The 70-70 Trial

Education is a world with a whole lot of theories. Intuitive theories at that. I’m sure it’s like this in most professions. We are seeing an issue. We reason out what the problem seems to be. We determine what the solution to our supposed problem seems to be. And we implement.

The problem with that is problems often have multiple causes. Solutions are often biased. Results have a tendency to be counter intuitive. For example, a paper recently published suggested that the increase of homework might actually cause a decrease in independent thinking skills. This probably isn’t a conclusive study, but recognize the idea that if students aren’t demonstrating independent-thinking skills, prescribing a problem-for-problem course of study for them to do on their own might not be the best solution.

This leads me to a trial that I am running in my classroom for a semester. I have four sections of geometry. I am going to leave two as a “control group” (very imprecise usage, I’ll admit) that will run exactly the same as they did first semester. The other two will run “The 70-70 Trial.” This is one of those theories that has gotten tossed about our district many times. It seems intuitive. It seems like it addresses a persistent problem.

The theory goes like this: If you go into a test knowing that 70% of the students have 70% or more on all the formative assessments leading up to the summative assessment, then we know that the students are reasonably prepared to do well on the test. If you give a formative assessment, and you hit the 70-70 line or better, you move on with your unit, business as usual. If you miss the 70-70 line, you pause on the unit until enough of the class is ready to go.

This seems reasonable to some, and ridiculous to others. Our staff meetings have seen some pretty intense discussion over it. Proponents lean on the logic. How can a group of students with high scores on formative assessments struggle on summative assessments? Opponents speak to the time crunch. When do you decide to move on? You can’t just keep stopping and stopping forever? You’ll never get through the material. Both seem like logical points…

But, as far as I can tell, no one has tried it to see what would happen. So, I figured that I had two classes that really struggled their way through first semester. It became very hard to energize and motivate these students because of how difficult they found the material. Perhaps shaking up the classroom management and unit design will add a bit of a spark. These two classes will be the focus of the 70-70 trial. I will use this blog as a way to record my observations and entertain any ideas from people who are looking to give me ideas to help this idea work.

This starts one week from today. I don’t know if it will work. I have my guesses as to what I think will happen, but I am going to keep those to myself. I absolutely want to see this work because if it does, that means my students were successful. My chief area of concern is what to do when 61% of the students score 70% or better (for example). By the rules of the trial, I can’t go on. I need a reteach day, but over half the class finds themselves ready to move on. What do I do to extend the learning for those students, while supporting the learning for those who need some reteaching and another crack at the formative assessment?

These are the kinds of things I will be looking for help with. Thank you for being patient and willing to walk this path with me. I will look forward to hearing whatever ideas you have.

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Thoughts on Proof… and showing your work.

Suppose I give a group of sophomores this image and asked them to find the value of the angle marked “x”.

G.CO.10 - #2

Consider for a moment what method that you would use to solve this problem. (x = 121, in case that helps.)

Now, suppose I asked you to write out your solution and to “show your work.” What do you suppose it would look like?

I was a little surprised to see what I saw from my tenth graders, which was a whole lot of long hand arithmetic. Like this…

2013-12-11 12.21.11

and this…
2013-12-11 12.21.42

One-in-three had a mistake, which, in the midst of grading about 90 started to become an entire class worth of young people who were making mistakes doing a process that seemed fairly easy to circumvent (and by tenth grade, seems fairly cheap and easy to circumvent without much consequence.)

So, I asked why they were so intent on doing longhand arithmetic. The responses were fairly consistent.

1. Our math teachers have asked us to show your work and that’s how you do it.

2. It’s easier than using a calculator.

I will admit I was not prepared for either answer. (In retrospect, I’m not sure what answer I was expecting.) When I was asking students why they resisted the calculators knowing that they lacked confidence with the longhand, they said multiple times that they could show me that they “really did the math” without demonstrating the longhand. Also, one girl wondered why I would be advocating for a method that, as she put it, “makes us think less.”

They knew that I expected them to provide proof of their answers. Most of them were perfectly willing to provide the proof.

This student is starting to suspect that proof means words. So, he used words to describe the process.

2013-12-11 12.21.31
The conversation was pretty engaging to the students. A variety of students chimed in, most of them willing to defend longhand arithmetic and the only “true” work to show. I had shown them a variety of different looks at the longhand (the ones picture here, among others… including some mistakes to illustrate the risk, as I see it.) Then I asked this question which quieted things down quite quickly:

“Okay… okay… you proved to me that you did the subtraction right. I’ll give you that. Which of them proved that subtracting that 146 from 180 is correct thing to do?”

