Grr. Really. I have some frustration I need to get off my chest, so this will be a “constructive venting” post. Constructive, because I really do have a point, and venting because I can’t believe the things that occur in math education.

First off, I have been really interested in how the curriculum of mathematics is designed and how that curriculum actually works against the learning of math as it is presented in our current textbooks. Honestly, I am really disappointed in textbooks right now. I have spent some time examining the vocabulary in different sections of our current books, and I realized that the books set up the topics of mathematics as if each chapter, each topic, is a disjoint Venn diagram. Chapter 2 has no relation to Chapter 3, and that is how math is done.

No wonder learners struggle with math and don’t make connections. We have DESIGNED math to be taught that way. Don’t agree with me? Just look at your textbooks and tell me if the linear vocabulary is identical to the quadratic vocabulary. Then ask yourself why not? 80% of the vocab should be IDENTICAL.

Let me show you what I mean. This is the current state of affairs:

 Linear Equations Quadratic Equations Standard Form Ax + By = C y = ax2 + bx + c Equation forms y – y1 = m(x – x1) y = a(x – h)2 + k y = mx + b y = (x – a)(x – b) m = (y2 – y1)/(x2 – x1) Vocabulary Rate of change, y- intercept, slope, rise over run, x – intercept x-intercepts, solutions, roots, axis of symmetry, reflection, translations, y – intercept

Hmmm, do you see ANYTHING that overlaps there? The only thing that is the same is the fact that the x-intercept and y-intercept are both used. However, in the quadratic unit, the y-intercept is rarely used, and the x-intercepts are normally called roots or solutions. Learners = 0, Notation = 1, point goes to the book.

Seriously, even look at the idea of the “Standard Form”. Why is it “Standard”? Because that is how we would normally want to write the equation. Lines get some weird, special form that relates to nothing else in polynomials, while quadratics and all other polynomials get something that makes sense and flows from one type to the next. Learners = 0, Complicated Notation = 2; the book is clearly leading.

Oh, that Point-Slope form looks pretty promising, though. It is very similar to the Vertex form of the quadratic. Well, kind of similar. It is the only form in the family of polynomials that uses subscripts, but that is okay, subscripts are super easy to learn. Oh, they aren’t? Subscripts confuse the heck out of learners? You mean there is a difference between a number in front of a variable, a number above and to the right of a variable, and a number below and to the right of a variable? Darn. And if the Point-Slope is supposed to come from the definition of slope, where did the subscripts of 2 go? Why ? I don’t get it. Why did things change? Learners = 0; Confusing notation = 3. Clearly the learners are getting the worst end of the deal here.

How can we make this better? Well first off, we need to realize that there is only ONE type of math we are doing; Polynomials. Start there, and build a set of math from scratch.

Damn. That sucks. That is a lot of work. That is akin to what Exeter has done or what Milton Academy has done. [Terrific article on Milton’s efforts here by the way.] And there in come the rub. Here we have the idea, but it takes a team of willing people who think alike to create. It takes time, energy, money, and effort. And then, it takes only one or two teachers to block the whole thing from implementation in a school or district.

Frick.

Double Frick.

Here comes the venting. I need to move beyond this so I will get something done.

I didn’t even realize this bothered me so much until I started thinking about why I was so paralyzed for the last two weeks. Very hostile. Very frustrated and angry, and then I started asking myself why. I think I have it figured out. I was at a department lead meeting two weeks ago, and the leader of the meeting presented my idea for lines. (okay, not mine, but Exeter’s). And the department leaders from other schools at the meeting said, “Why?” “Why would we do that?” “Lines are y=mx+b”. “That is stupid, the textbooks don’t teach it this way.” etc.

I. Kept. My. Mouth. Shut. It took heroic efforts, but I did not rip into anyone. I was good. I mean, they didn’t come up with any good arguments why it doesn’t work, shouldn’t work, isn’t a good idea. They just said, “ThatisnotthewayIwastaughtthereforeitiswrong <breath> thebookteachesitonewayandwecannot/shouldnotchange.”

THEN! OMFG. The pain wasn’t over yet. I was speaking to our assessment director after the meeting, and she was just as frustrated as I was, if not more. She told me a teacher actually contacted her and told her she was doing a bad job because she left the section of the book that taught lines as y=mx+b out of the curriculum guide. Why was that bad, you ask? Because that is the ONLY way that teacher teaches lines. That teacher won’t teach it any other way because it is the easiest way.

Frick.

Really? Sigh. You know that ONE teacher will end up on the next textbook committee, and that ONE teacher will end up railroading 4 other great teachers into buying a shitty textbook simply to shut that ONE teacher up. How do we fix that? How can we?

