Sep 092016

To my last post, “No more broccoli ice creamDavid Griswold challenged me with a very serious and thoughtful reply.

The phrase, “No more broccoli ice cream” came from this meme that I saved. I collect these memes, just because the provide interesting fodder for conversations about math in class.

textbook math is like broccoli ice cream

So who is Denise Gaskins? She is a home school parent who specializes in K-6 math (I am inferring this from her website and the books / content she talks about. I could be wrong about the grade levels.) She tweets and has a FaceBook page under the name “Lets Play Math.” It is clear she has a focus on making math fun, interesting, and engaging. At that age, my experience is that learners are very much into mathematics. I saw this bulletin board in a hallway last year.

2015-09-28 12.56.29

If you zoom in, you will see that almost every single one of those 4th graders said their favorite subject was math, or they enjoyed math, or they were good at math. 2 out of the 15 had no positive mention of math. A bulletin board next door to it had a similar proportion. When I saw this board, I wrote a post called “Where does the joy go?” This issue is one that I have been struggling with for a while. Why are young children excited about math, but junior or high school learners typically are not?

I believe it is because at some point, I and my fellow teachers stop thinking about math as joyful, and start thinking about it as “serious work.” We can’t have fun solving these equations, this is “serious business.” But that is true of all subjects in high school. It isn’t just math teachers, but English teachers, history teachers, and other teachers. We turn our subjects into these “serious business” topics that must be “mastered” and “assessed.” If you don’t pass the classes, then you can’t graduate, you can’t be successful in life without knowing “algebra.”

[yes, I used a lot of scare quotes in that paragraph because I do not want anyone to infer that there is an agreed upon meaning of those terms.]

Here is what David said in his questioning of my post:

I’m not sure I completely agree with this, or Lockhart for that matter. There are a lot of people who find joy and beauty in the curriculum, and there are lots of ways to encourage and celebrate that joyousness without throwing too much out. Will some people hate it? Sure. I didn’t like AP US History very much, though I liked my teacher. But I had friends who thrived there. And I’m okay with that.

Personally, I don’t think the ice cream metaphor is realistic. Math isn’t ice cream. Nothing is ice cream. No field or subject is as universally loved and delicious as ice cream, certainly nothing with any practical application. Math isn’t ice cream, it’s vegetables! So maybe “textbook math” is steamed broccoli and it’s up to us to add peas and roasted cauliflower and sweet potatoes (maybe even with some marshmallows on top) and even pickles, but the fact is some people don’t like ANY vegetables and some people like simple steamed broccoli the best and some people like ALL vegetables and, importantly, all of them are part of a well balanced diet. So our job is to be a math nutritionist.

The first paragraph I will not reply to, because it is his personal feelings and I don’t think there is anything there to discuss. It is real.

The second paragraph is the challenge. “No field or subject is as universally loved and delicious as ice cream.” But … I don’t like ice cream. I eat it maybe one time every four or five months, because my wife wants to share something.

And, guess what? Yes, math is as loved as ice cream at the lower grades. I have observed 4th and 5th grade classrooms where the learners are excited, joyful, and enthusiastic about math. The bulletin board above is anecdotal evidence.

I think we need to stop saying it is the subject that is like vegetables, and accept the fact that it is the way we teach the subject that turns it into vegetables.

Watch this video (it is 5 minutes) of these middle school learners struggle and succeed in math.

There is honest to goodness joy there.

They ate some yummy ice cream in that lesson. Why can’t we do that every day?

To answer David. Is math like vegetables? I think it can be. Is math like ice cream? I think it can be. The choice is mine.

If I get to choose whether math is more like vegetables or like ice cream in my classroom, I will choose ice cream (even though I don’t like ice cream).

I choose this not for myself, but for my learners. And David is right. Not everyone will love every subject. I am okay with that. But if I choose to present math like brussel sprouts instead of chocolate fudge peanut butter ripple, then I have denied some learners even the ability to choose whether or not they enjoy math.

And then, how do I make my pre-service teachers understand that it is a choice they can make too? [Wow, that is a whole different can of worms.]

So, is math like ice cream? For my classroom, for my pre-service teachers, the answer must be Yes.

