Aug 102017

The other day, Ann Arden posted about “Beyond tests in HS Math (part 1).” This was her very first ever blog post, and I am eagerly awaiting her second!

In Ann’s post, she set up a framework for understanding assessments which I had not seen before (someone please give me a citation if you have), and it really made me think about what I wanted to do with an assignment in the class I am teaching this semester. The image she posted and tweeted was this one:

She says she would change the top from feedback to assessment. I wanted to take it to another level, because I have both mathematics and science majors in my class. Because of that, want to address the actual standards of practice and content for both. In the ‘process’ and ‘product’ labels, I saw the conceptual and procedural ideals of mathematics learning in the CCSSM, which also has similar ideas (but uses completely different terms) in the NGSS. With that in mind, I updated the image to this, retaining the same positions of her quadrants.

What does that give me, however? Why?

Let me go back to Ann’s post. She said,

Quadrant 4: Assessment of a product /after the moment

In my experience, this is where most evaluation (and much formative assessment) occurs in high school math. The most common example of this is math tests. Students finish a section/unit of learning and write a test. This is usually done individually (more on this and group tests in an upcoming post). The teacher then marks these tests later in the school day or at home; away from the students. These hopefully get returned in a prompt manner, but can take a few days or longer sometimes as I can personally attest to. Most of the feedback is written and might involve short phrases, check marks and circles or other notations. Research has show that students have a difficult time interpreting this sort of feedback (e.g. Weimer, 2013’s review of Sadler, 2010). In addition, the delay between the “performance” and the feedback or judgment reduces the power of the assessment to serve LEARNING.

While tests are most common, formative quizzes and exit tickets are often also largely assessment of a product “after the moment” when the teacher responds the next class. For example, I routinely use “not-for-grade” quizzes. These quizzes are very short (usually 1-3 questions) and I give comment-only feedback. No grades, no levels, just written feedback. I also post solutions for these quizzes electronically so students can fully review solutions. Where multiple solutions are possible, I often post two interesting solutions and discuss in class. In addition to providing feedback to students, formative quizzes and exit tickets can also inform the teacher about next steps in instruction.

Notice something very important to the conversation (I put it in bold). In the first paragraph, is an example of a summative assessment, whereas in the second paragraph the assessment is formative. To give some grounding in how I will use those terms. Formative assessment is any assessment that is used to modify instruction either at the moment or in the future. Summative assessments are end of learning assessments that do not have an effect on instruction for this semester.

The question I had from this is, Can each quadrant have both summative and formative assessments?  What would that look like? What are examples that fit?

Quadrant 1, formative: Conversations about work being done (paper, VNPS, etc.), think, pair, share exercises, error analysis (teacher provides examples), My Favorite No, card sorts, etc.

Quadrant 1: summative:   <insert sound of crickets chirping here>  I am stuck.

Quadrant 2, formative: Journal writing, Reflect & Self-assess, correcting and re-evaluating turned in work, etc.

Quadrant 2, summative: Portfolio work that shows the process towards a learning goal, making a video which shows how to accomplish a learning goal, etc.

Quadrant 3, formative: In a presentation, keeping track of learning outcomes or goals through comments, question and answers or discussion during a presentation (thinking of a poster presentation here).

Quadrant 3, summative: <more crickets?> Again, I am stuck.

Quadrant 4, formative: A quiz, where the learners are encouraged to come in and relearn and retake or correct the quiz with grade replacement. <is this really formative? not sure> Exit tickets, collecting feedback on sticky notes, and other methods definitely are formative.

Quadrant 4, summative: An exam. This is a classic example of the end of chapter exam, or a quiz with no retakes.

I am not sure of all the types of assessments I put in the different quadrants. How would feedback being given in the moment on the process of factoring quadratics (for example) be summative? Is a quiz that is not fixed in the gradebook (as far as grade) really formative?

I find the structure of the quadrants helpful in thinking about when assessments are given and towards what end they are given to be helpful. However, the usual categories of formative and summative assessments don’t always fit here. Is this a problem of the terms formative and summative? Should we stop using them (fat chance, given the history and literature of assessment)?

Ann’s post really got me to think about assessment, and how I explain it to preservice teachers. I am not sure it is the last word, but I do know that I have had experience teachers explain to me that the only kind of assessment that is formative is “In the moment” assessment. Clearly that is false given the types of items listed in quadrant 4.