And that is my frustration? What can I do about that? It has eaten at me these last 2 weeks, and it shouldn’t. It won’t now that I have written about it. I have gotten it off my chest, and that helps.

How do I respond?

I push forward with my Exeter project. I write a paper and submit it to Mathematics Teacher. I stay focused on what I need to do to accomplish my goals.

And remember:

“Don’t say you don’t have enough time. You have exactly the same number of hours per day that were given to Helen Keller, Pasteur, Michelangelo, Mother Teresa, Leonardo da Vinci, Thomas Jefferson, and Albert Einstein.” – Life’s Little Instruction Book, compiled by H. Jackson Brown, Jr.

I have the time. I just need to use it better.

Where to start.

I have been thinking and working with the Exeter materials quite a bit in the last 3 months. I have come to see the value in the methods and the questions, and the way the questions cycle from lower levels to higher levels.

But I have to say I don’t see the Exeter curriculum as a magic bullet. It isn’t. There is no such thing as a magic bullet for math education. There is a lot of hard work. There are a lot of relationships to build with learners. There are many hours to put into lessons that engage learners to think deeper about the mathematical issues.

The Exeter Curriculum is a part of this process, not the end of this process. It is not something that will solve any problems. It is however, something that will help me, as a math teacher trying to improve my classroom, to engage learners, to develop deeper thinking, and to push the high standards of the Common Core into classrooms.

I am not confident of the efforts offered by the textbook publishers. Here are two examples of why:

If the CCSS is going to actually impact the classroom in a positive manner, we can’t take the same ol’ same ol’ materials and just slap on a new label. We need to structurally change and improve what we are doing.

That is where the Exeter Curriculum can come into play and help, and it creates the next problem I, as a public school teacher have. And this goes back to the first post I made, Exeter we have a problem. I had flashbacks of Apollo 13 as I wrote it because it is relevant. As the quote goes, “Houston, we have a problem” and the problem was absolutely centered in that little capsule. The experts who developed the program were on the ground and could go home safe and sound at the end of the day, but those astronauts needed to step out of their comfort zone and do something above and beyond.

As a public school teacher, I am in the same capsule. Our comfort zone has been stripped away and completely new standards pushed on us. We need to step up, or step out. It really does come down to that. The old guard who doesn’t want to change will be forced out through the new “evaluation” procedures that also have been forced down our throats by people who have no clue about education.

Okay, so the stage is set. Nothing I wrote above will change. Stop complaining.  What the heck am I going to do about it.

The Plan (or WCWDWT):

As part of our evaluation process I had to create a Professional Growth Plan. The plan I proposed and was approved was to take the Math 1 Exeter Curriculum and align it with the Common Core State Standards as well as simultaneously give the problems keywords and strands.

In addition, I have spoken with the two very nice and enthusiastic gentlemen from OpusMath.com who have the technical background to take the entire project, upload it to their website, and host the problem sets, alignment, stranding, keywords, AND make it all searchable, selectable and downloadable for FREE (and that is free as in air).

What Can We Do With This? We can create a database of problems that are rich. We can create a database of problems aligned to the CCSS that are searchable, selectable and downloadable for use in the classroom by math teachers around the world.

What can we do with it then? That hasn’t been explored. We have to create the foundation before we can build the building. I have spoken with someone at Exeter and they are interested in the project. Of course, they can not help much. It isn’t their burden to take on, it is ours (and now mine!).

I have another teacher at my school who has agreed to take on this with me. She is absolutely crazy to do so, which means I am completely insane.

One short lesson that came from the Exeter sessions was how they teach lines and what formats they use to teach writing equations of lines. It is amazing how often we, as math teachers, fail to build connections between different elements of mathematics simply because we feel like we have to hold to some form of “history” or “tradition” that is involved in math.

As an example, let’s examine the old standby to writing equations of lines, y = mx + b. Of course, it is the math teachers go to. Here in Washoe County several years ago, we spent a huge amount of time (and money) learning how to teach lines by using y = mx + b through the “Algebraic Thinking” company. We had foldables for learning y = mx + b, we had lessons using it, we had just about everything we needed to teach y = mx + b except the most important thing; learners who understood why we were memorizing such an arcane idea, why did we use m, why did we use b, and why should we or they care?

After all, look at one lesson from Algebraic Thinking. It uses “formula” 4 times, “Hopefully your students have not forgotten” once, “Your students are going to have to be meticulous, careful, and exact on when simplifying today” is an actual statement in the lesson, and they actually say in their handout:

There is a point-slope formula, but the last thing most students need is another formula to remember.  It’s a good idea to just stick to y = mx + b.  They have to know it anyway, so they can use it for everything.

