# 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.

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.

This is the third in a series of posts where I ask you for help understanding the idea of “college-readiness”.

The first post examines “college-readiness” as one of several competing ideas about the goals of a public school education.

The second post looks at math through the lens of reading and how some pretty influential people seem to support the idea of math and reading being combined for assessment purposes.

Math and reading both have the ability to act as gatekeepers. The ability to read is a classical form of empowerment. It was, for centuries, the primary way that powerful people controlled less powerful groups. In the last century a variety of different folks have suggested that math beyond basic computation is creating unnecessary barriers. (W.H. Kilpatrick was writing about this after WWIAndrew Hacker and Steve Perry are championing this cause in modern times.)

So, combing math and reading together could be troublesome. Kids are having to get through two gates with separate keepers in order to attain this “college-readiness” label that is so important.

And unfortunately, at least some data sets cast some doubt on whether or not our educational structures are properly preparing students to enter high school ready to take on the challenges of a system that is going to require to both read at a high level…

… and to perform mathematically at a high level.

So, we have reason to believe that the majority of our students are entering high school not proficient in math or reading… possibly both. (I am fully aware that this is just one measure… and one that isn’t universal beloved either.)

Let’s recap:

The College Board is writing college-readiness math tests that are written at about an 8th grade reading level to complement their college readiness reading and writing exams. And we are going to give this test to 11th grade students who have a better than 50% chance of entering high school below grade level in either reading or math.

Sounds like those of us in the education profession have a tricky task ahead of us.

I don’t want to make it sound like I’m the first person exploring this problem. Thanks to the work of those who are championing Universal Design for Learning (UDL) we are starting to discover that creating structures, supports, and access points for those who struggle either because of disabilities or lower prerequisite skills improves the experience for everyone.

The idea is that we can flex on the methods, the means, and the tools in order to keep the achievement expectations high. In math, this often means sticking to the learning objective without unnecessarily complicating the task by including areas of potential struggle. For example, students struggle with fractions. When you are introducing linear functions (a topic that doesn’t immediately depend on fractions), don’t use any fractions in your activities, materials, or assessments. In this way, you don’t create an artificial barrier to the learning of new content.

This thought process also applied quite readily to reading. If your students don’t read well, then why make them do it when we are trying to learn math? Dan Meyer suggests that there are student engagement points to be won by reducing, what he terms, “the literacy demand.” Rose and Dalton from The National Center on UDL discuss the variety of benefits of creating opportunities for our students to listen instead of reading. (One of which is creating better readers.)

And all of this makes so much sense…

… until our resident decision-makers decide that reading comprehension (in the traditional sense) is an essential component of a “college-ready” math student.

This puts math teachers between a rock and a hard place. On one hand, we can follow Dr. Meyer’s sage advice to reduce the literacy demand on our students. I’ve done this. He’s darn right about what it does for engagement.

But on the other hand, if the College Board knows what they are talking about (and I’d really like to think they do), do we risk eliminating an essential element from our courses if we work hard to limit the reading?

Perhaps this isn’t as tough a spot as we originally thought. Simply put, the work of the math teacher is complete when the student has developed the ability to solve a targeted set of math problems. This requires helping the students learn certain tools: Equation-solving, data collection and representation, strategic guessing and estimation… These (among others) are all essential problem-solving skills.

Most math teachers have mechanisms in place to support students who are in a variety of developmental levels on the  journey toward proficiency of any of those skills. They aren’t uncomfortable with a student who struggles to solve equations. We see it all the time. We know it’s our job to help support that student. So we do.

What if we looked at reading the same way? What is reading if not an essential mathematical problem-solving skill? A skill that our students are in a variety of different places on?

As math teachers, perhaps it is our job to teach reading.

In my next post, I’ll lay out what it might look like for a math class. I don’t mean a math class with a high literacy component. I mean a math class where the teacher and students all recognize that role of the math teacher is to help the students develop as math readers.

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…

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.

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…

# A word about teaching to the test…

It’s testing season here in Michigan, the dawn of a new age! Last fall’s paper and pencil MEAP is replaced with the this spring’s computer-driven M-Step.

And the M-Step is putting a lot of pressure on districts to reconsider what “test readiness” even means. What skills are required of a student who is having to transition from a question like this:

to something like this?

The “Performance Task” is another new type of test item. Teachers lead a scripted discussion through a content exploration. Students get time to discuss, explore, ask questions. Then they go off to answer a question based on the experience they just had. This all combines to create a startling amount of new experiences all in the same year.

Preparing educators for this newness was a large chunk of my work as my colleagues and I provided workshops to help explore this testing experience that is new for all of our students. As I had discussions with genuinely and appropriately concerned teachers and administrators, there were a few recurring concerns that bubbled to the surface.

1. What if a student knows the right answer, but has computer/technical skills that prohibit him/her from answering that question correctly (or in a reasonable amount of time)?

2. How are the non-traditional items going to be graded? (Non-traditional… rubric scored items, items with more than one correct answer, the drag-n-drop question like the number line one above, etc.)

Certainly there were lots of other concerns, but these two kept coming back again and again with good reason. We want to make sure that our students are getting recognized for the knowledge and skill they possess. This new test presents a new set of potential barriers.

So, how do you get around those barriers? Well, one way is to teach to the test.