At first, they weren’t sure what to do with that. Although, quickly enough they were willing to agree that none of the work got into explaining why 180-146 was chosen over, say 155+146 = x or something.

I tried to convey that by tenth grade, I’m really not looking for proof that students can do three-digit subtraction. I would very much prefer discussing why that is the correct operation. They didn’t seem prepared to hear this answer. Apparently we’re even…

To be fair, there was one example where a bit of the bigger picture made it into the work. Check it out:

2013-12-11 12.20.56

I learned a lot today. I feel like I got a window into the students who are coming to see me. I ask them to explain, to prove, to show their work. Many of them willingly oblige, they just see an effective mathematics explanation differently than I do. It might be time to help the students get a vision of what explaining the solution to a math problem really looks like.

I would very much like your thoughts on this.

Means Extremes – Balancing the Means and the Ends of Math Class

I have been seeing this play out in my geometry classes each of these past six school years. It’s been a tricky problem for me to figure out and once I started to see what was going on, it became even harder for me to communicate it. I think I am ready to try.

Each year, a fresh cohort of young people come in straight from Algebra I. In theory, I can assume that they are fully loaded with algebra skills and thought-processes that will support them through their study of Geometry. There is one glaring hole in their understanding which I attributed to the overall youthfulness of many of my geometry students. They are 13-15 years old and, for the most part, their math experiences lack a significant diversity. So, I am able to give them a pass on some of the ways they are still developing as math learners.

But then I saw the same deficiency in my calculus students that I am only seeing now because this is my first year teaching calculus. Suppose that I give a geometry student this problem and ask them to find the angle measures of each angle.

Taken from Holt Geometry - Pg 181

Taken from Holt Geometry – Pg 181

Or suppose I gave my calculus students this problem.

Soda Can Problem

Let’s add in as evidence that the Geometry students are used to application problems that look like this:

Taken from Holt Algebra I - pg 478

Taken from Holt Algebra I – pg 478

And that the calculus students have spent a lot of time looking at pages like this:

Taken from Holt Algebra II - Pg 580

Taken from Holt Algebra II – Pg 580

The primary difference between the work I’m asking my Geometry and Calculus students to complete and the work that they are used to in the Algebra I and Algebra II is in the latter the equations are provided and in the former, the students are required to write the equation.

This is no small point. No side conversation. I am not splitting hairs. I am convinced this is a big difference.

Let’s go back to our Geometry problem.
Taken from Holt Geometry - Pg 181 First and foremost, recognize that there is very little natural or intuitive about this set-up as a whole. There is very little reason why angle measures are represented with algebraic expressions. The variable “x” doesn’t represent any actual value and so, the students are left to their abstract understanding of how equations are built in order to solve this problem. Their previous experience hasn’t really prepared them for this. Overwhelmingly, their mathematical experience leading up to this point has trained them to know how equations are solved.

Let’s expand this to our calculus team (of 23) of whom I noticed only about 8 or 9 who seemed comfortable modeling volume and surface area with equations and then engaging the formulas. So, even among our most talented high school students, there is a problem with the use of equations as modeling tools. Once they have them, they can operate with them wonderfully, but they struggle when it comes to writing them to specifically to match a specific situation. And beyond that, checking the accuracy of the model and then making sense of the product once they are done.

Herein lies the major issue: the paragraph above highlights a variety of skills that students (at all levels, from what I can tell) struggle with. They seem to struggle with them because they aren’t practicing them. But those are the skills that actually make mathematics worth doing to EVERYONE. The ability to do complex arithmetic on a rational or logarithmic expressions is something that is going to come in handy to people for whom formal mathematics is going to extend into their post-secondary lives. This isn’t a high percentage or our students, but these skills constitute a high percentage of the problems in our textbooks.

On the other hand, being able to recognize a situation as linear, quadratic, logarithmic or rational and have a sense of how to model that in order to make some predictions? That is something that could be valuable to a higher percentage of people outside of school.

I think that we need to recognize that the specific skills that we are teaching our math students are a means, not an end. They are the tools, not the final product.

The real goal is for the students to explore a situation, recognize the mathematically significant parts and use their math tools to model the situation strategically to help them achieve their goal. In addition to our student being better, more confident, flexible and patient problem-solvers, it seems like we’d also hear “when am I ever going to use this?” a whole lot less.