Every calculus teacher reading this just threw up a little in their mouth. Right, let’s not use the easier formula. Let’s go for the formula that takes the MOST amount of effort, and ram that down their throats through memorization. Glencoe, McDougal-Littel, and all the other textbooks do the same thing.

Let’s examine some ideas based on this y = mx + b from a pedagogical point of view, so let’s start with a problem. “Write the equation of a line with slope 3 and goes through the point (–2, 5).” Straightforward enough, right?

1. First off, the learner has to memorize that the slope, 3, corresponds to the “m” in the equation y = mx+b. Why is it “m”? No one really knows. Seriously. Follow that link. There are several explanations there, and none of them are definitive.
2. So the learner now substitutes the 3 in for the arbitrary “m” and gets y = 3x + b. Great.
3. Next the learner has to substitute in the x and y co-ordinate of the (-2, 5) to get 5 = 3(-2) + b.
4. Now some algebra, multiply and add to get 11 = b. (notice there are 2 steps there)
5. Next the learner must substitute ONLY the “b” value back into the original step 2, but not the values of the (-2, 5) that they just substituted. Why? because the foldable says, just do it. (not really the answer I give in class. An equation has to have a “y” an “=” and an “x”, so we can’t substitute those again.)
6. Then write the final equation, y = 3x + 11.

Summarizing, that is 7 steps, 2 of which are substitution, and one hidden step that is a NON-substitution that confuse the heck out of learners. All of this was done because we don’t have a really good reason to use y = mx + b except the textbook publishers have asked us to.

Also notice, THIS WILL NEVER COME UP AGAIN IN ALGEBRA 1 OR 2! At no point will a learner use this method for quadratics, absolute values, cubics, or anything else. It is an island unto itself that is only applicable in linear equations.

If we then ask the next question, “Take the equation written above, and write the equation of the line perpendicular that goes through (6, –2) and then also parallel to the original through (3, 7).” You have a learner using y = mx + b for about 5 minutes substituting and resubstituting over and over again. 5 minutes that is, if they are really good at it. Most learners give up because it is difficult and confusing.

I want to shake this up and look at how an Exeter teacher would ask their learners to write the same exact series of equations.

Write the equation of a line with slope 3 and goes through the point (–2, 5), then write the equation of the line perpendicular that goes through (6, –2) and then also parallel to the original through (3, 7).

This should take us about 45 seconds.

Why the difference? Because we are not going to start from y = mx + b.

Let’s start from the vertex form; y = a(x-h) + k.*

Go.

The first equation is y = 3(x–2) + 5 so therefore: y = 3(x+2) + 5

The perpendicular is y = -1/3(x-6) + –2 so therefore: y = –1/3(x-6) – 2

The parallel is y = 3(x-3) + 7

Okay, I exaggerated. It took less than 45 seconds. Notice that the learned and memorized information is the same for both situations (the y=mx+b and the vertex form); perpendicular slope is the opposite reciprocal, and parallel slope is identical.

But look how little effort goes into actually writing the equations. It is ONE substitution from the vertex form to the equation.

Want to graph it? Sure, start at (-2, 5) and go up 3, over 1. Okay, if you really want, I will let you distribute and add. 2 steps, piece of cake.

Wait, let’s look again at that vertex form of the line. y = a(x-h) + k. It looks surprisingly familiar. It looks like the vertex form of the quadratic, or the absolute value. Let’s line them up and see:

• y = a(x – h) + k  : Linear vertex form
• y = a(x – h)2 + k  : Quadratic vertex form
• y = a|x – h| + k  : Absolute Value vertex form

I could go on, but I think the point is made. Using this form to write the equation of a line is faster, less to remember, makes connections to other units in math, and overall allows a learner to understand what they are doing instead of memorizing steps.

There are still some things that have to be memorized, for instance: What does the “a” do in front of the parenthesis? It makes it steeper. What if it is negative? it flips it over the x axis. What does the h do? What does the k do?

But notice, the answers to those questions are the same in the linear unit as in the quadratic unit as in the absolute value unit, or cubics, or … you get the point.

In the end, I have to ask the question, Why do we torture our learners with y = mx + b. It is arbitrary, and doesn’t make sense. It is hard and requires far more effort, and it is stupid.

Let’s start teaching math better. Throw away the textbooks. Seriously. Take the Algebra 1 textbooks and cut out the problems, throw everything else away. Now reorganize the problems so they make sense and build off one another.