David responded on Twitter with these series of tweets. I think they add a great deal to the conversation.


Thank you David for making me think.

Apr 132013

I have been mulling this question over for a while now, since last summer at least. It is a offshoot of the time I spent working with Exeter materials at an Exeter summer institute, and if anything the question has grown in my mind to the point where I must answer it for myself and act on it.

Here is the newest version of my question: If much of what and why we teach math the way we do is arbitrary, then why not change to make it easier to learn?

Now of course, there is a HUGE set of presuppositions / assumptions just in asking the question. First, I assume that much of what and why we learn math is arbitrary. Well, I don’t think I am that far off the mark. Let’s look at Algebra 1 as a course first. Honestly I am in good company with this thinking.

Grant Wiggins (author of Understanding by Design) and his thoughts on Algebra 1

I agree. Algebra 1 has a huge failure rate because it is very abstract, meaningless content. We don’t really ever see why we are doing it, we are just learning to manipulate variables and constants around. Grant gives a small part of Lockhart’s Lament (pdf), and it is worth linking to (and reading) completely. Again, the ‘why’ of ‘why do we teach it this way’ is completely arbitrary. Which is why we get political science professors arguing that Algebra is unnecessary because it is hard in the NYTimes. That Algebra is a gateway topic is not in question. It is. The content is essential for future jobs and future success.

If we look a the content, we see arbitrary stuff all over. Heck, just look at the old y=mx + b. Why “m”, why “b”? There is no good answer. Do a google search and get 33 Million hits, none of which can definitively tell us why. I find the answer from the Drexel University Ask Dr. Math to be the most grounded answers, which you can find here for m, and here for b. And I LOVE the answer given here by math historian Howard W. Eves in Mathematical Circles Revisited (2003), where he suggests that it doesn’t matter why “m” has come to represent slope.

“When lecturing before an analytic geometry class during the early part of the course,” he writes, “one may say: ‘We designate the slope of a line by m, because the word slope starts with the letter m; I know of no better reason.’ ” via

I totally agree. In other countries they use other variables for the same meaning (scroll down), so clearly the “agreement” that we all must use the same convention is not universal. There are so many conventions in math that are purely arbitrary. Since they are arbitrary, we must feel free to throw them away when they interfere with good learning and teaching.

So the second question, and a very important one, is: How could we teach math differently to make it more understandable?

One thing I think is important is to connect the vocabulary / language / processes of linear functions with other polynomials / transcendental functions. After all, look at the amazing similarity and simplicity of understanding the transformation processes.


Don’t believe me that every single function listed has exactly the same transformation rules? Try this little GeoGebra applet I whipped up. Think about that for a second. When I have shown this to math teachers I get two reactions, “Well duh” and “OMG, I never thought about these like that.”

The teachers who see this as obvious are the teachers who are much more experienced and have taught for many years and have spent the time looking at the math. The crazy thing is that very few teachers have told me they teach this. Why not? Because it isn’t how they were taught, it isn’t how the books phrase it, it violates the conventions of math teaching. So they know it, but ignore it.

And don’t get me wrong. I am not suggesting this is where we stop teaching, we use the exact similarities as a springboard to bounce into the other types of functions. If the learner of math knew this with strong understanding, then the rest of algebra becomes a close examination of each type of function (which is all the different algebra courses are anyway.)

The Common Core Curriculum has mixed up the order of teaching these functions, but the fact that all algebra is just an examination of the skills (which are essentially the same, find ones by multiplying by the inverse, find zeros by adding the inverse) needed to solve, graph, and understand how each function is used.

The last question I have is: Why do we, as teachers put up with this, and what are we going to do about it?

I think the CCSS gives us the perfect opportunity to demand better from textbook publishers as well as our professional development opportunities. We, as teachers, need to be willing to throw the ‘conventions’ away and teach better.

Will it make a difference to the failure rates of algebra 1? I don’t know. but how can it hurt? How can it hurt to strongly connect all of algebra through trigonometry with an unbreakable thread so learners know that what works for one type of functions will work for every other type of function too. It shatters the concept that Chapter 3 doesn’t relate or have anything to do with Chapter 7. That is what learners think now.

