The Struggle for New Math Instructional Technology

This article by Jack Smith IV ( says what I’m thinking better than me. Matt McFarland (Washington Post) has a similar argument has a similar argument most focused on the price (which mysteriously hasn’t followed the trends of any other piece of personal technology by staying quite high).

The argument: Texas Instruments has some kind of a stranglehold on secondary mathematics here in the United States and it certainly isn’t that it is the best tool for the job.

The arguments that I most often hear from teachers in support of sticking with TI are that it is the tools that have been in place so long that they are the easiest for teachers to teach. It’s just easier to have the students all using a single tool that the teachers are really familiar with.

In an article written in The Atlantic, Alexis C. Madrigal argues that there is very little need for an update in the technology if there hasn’t been a corresponding update in the math content requiring the technology. “After all, the material hasn’t changed (much), so if the calculators were good enough for us 10 or 15 years ago, they are still good enough to solve the math problems.”

And also that these tool are the tools that are allowed on the SAT or the ACT. For some teachers that I talk to, this is kind of a big deal. And there is a reasonable logic to it. Those tests are pretty important (especially in Michigan, where school accountability for math is connected to student outcomes on the SAT). So, why confuse a student’s brain with a variety of tools that, when the rubber hits the road, they won’t be able to use?

It’s worth noting, however, that College Board (the designers and publishers of the SAT as well as the Advanced Placement exams) have a particular target audience in mind. In states where the SAT isn’t the state-mandated accountability measure (meaning every single high school junior statewide will take the test), which students are likely to take a test designed by College Board? Students who are college bound and/or enrolled in advanced placement classes. For many, many of these students TI Calculators have “worked out fine”. Also, they’ve likely been the only tool made consistently available. And it’s better than graphing all of your parabolas by hand.

But, have you ever watched a group of students in an algebra class for the second time trying to remember those key sequences and explore images on those tiny, granulated screens? And by now, they know there are technologies out there that are easier to use.

I am certainly not accusing College Board of being deliberately prejudiced. I am not accusing TI of being deliberately prejudiced. I’m simply accusing them of having goals that are different than “100% of math students will learn math at the highest possible level”.

But that’s my goal. And it should be the goal of all of our classroom teachers. And the question should be whether or not TI calculators fit within that goal.

Well, in Michigan, there might be a couple of potential cracks in the shell through which the light of better technologies (like Desmos and Geogebra) could potentially shine through to teachers previously unwilling to explore them.

The Redesigned SAT, which launches in March 2016, has a math portion that includes two sections. There is a significant portion that prohibits calculators completely. Then, there is a section that allows calculators, but the College Board includes questions “for which calculators would be a deterrent to efficiency.” D’ya catch that? That’s our window. This allows for an entirely different message than simply “TI’s are allowed on the SAT, so we’ll use them.”

You could reasonably guess that AT LEAST half of that test should be able to be done with no calculator at all. This could potentially leave a huge hole in one of the most stubborn supports for the TI-status-quo. Now, we are potentially keeping these clunky, expensive devices as the primary tool because of possibly 25 questions that they’ll use it on for a single testing sequence. That’s a much tougher sell.

The other main issue (teacher familiarity) is is a matter of exposure. The teacher prep courses could support this for our preservice teachers. (In 2004, when I was a preservice teacher at WMU, I had one “instructional technology in math” course that has a TI-89 or Voyage200 as a required purchase. Luckily my girlfriend (now my wife) had one.)

Instructional technology in math should be way, way more than a semester-long how-to on a single TI device. Desmos and Geogebra and other available technologies that are free for use in the classroom provide valuable opportunities to give students different experiences with problem-solving in mathematics. But teachers, like all people, are going to stick with what they are familiar with.

And most of these young math teachers (most born in the early 90’s), we raised in the classroom as math learners with TI calculators. Given no other experiences, they aren’t going to have just their familiarity to build from. Especially when, for most of them, the TIs “worked out fine.”

As for the teachers already in the classroom, that’s part of the work of math-minded instructional techs… like me. I’ve got two sessions coming up that are focused on Desmos as a tool to engage math learners.

The SAT did it’s part and I intend to do mine. Our students deserve better than TI or nothing.