Any suggestions? Additions? Criticism?  I think this model of thinking about assessment has opportunity for understanding the different types, but it needs to be fleshed out more.

Aug 082017

This question has been on my mind since I finished reviewer the submissions to the NCSM National Conference. This was my third year as a reviewer, and this year I noticed something very …. different. I was assigned 17 out of 18 articles in the “Equity” strand, so I read many submissions that were supposedly in the same strand. I say ‘supposedly’ because, in my opinion, they weren’t. Not at all.

Of the 17 submissions, four submissions were straight up, this is how you (as a teacher) differentiate instruction for learners with learning disabilities. Is this equity? I am not sure. I suggested they be moved to the instruction strand, because I do not consider a narrow focus on differentiation to be equity.

Another six submissions were about differentiation, but not narrowly focused. These submissions were about how to differentiate summative or formative assessments, or instruction, or discussion, so that all learners would have the opportunity to access the course content. These are more along what I consider equity, but I still have reservations.

The last group were very much in line with my idea of equity, and were concerning how to modify teacher practice to allow for under-represented groups to access course content at a high level.

These submissions challenged me to think about what is Equity in mathematics education.

After spending three days at TMC17 thinking and discussing Equity with Grace Chen as one of the leaders, I am still not convinced that the four submissions are about equity.

I think that addressing equity is more than a narrow focus on learners with disabilities, but must be a larger discussion about the inequities that exist in our classrooms. However, the IDEA was enacted because there were severe inequities in how special education learners were treated in classrooms.

But, is a narrow focus on special education learners equity?

I still say no.

Equity in the mathematics classroom is not about differentiation, but about teaching the content in such a way that each learner identifies with the content in such a way they are able to see themselves in the material. More importantly, the learner is then able to use the content in such a way that they take it and change their world with it. For example, in this AVID video, education is definitely something the learners here are using to change their world. 

This ideal of equity comes out of Freire and Gutstein’s ideas on equity. It is not focused specifically on a race or ethnicity, however this ideal does focus on giving learners who have been historically disenfranchised a connection with content they have been denied in the past.

Will all learners benefit from this? Sure. But learners who are from disenfranchised populations will benefit the most. They have been denied access to a curriculum for a long time, and gaining access to it in a way that will allow the learners to enact change is powerful.

This, to me, is what the definition of equity includes. It certainly is not an exhaustive definition, and I need to think and read more to expand it. It is my starting place, however. Give each learner the ability to connect with the mathematics content in such a way they can gain mastery over it and use the content to change their world.

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.

Jan 152017

One question that comes ups often with math majors in the program is “Why do I have to take a computer science class?”

I am not sure where the official requirement comes from, but I can say that I am extremely thankful I had a computer programming class in college. It was over 20 years ago, and it was Pascal programming, but I am very happy that I still remember the skills I learned. I don’t remember anything about Pascal, but over the last 20 years, and especially the last three weeks, I have used the heck out of those skills.

When I was in business, the programming skills allowed me do some serious Excel sheets and data crunching that got me noticed and promoted.

As a teacher, those Excel skills allowed me to strip data from the PDF reports and turn those into useful files that we could actually mine for relevant data on our learners and their learning. Those skills also allowed me to learn basic HTML and CSS coding to build websites over a Christmas break and create multiple websites.

Now, as a master teacher I have spent several weeks building a very complex database in Access to manage our check in and check out process with the hundreds (soon to be thousands) of items in our teaching supply store room. To do this, I have had to teach myself Visual Basic, Access structure, as well as some basic SQL database language.

Now don’t get me wrong.I do not have anywhere near the skills to be paid to program in any of these languages, and it is taking me 5 times longer than a real programmer would take. But, because of that Pascal programming class 26 years ago I have the ability to learn the new skills, new languages, and troubleshoot the really bad code I am writing and make it better.

Why should today’s learners learn coding? Because if this dinosaur can reap these benefits out of the class over my career, then imagine what benefits our learners today will reap over the next 25 years! It only gets more important and more essential from here.

Dec 012016

We are starting to gear up for TMC17, which will be at Holy Innocents’ Episcopal School  in Atlanta, GA (map is here) from July 27-30, 2017. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc ( It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, however, this year we cannot guarantee that everyone who submits a proposal will be accepted. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Thursday, July 27 and 48 one hour sessions that will be either Thursday, July 27, Friday, July 28, or Saturday, July 29). That means we are looking for somewhere around 70 sessions for TMC17.