Oct 132012

Grr. Really. I have some frustration I need to get off my chest, so this will be a “constructive venting” post. Constructive, because I really do have a point, and venting because I can’t believe the things that occur in math education.

First off, I have been really interested in how the curriculum of mathematics is designed and how that curriculum actually works against the learning of math as it is presented in our current textbooks. Honestly, I am really disappointed in textbooks right now. I have spent some time examining the vocabulary in different sections of our current books, and I realized that the books set up the topics of mathematics as if each chapter, each topic, is a disjoint Venn diagram. Chapter 2 has no relation to Chapter 3, and that is how math is done.

No wonder learners struggle with math and don’t make connections. We have DESIGNED math to be taught that way. Don’t agree with me? Just look at your textbooks and tell me if the linear vocabulary is identical to the quadratic vocabulary. Then ask yourself why not? 80% of the vocab should be IDENTICAL.

Let me show you what I mean. This is the current state of affairs:

Linear Equations Quadratic Equations
Standard Form Ax + By = C y = ax2 + bx + c
Equation forms y – y1 = m(x – x1) y = a(x – h)2 + k
y = mx + b y = (x – a)(x – b)
m = (y2 – y1)/(x2 – x1)
Vocabulary Rate of change, y- intercept, slope, rise over run, x – intercept x-intercepts, solutions, roots, axis of symmetry, reflection, translations, y – intercept

Hmmm, do you see ANYTHING that overlaps there? The only thing that is the same is the fact that the x-intercept and y-intercept are both used. However, in the quadratic unit, the y-intercept is rarely used, and the x-intercepts are normally called roots or solutions. Learners = 0, Notation = 1, point goes to the book.

Seriously, even look at the idea of the “Standard Form”. Why is it “Standard”? Because that is how we would normally want to write the equation. Lines get some weird, special form that relates to nothing else in polynomials, while quadratics and all other polynomials get something that makes sense and flows from one type to the next. Learners = 0, Complicated Notation = 2; the book is clearly leading.

Oh, that Point-Slope form looks pretty promising, though. It is very similar to the Vertex form of the quadratic. Well, kind of similar. It is the only form in the family of polynomials that uses subscripts, but that is okay, subscripts are super easy to learn. Oh, they aren’t? Subscripts confuse the heck out of learners? You mean there is a difference between a number in front of a variable, a number above and to the right of a variable, and a number below and to the right of a variable? Darn. And if the Point-Slope is supposed to come from the definition of slope, where did the subscripts of 2 go? Why ? I don’t get it. Why did things change? Learners = 0; Confusing notation = 3. Clearly the learners are getting the worst end of the deal here.

How can we make this better? Well first off, we need to realize that there is only ONE type of math we are doing; Polynomials. Start there, and build a set of math from scratch.

Damn. That sucks. That is a lot of work. That is akin to what Exeter has done or what Milton Academy has done. [Terrific article on Milton’s efforts here by the way.] And there in come the rub. Here we have the idea, but it takes a team of willing people who think alike to create. It takes time, energy, money, and effort. And then, it takes only one or two teachers to block the whole thing from implementation in a school or district.


Double Frick.

Here comes the venting. I need to move beyond this so I will get something done.

I didn’t even realize this bothered me so much until I started thinking about why I was so paralyzed for the last two weeks. Very hostile. Very frustrated and angry, and then I started asking myself why. I think I have it figured out. I was at a department lead meeting two weeks ago, and the leader of the meeting presented my idea for lines. (okay, not mine, but Exeter’s). And the department leaders from other schools at the meeting said, “Why?” “Why would we do that?” “Lines are y=mx+b”. “That is stupid, the textbooks don’t teach it this way.” etc.

I. Kept. My. Mouth. Shut. It took heroic efforts, but I did not rip into anyone. I was good. I mean, they didn’t come up with any good arguments why it doesn’t work, shouldn’t work, isn’t a good idea. They just said, “ThatisnotthewayIwastaughtthereforeitiswrong <breath> thebookteachesitonewayandwecannot/shouldnotchange.”