Great, now photocopy them, scan them in, and post online for everyone to use.

That is essentially what Exeter has done. Why don’t we just start using their materials instead of buying the textbooks to begin with? The textbooks suck. We know it. Start acting on that knowledge.

———————————-

*Why the vertex form and not the point slope form? I ask what is special about y – y1 = m(x – x1) the answer is nothing. Notice the subscripts? They confuse the heck out of learners. Where does this come from? The definition of slope. So why use the arbitrary “m”? Why not call it slope and therefore is “s”? There isn’t a good answer, but using “a” for it at least makes sense given the other vertex forms.

After all, if we are going to be arbitrary, at least let us be arbitrary consistently.

Okay, all along I was promising a massive file upload for all the readers who want the Exeter materials. I will explain what each group of files are for as I go.

All files are in WORD or PDF format, and all are in a zipped folder. Downloading and unzipping the folders will speed up your access tremendously. All in all there are 44 megs of files here. That does not sound large, but word files and pdfs are incredibly small these days!

[The placement tests were posted on Exeter’s website, but they didn’t realize they were made public. Links to that page have been removed. If you are a teacher and would like the files, let me know.]

The progression at Exeter begins with a Placement test to determine what course the learner should be enrolled. These are released Placement tests from Exeter:

Released Placement tests
After being placed in the correct course, the learners then start in on the problem sets. I have 2 years archived, but I would love more if someone has them.

Problem Sets 2011-12

Problem Sets 2012-13
The current year is 2012-13, so the archive is an August download of the new materials, including the change logs. If someone has the change logs for the 2011-12 or the files for previous years, I would add them also. The live location for the current year’s materials can be found on their site.

I do have solutions to the 2009-10 problem sets (I was given these without the actual problem sets) and solutions to the 2011-12 problem sets. Will I post them? No. I know I would not be happy if a teacher posted solutions to all the problem sets I created. That is the one thing I won’t post.

During the class the learners are in, they will do hands on activities, and use Geometers Sketchpad to explore math. The Instructor of the Exeter sessions I attended was nice enough to share these. They are all written by Exeter teachers, so no poaching and claiming them for yourself. Please attribute them accordingly.

Hands On Activities 2011
(Both Word and PDF documents!)

GSP Document and Sketches
If you are looking at these thinking, “Dang! That is a lot of material to go through!” You are absolutely right. The 2 docs for Alg 1 are 59 pages combined, the Geo doc is 62 pages, and the Alg 2 Hands on is 69 pages. Right there are enough docs to keep a person busy in class for a long time, and you would be learning terrific math as you go. In the GSP Document and Sketches folder, there is a document called “2011 gsp.doc” It is 101 pages of GSP constructions.

So the learners are working problem sets, they are working activities and extending their learning beyond the problems and being active with the math. Now it comes time for some assessments.

Math 1 tests
Math 2 tests
Math 3 tests
These are all in word format, so you can edit and use them in your classroom if you like. These tests give you some idea of how Exeter assesses their learners. Something you should know is that every one of these assessments are open notes. Every problem set they have worked is available to them on the exam.

Finally, the year is over, the faculty get  together and evaluate the problem sets. What worked, what didn’t, what can be improved. And the writing committee collects all those comments and distills them down into a commentary on the problem sets for the rest of the staff. Then the rewrites happen, and the new problem sets are published, and the cycle starts all over again.

Commentary 2011-12  [if you would like the commentaries, and can demonstrate you are teacher, please email me or comment and  I can email them to you. The files have been removed at Exeter’s request.]

And there you have it. This is the cycle of development of the Exeter curriculum and materials. The vast majority of the work is done by the writing committee, compiling the commentaries and editing the problems. That is a huge task, and I would love to have a serious discussion with someone at Exeter just about that. Heck, I would spend a week with them just asking questions about the writing of the questions, let alone working and thinking about the problems themselves.

I hope this is of some help to other teachers out there.

During the summer, the teachers of the math classes at Exeter get together and review the problem sets and compile a series of documents they call the Commentaries. These Commentaries are then used by the Writing Committee to review, edit, and modify the problem sets to make them better for the next year.

First of let me just say, Wow. Exeter’s commitment to the constant improvement of their curriculum is amazing. Contrast that to the situation we have in the public schools. The district spends millions of dollars on a textbook from Pearson, McDougal, or Holt and then we are stuck with those textbooks until the next textbook adoption (every 7 years in NV, unless the budget delays it.)

