Feb 072017

Math majors who are interested in teaching are the toughest group of learners. They really are. They are in a mixed science / math class, so they band together. They reinforce each other’s beliefs that the way they have been taught math is a great way, because they have been successful in learning math that way. They then fight against any notion that math can be taught any other way than the way they have been taught.

The struggle is real.

Except last week, in an introductory class, I had a break through. One of the learners asked if a better way would be to have a learner go to the board and do a problem.

I had an aha moment. I asked THEM what they did when a teacher had a learner at the board. They unanimously agreed they tuned out.


Then, I asked how many of them tuned back when the teacher took over.

They agreed that maybe 30% of them did tune back in. The rest (these are all science / math majors who were successful, mind you) said they just relaxed and let the teacher work.

Next, I asked, “If you are the successful learners, how many of the rest of the class tuned back in?” The agreement was unanimous, no one.

My last question sealed the deal for them.

“If only 30% of the successful students tune in, and none of the unsuccessful students tune in, why do you think the way you have been taught was successful?”

The silence was deafening.

That small exchange finally made them think about what success and failure is in teaching.

Success is not the teacher working and the learners listening.

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

Jul 162014

I have been thinking a lot about growth mindset lately (really, what teacher is not.) But I have really been trying to come up with positive, constructive ways to model and use it in the classroom as a way to change the learners beliefs.

One way I came up with using it is to have some statements that I can use consistently when learners are struggling in class. My personal challenge when dealing with the fixed mindset is what to say, how to constructively come back with something that will start impacting beliefs. As a teacher, we hear it, but how do we respond? It has to be consistent or we lose their focus. These cannot just be quotes on the wall, but statements delivered with conviction face to face to have an impact.

So, here are some statements I hope to use in my classroom. I am going to print them out and post them where I can see them every day in the morning and before every class to remind myself to use them until I don’t need the list any more.

Learner Says or is Doing: Teacher (ME!) says:
Learner is struggling with material “If it was easy, I would not waste your time with it”
Learner whips through problem, too easy “I apologize for wasting your time, I will find something more appropriate for you.”
“This is too hard.” “What strategies have we discussed that could help you get started?”
“This is too hard.” “It is difficult now, but so was adding in elementary school. You overcame that with effort and you will overcome this with effort.”
“I am not good at this.” “The more you practice the math, the better at it you will become.”
“This is easy.” “I am glad you understand this, can you develop a more complex idea with it that challenges you?”
“This is as good as I can do.” “You can always improve, as long as you give it some more effort. What other strategy have you not used yet?”
“I made a mistake, I can’t do this.” “Mistakes are how we learn. If it was easy, you wouldn’t be learning anything new.”
“This is good enough.” “Is this your best work to show your learning?”
“I didn’t get it on the first try, so I won’t.” “So your plan A didn’t work out. Good thing there are 25 more letters. Start on plan B.”
“You are just too hard on us. We can’t do it.” “I’m giving you this assignment because I have very high expectations and I know that you can reach them.”


The goal here is to have a bank of statements that reinforce growth mindset that are easy to memorize, adopt, use and believe in so that every day I am consistently changing the dialogue in the classroom. I have found that it is easy to get sideswiped by a comment and not have a positive response handy. My goal is to fix that.


Any suggestions? Additions? Changes?



Some resources for Growth Mindset I will also use come from:

http://mathmamawrites.blogspot.com/2010/07/day-one-of-class-beliefs-about-math.html Sue is an amazing writer and teacher. Her take on this is invaluable

http://mathhombre.blogspot.com/2010/07/growth-model.html Just download John Golden’s Implict Theory of Mathematics Learning worksheet now and give it out the first week of school. I am, and you will be glad you did it too.

http://practicalsavvy.com/2012/01/31/inspiring-quotes-demonstrating-the-growth-mindset/ These are great quotes, but quotes around the room won’t cut it. It has to come from my mouth, every day.

Jun 292014

This post is really for a friend of mine who is very much math aware and capable, but does not teach math. He has twin sons who are absolutely amazeballs smart, and through talking with him one day I espoused my approach to algebra. He was intrigued and wanted more info. This is my attempt at more info.  I figured that others may find value in it, so I am publically posting it for all to see (and read, hopefully.) Please rip it to shreds if you feel I am in error or made a mistake. I want to do better, and I can’t if I am doing something wrong.

So let me begin with where this came from. At the NCTM Las Vegas regional conference in 2013 I was introduced by an elementary teacher to what she called 13 Rules that Expire.