“Right and Wrong” vs. “Good, Better, Best”

So, I’m finding that I am becoming hooked on showing student work back to the students for them to explore as an opportunity to develop a deeper understanding of mathematical topics. But I’m learning there are a couple ways to handle this situation.

I could show this picture…

Geom #4-D

… and ask if the reflection is right or wrong. Which would activate a certain type of thinking. But it is a short and stifling line of thinking, especially when you consider that most hand-drawn student work isn’t perfect so “wrong” is the most likely answer. Then it risks becoming an annoying knit-picking session, which might have a negative effect on engagement.

A different approach would be to acknowledge that which we all know (that nobody’s perfect) and ask the question differently. Suppose I show the follow four photos together…

Best Reflection

… and ask “Which of the reflections is the best?”

That question opens up a lot of potential thought-trails to wander down. As I did this activity with students today, the class settled on three criteria for rating these reflection attempts. The first is that the image and pre-image should be congruent. Attempt A won that battle (with B at a close second). The next thought was that the image and pre-image should be the same distance away from the line of reflection. Attempt B was the closest (with D pretty good, too.) Finally, the students thought that the segment connecting the image/pre-image pairs would should be just about perpendicular to the line of reflection. Attempt B took that contest quite comfortably.

The students concluded that attempt B was the best reflection and I was able to confirm that by showing them this photo, which seems to agree quite strongly with that conclusion.

Best Reflection - Act III

If you want to try this activity, go ahead. I’d love to hear some feedback on how it went in your class. I’m still dealing with some quality control issues with some of my multi-media projects, so I apologize for that. I didn’t notice it to be too distracting when I was showing it to the students, but there is still room for improvement.

Best Reflection – Act I

Best Reflection – Act II (reflection distances)

Best Reflection – Act II (reflection angles)

Best Reflection – Act II (segment lengths of image and preimage)

Best Reflection – Act III

Mathematical Modeling: The Glass is Half-Full

Where is the horizontal half-way line?

Where is the horizontal half-way line?

By the time they get to me, the students in my calculus class have been given a chance to master a whole lot of math. Typically, though, they haven’t been exposed to many situations where the main challenge of the task is figuring out which types of mathematical tools will best model a problem, and thus, drive the method used to solve the problem.

Take, for example, the problem described here. This is a wonderful, challenging little task that seems so fascinatingly simple and yet becomes quite complicated quickly.

The task: draw a single horizontal line on the cup that represents the half-way line by volume. Almost as soon as I asked the question, you could see the wheels start spinning in the heads of the students.

"Well, it looks sort of like a cone, but it doesn't have a point."

“Well, it looks sort of like a cone, but it doesn’t have a point.”

Some took measurements and prepared to “calculate” it. But did they need a formula? How would they find it? What the heck kind of shape is this thing anyhow? How can we be sure out measurements were accurate?

Making sure measurements are accurate

Making sure measurements are accurate

Some were going to draw it onto paper. But how to they model it? Can they use a 2-D cross section? Which cross section do we use? What do we do with it now that we have it?

Modeled as a two-dimensional shape

Modeled as a two-dimensional shape

Some figured to guess and check. I mean, the half-way is probably going to be somewhere in the middle. There’s not THAT many different values, really. But what do you guess? Do you have to guess more than one thing? How do check your guess to see if it is right?

And all groups had to deal with the question: When you have finished up the work on the page or on the calculator, how do you accurately transfer it back to the cup?

So, let’s get it out into the open: being able to mathematically model a problem that exists in your hands with math that has always existed in a book is not something that comes naturally to most people. In addition to the content, students need to practice modeling the mathematics. They need to learn what the book math looks like when it is in their hands. Jo Boaler (Stanford Math Ed Professor) has a wonderful line about this.

“…students do not only learn knowledge in mathematics classrooms, they learn a set of practices and these come to define their knowledge. If students ever reproduce standard methods they have been shown, then most of them will only learn that particular practice of procedural repetition, which has limited use outside the mathematics classroom” (pg. 126, see bottom for proper citation).

It’s as if the math experiences we are giving most students in class are the kinds of experiences that will do them the least good outside class.

Authentic mathematical modeling requires giving students a task with a simple and clear goal and then letting them decide what kind of math will help to complete the task. The decision is an incredibly important part of the process. They must recognize the variety of available tools, have to choose which ones will work, and of those ones, which one is the best choice.

The #MTBoS is doing a great job of providing tasks of this kind of a nature (Like here, for example… or here, as another example). They are plentiful, well-done, and FREE! Give them a shot and see what happens. We owe it to our students to provide them opportunities to take the math off of the pages of the book and let them see what it looks like when it shows up in their hands.