What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 16, 2017 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC17 – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Daniel Forrester, Megan Hayes-Golding, Cortni Muir, Jami Packer, Sam Shah, and Glenn Waddell

Nov 062016

In my last post, Why I won’t use Direct Instruction, I was provocative and challenged some of the typical thinking about math instruction. The post generated some terrific conversation, both here and on Twitter, and although I have not changed my mind for my own classroom use, I do admit there may be times when DI has a function and purpose.

The grad class I am in has moved on, which is how classes work, and we are discussion Cooperative teaching this week. The textbook is very focused on English and Social Studies, which leaves the math and science people out a bit, but it does discuss the Jigsaw lesson plan at some length. The Jigsaw is a good strategy, and it is useful in math class for sure, but there are so many others!

To create a better list, I asked the #MTBoS for their favorites. I won’t embed all the tweets, but will give attribution to every person who submitted and idea or link in the idea. I want to feature the lesson plan ideas and the links to them. I have not used all of these. Heck, I don’t think I have used half of these! But the collection is amazing, and although some of the ideas don’t have details, you can figure out the idea from the names of some.

The really nice thing is that these are all cooperative lesson strategies from math teachers for math teachers. If you want some ideas on how to incorporate these well tested strategies, here you go:

  1. Speed dating: Me! @gwaddellnvhs, Mary-Ellen @MathSparkles; This was one of the two I suggested. I really like this method of getting learners collaborating with a purpose.
  2. Add it up or Placemats or 4Sum or Add ’em up: Me! @gwaddellnvhs; Heather Kohn @heather_kohn; S @reilly1041; Kate Nowak@k8nowak; This is another strategy I offered in my original question. I think I got it from Kate originally, forgot the name, and then called it Placemats because of a way to set it up using butcher paper. Same idea, different names.
  3. Participation Quiz or Partner Quizzes: Martin Joyce @martinsean; Rachel @Seestur ; Used these often. Very engaging way to get everyone focused. The tricky part is creating the teams for the quiz, but that is achievable.
  4. Clipboard of quotes & actions that support each other. Update whiteboard, then go over: Martin Joyce @martinsean
  5. Whiteboard Game: Lisa Bejarano @lisabej_manitou
  6. Problems around the room: Lisa Bejarano @lisabej_manitou
  7. Also a big fan of whiteboards where students keep answers secret and then they All “flash” at the same time: Mary-Ellen @MathSparkles
  8. Pass the Pen: Madelyne Bettis @Mrs_Bettis
  9. Work on the Wall: Madelyne Bettis @Mrs_Bettis
  10. Ss work prob on board while 2nd Ss “calls” it like a baseball announcer: Mary Williams @merryfwilliams, The boys get into it with the Bob Costas enthusiastic voice, “and he is STRIKING OUT LADIES AND GENTLEMAN!!” Most of the time they are really positive though – all the sports enthusiasts enjoy announcing 🙂
  11. Ghosts in the Graveyard: Mary Williams @merryfwilliams
  12. Sage and Scribe: Briana Guzman @brianalguzman
  13. Quiz Quiz Trade: Briana Guzman @brianalguzman
  14. There can be only one (marker): Nathaniel Highstein @nhighstein
  15. Having round tables in the classroom: Rachel @Seestur Rachel really enjoys having the round tables so learners have to look at each other while working. It makes total sense to me!
  16. Tarsia Puzzles: Sheri Walker @SheriWalker72; Paula Torres @Lohstorres1; In case you don’t know what Tarsia puzzles are, Tarsia is a FREE software package to make puzzles out of sets of problems. They are really cool, and when you require them to be worked in partners, can be a great way to incorporate cooperative learning in a different way.
  17. Card Sorts: Beth Ferguson @algebrasfriend; Card sorts have been around a while, and they are highly effective. I used them in AP Stats as well as algebra. Desmos recently incorporated card sorts into the Activity builder, so you can get awesome electronic card sorts now too!
  18. Row Games: Kate Nowak  @k8nowak; Beth Ferguson @algebrasfriend; I have used Row Games too. The best part is the link takes you to a folder owned by Kate that has 3 pages of mostly word docs of teacher created games. This means you can edit and change them to make them better for your class! Also, it would be awesome if you shared back your creations to help others.