THEN! OMFG. The pain wasn’t over yet. I was speaking to our assessment director after the meeting, and she was just as frustrated as I was, if not more. She told me a teacher actually contacted her and told her she was doing a bad job because she left the section of the book that taught lines as y=mx+b out of the curriculum guide. Why was that bad, you ask? Because that is the ONLY way that teacher teaches lines. That teacher won’t teach it any other way because it is the easiest way.


Really? Sigh. You know that ONE teacher will end up on the next textbook committee, and that ONE teacher will end up railroading 4 other great teachers into buying a shitty textbook simply to shut that ONE teacher up. How do we fix that? How can we?

And that is my frustration? What can I do about that? It has eaten at me these last 2 weeks, and it shouldn’t. It won’t now that I have written about it. I have gotten it off my chest, and that helps.

How do I respond?

I push forward with my Exeter project. I write a paper and submit it to Mathematics Teacher. I stay focused on what I need to do to accomplish my goals.

And remember:

“Don’t say you don’t have enough time. You have exactly the same number of hours per day that were given to Helen Keller, Pasteur, Michelangelo, Mother Teresa, Leonardo da Vinci, Thomas Jefferson, and Albert Einstein.” – Life’s Little Instruction Book, compiled by H. Jackson Brown, Jr.

I have the time. I just need to use it better.

Jan 242010

I attended the Southern NV Math and Science Conference  over the weekend. I presented part of the Advanced Algebra Applications course, but more on that in another post.

I also attended a session given by one of the major textbook publishers on 21st Century learning with their products. I was told by my boss to be quiet and let the guy speak because I was interrupting him so much. I have 3 main problems with what this publisher rep was telling / showing us about the upcoming editions of the math textbooks.

  1. The rep was really excited about the 21st Century tools his books will have. I mean they will have VIDEO to teach the problems. Video featuring actual high school age learners, CREATED by actual high school learners. And avatars to explain the math. You can’t get more 21st century than that, right?
  2. The problems are all going to be “relevant” to today’s learners. Really. They are going to re-write all the problems and they will be relevant and therefore more interesting. Honest.
  3. Finally, the research shows that learners are not reading the explanations because the math comes first. If the words came first, then the learners read the words, and they learn more of the math. (Think 2 column explanations, where traditionally, the math is on the left and the explanations on the right.)

Those are my observations. Here are the problems I had with them.

  1. The video they showed was of 2 high school age learners discussing exponential growth of sharks. Right, they made the video with no help, no script writing by adults, no props brought in by a major company. Bullcrap. It was so obviously staged, the language was directly out of a textbook, stilted and lame, and the average HS learner would spot it as a joke immediately. As to the avatar, having a 4th grader explaining exponential functions is very disconcerting. The voice was horrible, and the graphics were even more lame.
  2. Is taking a problem like this:
    • “John goes into the store to buy garden hose for his cousin. John spends $85 total for 4 hoses and 1 spigot. The spigot cost $15. How much did John spend?“ and re-writing it to say:
    • “John went to a concert and spent $85. He bought 4 shirts and 1 cd. The cd cost $15, how much were the shirts?” really relevant?
    • No, seriously. He went on for 10 minutes as to how this is so much better because the kids actually go to concerts. 
    • Like the kids wouldn’t just say, “Dude look at the sign posted on the shirts! If they don’t have a sign, don’t buy them!” How incredibly lame.
  3. Finally, and this one really upset me, he said the research shows the kids need to read the explanations first. Research has shown they do better when they can read the words first, then the math. And that makes TOTAL sense to me. BUT, they will only do this to the LOW LEVEL TEXTBOOKS! Yup, it is good for all learners, but they won’t do it for all  learners, just the books focused on Special education and ESL.

What total crap. I asked him why, if the research shows it is advantageous, they won’t do it for all their books. The answer was, teachers won’t buy the books because they want them to be traditional.

Yup, teachers want what is best for them, not their learners, so the textbook company caves and follows research only for the low level books. Even though the research shows it is good for all learners. I would like to see the research, but I have no reason to doubt it.

And this is a company I like. They have the better online resources out of the 3 books I use. Their software is better, and they are trying more than the other companies. That does not bode well for our ability to get change in the textbooks we use.