13 rules that expire

There are some definite Algebra concepts on here like number 2) Use keywords to solve word problems and number 3) you can’t take a bigger number from a smaller number or what really kills me is number 8) multiply everything inside the parenthesis by the number outside the parenthesis.

Yea, right. Try that with f(x)=2x+5.


And from there we have Nix the Tricks. A more rigorous treatment of the stupid things we as math teachers do to mess up learners in the guise of teaching them to get an answer instead of understand the mathematics.

So I sat back and tried to come up with some rules that did not expire. Some essential rules that always work, that always build understanding and not destroy it. I ended up with 3.

1. When solving an equation, you can do absolutely anything you want, as long as you do it to all terms of the equation.

2. When working with expressions, you can only change it by adding or subtracting 0 (by using additive inverses), or when solving equations, you create zeros by adding or subtracting 0 (by using additive inverses).

3. When working with expressions, you can only change it by multiplying by 1 (in any form), or when solving equations, you create 1 by using the inverse functions/operations.

That’s it. When I am teaching math, I stress the idea of creating a zero or using a zero. When demanding written explanations, I demand they say that 5+-5=0 in their explanation. So, a nice short example.

Find the function that is the inverse of y = 1/3x – 4

Work                             Why did I do what I did?

x = 1/3y – 4                    Because the first step in finding the inverse is switching the x & y

+ 4        +4                     Add 4 to both sides because –4 + 4 = 0 (additive inverses = identity)

x + 4 = 1/3y                    result

3(x+4) = 3(1/3)y              Mult by 3 because 3(1/3) = 1, (multiplicative inverses = identity)

3(x+4) = y                       finished, but should check it by …….

So there is an example of the work I require. I want them to be using the language of inverses and identity. Why does the square and the square root cancel each other? Because the exponent of 1/2 (the square root) and the exponent of 2 (the square) when multiplied equals 1, the identity.

Along with this, there are some forbidden words and phrases in my classroom. One is above, “Cancel”. I do not allow my learners to use it. At all. Ever.

Why? Because I have seen all of the following described by the word:  -5 + 5 cancels to make 0. 3/3 cancels to make 1. Log10^4 cancels to make 4, sqrt(5^2) cancels to make 5 and on and on and on.

If those all “cancel” then that word means nothing, and it does not mean anything at all. It is just a word used to hid the mathematical knowledge of inverses and identities.

Another phrase that I will not allow is the common answer to the question, “Why did we add 4 in the first step above?” They typical answer I get is, “Because we want the y by itself.”

Huh?  That is not why we added 4. We could do anything we want in the world. We could have added 6, or subtracted 3, or taken both left and right side and made them exponents with a base of 7.5. We added 4 because –4 + 4 = 0. That zero is important. I also write it down. I think most teachers do not. Or they draw a line through the –4 and 4. What does that line mean?  I tell them it means a 1, and did we make a one? No, we made a zero.

It takes some time to unlearn the bad habits, but this is a Nix the Tricks kind of endeavor. In the end, I think I am helping the learners understand math better and more deeply. At least, I hope I am. Only time will tell.

Aug 182013

So the other day I am giving my Algebra 2 STEM learners (this is our Honors designation” the big picture of Algebra 2. You know, so they know where they are at and where the class is going. This is what I had written on the right of the screen.

2013-08-15 11.41.37 2


After writing this down, I did the following exercise:


What do you notice about this list?

Here is a list of everything I wrote on the board:











Then I asked what do they wonder. This took some time, and I was REALLY glad I gave them lots and lots of processing time. Some of the later wonderings took some time to develop.

THEN I asked for a show of hands how many people wondered the same thing. It was amazing that most of the learners wondered the same thing about the important math. I was very pleased.


I think the list of “vertex functions” will be written on the board as we go repeatedly so they always have that reference of where they are at in the process.

What do you Notice, What do you Wonder? is a very powerful tool to get the learners involved. Max Ray at The Math Forum at Drexel has done a lot of work (even applied for a trademark for the Notice / Wonder phrasing) in this area.

The “What do you notice?” and “What do you wonder?” questions are great ways to pull even reluctant learners into the process of engaging with the material. I know I will be using it again and again.

Jun 242013

This is the most recent 5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay Stein.

Great conversation and discussion all around. Some of us are going to try to come up with a task from Algebra 2 to try to create a conversation around. Next conversation is Monday, 1 July.