In the meantime, we complain about the books because we know there are better ways to teach and better ways to work with the material, but we are bound by textbooks that are bound to disappoint. Today I told my department that it would not bother me if we threw all the Algebra 1 textbooks away. It shook them up and made them think a bit about why we teach the way we do. But I digress.

Let’s take a look at a couple of problems and see how the process develops new questions, or if not new questions, new understandings of the questions.

M1:26:4 from the 2011-12 problem set

I chose this problem because it is a pretty standard type question, used in Algebra 1 to work with systems of equations. It has multiple questions underneath, and has the zinger question in part e that challenges the learner to figure out some answers without algebra.

The Commentary on the question is:

The Commentary for the question suggests some methods of solving, and points out the fact that the Algebra is the best way of solving the question. The Commentary does not mention part e, which has a lot of mathematical exploration involved. But the Writing Committee clearly felt something was going on in part e that was not successful, because the 2012-13 question is now:

Identical question, but the committee dropped the exploration question to focus on the mathematics and the generalization found in part d.

Here is another example where the question is really straightforward and does not change from one year to the next, but the commentary is terrific in guiding a discussion.

M1:2:5 (Both years)

Very basic question, but the commentary opens up a very different scenario with the material.

Wow, look at that. High school teachers I have known sometimes fall into the “why should we teach something so basic, that is a middle school standard and they should just show up knowing it, but I guess we can review it” trap. It is a trap, and it sucks you in and destroys you if you let it.

Here the Commentary shows that the trap opening, “It is surprising that some students have so much trouble…” but they don’t fall in. They point out to their teachers to look for the shy, the quiet learners and ask questions the quiet learners may not ask but desperately need. Very nice.

What the Commentary is clearly for is to show the TEACHER what traps are possible with the material and to develop better questions in the treatment of the material. Imagine a brand new teacher at Exeter with the problem sets getting the Commentary. They can work the materials easily, otherwise they would not be teaching math, but the Commentary is what allows the new teacher to develop the questions that need to be asked in class.

It is definitely a tough proposition to write the commentaries and get the information from 20+ teachers and simplify those comments down to a short paragraph. But very worth the time and effort to do so. If you are interested in the Exeter problem sets, I recommend you read the commentaries as well.

Below you will find the commentary folder for the 2011 – 12 Problem Sets and the 2011 – 12 sets. The Exeter website has only the updated 2012 – 13 sets with change log.

Commentaries for 2011-12 [link removed at Exeter’s request. If you can demonstrate you are a teacher and would like them, let me know.]

While we did many problems and activities at the Exeter training, we discussed how the Exeter instructor conducted his exams very little. We did have a short discussion one day on assessments, and he did give all of us some files with sample exams. He does not use these exams any more, so he felt comfortable giving them, and he gave permission for me to post them.

First off, here are the ground rules for all Exeter math exams. They are open book, which means, open notes. The learners have access to their entire work history that has been done to date in the course.

Part of the exam is to be done with no calculator, and part with calculator. In addition, the assessments are designed to fairly assess the learners thinking and problem solving skills.

First, I’ll look at some problems from the Problem Sets in Math 1, then I will compare them to the assessment questions that are similar.

M1:2:5

M1:3:2 & M1:3:5-7

Math1:4:9

And here is 1 – 4 from the first exam, no calculator allowed section.

(I didn’t select the algebra solving problems, just looking at the number line questions for these.)

I only went through page 4 on the Problem sets, and notice that question 4C is exactly the same as the problem M1:4:9. There is nothing really tricky here, just “can you take what you learned and find it in your notes (4c) or can you extend what you know from your notes to a newish situation (4a & b).”  Problem 1 on the exam matches well with the questions from the problem set as well.

So nothing really tricky, just an honest assessment of knowledge. Now granted, this is only the first exam, not an exam from later in the year where the material is tougher, but it does set a pedagogical pattern for the assessments.

In the first test of the year, there are 5 questions w/o calculators and 5 with. If we look at the 5 with calculator the pattern holds.

Notice the heavy reliance on the problem solving, and the methods of problem solving discussed previously.

The exam treats the problem solving very straight-forwardly and even suggests to the learner they SHOULD be using the table system. Then, in question 10, a little extra credit is dangled for taking the problem from guess and check to the generalized answer.

Nice.

All in all, I would say there is nothing ground breaking or earth shattering here. It looks like the Exeter teachers assess their learners like I assess my learner, and like many / most / all teachers are taught to create good assessments.

The lack of multiple choice is the real difference, but the reliance on MC exams is a crutch because of huge classes (30+) and the stupid demands of high stakes testing foisted on public schools. Doesn’t make it right, just makes it what the ‘reformers’ want.