“Teaching to the test” as a phrase is usually not seen as a good thing. Although, I can remember a conversation with a district leader a while back who was quite proud of the fact that curriculum decisions for their math department were based almost solely on what was on the ACT… he refrained from using the term “teaching-to-the-test”.

This pride fits in with recent statement from a colleague. “Teaching to the test is only a bad thing if the test is no good.”

Now, that having been said, there are a lot of people right now in Michigan who are feeling like the M-Step is no good and they might be right. As I write this, we’ve just completed our second day of testing, so I’m going to reserve judgement until we’re more than 48 hours into this new experience.

But regardless, that sample question from above…

… isn’t a ridiculous question. It lends itself to considering if teaching strategically in such a way that students could do that (which is motivated primarily by the fact that a high-stakes, state-level standardized summative assessment is going to ask them to do it) is a bad thing? I would suggest that this is a task that most 8th grade math teachers would want their students to be able to do. Where the conflict arises is when teachers or districts aren’t inclined (or able) to provide exploration or assessment of this skill in a computerized way. The test becomes a motivation to have to change course. So we find ourselves faced with “teaching to the test.”

So, if we’re going to be “teaching to the test” anyway, what situations can we design for our students that while we are probably only doing ten because they will be on the test, are still, in the end, high-quality experiences for the students to have?

One suggestion comes out of this blog post from Fawn Nguyen who is trying to help the students interpret a scoring rubric. That was one of her goals, because her second goal was to help the students focus on a set of skills they will need to be successful on the test they are going to take in May.

The skills she’s referring to: “For them to attend to the same thoroughness and precision in their own solution writing when it’s their turn…”

While Ms. Nguyen is openly “teaching to the test”, that skill she’s highlighting is a valuable one. I encourage you to read the post. The tone of it doesn’t sound like someone who begrudgingly throws test prep items at her students, but rather a teacher who saw an opportunity to weave test prep into an experience that ultimately led to student feedback like this:

I believe this was helpful because when I take the test, I will be more aware of the questions and what is expected of me. I will make sure to always back up my answers with evidence.

I’ve talked to a lot of teachers (math, ELA, social studies) who’ve expressed frustration that their students won’t “always back up their answers with evidence”. Ms. Nguyen provides us an example of how a test prep experience can be used to further broader goals.

And I think, in the end, given the stress and strain that these testing situations put on all of us, it’s nice to see an example that reminds us that teachers still have ability to make decisions that end up as net positives for our students.

# Should we test them? or not test them? (Hint: Those are the wrong questions…)

This time of year in Michigan, standardized testing is on everyone’s mind. This year is a little more frantic than others, I’m afraid. Michigan’s Department of Education is piloting the online version of a new state test. In my role, I’ve spent a lot of time contributing to support all over the state trying to help concerned educators prepare themselves and their students for this transition.

Besides that, every Michigan HS Junior is given a chance to take a college entrance exam. Since the inception of this practice, this exam had always been the ACT. Then, just after the new year, news broke (somewhat out of nowhere) that Michigan was going to switch to giving the SAT.

Now, each of these tests has a whole machine surrounding it. There are practice tests, prep sessions, special activities put in place to prepare for both the look and feel of the testing items as well as for the content. In many ways, “teaching-to-the-test” has become a foregone conclusion. This is especially true in districts who are working very, very hard to improve the test scores for evaluative or punitive reasons. (More on this later.)

All of this fervor can really make a guy wonder whether or not this testing is worth all the hassle. Are we doing good things to education by instituting all of this testing? Are frequent assessments the right way to go?

And I think those are the wrong questions.

Consider a person who is hyper-interested in his or her weight. Like dangerously so. This person steps on the scale several times a day. And makes aggressive changes for the sake of gaining or losing 10 or 20 lbs very, very quickly. On what can we blame this problem?

Lots of things, I suspect. This person may have a background with some experiences that need to be reconciled. This person may have anxiety issues that need to be resolved. This person may be in a personal or professional relationship that puts unrealistic pressure on his or her appearance. This person may spend too much time focusing on and associating with other people who have similar habits. There is a culture that he or she has become a part of that feeds into this point-of-view.

The scale that he or she steps isn’t the problem (provided the scale is accurate). The scale is the tool that provides the information. The information is then getting abuse by the recipient. The solution to this problem is NOT to dispose of the scale. The abusive recipient is still going to seek for information and they will find it in perceived tightness of clothes, or calories consumed or duration of a workout.

What we need to adjust is the response to the information.

Right now, we are asking our schools to step on the scale too frequently, and making too much of the number we get.

Look, changes in policy and curricula often show effects slowly. And we should look for slow, sustained improvement. This is evidence of a cultural change that is becoming a new standard. Just like the dieter who is looking to lose 50 pounds. A two-year journey losing a three or four pounds per month doesn’t make for an exciting, story. It doens’t get you on NBC Shows, but it’s healthy. It’s sustainable. It is evidence that a real change has taken place.

It is with this in mind that I will assert that the tests aren’t the problem with our educational testing culture. While I suspect that we are testing too much. If the person from the analogy has a scale in every room, then it might be useful to reduce that number to one, but it would be wrong to blame the tests. They are giving us information. And we in the educational community aren’t handling the influx of information very well.