May 152013

The one problem with having a summer list is that I always want to add to it. I try not to, but things come up that I cannot say no to.

This is one of those things. It is a course taught by Stanford University professor Jo Boaler. If you don’t know about her, she has gone to bat for math teachers and taken some professional licks for it, and she is the author of the fantastic book “What’s math go to do with it.” (Amazon, B&N). I can not say how much I liked this book, and then to have the opportunity to take a class from her for free; well, that is too good to pass up.

The class is: EDUC115N: How to Learn Math.

Many of my Twitter PLC has already signed up for it, and I just sent the link out to my department and hopefully many of them will sign up too.

I hope many people will sign up for it so we can have some great conversations in the class.

Here are the topics the course will cover, which makes it even more inviting and exciting! See you there!


1. Knocking down the myths about math.
Math is not about speed, memorization or learning lots of rules. There is no such thing as “math people” and non-math people. Girls are equally capable of the highest achievement. This session will include interviews with students.

2. Math and Mindset.
Participants will be encouraged to develop a growth mindset, they will see evidence of how mindset changes students’ learning trajectories, and learn how it can be developed.

3. Teaching Math for a Growth Mindset.
This session will give strategies to teachers and parents for helping students develop a growth mindset and will include an interview with Carol Dweck.

4. Mistakes, Challenges & Persistence.
What is math persistence? Why are mistakes so important? How is math linked to creativity? This session will focus on the importance of mistakes, struggles and persistence.

5. Conceptual Learning. Part I. Number Sense.
Math is a conceptual subject– we will see evidence of the importance of conceptual thinking and participants will be given number problems that can be solved in many ways and represented visually.

6. Conceptual Learning. Part II. Connections, Representations, Questions.
In this session we will look at and solve math problems at many different grade levels and see the difference in approaching them procedurally and conceptually. Interviews with successful users of math in different, interesting jobs (film maker, inventor of self-driving cars etc) will show the importance of conceptual math.

7. Appreciating Algebra.
Participants will be asked to engage in problems illustrating the beautiful simplicity of a subject with which they may have had terrible experiences.

8. Going From This Course to a New Mathematical Future.
This session will review where you are, what you can do and the strategies you can use to be really successful.

May 112013

I am trying to put some order to my to-do list this summer, as well as create some structure for all the work. I want to avoid creating a broken, fragmented summer where I accomplish nothing but spend a lot of time spinning my wheels.

First off: Trainings, conferences and travel I am planning.

1. A training given by my district on CCSS changes to Algebra 2, 10-12 June. Kind of important since I signed up to teach the new CCSS Algebra II STEM changes at our August Mandatory PD session. It will be a great three days in June, and a great way to start the summer.

2. The Silver State AP Institute in Las Vegas 24 to 27 June: Really looking forward to this institute given by Josh Tabor to the experience AP Stat teachers. It says he is going to work with Fathom a lot, which is good, I have no experience with Fathom. I am interested in seeing what the new additions to GeoGebra can do as far as stats teaching, so I will work with both programs and see what I can do to crossover.

3. A fun trip to Chicago for some family time around the 4th of July, and then a week later a motorcycle trip to Portland and Montana. That will be a blast! It is always good to see mom, sister & family in Portland as well as family in Montana.

4. TwitterMathCamp 2013 from 25 to 28 July! Yay. Last year it was amazing and hands down the best PD I have ever done for myself. I am very happy to be going again. Looking forward to developing relationships with more teachers and building stronger relationships with the ones from last year!

5. In addition to all that, I signed up for a MOOC on Coursera on the Philosophy of Mathematics starting in July and going through August.

Whew, that is a lot of traveling, and it will definitely keep me hopping. But in addition to traveling and learning, I want to really dive deep into a some books and synthesize some ideas.

My reading list, in no particular order is:

The Art of Problem Posing by Stephen Brown and Marion Walter

5 Practices for Orchestrating Productive Mathematics Discussions by Margaret Smith and Mary Kay Stein

Common Core Mathematics in a PLC at Work: high school by Gwendolyn Zimmerman et al.

Common Formative Assessments: how to connect standards based instruction and assessment by Larry Ainsworth and Donald Viegut

I have some other sitting on my bookshelf that I want to revisit, but those are the 4 that I absolutely want to get through this summer.

The thread I am working on is connecting classroom practice to better questioning, learning how to ask and guide better questions, and then teach other teachers in my department how to ask better questions.

This is a big chunk, but I think it is important to developing a better math program and math classroom.

I will post here on the books and conferences.

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.