Below you will find a zipped folder for download of some out of date exams used that used to be used at Exeter in Math 1.

This is a post that can not be written without a better understanding of how Exeter structures its school year. First off, they are on a 3 term schedule; 3 ten week (approx) terms per year. This is a FABULOUS schedule, and I can speak from experience. It is a similar schedule to what Knox College (where I attended college) uses .

The benefits of the 3 term schedule are many, but some of the best are the sense of urgency it places on the learner and the ability to be more flexible in scheduling. The sense of urgency is terrific. You have 10 weeks in your class, and you are counting down right from week 1; “9 weeks left, 8 weeks left … oh crap, finals are next week.” You are never allowed to kick back and think, “nah, I have time, I don’t need to worry about that project / assignment / test yet.”

The second benefit, flexibility, is the one thing that truly is what I want to discuss here. I asked our instructor how many new learners Exeter gets each year. I was thinking that they probably didn’t have many transfers in after the first year, but that was not true. The Freshman class starts around 250 each year. Then, each year, approximately 50 learners are added to the class until the Senior class graduates with around 400 learners. [Of course, these numbers are not set in stone, they were given as an example only.]

That means that Exeter has a huge problem each year. The Freshman class is coming in with a wide range of ability levels, and then every year after that they get more learners who should be at one level, but in reality may be way way below that level or even above that level. This then is their dilemma, how to place the learners in the correct course for their level of ability.

First off, they have a strong system of placement exams for the learners. Wow. Imagine that, a learner placed into the class they should be instead of the class that all Freshman or all Sophomores should be in. Of course, given Exeter’s commitment to sharing what works, they have put some old placement exams online for everyone to see and use. Sweet! [Turns out, they weren’t old placement exams, but an internal site that was made public on their end.]

Secondly, they teach all three of the Math1 and Math2 classes each term. If Math1 is broken into 3 pieces, which I will call 1a, 1b, and 1c, then during term 1 of the Freshman year, all three classes, Math1a, Math1b and Math1c AND Math2a, Math2b and Math2c are taught all 3 terms. Wow.

This means a learner might be placed into Math1c as a freshman during Term 1, not every learner is required to start with the first Math1a class. This is HUGE! Now you don’t have to worry about the bored learner sitting in class causing problems waiting for the material to catch up to them.

Okay, can this be used in the public schools? After all isn’t that the point of this? I am not sure.

This kind of system works for a private school (and Knox is a private school as well).  because the system is not set up to provide an education for farm laborers. You do know that is why we have long summers off, right? Because the kids in school were needed to work in the fields during the summer. That hasn’t been true for 30 years, but we still have that long summer break. Sue provided a link to the myth of the summer break below in the comments. It is worth reading and following up on.

It certainly could not be adopted because of the extremely underfunded nature of public schools. We are lucky to have 1 math teacher per 30 learners, while a private school can charge as much as they need to in order to cover the extra teachers to teach the same course every semester and hold the ratio down to 1:12.

So the short answer is no, we can’t do it like Exeter can. That is a cop out though. How can we do something similar?

How can we personalize the instruction so learners who are advanced can forge ahead?

How can we implement and use placement exams to tailor the instruction?

Kahn Academy is getting some traction on this, even though the videos contain mathematical errors and mistakes because it is a way to personalize instruction. Can we, as innovative, driven math teachers figure out a way to do it better?

I am willing to bet yes. Once we are willing to throw away the books and embrace the standards, then we can simply say, Johnny knows standard 1, 2, and 3, so an A on those, but he needs to work on standards 4, 5, and 6.

Next we need a way to assess well, and finally we need a way to locate those standards out all of the standards that are part of the problem sets.

Once we have these elements, then we can at least try to individualize.

In this post I want to show Exeter’s problem solving strategy. This is important, because it is SO different from how a problem like this is typically approached.

First off, the problem I am going to model is M1:21:11 [Math 1, page 21, problem 11]

11. Alex was hired to unpack and clean 576 very small items of glassware, at five cents per piece successfully unpacked. For every item broken during the process, however, Alex had to pay \$1.98. At the end of the job, Alex received \$22.71. How many items did Alex break?

In a typical Algebra 1 class we would try to get the learner to see the equation is:

.05(576-x) + 1.98x = 22.71

In fact we try to get the learner to jump directly to the equation from the problem by deconstructing the sentences, and then solve the equation. x = 3, by the way.

Now, let’s see how Exeter expects and demands that ALL of the modeling problems are handled.