Don’t forget Geometry when teaching Algebra

Right now, Michigan educators are trying to sort out the implications of the state switching to endorsing (and paying to provide) the SAT to all high school juniors statewide instead of the ACT as it had previously done. The SAT relies very heavily on measuring algebra and data analysis. This leaves plane and 3D geometry, Trig and transformational geometry under the label “additional topics in mathematics.” The new SAT includes 6 questions from this category. Compared to closed to 10 times that from more algebraic categories.

This comes up in every SAT Info session I lead. Should we just stop teaching geometry? All Algebra? Could geometry become a senior-level elective? If it’s not going to be on the SAT, then what?

These questions reflect a variety of misconceptions about the role of testing in curriculum decisions. In fact, these are the same misconceptions that are driving an awful lot of the decisions that are being made. And in this case, I think there’s more at risk than simply over-testing our students.

It would be a real shame to see Geometry become seen as an unnecessary math class. Because it’s not.

And to illustrate that, I’m going to tell you a story about a conversation I had with my daughter. She’s 6.

She wanted to try multiplication. She’d heard the older kids at school talking about it. So, I taught her about it and let her try some simple problems to see if she understood. And she did, for the most part.

So, I started to not only make the numbers bigger, but also reverse the numbers for some problems that she’d already written down. She had already computed 2 x 3, and 4 x 2, and 5 x 3, but what about 3 x 2, and 2 x 4, and 3 x 5. Were those going to get the same answers as their reversed counterparts?

She predicted no. So, I told her to figure them out to show me if her prediction was right or wrong. To her surprise, she found that switching the factors doesn’t change the product.

“Daddy… why does that happen? It should change, shouldn’t it?”

Translation: Daddy, how do you prove the Commutative Property of Multiplication?

How would you prove it?

The geometry teacher in me thinks about area when I see two numbers multiplied. We can model (and often do) multiplication as an array. It’s just a rectangle, right? A rectangle with an area that is calculated by it’s length and width being multiplied.

What happens if we rotate the rectangle 90 degrees about it’s center point? Now it’s length and width are switched, but it’s area isn’t. Because rotations are rigid motions. The preimage and image are congruent. Congruent figures have the same area.

If l x w = A, then w x l = A. 

Geometry helps prove this. Geometry also helps support a variety of other algebraic ideas like transformations of functions within the different function families. Connections between slope and parallel and perpendicular lines. There’s also the outstanding applications of algebraic concepts that geometric situations can provide. Right triangle trig, for example, is often a wonderful review of writing and solving three variable formulas involving division and multiplication. (A consistent sticking point for lots of math learners.)

As a teacher who spent years watching how Geometry presents such an environment for real, effective and powerful mathematical growth, eliminating it will leave a lot of holes that math departments are used to geometry content filling.

Gallons and Gallons of Pennies

Sometimes, “real-world” problems just go ahead and write themselves. And I say take advantage. Why be creative with the actual world can do the heavy lifting for you, right?

This floated across my Facebook feed. Pretty sure you’ll see where I’ve made some edits to the original texts.

Pennies Problem

Sequels could potentially include:

If Ortha wanted to exchange them for quarters, how many 5-gallon jugs would she need? You could do the same with nickels or dimes.

What would be the mass of each penny-filled jug?

What do you think? What other questions could come off this wonderful set up?

Exploring Reflections with Desmos Activity Builder

Geometric transformations take up a good chunk of the first quarter of Geometry (at least the way that I had it sequenced). The tricky part of teaching transformations in Geometry is the delicate balance between the non-algebraic techniques and understandings and their algebraic counterparts.

For example, consider the two following statements.

Two images are reflections if they are congruent, equidistant from a single line of reflection and oriented perpendicularly from said line of reflection.

The reflection of A(x, y) is A'(-x, y) if the line of reflection is the y-axis and A'(x, y’) if the line of reflection is the x-axis.

As a geometry teacher, am I to prefer one over the other? In my experience, they both present challenges.Students are often a little more enthusiastic about the first (which is why I start there), but can often be more precise with the second. The second requires the figure to have vertices with coordinates and the line of reflection be an axis. The first requires the figure be drawn (and fairly accurately, at that.)

So, in my attempt to learn how to use Desmos Activity Builder, I wanted to produce something useful. So, I made a Desmos Activity to bridge the transition from a visual, non-algebraic understanding of reflections to an algebraic one.

Full disclosure: This activity presupposes that students are familiar with reflections in a visual sense. It isn’t intended to be an introduction to reflections for students who are brand new to the topic.

Feedback, questions… all welcome.

The two key ingredients of real problem-solving

A quick word about dissent.