Additionally, David Wees tweeted out the following people, but didn’t give more info. I suggest contacting them directly for more information.

David later did followup with this link to TEDD (Teacher Education by Design). I poked around their site. Looks promising!

Amy Lucenta also was kind enough to let us know her ideas are found in her book from Heinemann Publishers.

I hope this helps, and if you have any other cooperative learning ideas, drop them in the comments please!

Oct 162016

This post is born out of a PhD class I am taking called “Models of Teaching.” It is a great class, but one of the requirements early in the semester was to write how I would use direct instruction in my classroom. I refused. I wrote a lengthy screed against DI. I attacked it, aggressively. What you have here is an edited, cleaned up, and less aggressive post born out of that assignment.


As a first year teacher, I was explicitly told by a principal to use direct instruction. He very carefully outlined what he expected any class to look like, and what the learners should be doing at every stage, every minute.

When that year was over, I left that school without a second thought. To deprofessionalize teaching to such a degree that someone could outline any class, any day, any lesson to the minute is reprehensible and borders on educational malpractice.

If you get the sense from this that I do not value direct instruction very highly; good.

I mean, really. Look at the way people think about education and specifically math ed. I think using comics as indicators is a great idea, because comics take a shared experience and pokes fun at it. Comics make us laugh through the pain, and there is a lot of pain in education.

Baldo, I cant believe school starts tomorrow

At the younger grades, we definitely see excitement for learning, but at some point, we beat that excitement out of kids. Why? This is a question I have asked repeatedly here, but I think DI has a lot to do with it. I mean, DI is a common way to teach math, as well as other subjects. Can we blame learners if they are bored, frustrated, and unexcited about classrooms that are taught through DI? And they are all 3.

math class is like a 40 foot long colon

Really? The punch line in this Baby Blues makes me cry. Literally. This is what the general public finds funny about math class?! But it isn’t just these comics. It goes on. And on.


The common theme of memorizing is so frustrating.


I am not advocating for “learn what you want” or unschooling, but certainly we can figure out ways to build in learner interests, right?


And DI just take us to the point repeatedly. “Oh, you weren’t paying attention while I was sharing what you were supposed to be learning? That is your problem, not mine.”

dennis-the-menace-back-to-the-salt-mines  dennis-the-menace-principal-not-warden

Yea, nothing more needs to be said here. Sigh. These were published in October. Of 2016. These are current. It makes it just that much more sad.


This Zits comic pretty much sums up the idea of Direct Instruction for me. It is clear that Jeremy (the teenager) has teachers who use DI pretty much the entire day. He is just consuming the knowledge of the teacher, puking it back for the test, and starting over each day.


And this focus on memorizing, and storing the teacher’s knowledge leaves learners doing what Paige Fox is doing here. Focus on the test, not learning. As long as the test comes out okay at the end, then all is good. Same issue Calvin had above.

But my objection to DI goes beyond the fact that it creates a horrible perception of classrooms. The philosophical underpinnings of direct instruction follow from Behaviorism and the work of B.F. Skinner.  Skinner, in his book “The Technology of Teaching” introduced wonderful machines that replaced teachers. In the behaviorist world, teaching is only necessary to introduce proper conditioning, and you do not need professionals to create those behaviors. Machines, called appropriately enough, “Teaching Machines” can replace teachers wholesale.

teaching machines by skinnerJust read the question, mark the response, check the response to the key, move a lever left if correct and right if wrong. Finish the lesson and repeat until they are all correct. This is the legitimate end result of behaviorism and the deprofessionalization of teaching. We see it in such sites as Con, er, Khan Academy, where the boring and mistake prone  lectures are used to give a false impression of learning. This kind of approach to teaching and learning is why at least one US Senator has suggested doing away with college professors and just have students watch Ken Burns videos to learn about the Civil War. Not joking. This is real. This is the direct benefactor of behaviorism.

In short, there is not enough alcohol to burn this chapter from my memory. [I leave this sentence in here from the assignment for a reason. Yes, I really did turn this sentence in, but also because it shows just how strongly I feel about this issue.]