First off, we will be making a table. The headings in this table are mandatory and can not be short cut. The learners must label the table thoroughly so that it makes sense. Remember, this is the same problem as above. I am going to paste in my table all filled out, and then explain the essential elements.

 Guess: # of broken bottles # of unbroken \$ Paid for unbroken \$ subtracted for broken Amount paid Goal Check 0 576-0=576 .05(576-0)=28.8 (0)(1.98)=0 28.8-0=28.8 22.71 no 5 576-5=571 .05(576-5)=28.55 (5)(1.98)=9.90 28.55-9.90=18.65 22.71 No 3 576-3=573 .05(576-3)=28.65 (3)(1.98)=5.94 28.65-5.94=22.71 22.71 YES! B 576-B .05(576-B) 1.98B .05(576-B)-1.98B = 22.71

Okay, there we have. A decent example of what a modeling, problem solving solution would look like. At the beginning stages of Math 1, they would not demand the last row, the equation row. But quickly they would ask the learners to start generalizing their solution.

The guesses column are not set in stone. The guesses are going to be the learners guesses. They are going to guess whatever they want. I started with 0, because maybe he didn’t break any. Then I saw that was too high to my goal, so I figured Alex broke a few. Then I was too low, so I picked one in the middle.

Now, let’s examine what the columns mean. It is clear from the headings that each column has a very specific purpose and is clearly labeled. What are we guessing? We are guessing the number of glasses he broke. If he breaks 5, then he didn’t break 571. How do we get that, we subtract. Each column must have in it HOW they get the number, not just what the number is. And so on.

Notice that by the time the learner reaches the answer, they have worked several times the process, they know the multiplications, the subtractions, and they have the solution worked out. Where does the variable go? It goes into the spot where numbers change. What do we call the variable? Don’t care, use a letter that makes sense to the problem.

How do they start this process? The first problem that is a modeling / problem solving problem is M1:9:4. It looks like this:

Notice that they start by giving the table and even filling out the first row. The problem I worked above, didn’t have that level of detail. The learner had to provide it. That is the point.

EXETER MODELS AND LADDERS THE LEARNING UP TO THE LEVEL THEY WANT.

Yea, I shouted that. We have this impression that Exeter is so fabulous, that they don’t have to ladder or work with learners. We think that the learners just will magically go *poof* and be able to do all these things that we struggle with.

Guess, what, they struggle with similar things there as we do in our schools. It might be easier because of smaller class sizes, but the root problems are the same.

Okay, off my soap box.

The Algebra 1 activities have some problem solving activities, and they even are sneaky by giving a blank table with fewer columns than the learners need! The learner is pushed to make the table for themselves.

Think about this type of problem solving for special ed, or EL Learners. They have the numbers set up, they can see where the Letter for the Unknown goes, because it is the only number that changes when they are doing the problems. Wouldn’t this method help them out so much?!

Think about your average learner who struggles with parsing the language of the problem. If they work 10 or 20 of these as starters, as homework, as in class activities, do you really think they are going to stress about a word problem?

Nope, they are going to say, “Mr. Waddell, these are easy, can we move on to something harder?” And you know they will.

Think about the really advanced learner. They are going to resent the table after a short time, but they will go to the generalization much faster because of it.

Can you think of any downside to this method of problem solving? I can’t. I have done Algebraic Thinking’s “SOLVE” method, and other methods. None of them are as straightforward and easy to put together as this method. We could spend THOUSANDS of dollars on professional development on problem solving, and none of that money would come close to the success of just creating a table, labeling, and working it out step by step.

Guess and Check. That is what Exeter calls it. I call it just downright successful for every level of learner.

Before I begin going through this problem I have selected, I want to link you to where the PDF’s of the documents can be found. Notice these are NOT the problem sets, these are activities that ARE used in class, but also are pulled together for teacher use at the Phillips Exeter Academy Summer Math Institute. Just so we are all on the same page on where these are coming from.

Also, at the end of this series of posts (and I have mapped out at least FIVE more of them) I will post these resources in WORD format instead of PDF found on Exeter’s website. I asked the instructor and he gave me permission to post them as long as credit is given to Exeter. Have I said how much I like the Exeter Academy, their curriculum and how much respect I have for their willingness to share?

Okay, so let’s jump into the problem. Before I begin, I must say that we spent 3 days working this problem I am presenting here. We did a bit of it one day, a bit more another, and finished it as a third. We did parts as a starter problem, parts as an activity and parts as a “could we do this with this problem?” extension. This translates very well to the high school curriculum because it allows for a stepped pacing, starts off slowly with multiple entry points for all learners to accomplish, and them moves them slowly up the ladder to high levels.  Oh, it is essentially the Common Core.