During a recent conversation with a teacher-friend I we stumbled into an area of conversation that allowed me to see dissent through the lens of leadership and problem-solving in a way that I hadn’t before.

Acceptance of dissent isn’t a new idea in leadership. Lots of writers talk about the need for leaders to appreciate it… here’s an example.

“Defining effective leadership as appreciating resistance is another on of those remarkable discoveries: dissent is seen as a potential source of new ideas and breakthroughs. The absence of conflict can be a sign of decay.”

– Michael Fullan (From Leading In A Culture of Change, 2001, pg 74.)

We were talking about problems that tend to have some pretty zealous advocates. For the sake of exploring a concrete situation, I’ll choose one for an example. How about student retention? This is a topic that can bring some energy out of some folks. It’s an important conversation, too. What happens when a student finishes a school year without meeting the minimum expectations to complete the grade/course they are in?

To push them forward would mean pushing the student forward into academic challenges that they likely aren’t prepared to tackle.

And making student repeat grades has just not been an effective solution according to ASCD, Education Week, John Hattie, etc…

So, when a district sits down to really solve this problem, they need to accept that they probably are going to need to choose a third option. Carelessly moving the student on is probably a poor choice. Making the student repeat the grade is also a poor choice.

The better option, the third choice, the one that will work better, is likely going to have to be crafted on site and with the resources available helping to guide the process.

This is where I began to see the need for two very distinct groups of people.

One group of people creates the boundaries… I’ll call them the idealists. These are the people who say, “We can’t retain them. We can’t. I don’t care what we do, but we aren’t retaining them. It doesn’t work.” Every issue has these people. Most of us can become these people when the issue at hand strikes us right. Luckily, they seem to be essential to the process. They also happen to be very frustrating to people who either disagree or just don’t see the issue as important.

The main issue with these folks is that zeal often doesn’t really solve problems. It creates boundaries for the solution, but (in the case of our example issue) simply eliminating retention doesn’t actually solve the problem of students falling behind. It just eliminates a series of potential solutions.

So, we need to bring in the dissenters… I think of them as the holders of the “yeah, buts…”

“We can’t hold them back.”

“Yeah, but they are still behind in their learning, so we can’t just move them on.”

Now… at this moment… as long as neither the idealist or the dissenter storms out of the room, the real problem-solving work can begin. The boundaries are set, the reality checker is in place and now the focus can turn to the ACTUAL problem In the case of promotion v. retention, it’s the fact that students are making it to the end of the school year not ready to move on.

And that takes some deliberate focus and patience. The zealous boundary-setters don’t want to hear about “yeah, buts…”. The dissenters tire quickly of the perceived inflexibility of the idealists. But I’m not sure real solutions to tricky, messy problems are more likely than when these folks can unify around a common goal.

American education (shoot, American culture as a whole) has a whole variety of problems that we are having trouble solving because the zealous idealists and the persistent dissenters have such a hard time embracing the valuable contribution that each other makes in the course of creating real solutions.

But real solutions… solutions that are effective and sustainable… probably require the active presence of both.

Changing the conversation about testing and data

What if I told you have I know of schools that run through their first grade students through just over an hour of math and reading exercises while recording their results to get a sense of their strengths and weaknesses? These exercises are done a little bit at a time in the first three weeks of school. They do this so that they can make accurate decisions about the ways that each of these students will be properly challenged. This way, each young person gets exactly what they need to grow as learners.

What are your thoughts about these schools? Would you say they care about their students? Would you say that this is a nice approach to education?

Pause here for a moment…

I’m going to start this blog post over again. This time I’m going to tell the same story in different words. I want to see if different words paint a different picture of these schools. Keep in mind that the second set of statements are equally accurate.


What if I told you that I know of schools that will give their first grade students eight different standardized tests by the end of September? They do this so that they can record a bunch of data about the students so that they can group them based on the data on those tests.

Sounds a little bit different, doesn’t it? Eight standardized tests sounds like a lot. (Even if the longest of them 8 minutes long. Some are as short as 1 minute.)

So, we’re faced with a decision. Is the first one unrealistically rosy? Or is the second one unnecessarily cold? Your bias will determine which of those viewpoints speak to you most. My bias certainly is.

What isn’t based on a bias is that “standardized test” and “data” have become hot-button, divisive words. And there’s been some backlash. That backlash is captured by posters like these.


