Common Core: The Blessed Textbook Conflict

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We started working because the textbook stopped.

When we started the process to realigned our curriculum to the Common Core, we noticed that our textbook, which previously aligned to the Michigan Merit Curriculum, stood no chance against the Common Core State Standards. This forced us to make several decisions: First, were we going to replace the textbook with a new one? Second, were we going to keep the textbook and, in a sense, align the CCSS to the book?

After much discussion, we decided to do neither.

And it was the best decision we ever made.

It forced us to meet, research, collaborate, decide, create, experiment, reflect, analyze, adjust, and all sorts of other verbs that show up on the top of Bloom’s Taxonomy.

We are fighting through this year. We have reflected on the lesson’s we’ve learned. It hasn’t been easy. But our geometry team, which includes three teachers, has invested in a product that has resulted in some of the most intense and effective professional development that has forced us to have real conversations about student engagement, assessment, grading procedures, class structures, and all sorts of other goodies.

And none of it would have happened if we went with the textbook.

“Useless” Math Class – Misconception #1

In my previous post, I commented on how struck I was upon reading an embittered writer’s rant about having to take a “useless” math class. I mentioned four main misconceptions that we math teachers have allowed to take root in the modern academic mindset. I will now address misconception #1:

“Math is about numbers. Writing is about words. Wordy people write. Numbery people math.”

I want to start by saying that writing and math can both be treated as stand-alone topics. You can study math or you can study writing… but most of us don’t. We are typically doing either one of those things about something else. Perhaps my situation revolves around a car, but I may need to use math and writing to deal with this situation. At that point, math and writing are vehicles (pun definitely intended), but the car is still the focus of the situation.

Now, it has been said that math is a language of its own. (In fact, here is a book about it, if you want to read it.) I understand the point behind such a statement. However, in the end, math doesn’t have it’s own language. Sure there are mathematicians who can cross cultural barriers by writing everything in set notation, but among “non-mathy” folks,  mathematics requires a common tongue. In this way, math is just like anything else.  “Два яблока добавил еще два яблока в четыре яблока” is meaningless to anyone who doesn’t speak Russian and it wouldn’t matter if that was a math statement, a religious statement or a question about sale on ground beef.

Beyond that,  consider what kinds of mathematics “non-mathy” folks are talking about in everyday conversation. Nay-sayers are correct when they say that it probably isn’t going to be formal mathematics. But, it will be important. The cop gives directions to a lost citizen. The salesman explains a payment plan to a potential buyer. The marketers discuss the exposure rates of a type of advertising. The psychologist explains the results from a study to a therapy patient. The professional athlete figures out how to invest his or her signing bonus. (By the way, these types of communication fall into “mathematical literacy” as discussed by Jan De Lange in this paper.)

Unskilled wordsmiths tying to have those conversations are going to leave with more questions than answers. Those aren’t necessarily simple conversations to have. They require patience, dialogue and WORDS. Good words. Accurate words.

And where do people get a chance to discuss complex mathematical situations? In “useless” math classes… like the ones I teach!

When done properly, math class provides an opportunity for students to struggle and stumble over the wording of math situations. To listen to an explanation and respond with a targeted question. The students develop the use of specific vocabulary and learn when to use it.

So, why do wordy people have to take my “useless” math class? Because we need you. We need you folks who are good with words to help us figure out how to explain the math to the rest of the world. Sure, there are “numbery” people there to figure the tough mathematics out, but without the “wordy” people we are left with Sheldon Cooper to explain the math to the rest of us.

Common Core Geometry: An update one semester in

End of first semester provides a chance to reflect

Photo Credit: Flickr User “Neil T.” Used under Creative Commons.

This is our first go-’round with the Common Core Geometry. Without a usable textbook, our local geometry team has been responsible for most of the content. So, where are we?

We completed three units: Unit 1 was an introduction to rigid transformations. Unit 2 used rigid transformations to develop the idea of congruence, specifically congruence of triangles. Unit 3 began to formalize the notion of proof by using angles pairs and rigid transformations to discuss parallel lines cut by transversals, isosceles and equilateral triangles and parallelograms. (We should have finished one more unit, but that always seems to be the case…)

So, how did the first semester go? Well, the algebra-writing relationship is an interesting one. In previous years, our first unit was dedicated to writing and solving equations based on geometric situations. To see if the students thought that those two angles are congruent, we would put an algebraic expression in each angle and see what the students did with it. It turned largely into Algebra 1.5 with the mysterious “proof” added in for good measure. It didn’t seem to make much sense.