I am going to put this below the fold because it gets long. No, really, lots of pictures, and it is long. It is completely worth the read though, and you will see how this fits into algebra and geometry.

Before I get into some examples of what we did for the 4 days of Exeter training, I want to discuss what the overall philosophy of pedagogy that was modeled exhaustively. Never once did the instructor actually SAY, “This is how we designed and planned the problems.” Instead, he set up the situation where we worked problems, and through my experience at #TMC12 and paying attention to how he phrased things, and how he moved from one problem to the next I was able to work it out.

Then I asked him point blank if I was right. I was. We had a short (very short) discussion on it, and then we did more math.

In describing the system / plan / organizational structure below, I want to make it clear these are my words, my descriptions, and my labels about their essential methods.

I am going to use the following descriptors for my understanding of their system; the setup & modeling, naming, and extension. I will work through each one of these separately, and connect them to some problems found in the Exeter problem sets as examples. When I am talking about problems found in the sets, I will use the following notation: M1:1:3 means Math 1, Page 1, Problem 3. All problems can be found on Exeter’s site here.

• The set up & modeling: M1:15:6, M1:17:3,4,5,6, M1:18:1,3,4 M1:19:2,3

All of the problems listed above have one thing in common; they all are slope problems and yet NEVER ONCE mention the word slope. They use “rise” and “run” or some other variant. They discuss the change of one thing and the change in another, they talk about stairs or setting up a table of values or walking at a continuous rate or … you get the picture.

All these problem (and more, I just pulled out a sample) model the idea of slope of a line in a REAL WORLD basis. It has the learner calculate slope 5 different ways, from 20 different types of constant change. It clearly equates the idea of slope and rate of change, and puts emphasis on units and context. Never once is slope mentioned. No definition, no definition of a line, no y=mx+b, nothing. Just; here is a situation, figure out the answer. And the figuring isn’t all that difficult. It just asks the learner to understand what the rate of change is for each real world problem.

• The naming / defining: M1:19:4

AND THEN they spring the definition of slope on the learner. It is kind of off handed, “hey, you know that thing you have figured out how to calculate in all those different situations, it has a name, it is slope btw. How cool is that, we now have a name for the idea we have been working with the last two weeks.”

I asked our instructor if he would ever say the word slope before they worked this problem, and the answer was no. Let’s think about that for a second. This problem is on page 19, which essentially means (but not necessarily) the 19th day of class. The learners in this class have been working with linear equations, problem solving with linear situations, and slope problems for about 10 days of the last 19 and the teacher JUST NOW UTTERED THE WORD SLOPE!

This is essentially 100% backwards from how we are taught to teach, and completely and diametrically opposite the textbook approach. We use the word, give the word a meaning, try to get the learner to memorize the meaning, create foldables to help them memorize the meaning, and then are frustrated when the learner forgets the meaning.

Exeter has the learner work with the meaning, solve problems with the meaning, completely understand the meaning, demonstrate they know the meaning through 15 different applications of the meaning AND then they say, “Oh, btw, that meaning has a name.”

Which method do YOU think is better? I know the answer for myself. Having tried and failed the last 5 years at getting learners to memorize meanings for words, it is a hell of a lot easier to get them to memorize a word when it is being attached to a meaning that is well understood.

• The extension: M1:19:5,6,8; M1:20:9; M1:21:3,4,11

So now that the learner knows the word, the world opens up because instead of long problems that model real world situations the questions can become much more abstract. But they don’t! That is the point!

The problems just step up the level of thought required to a new level. Now the instructor can create activities that challenge the learner to think about slope and y-intercepts in a more thoughtful way. For instance, one hands on activity has the learners working with geoboards and thinking in depth on what it means to attach certain adjectives to the slope, or certain numbers to a slope.

This extension piece can not be stressed enough. It is not working problems 2-30 even from section 3.2. It is; explain what the slope of a line looks like as you take the value of the slope from 1/5 to 5/1 through all integer steps of both numerator and denominator. Then explain WHY your explanation in the first part makes sense.

This is the pedagogical pattern used at every level, and for every topic. Setup the topic simply, show how the topic models a real world situation and work with that topic for several days in several different ways, then define the topic in a very straightforward manner, and finally extend the topic to new, novel, and more complex situations.

It is important to understand this progression of understanding for the next post, which will take one very straightforward idea, and end up in a place where we can calculate the volume of a 3 dimensional derive the formula for calculating the volume of any 3 dimensional parallelogram. Yea, it really can be that easy.