Sep 072016

The struggle to understand why we teach K-12 mathematics in the order we do, and the content we do is real. I have wondered about this for a long time, and really have never found a good answer.

I threw out the idea of teaching y=mx+b as the only way to write lines (even though the district materials at the time said it was all we needed). I took a lot of heat for that decision from some people. I was told I was completely wrong; by teachers. I stuck to my guns because y=mx+b is a stupid way to teach lines. And in the end, I was told by other teachers that I influenced them to change too.

But really, K-12 mathematics education is nothing like this:

Mathematics as human pursuit

Think of Lockhart’s Lament.  You read Paul’s words, and you are hit by the poetry he sees in math. It is also 25 pages long. I read somewhere that Lockhart’s Lament is the the most powerful and often cited mathematics education document that is never acted upon. What does that say about us, as educators, who cite it?

Lockhart is passionate about math education, and he feels that the current state (in 2002) of math education is in trouble. His words may be as apt today as it was then. On page 2 he writes,

In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

How much impact has Lockhart had on mathematics education? Often cited, rarely used or implemented. And yet, my Twitter feed and Facebook still have things like this pop up regularly.

Mathmatician is like a painter G H Hardy

What beautiful words representing fantastic ideals. Are you starting to see the cognitive dissonance I am feeling today? Too bad none of these ideals are found in our textbooks or our standards (and don’t get me wrong, I am not hating on the CCSS-M here). In fact, much of school mathematics is exactly how Seymour Papert described it here.

Papert - outwitting teachers as school goal

It is mindless, repetitive, and dissociated.

So as I was thinking of the question of “Why?”, I stumbled upon this article. Why We Learn Math Lessons That Date Back 500 Years? on NPR. To find out the answer is pretty much, “Because we always have,” is sad, disappointing and frustrating. We have taught it this way for the last 500 years, so we will continue to teach it this way for the foreseeable future.

I call B.S.

Seriously. We need to rethink how we teach math in a substantive manner.

We are part of a system that is not allowing learners to find the joy of mathematics, but the drudgery of mathematics and of learning. And this is not new. Not by any means. Edward Cubberly, Dean of the Stanford University School of Education around 1900) said,

Our schools are, in a sense, factories, in which the raw products (children) are to be shaped and fashioned into products to meet the various demands of life. The specifications for manufacturing come from the demands of twentieth-century civilization, and it is the business of the school to build its pupils according to the specifications laid down.

The fact that the specifications of education haven’t changed in hundreds of years is a problem (see the NPR article). It may even be THE problem. I am not so confident to claim that for sure, but it is definitely A problem.

At what point do we, as teacher leaders, rise up and demand this change. We see the damage. We see the issues. We must start demanding the curriculum be changed to meet the needs of our learners. I am not sure that the CCSS-M is that change. It seems like it is codifying the 500 year old problems that we are currently doing.

But it doesn’t not have to.

The Modeling Standard is gold. It is also 1 single page in the entire document.

I will just end this rant with that thought. Oh, and this thought. No more broccoli flavored ice cream.

textbook math is like broccoli ice cream

Nov 062014

I haven’t posted in a while, mainly because I am just so happy with how my classes are going. I will focus on Alg 2 here, because these awesome learners are just knocking my socks off.

I am in the polynomial unit, knee deep in graphing, and increasing, decreasing, relative mins, relative max’s, absolute mins, etc. This is the problem set we were working on today in class:


Here are the questions I ask (docx format) for every single graph, from lines all the way through sin & cos at the end of the year.

Yes, some of these are going to be Does Not Exist. That is okay. Just because we don’t need to think about asymptotes with cubics does not mean we shouldn’t ask about them.

A little back story before I say something about my learners. I used to teach the textbook. I admit it. I sucked, horribly. My learners did not connect anything with anything and they did not see how to connect topic from one unit to the next. I was frustrated. So I first came up with my list of functions in (h,k) form, wrote it on my board and changed how I approached algebra.

