students for their final exams, I fielded questions  from students about how to access the study guide, the best way to study at home, and how many class periods would be dedicated to finals exam review. My response was consistent with past years.  “Every one of you is aware of what we discussed in class.  You are also aware of how to access the resources we used in class.  To study for the final exam,  consider our conversations and reflect on them.”

When the final exams came, I chose to spend an extra day administering the exam over two full hours so students could consider their responses thoughtfully without time constraints.  After I distributed the exam on the first day, students responded “It’s only 8 questions.  Why would it take 2 hours to complete?”  Shortly after exams were over and students had a chance to reflect, I asked them what they thought of the exam and their responses reaffirmed my final exam strategy.  “It wasn’t hard but my answers were not the same as those of my friends.  We knew you would not accept a simple ‘yes’ or ‘no’ so we really had to think about all the options and justify our responses.

To me, final exams aren’t for the sole purpose of determining what knowledge a student has attained and what facts that student has memorized.  They are an integral part of the learning process.  For many decades, we have asked Webb’s Depth of Knowledge (DOK) 1 and DOK 2 questions: those questions with a single correct answer that leads students through a single path of thinking.  Now, we are focused on  DOK 3 levels of thinking, so we know there is a better way to evaluate student learning.  Yet, we still fall back to the familiar ‘yes’ or ‘no’ style of questioning in many of our final exams.

I was on Facebook the other day and saw a post that made me stop and smile.  “Don’t ask kids what they want to be when they grow up but what problems they want to solve.”  This is a perfect example of changing a question from a one-word DOK 1 answer to a thoughtful, considerate DOK 3 response.  What problems do we want our students to solve?  I don’t want them to solve a problem in which the answer is known.  We need to stop training our students to accept DOK 1 questions as the norm.  Life doesn’t contain these simple problems, and neither should education.  Any question that ever told me anything about a student was a question I didn’t know how they would answer it and more importantly, how they would explain their answer.

While working with one of my teachers responding to an email from a disgruntled parent about the lack of “math” being taught we decided to dig a little deeper.  The parent described how he retaught his daughter using proper math and how she now completely understands everything we were trying to get through to her.  We did what any good teachers would do.  We asked her to explain her understanding.  She could explain parts, but after a few steps, she could not explain the next procedure and, most telling,  couldn’t explain her goal.  DOK 1 questioning that we have been doing for years promoted a recipe style of mathematics.  Our students don’t retain it and, most importantly, they don’t like it.

Webb’s DOK is not just a method of questioning for the sake of questioning.  It is a way to help students attain a higher level of understanding.  At the same time, it is a method that encourages student engagement.  Encouraging students with questions that make them consider multiple options and different perspectives to solving mathematics will only aid them in enjoying what they are doing, which in turn will make for better mathematicians.

It’s not about just teaching for recitation.  It’s about giving students the tools to solve the problems they want to solve.

Walter Wick bringing a New Perspective to Math...

On a Sunday afternoon that the family finally had a free moment, my wife took the family to the Woodson Art Museum to view the Walter Wick exhibition. Walter Wick is the photographic illustrator of the I SPY series, and the author/illustrator of the Can You See What I See? series.  Besides see some amazing photography and reflecting on a bit of my own past, I had so many moments of game changing instructional opportunities.  

The first came from a pair of photos called Balancing Act, of which one is pictured at the right.  The photo shows many objects seemingly placed at random all balancing on a single piece of LEGO.  Mr. Wick mentions the process of getting everything to balance took over a week with much trial and error and several crashes along the way.  What I saw was the amount of math that could be extracted and then performed from such a starting image.  From 7th grade ratios and proportions to symmetry all the way through the upper levels of mathematics.  What intrigues me the most is that almost every student will have objects similar to these sitting in their home.

Next to this photo was another called Sorting and Classifying from the I SPY School Days.  From my experience teaching Geometry to planning earlier math lessons the concept of a Venn Diagram is not the easiest concept to grasp when applying it to mathematics.  However, what if the class started with a photo of Sorting and Classifying followed by the simple question, "What is the purpose of the rings?"  Instead of teaching students what the Venn Diagram is, allow students to discover its' purpose and what they sort in this situation.

The final photo thatI felt it was important enough to share is Mirror Maze.  This photo is created by using mirrors in the shape of an equilateral triangle to make the maze.  I sat in front of this photo for at least 15 minutes just following the reflections and identifying where I felt there could be inconsistencies while also looking for justifications of the inconsistencies.  This is the type of thing that would make Geometry much more intriguing.  The number of places it could fit in during the year is almost limitless.

All these are just pieces to a puzzle I have been trying to solve in my head and in the classroom for some time.  Students have a limitless amount of stimulus throughout the day that take their attention away from the classroom.  However, rarely do they find something that they could just stare at and be intrigued.  The other piece to these photos is not only the depth of the mathematics but the access to many other levels of math.  For example, most of a typical Geometry course could be made up with just these three photos and connecting the concepts between them.

My thoughts now settle on the art that I am missing to further enhance mathematics.  On a side note and for our M.C. Escher enthusiasts.  Check out Going Up and Tricky Triangle.  These are not drawings, which often lead students to find M.C. Escher "cool" but not with the same curiosity as something real.  These are photographs of real objects.  Go ahead and find the intrigue.

                                                   

Why a Retake doesn't enhance Learning...

Recently, a post on a listserv I am part of was discussing retakes at the secondary level and how different schools have structured them. The thread was well responded to and well thought out. We had responses from giving them in every case no questions asked to not ever allowing a retake. Some schools require certain things before a retake can begin, others allow a certain maximum number of retakes in a semester.

Throughout the thread, I read, but didn't respond. I wanted to know if my thoughts about retakes were unique, or being rather reflective, on the right track. In my experience, retakes have been frustrating to say the least. The culture of the my school has changed so drastically since the advent of the retake, it puts in question the core concept of should retakes happen. It has been pointed out to me that I never statistically determined that the retake is the cause of the cultural issues. Nope, I haven't. Personally, I don't feel I need to. When student's respond on the first assessment of the year that they "just want to look at it and take the retake tomorrow," I don't need a statistical analysis to determine there is a cultural issue and retakes, at minimum have something to do with it.

My beliefs about retakes are simple. Not only should they happen but they are an essential component of education.  My guess is you may be a little surprised at that statement.

Retakes and RTI need to go hand in hand.  If a student is retaking an exam, the question really needs to be why, not what percentage it should take in the grade.  We should also be asking questions such as:  What did the student do to not earn a passing grade?  Do they have prior gaps preventing them from learning?  Do they have poor work habits?  Is our test based on work habits or learning?  How can we design their day to best help enable them to succeed?  Could the issue be a home issue and this is a one time instance?   Or, is there a different underlying issue?

 

I feel it is too easy to pull the student responsibility card and much harder to look at it from an individual student perspective.  It is important to remember that these are kids we are working with even though in most cases we want them to act as adults.  This doesn't mean there are no consequences from poor choices.  On the contrary, it means the complete opposite.  If a student legitimately didn't do anything prior to the exam the true reaction should be they don't take the original assessment.  Wouldn't this be more effective than allowing the student to fail initially.  The retake needs to be used as a teaching tool, not a gift.  Retakes don't lead to learning.  If a retake needs to happen what is the reason.  That is where the learning occurs.  Should they happen, yes...after we have figured out what the root cause of the lack of learning is.  Otherwise we are just perpetuating a viscous circle.