We have moved away from this for two reasons:

First, proofs are about expressing relationships in writing. It seemed silly that students would learn most of the geometry concepts through an algebraic lens and then we had to change gears to try to develop proof. We thought to start with the writing to smooth out the transition. Then, we could add the algebra back in later. The students have been through three straight algebra-based math courses leading up to geometry, so that will come back much more easily.

Second, we thought that another “same-ol’-math-class” might be what was sapping the enthusiasm and engagement. So, we have gone heavy on writing, creation, and visual transformations. The students have responded well. We’ll see what happens when the algebra makes a comeback in the second semester.

But, improvement is definitely needed. We weren’t prepared for the shift in the needs of our students in a non-algebra class. Our students aren’t learning with a great deal of depth. They are having a tough time writing with technical vocabulary. They are having a hard time making connections across topics. These are problems we are going to have to address moving forward. We were algebra teachers. I know dozens little tricks for solving all sorts of different equations. I don’t know a single one to help a student remember the different properties of a parallelogram.

I think we have done a nice job creating activities that will engage the students, but now we need to make sure the content depth is appropriate as well. It might be as easy as changing the questions that I ask. I can think right now of an activity regarding parallelograms as one that I didn’t draw out nearly enough depth from them. I didn’t facilitate the student thought and discussion. It slipped back into teacher-led note-taking. They struggle with the parallelogram proofs on that test. No surprise.

Please. If you have ideas, resources, processes, thoughts, lessons, handouts, anecdotes, or other helpful offerings, I will be accepting them starting now. Load the comment section up and be prepared for follow-up questions.

 

Oh, and if you are interested in reading a semester’s worth of my previous reflections on our new common core geometry course, here they are:

From Jan 7 – Vocabulary: The Common Core Geometry’s First Real Hiccup

From Dec 12 – Why you let students explore and discuss – an example

From Dec 7 – When measuring is okay…

From Nov 16 – Proof: The logical next step

From Nov 15 – When open-ended goes awesome…

From Nov 2 – Improvement under the common core…

Vocabulary: Common Core Geometry’s first real hiccup

So, my student’s last Geometry unit test force a realization. Computational math classes, which my students have all had up until this point, generally are not focused on vocabulary.

I’ll give you an example:

Suppose a typical Algebra I teacher asks his or her students to solve 3x + 7 = 28. How would they do it?

Well, for the most part, they will probably add seven to both sides of the equal sign. Then, after dividing both sides of the equal sign by three, the answer would be x = 7.

Before this year, I would have been content to accept that as a “full-credit-answer.” My students would have never been expected to know that the reasons that those steps are effective. Namely, the subtraction and division properties of equality.

However, on this most recent unit test for Common Core Geometry, each of the six questions required written explanations. Written explanations require vocabulary. Sometimes a lot of it. And it needs to be used properly.

The first real hiccup in our version of the Common Core Geometry is that this math teacher is not used to having to facilitate the deep understanding of technical vocabulary. I’ve taught skills, procedures, and technical reading, but I’ve never required my students to need to know the vocab as well as they do now. Previous geometry courses that I’ve taught have still been so algebraic that if the students didn’t learn the vocab, they could still get by doing the crazy amount of algebra.

This year, the vocab has taken center stage… and the students aren’t the only ones needing to adjust.

Please add to the comments any tips or pointers you might have in helping develop deep and flexible understandings of math vocabulary.

 

What 2012 Has Taught Me #1 – Public Opinion

I teach in Michigan where “right-to-work” is brand new. It hasn’t even taken effect yet, that’s how new it is. Before that, the public education sector has undergone many changes to the retirement system, laws capping benefits, curriculum overhauls complete with new standardized tests, etc. The transitions are swift and seem to be ongoing.

The opinions on this are as diverse as they are extreme. From the left, it’s called “attack.” From the right, it’s called “reform.” Whatever you call it, it seems like everyone and their mother has an idea of how I should do my job, what my students ought to look like when I’m done, and how I should get compensated for the time I take to do it.

It also seems like an awful lot of educators are getting very caught up on all that attention. The fact that so much is being made of what is wrong with public education is distracting to many. At times, it is distracting to me. But 2012 has taught me this:

The job of a teacher is not to sway public opinion. In fact, it is time we stop being surprised how the public views us. It isn’t new. It isn’t even a problem just here in America. Shoot… scrutinizing society’s educators isn’t even new in this millennium.

Sometime in the 6th decade after the birth of Christ, a Jerusalem man who came to be called “James the Just” wrote a treatise after the stoning of his friend, a Christian deacon named Stephen. In his treatise, James the Just writes:

My brethren, let not many of you become teachers, knowing that we shall receive a stricter judgment.

Translation: Be cautious in becoming a teacher. You will be held to a higher standard. You will be scrutinized for more of your actions. If you can’t handle that, teaching isn’t your calling.