2014-08-10 15.43.46

That was a win. But, then I was frustrated because every time I changed the graph, added an exponent, I had to teach a new set of vocab, but everything was the same; so why was I teaching new stuff? Why couldn’t I teach all the vocab up front, and then just explore the heck out of each function family?

Short answer was, I could. So, I did. That is where the form above came from. I introduced it last last year, and used it and modified it and tweaked it and the learners responded.

Enter this year, this class. I have everything set on day 1. We entered the year thinking about connections and planning our math and discussing end behaviors of lines (wow, that was easy, hey, they are always the same!, etc). Then quadratics, and we completed the square to get vertex forms, and we factored, and saw how intercept, standard and vertex forms were all the same function, and and and.

Enter polynomials.

We have done them from standard form, and done the division to get intercept form, we have broken these guys down every which way. I have tossed them fifth degree and fourth degree polynomials, they didn’t even blink. “Oh, so this just adds a hump to it.” I have explored more in polynomials this year than ever before.

And, since it is a constant review of prior material (“If this works with quartics, will it work with quadratics too? Yes”) I am constantly cycling and eliminating the mistakes my learners made in previous sections and on previous exams.

Which brings us to the problem set above. That is a killer set. The 4th and 5th are tricky, and they struggled. Until one of the class members said, “Don’t all we have to do is distribute them and so it is just a bigger distribution problem?”

Done. And. Done.

Now, of course there is a nicer way to do it. Substitute “u” or some other variable in for (x-3) in the fourth problem so you are multiplying binomials first. It saves time. BUT, it was not necessary to show it. They know distributing, so distributing is what works and they rocked the socks of of it.

So, why have I not been posting much? Because I have been enjoying the heck out of teaching. These learners are taking these ideas and running with them.  And I love it and them.

May 022013

I have had this long term struggle going on in my head this year that we really don’t do a good job making connections between material in our classes, and that lack of connection is one reason why “transfer” (ala. Grant Wiggins and Understanding by Design) doesn’t occur as  frequently as I would like.

Well, I am not going to talk about it any more. I have the beginnings of a plan to enact. There will be many steps to this plan, but I think the starting point needs to be simple to enact and creates some opportunity for connections to be made.

Every test in my department from Algebra 1 through Trig/Precalc must have a couple of different kinds of problems on it. This is step 1 I am implementing next year.

The first type is a literal equation. Of course, as a stats teacher my first thought was M=z*root(pq/n). Perhaps at the algebra 1 level we won’t start there, but we can select most of the formulas needed in geometry and use them as literal equations and every quiz and test solve for a different  variable of one of the formulas. And, here is the kicker, EVERY time, the learner must explain why they are doing the operation. Justification is mandatory. If we look at the Margin of Error formula above, there are 4 different questions to be asked. That is 4 quizzes or tests that one question can be used.

The goal is get learners to think of literal equations a part of algebra and the justifications as the same thing as every other problem. By the time they reach AP stats, they will have seen this equation repeatedly and know how to manipulate it as a literal, not just with numbers in it. We need to connect AP Stats to Algebra 1.

Next, every test at algebra 1 level must have some form of the following question:

Evaluate (x – (x+h))/x with x = 2 and h = 3. Yes, I know it reduces to h/x, but as we move forward with notation, it becomes:

Evaluate [f(x) – f(x+h)]/f(x)  with f(x) = 2x+5, x = 2 and h = 3.  As the years progress the function can be moved from linear to quadratics to absolute value to cubics or rationals.

Finally, truly stress and monitor that verbage “rate of change of” every time the word “slope” is used.  The learners need to hear and write over and over the “rate of change of” the line in algebra 1, geometry, and algebra 2.

The goal is to create a common language / strands through all math courses and chapters that lead to AP calculus and AP statistics. All learners need to be exposed to the language of statistics and calculus repeatedly throughout their education so it is not different at the upper levels.

So those are the three things I can and will implement next year, without fail.

What am I missing?

Any other language to implement? Any other formulas / concepts that can be used at the lower levels of math that lead directly to the upper levels?