These are harsh words. I freely admit that. I have very few, if any, kind things to say about direct instruction. I stopped teaching this way after my second year in the mathematics classroom. I would never go back, nor would I ever try to teach this way again.

It is painfully boring for the learners, and it is equally painful for the teacher. The fact it is completely ineffective to teach or learn higher order processes and skills makes it doubly not worth using.

Direct Instruction is the worst of all teaching methods, and continuing to use it just reinforces the boring nature of what learning can be. It doesn’t have to be that way! It really doesn’t.

When I write lessons, whether it was for high school or for the college classes I am teaching now, I start each lesson with these questions (replacing math with teaching now):

Am I:

–Assisting learners in creating THEIR own math understanding?


Forcing learners to curate and consume MY math understanding?

My goal is clear. I want every learner to move beyond my understanding quickly and efficiently. That can’t happen with DI. DI is a way to force learners to store my knowledge and understanding.

And, we need to figure out ways to stop asking learners to store our knowledge and instead celebrate their own. There are many constructivist teaching models. We need to use them. Find two or three that resonate with you and practice them. And then, celebrate the accomplishments of learning for more than 2 seconds.

Calvin is sad for a reason.


Oct 122016

I had the opportunity to read a preprint edition of Malke Rosenfeld’s new book, Math on the Move, and here are my thoughts.

First off, let me start off with what this book is not. As educators we have probably sat through a professional development where someone told us that in math class, we can appeal to the “kinesthetic learning style” by having the learners up and moving around the classroom. We can appeal to “kinesthetic learners” by having them move their arms, or by doing gallery walks. I have sat through several of these. [yes, I put that phrase in quotes on purpose. I do not believe in ‘learning styles’. Multiple Intelligences, yes, learning styles, no.]

Rosenfeld’s book is not this. No where near this. This book is not about “kinesthetic learning” this is about making connections in mathematics through motion, body, and dance for elementary school learners. It is an amazing concept to think about. I really appreciate that on page 2, she says, “not all of dance is mathematical and not all math is danceable.” That sets the tone for the entire book. Rosenfeld looks for the strengths in using movement, and using the body as a thinking tool. This is a powerful idea, and the first chapter of the book is about what doesn’t and does count as using the body as a thinking tool. I loved the deep thinking this chapter provoked, because it made really think about dance and movement with respect to math.

And, let me be honest. My knowledge of math through motion is very limited. My idea of dancing is more aligned with this guy than anything that someone else would consider “dancing.” Honestly, I wondered for a moment if someone had recorded me actually dancing when I saw this gif.

dancing-gif via

But, despite the fact I am both musically and rhythmically challenged, I have always thought there was opportunity to connect math and movement. I have never figured out how, but I have been intrigued by the idea. After reading the table on page 17 I realized why.

table of nouns and verbs about math movement

The verbs of math are aligned with the verbs of dancing. The nouns of math are also aligned in large part. Looking at the list, and knowing, intellectually, about the ideas of dance, it is easy to understand how strong the connection is. Through examples of learner work, QR codes showing video of learners moving, multiple lesson examples, pictures, role playing examples, and well developed explanations, Rosenfeld shows me how to implement dance in a very constructive way in the elementary classroom. By the end of chapter 3, I was willing to try it with elementary kids tomorrow. That takes a lot for me to say, because I am secondary through and through. Little kids scare me. But I am so excited by the opportunity I see after the first three chapters of lessons that I am willing to try them. They are so interesting!

I think the real power comes later in the book when the 6 stages are developed further.

  1. Understand
  2. Experiment
  3. Create
  4. Combine
  5. Transform
  6. Communicate

These stages allow learners to move from the understanding of a concept and goal to the creation of a multi-step dance pattern and ending with the discussion and communication of the idea through a presentation of the dance. The last half of the book has QR Codes on almost every single page with video link examples. The depth of knowledge these can provide is stunning.

All in all, the more I read and find the joy in mathematical dancing, the more opportunity I see to push this into the upper levels. There is so much more that can be done with this idea beyond the boring and basic. It might even make me a better dancer! Well, no. It isn’t a miracle book, just a really good math book. It is authentic movement, not the usual fake stuff we see.

I think it is time to bring real motion in to math class, get learners moving in purposeful, meaningful ways, and leverage that motion into strong mathematical knowledge.

If you want to read a chapter for yourself, check it out on Heinemann’s website.


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.

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