Sharing encouraged by Marie Rippel

The message is that our young people are more than a few data points. And that, no matter how much data that we collect, there are important elements to these young people that no test can reveal. That is absolutely correct and if you don’t agree, I’m curious to hear your argument. Post it in the comments and we’ll explore it together.

But that doesn’t mean that the poster (and the related sentiment) are safe from push back. First, there are some things on that list that tests actually could measure. It would be fairly reasonable to collect some data on “determination”, “flexibility”, and “confidence” provided we could all agree on the definitions and manifestations of them.

But secondly, that poster includes items like “spirituality”, “wisdom”, “self-control”, and “gentleness”, which are items that different groups would argue aren’t really the job of the American public school system. That isn’t to say that these groups wouldn’t consider these valuable qualities, just qualities that the schools aren’t on the hook for teaching.

To me, this is an important point. Because there’s a variety of other things your garden variety standardized tests don’t measure. For example, they don’t test a student’s ability to drive a car, their ability to write a cover letter or resume, or their ability to cook a decent meal.

These fit largely into the same category as the items on that poster. Important qualities that are common among successful people, but not qualities that are tested on any of the standardized tests that the students take in the K-12 education. Yet, I’ve never heard anyone use their absence as a support to discount the value of the tests. What makes these qualities different that the ones on the poster?

It could be that the American public and teaching professionals agree that those things are not the job of our public schools. It isn’t their job to teach young people how to drive a car or write a resume, or cook a meal. So, clearly we should be inspecting their ability to do so.

So, would I be safe in assuming that if we could all agree on the job of public schools, then some of the fervor over tests would cease?

Would the authors of that poster be more satisfied if we were collecting data on students compassion?

What are the jobs of our public schools? Frame your answer from the context of what should all students be expected to do when they leave the educational systems after spending 13 years in it.

And how are we going to know if the system is doing it’s job? Listening to a discussion regarding those questions sounds like a huge upgrade compared to listening to hours of endless back-and-forth about whether or not to test, how to test, which tests to use, or what to do with the results of the tests.

What are our goals and how are we going to know if the system is doing it’s job? It’s fine by me if testing students isn’t part of that. But, our educational system has a vital job to play and somehow or another, we need to develop a way to inspect what we expect from the system.

Perhaps the first step of that is coming to consensus on what we expect.

Mathematical Reading – Wrapping my thoughts up.

I have spent the last couple of posts discussing the value, need, and potential of considering mathematical reading an essential learning target in all math classes.

Typically, this isn’t a tough sell in the elementary world because elementary teachers are teachers of all things anyway. They teach reading, writing, science, math, (and in some cases, art, music and phys ed, too.)

Secondary folks, on the other hand, tend to exist is a more compartmentalized world. This is largely a product of the increase in sophistication and depth of the content as the public education sequences progresses toward graduation. It is simply unreasonable to expect educators to have a teachers-level knowledge base of biology, economics, civics, algebra, and literature, as would be required if freshman year structurally looked like first grade. Compartmentalization (or silos as is becoming a popular term) has downsides as well. And many of those downsides can be wrapped up in the all-too-often uttered phrase “It’s not my job.”

And in my years in education, I’ve heard “it’s not my job to teach reading” from math teachers many times. And I forgive them for saying it. Math is a world that communicates differently. Graphs, charts, symbols, equations… we do that stuff so that we don’t have to read.

And they have a point. Consider these mouthfuls:

“The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse”.

“The slope of a linear function is the coefficient on the independent variable when the function is written in slope intercept form.”

There’s a reason people (both mathematicians and students, mind you) look to use notation to represent those two statements. it is quite a bit easier for a student to say “well, y = mx + b… slope is the m.” And what’s more, that statement will work effectively more often than not. So what’s the problem?

Through the lens of solving math problems on a test, there probably isn’t much of a problem. But consider reading to be an essential problem-solving skill, then there’s a risk to consistently easing the reading burden. We might be navigating our students strategically away from something they’ll need.

And while this thought process was instigated by the releases based around the redesigned SAT, I wouldn’t simply use the test as the primary motivator for updating our math classes. I would prefer to examine what message the College Board is trying to send by insisting that their materials insist on such a high degree of literacy for all subject areas, even considering that they have a reading and a writing test already.

And the message might be worth listening to. And possibly not. Remember, the very first post in this 5-part series started with the words, “This post has questions. No answers in this post. Just questions.”

Now it’s on all of you to help answer my questions and there was a lot. Ready? Go!