Aug 022012

I am rather late to the gate with these thoughts, in large part because I drove (well, rode my motorcycle) to St. Louis from Reno, which took 4 days to get home. More on that in a different post when I do the math from the trip video.

If you ask one of the teachers from my department about me, one thing almost all of them will say or agree with is that I hate paper. I hate paper note, paper suggestions, paper anything. I ask my department to submit everything to me through email, because I lose paper notes, throw away binders, and just downright hate keeping track of paper.

With that said, I admit I took 11 pages of notes, kept track of them through 4 days of a conference, and 2000 miles of motorcycle riding. This event meant that much to me, and that is saying a lot.

I am going to start on day 1 and work through all my notes. I am composing this more for myself than anyone else. Like all of my reflective posts, it really doesn’t matter if anyone else ever reads them. This is the place where I reflect and compose my thoughts for my future self (because if I do it on paper, it gets thrown away!)

Wednesday – A personal exploration:

To say I was nervous on Wednesday would be an understatement. I had just ridden 1900 miles over three days to meet with a group of people I had never met and only talked to through twitter, emails and blogs. Yea, I was nervous. You see, I am an incredible introvert. I attended a NCTM conference my first year as a teacher with 2 other teachers from my school. At the end of the conference, I had spoken with exactly 3 people in a meaningful way, 2 of which came with me. I attended sessions and sat in the back. I wandered the floor and just kind of nodded and said the minimum I had to.

At that point I realized that if I was going to become a half way decent teacher, I had to overcome this fear of putting myself out there. Even riding my motorcycle to the event is a sign of my hesitance to talk to people. I was riding alone, where the only conversation going on was inside my helmet. If I flew I might have to actually talk to someone (although I usually don’t when I fly, either.) This has been my most difficult challenge as a teacher, because it is so incredibly easy to just stay in my room and never reach out. I used to actually have it on my to do list every week, “Make contact with other teachers.” I don’t any more, but the tendencies remain.

So I walked into the hotel dirty and sweaty from riding in the heat and the first thing that happens is Lisa H. tries to give me a hug! Okay, ice broken, I can deal and grow.

After a shower so I was feeling like a human being again, a group of us walked up to the nearby mall and had a nice dinner at a southwestern restaurant and chatted. It was nice, and it gave me a chance to get to know people in a very casual way. It definitely was the right thing to do. I did think about staying in the hotel and sleeping, but decided otherwise. I am glad I did.

The actual events of the week after the break (warning it is long & detailed):

Continue reading »

Jan 282012

One of the struggles I have as department chair is coming up with what we should be doing in our PLC time. I decided that we needed to start examining the CCSS much closer, and really start implementing things this year that we are going to be asked to implement next year.

Why? Because if we can get started on implementing some basic ideas of the CCSS now, then there will be less work to do later. It makes sense, and we jumped in with gusto.

…  crickets chirping …

Okay, perhaps we got off to a rocky start at first. But the entire department started coming together and thinking about what the Standards for Mathematical Practices really mean, and we all agreed pretty quickly that #3, Construct viable arguments, is going to be the easiest to implement right away, as well as provide some very useful impact in our classrooms. The same thing goes for 5 and 7, use appropriate tools and look for and make use of structure.

All of the 3 PLCs in my department started on a plan in their content areas (Alg 1, Geometry, and Alg 2) to begin to make plans and more importantly, IMPLEMENT the plan in their everyday teaching. I know I have seen a positive impact already.

Here is what I have implemented.

First, I model every problem I do with words. Here is what I mean by this statement on a very easy example, with some steps from the left side left out, nothing from the right.

2x + 5 = 5x – 4     [the problem]

+5 = 3x – 4     I subtracted 2x from both sides of the equation because 2x –                              2x= 0, and I only subtract from the 5x bcs they are like terms.

9 = 3x               I added 4 to both sides because –4 + 5 = 0, and like terms.

3 = x                 I divided 3 to both sides because 3 / 3 = 1, which gives me x all alone and the answer.

I noticed an immediate change in the lower level learners. They immediately started nodding their heads and following along. Of course, this was Alg 2, so the example we were doing was a rational radical problem, and the words were the only way they could understand what x^(4/5) means.

