The Struggle for New Math Instructional Technology

This article by Jack Smith IV (mic.com) says what I’m thinking better than me. Matt McFarland (Washington Post) 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.

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.

Ready?

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.

College-Readiness, math, and reading…

Consider this math problem:

In 1974, the state that had the highest population density was New Jersey with a population density of 1305 people/sq. mi. In the years that followed the decline of the auto industry, the populations began to shift away from the major industrial centers (like many of the cities in New Jersey). By 2015, New Jersey’s population density had dropped to 1210 people/sq. mi. If New Jersey has a total size of 8700 square miles, how many fewer people live in New Jersey in 2015 than in 1974?

Okay…

The math of this problem isn’t really that sophisticated. Take your densities, multiply them both by the area to get total populations. Subtract the bigger population from the smaller population and there you go.

However, what is making me struggle with this problem is the undeniable literacy component. When I consider the reasons a student might get this problem wrong…

1. Made computational errors
2. Did computations correctly, but computed the wrong numbers.
3. Didn’t know how to set up the computation because they got lost in the vocabulary or notation.
4. Didn’t know how to set up the problem because the task was unclear.
5. Got frustrated and skipped it because the US Census keeps records of state populations that can be referred to instead of having to crunch numbers.

From an assessment standpoint, what does a correct answer from a student reveal to us about what that student understands and is able to do?

1. We know that student can work with rates and units in context.
2. We know that student can multiply and subtract strategically and accurately.
3. We know that student has the ability to accurately comprehend a piece of reading equivalent to about a seventh or eighth grade level.

I want to talk about that last one. The reading one.

If that question appeared on a math test, what would be the value of exploring their ability to read? It seems like we have tests for that. Won’t they reveal those things? Shouldn’t the result a math test be based simply on a student’s ability to do math?

Well, I’m going to go ahead an add a wrinkle. Michigan just adopted the SAT as the state-sanctioned offering to the legal requirement that all juniors in the state of Michigan will take a college-readiness test before they leave high school. (From 2008-2014 it was the ACT.)

The literacy component of the SAT Math test is quite heavy. The problem that I highlighted (which I made up… along with most the data in the problem) resembles the SAT Math questions pretty well.

So, the College Board (authors of the SAT and most the AP Tests) seem to be making the statement that college readiness includes the ability to read. I’m not sure there would have been much argument for that in general, however, there are SAT portions for reading comp and an essay. The literacy bases seem covered.

So, why put such a high emphasis on reading in math?

Perhaps The College Board is making the statement that math proficiency includes the ability to fluently read mathematical scenarios.

I’m of two minds on this issue, so I’d really like some reader participation in the comments. I’m not at all attempting to challenge the value of reading, but some students really struggle with reading. Does a reading struggle apply a ceiling to future math growth?

And if there is an essential connection between math and reading, what role do math teachers play in teaching reading? Should we be developing strategic interventions for math-based reading?

I hope you’ll feel comfortable adding a comment, idea, or question that I’m not thinking about.

In my next post, I’m going to further break down this idea with respect to my limited understanding of Universal Design for Learning.

What does “College Readiness” really mean for math class? I mean… really…

This post has questions. No answers in this post. Just questions.

I just spent the better part of a day exploring the SAT test (which Michigan has recently adopted as the test that all high school juniors will take as a “college readiness” test.)

I also read this Slate article which speaks to people who feel like they aren’t “math people.”

As I continue to listen to various groups chime in with that they think kids need with respect to the mathematics portions of our various educational systems, there seem to be a few ideas that are coming out.

The first two are usually close-to-unanimously agreed upon.

1. All young people… ALL… young people have a God-given right to a high-quality education.
2. A high-quality education includes a significant amount of mathematics beyond basic numeracy.

Beyond that, the overarching ideas push into value-based philosophies about what the author or speaker believes are in the best interest of American young people. These are definitely thoughts where two reasonable people could find areas of disagreement.

3.  An education earns the title “high-quality” when the receiver can use it to                    successfully take the desired next steps after it’s done. “Next steps” are                      generally considered one of the following: A. going to college, B. going to                    work, C. going into the military, D. starting a family or E. any combination                    thereof. No one necessarily more noble or challenging than the others.

4. The overseers of those areas are the authority on what is required to be                     able to successfully join those communities. College professors, business                   leaders, military leaders, and church leaders all have a reasonable                             expectation that they would influence the courses of study that lead into                     those individual arenas.

If we push these thoughts to the next level, it becomes reasonable to assume that college professors, business leaders, military leaders and church leaders are going to quite often disagree on the necessary requirements for an education to be considered “successful”. What’s more, there will be large segments of the general public who would prefer one or two of those get a larger say than the others. Certain groups of people would prefer that college professors should have the last say. The incredible number of religious private schools speak volumes to our public’s desire to allow their particular faith to have the ultimate say in the educational program.

And math doesn’t get a free pass in these disagreements. You will see wide differences in the types of mathematical content that are preferred as well as the instructional and assessment methods. The nationwide introduction of Common Core has demonstrated that math curriculum can trigger some very negative responses from significant portions of the American public.

And the fact that nationwide, our school systems are “public” empowers the general public to have a say. In fact, this is often a tricky balancing act for education professionals. There is a certain amount of background knowledge necessary to make sound decisions within the field of education. But, practically every single adult walking the streets has a decade (give-or-take a year or two) of experience within the field of education. There is a certain amount of boldness that that kind of familiarity breeds.

Result: Everyone has an opinion on how education should look. And that opinion is largely based on the relative satisfaction that the opinion-bearer feels when he/she reflects on his/her past experiences.

That is a very good thing with some inconvenient consequences. One of those inconvenient consequences arises when legislators get involved. The American public has an oddly-trusting, yet often cynical relationship with their elected officials. Most people have very few good things to say about them. But when it comes to their own personal beliefs about society, getting their views enshrined into law become the highest priority. You can see this on both sides of the political spectrum. These institutions disgust us and we don’t trust them, but we want them on our side. It’s an odd paradox.

This paradox extends into the fields of education where, at least here in Michigan, the state government has made several plays that are tipping the scales in terms of which of the aforementioned groups is getting the state favor. As a result, “College-and-Career Readiness” is becoming cliche.

But our familiarity with it doesn’t mean that we have any idea what to do with it. Moreover, it doesn’t even mean that the general public has agreed upon definitions of “college ready” or “career ready.” The state has just solidified those two arenas as the goals.

It’s our job as educators to figure out how we are setting up our classrooms, schools, and districts to maximize our impact on young people toward those goals.

In my next post, I am going to lay out the primary issue that is creating this inner conflict I feel…

Reading.