(By the way, that quote was taken from the opening of chapter 3 from The New Testament Book of James. You can read about his friend Stephen in the 7th and 8th chapters of the New Testament Book of Acts, if you’d like.)

I can remember during my undergraduate studies in the college of education being told about the nobility of teaching, the challenge of teaching, the skill of teaching, and so forth. No one was preparing the students for scrutiny of teaching. A scrutiny that isn’t new. A scrutiny that shouldn’t be surprising. This is the way of it.

I do not believe that people simply do not like teachers. But we have all been taught. We all have experience in this field. Everyone can tell you with the utmost conviction a favorite teacher and a teacher who wasn’t fit to park in the parking lot. In reality, the favorite teacher might have been a dreadful educator and the hated teacher may have been very skilled and successful.

When the entire society has the means to form an opinion, the scrutiny will be intense.

2012 has taught me that there is no sense fretting over it. The more I fret over it, the more energy I am devoting to worry that won’t make me better at my work. Because if I am trying to stop something that was being commented on matter-of-factly almost two millennia ago, it doesn’t seem like I’m going to have much luck in changing it.

The Boiling Water Problem – Assessment on Functions

Boiling Water – 2 Cups – Full from Andrew Shauver on Vimeo.

 

Our transition to the common core has forced me to reconsider how I assess what student know and how much they understand.

A fairly standard content expectation from the Michigan Merit Curriculum might read, “Students will be able to identify what family a function belongs to and can analyze transformations to the parent function that will yield…” you get the idea.

A lot of questions come up in my mind when I think about assessing that standard. The first is how am I going to assess it? The second one is how can we make that knowledge useful to the students. The answer to the second question is the input, if the output is to be meaningful learning. The answer to the first question is the basis for how I design the situation that leads to the learning.

So, I submit for internet scrutiny the video above. I asked the following questions.

What is a possible independent variable? What would be the dependent variable? Do your choices create a function? Why or why not? If so, what family does your function best fit in? What would the parent function be? What transformation happened?

I found that there are ALL SORTS of misconceptions about variables, functions, parent functions, transformations.

I also found that the students are pretty good at crafting an argument using a formal math definition as support. They wrote pretty coherently as a group, although some were resistant to write with depth. That is nothing new, I’m afraid.

Ordinarily this type of thing would be an end of the unit project. I’m trying to use it as an exploration that is instructive and provides a chance for collaboration, flexibility, and individuality. Plus the fact that it serves as a formative assessment doesn’t hurt.

 

Feel free to load the comments with suggestions, constructive critiques, alternatives, and other ideas.

 

Improvement under the common core

Problem from the board

An introduction to mathematical proof

There are a lot of reasons that I am happy with my school’s transition to common core (I say that as if we had a say in the matter. We didn’t. But we have chose to make the best of it, at least in Geometry.)

In previous years, we have stuck to a textbook that was aligned to the Michigan Merit Curriculum. We used the book as the backbone and central nervous system of our class. Vocabulary, notation, structures, and other things were decided by what the book used. The book wasn’t bad.

The Common Core doesn’t align at all. It forced us to scrap the book.

It may have been the best decision we could have made.

Our recent liberation from the textbook has been a huge benefit to our students. “The proof” is a hallmark of most geometry curricula. Most text books push the structure of “two column proof, ” (you can see an example here if you aren’t familiar with two-column proofs).

The biggest problem with two-column proofs is the lack of engagement. They seem unapproachable, unnatural to students. The students aren’t sure how to work with them and because of their rather unique look (can you think of any other form of writing that looks like that?) it is tough to connect the two-column structure to any of their previous experiences.

But what is the structure doing? It is attempting to mimic a deductive reasoning structure. But, see, students are familiar with deductive reasoning.   Most students who try to convince me that I shouldn’t make them take a test because of an absence the previous day will use a deductive argument to do it. (“Mr. Shauver the school rules say that I get a day for every day I missed. Today should be that day!”)

So, the problem with students not being able to “do a proof” could be more of a problem with the two-column structure than with their ability to support a mathematical statement.

Today, I gave them the problem at the top of this post. I gave them an image with four facts. Their task was simply to explain why the two triangles were congruent, but to write two different explanations. The first used rigid motions to support their argument and the second used the definition of congruent triangles. I told them that they could write “paragraph-style.” I have some who choose bullet points, but not many.

Engagement was over 90% across the board. Students weren’t all confident in their answers, but they all had answers. They had all written a few sentences for each. And it didn’t take a lot of wrangling to get those sentences. In previous years, I might have had half the class try and the rest practice their avoidance behavior, claiming they didn’t know how, “collaborating” with neighbors, asking to use the restroom.

Here’s to hoping those days are gone.