I asked them to do one problem exactly as above in class in the last 5 minutes, and I had over 3/4 of the class finish the problem in the time allowed. They did the work! More importantly, they were understanding the process. Writing out the because to every step gave them the structure to understand not just WHAT they were doing but WHY. They could explain.

Then we came to homework. I gave them the following assignment. 78-80, 83-85, 88-90, Pick ONE problem from each section, and explain every step as above.

I expected tons of complaining and whining. I did not get it. The assignments turned in were very good, with mostly decent explanations of why. There were some higher level learners who thought that the WHAT was enough. Next week I will hand them back and disabuse them of the notion.

The surprising thing to me was the number of ELL and SPED learners who turned in homework for the first time all year. Maybe because it was at the beginning of the semester and they turned over a new leaf.

But I am thinking that partially it is because they got to choose which problem they did, and they were explaining WHY they were doing what they were doing. It is still early in the semester, and I am not sure if it will continue. I will update more on this issue though.

As a first step to CCSS, I think it is a good one, albeit small. I will also report on some things other teacher in my department are trying. CCSS is slamming down on us hard, and we have to get ready now, not wait for our district to tell us what to do.

Jan 242012

My last post was about an idea to use old scantrons as a visual aid to build knowledge of the binomial probability formula before the learners actually were introduced to the formula.

Short post: it worked, I think.

Long post: I passed out the scantrons, which immediately brought forth a groan. We just had the final exam last week, and I was already giving them a quiz! Once we got over that part of it, I asked a question.

I was giving them a 1 question quiz. What is the probability they would get it right? All they had is a scantron, no other papers or anything else, so they asked if they were just supposed to guess at the answer. My response was “yes” and quickly they had in hand that the probability was 1/4. It always surprises me how long it takes to get to that point with some learners though.  It is not as quick as I would think. But we all got there within a minute or two, which is faster than normal.

Next, I told them they were taking a 5 question quiz. And I asked the question, “what is the probability you will get a passing grade on the quiz?”

Now the frustration started. They wanted initially to just say (1/4)(1/4)(1/4) = 1/64.  Not so. I killed that pretty quick. But before I did, I wrote it in exponent notation, so they would be comfortable with the idea of exponents having a meaning in the problem. It worked.

The class started guessing lots of things then. (1/4)^3(3/5). That one was creative, accounting for the 3 out of 5 questions. I did not tell them what was right, but we had in the end 5 options the class thought were possible. One of the options was (1/4)^3(3/4). Pretty close to the basic part of the formula, just missing the fact they missed 2 questions, not 1.

So I asked them which of the options written down actually referred to something bubbled on their scantron. After all, 1/4 means something physical related to their scantron. They quickly ID’d the right equation as meaning something consistent,  and very quickly said they needed an exponent on the 3/4.

The last step was asking them how many different ways to get 3 right out of 5. I did the standard counting and after drawing 3 different options someone said, “there must be an easier way.” The class suggested permutations and combinations, and we quickly settled on the combination of 5 C3.

Done. In 15 minutes, we constructed the Binomial Probability formula using nothing but a stack of old scantrons. I then wrote the formula down for them, and they explained what the pieces were for. They made the connection much quicker than before, and I was really pleased with how fast they could plug numbers into the formula.

I remember last year just screaming inside because they could not get the difference between the probability of the problem as a whole, and the p and q values in the formula. They understand that now.

Success, probably. I like it.  Below is a jpg of the notes I made while doing the exercise.

 click to enlarge.

Aug 112011

I was not familiar with the acronym T.P.O.V. either until a couple of weeks ago when I was introduced to it at the NCTM’s Reasoning and Sense Making Institute. It was the presentation by Timothy Kanold that introduced the phrase, “Teachable Point of View”, and he brings it up in his book in Chapter 1: The Discipline of Vision and Values.

What brings me to write about this chapter specifically is I have an interview next week for department leader, which is the de facto leader of our PLC. I am going to be (because success is the only option) our department leader, and if that is the case, I better be prepared for the interview and the challenges that come later.

First thing, what is my vision for our math department? What are the values that support that vision? What is my TPOV?

First, let me start with the values of my school. We have 6. Responsibility, Respect, Honesty, Courage, Loyalty and Success. These are the 6 values of the school as a whole. Anything I do as department leader must fit into these values somewhere.

My Vision for the North Valleys High School Math Department

By 2014, the NVHS Math Department will have the highest scores on the District end of course exams, and we will continue to meet or exceed all district mandated requirements for the NV Proficiency exams.

How we will do it:

  1. We will implement Reasoning and Sense Making type problems at all levels beginning 2011-12.
  2. We will increase the number of RSM type problems over the next 3 years.
  3. We will implement the CCSS with fidelity across all levels, starting in 2011-12 (ahead of the district implementation).

There you have it. That is my vision for my math department. We are currently in the middle of the pack on the end of course exams, and we are meeting AYP proficiency goals. We were on a year 4 needs of improvement classification, but through a lot of hard effort, we pulled ourselves out (our consistent red cell was mathematics ELL and Special Ed.)

It will definitely take some Courage and Respect to implement this vision, but implementing will lead us to Success.

One reason I think this is achievable, is that out of the 700 teachers at the RSM Institute this summer, only 5 people from my district attended, and only 2 of them are high school teachers. Both are from my school. We have an advantage in learning, and we can leverage that advantage to overcome the obstacles that stand in our way.

Any comments or feedback on this vision statement? My interview is on Tuesday morning next week, so I need to refine this and sharpen it quickly!

Thank you!

Aug 022011

In Montana, there is a town called Haugan on I-90 near the Idaho border with an interesting building. As you approach, you see signs like this every couple of miles. [click on the pictures for a higher res image]


Outside, the place looks like any tourist trap.


Do they really have 50,000 silver dollars inside?

DSC01898 DSC01895 DSC01893 DSC01892

And at the back of the bar they have a sign saying what the current count is. I visited twice, 1 year apart.

In July 2010, it looked like this:                    In July 2011, it looked like this:

DSC01891 DSC00182

Any Questions?

[Please leave me some challenging questions in the comments. I am looking to develop this into a classroom material lesson. I also have more pictures of the inside, I just posted a few of them here.]

Aug 022011

This will be the last post I make on the specific topic of the 3 days spent in Orlando at the Reasoning and Sense Making Institute. I will expect that I make many more posts on things I do in the classroom that are based on the ideas I have learned, however.

A collection of all the posts, organized by days.

Day 1:

Am I creating more “Clever Hans’” – Notice the plural on the Hans. My answer in general is no, but that may be overly simplistic. It is something that must be fought every day. The rest of the post is about Dan Meyer’s challenge to reform math education is even more important!

Day 2:

Reasoning and Sense Making day 2 Most of the great resources I found are in this posting, along with some fabulous ideas for Algebra and factoring.

Beth Chance, Henry Kranendonk, and an overwhelming task in statsBeth Chance and Henry Kranendonk went through how to teach a lesson using Reasoning and Sense Making. Very powerful to actually have someone demonstrate it.

Day 3:

PLC’s at work with Timothy Kanold – A very powerful look at what the impact of PLC’s can be, and how to evaluate your own PLC. The best quote I wrote down that I must bring back to my department and ask ourselves daily is:

Do we want to be known as a school where the math department’s decisions are based on best practices and evidence or as a school where the decisions are based on opinion?

Landy Godbold & “Given That” shown visually – This session started off slow, but ended with a bang! Great ideas were taken away from this session and will be used in class.

Viva Hathaway & stats with food! – Viva is an epic teacher. We didn’t spend enough time doing what I would have liked, but her personality and energy is really the motivating factor in the room. I will attend more of her sessions if given the chance.

Random thoughts and musings that don’t fit above.

Tweet Archive of #nctmrsm11There were very few teacher tweeting at the event. Internet was an issue, power was an issue, and it made things difficult to do create electronic versions of the talks. But, some powered through! I won’t make any claims to the accuracy of the archive, but there it is.

1. One of the goals of the Institute was for the participants to walk away with an action plan to implement RSM in our classrooms. Notice that nowhere did I speak to my plans! I have one. I wrote down two pages of things I will do, but I didn’t want to spell that out yet. Part of the issue is I was blindsided by some news at the institute about another teacher. I can’t go into details, it isn’t public yet. But it radically altered my plans and what I need to do.

2. I noticed that as the institute went on, I became a better participant. My notes became better and my thinking sharpened. Just look at the postings above. Day one to day 3 shows a marked distinction in the quality of postings, and what I took away changed as well. This is only the second conference / institute I have attended, and I need to attend more. Before I do, though, I need to go back and read these postings to develop that sense earlier rather then later.

3. Finally, all this was money and time wasted if I do not actually put forth the time and energy throughout the school year to actually DO reasoning and sense making in my classroom. It must become a habit for it to be successful (Beth Chance), it must be implemented with fidelity throughout the department (Tim Kanold), it must be done with energy and enthusiasm (Viva Hathaway), and it must be done with tools that the learners will relate to and engage with (Dan Meyer). That is the real takeaway I have from the 3 days I spent in Orlando.

Aug 022011

At the NCTM Reasoning & Sense Making Institute, I attended the session given by Beth Chance and Henry Kranendonk, two of the three authors of the Reasoning and Sense Making in Statistics and Probability book.

I have that book on my bookshelf at school, and I have looked at it, but have not tried any of the lessons in it. I read the lessons at the end of last year, and said to myself, “Are they crazy!” “I don’t understand what they are getting at!” and “Maybe next year.”

After their session, where they essentially went through several of the lessons in fairly good detail (the Old Faithful Data and the Will Women Run Faster than Men in the Olympics as well as others) I am much more confident of doing the lessons successfully.

They had a handout of the material, which I scanned and you can download here.

I apologize for the missing data on the women column, but my copy was very faint also. However, if you go to the book link above, the “Read an Excerpt” is the full chapter from the book in PDF form! Yay!

I am not here to sell their book though. I feel the book is worth buying, now that I have spent better time with it, but the front side of their handout scared me silly.

Go ahead, download it now. I will wait and get some coffee while you do.


See what I mean? Under “Habits of Mind in Statistical Thinking” they have 24! bullets under 5 sections. Here is my problem with that list. A habit is something you do unconsciously, because you have done it so often and so repeatedly that it becomes second nature; like breathing.

How can a list of 24 bullets become second nature? The first and second time I read this list I just got a sinking feeling in the pit of my stomach. If this was successful Stats teaching, then I have been failing. Clearly something has to change in the way I teach Stats and what I teach in Stats.

And then it dawned on me. What must change is the quantity of Reasoning and Sense Making I do in the course. After all, let’s examine the list, not for the whole statements but just the root verb stems.

describe, analyze / analyzing (2x), looking for, making deductions, choosing, creating, considering, drawing conclusions, comparing (2x), evaluating (2x) questioning, applying, noticing, identifying, understanding (2x), connection, considering, determining, justifying, and looking for

When you get rid of the other language, the details of what the statements are, and focus on those verbs right there, it is easy to see how a habit can be formed. It is not hard through repeated effort to construct those habits in learners.

See, all those verbs have something in common. They all have Reasoning and Sense Making (RSM) as their goal. Anything I do in class that does not have the goal of RSM is officially a waste of time. There it is. There is the big takeaway from the session.

I just figured out what every lesson I teach in AP Stats needs to revolve around. Now I just have to rewrite all of my lessons to focus on that. That will be the subject of many more posts.


As an aside, Beth used some dice and applets in the session that were very cool and useful. The Dice showed an 11 in 6 out of 9 throws. I searched for something that would do that, and found them here and here and here again. These are very useful to generate some statistical thinking skills in the learners.

The applets she used can all be found on the website. There are additional materials on this site from other presentations they have given. I may end up using her applets almost exclusively, just because they are all on one site. It makes life easy.