Reference

Boaler, Jo (2001). “Mathematical Modelling and New Theories of Learning.” Teaching Mathematics and Its Applications. Vol. 20. No. 3, p 121-128.

The Essentials of Project-Based Learning

On Thursday, I will be facilitating a collaborative session for teachers on Project-Based Learning, which, to be fair, is a topic that I have never considered comprehensively. I’ve never been asked to define it. I’ve never been asked to explain it and, quite frankly, until the organizer of EdCamp Mid-Michigan approached me to facilitate the session, I never considered myself a practitioner of it. I just tried to design lessons that were engaging and built authentic, lasting learning.

But, this EdCamp facilitator, who’s name is Tara, has known me for quite a while. We did our undergrad together. We did our Masters work together. She’s been on an interview committee that almost hired me. She’s aware of my shtick. Maybe she wants me to facilitate this through the lens of how I work. If that’s the case, then this takes on an extra degree of difficulty because I still feel like I have a lot of development left to do before people start emulating my approach.

Besides that, my approach isn’t really that complicated. For any activity I consider offering to the students, I simply try to ask a couple of questions.

1. What can I do to maximize engagement? (I’ve seen plenty of ideas that I have had flop simply because the students aren’t drawn in.)

2. What can I do to facilitate collaboration? (The most effective use of class time that I ever see is when a group of students are effectively working together to solve a problem.)

3. What can I do to create a problem that will provide a variety of ways to solve it? (If there is only one real method to solve it, then as soon as one student gets it, the “collaboration” will become that student communicating “the way to solve it” to everyone else. My favorite evidence of this is when I hear a few “I don’t get how they did it, but this is how we did it and the answer came out pretty close to the same.”)

4. What can I do to make sure that the solution(s) are approaching effective learning outcomes? (While it can be interesting to occasionally have a group find a way to solve (or estimate a solution to) a problem effectively in ways that circumvent my desired learning outcomes, if I am continually putting together problems that don’t push the curriculum the community is trusting me to teach, then my students will be missing something.)

My favorite examples of this process working well are The Lake Superior Problem, The Pencil Sharpener Problem, The Wedding Cake Problem, and the Speedometer Problem. In each case, engagement and collaboration were high. Multiple solutions were present and most of the solution techniques required the students to make sense of mathematical procedures that could be correct or could be incorrect. “Why this formula is better than that other one.” Or “neither of our answers are perfect, this this one is better because this process has less of a margin for error.” Stuff like that. These conversations provide great opportunities to move beyond the memorization of a formula and to push toward sense-making of how that formula is used effectively.

While I’m not sure if this is “by-the-book” project-based learning, I have seen improvements in my students’ learning through the lesson design process I described above. Does anyone have any advice for me? Questions for me to think about? Corrections? Obvious holes in my logic? I appreciate this community because of your willingness to share, so feel free to do so.

Learning from Playing Around

These past two weeks have been an awesome time of learning for the students we’ve been working with, but I’ve also done a bit of learning myself.

I’d like to have my students love math and science and naturally be interested in it. But they’re kids. They would prefer to play. People get the most out of that which they put the most in to. If given the chance, students will put a ton into playing around.

These past two weeks I’ve been working with upper elementary-aged students. I normally teach high school students. I’m not sure if the age difference changes anything. The stuff they want to play with might be different, but not the desire to play.

And after 7 years of teaching math, there’s something appealing about a situation where students will be voluntary and enthusiastic participants.

Play Science 1

I have just spent two weeks watching students play with two activities. The first was an activity called “Table Timers” where they were challenged to design and construct an apparatus on a table top that reliably moved a marble down an inclined table in ten seconds. Second, The Helium Balloon Problem challenges students to keep a helium balloon rising, but to have it travel as slowly as possible upward. Not every group worked well and not every group achieved these goals, but the engagement level has been high. I suspect this is because they we allowed to play.

Here’s what caught my attention the most: In the midst of their play, the students demonstrated some authentic problem-solving techniques. They had to identify the major challenges to their goal, which they often did. They had to brainstorm possible ways to overcome the challenges, which usually took the form of raking through a tub of blocks or looking through the supply table. They discerned which seemed like the most realistic and then test. Following a test, they discussed what happened, why, and revised. And the students were often quite excited when they got the right answer (knowing themselves that it was right and not relying on me to tell them).

That’s a pretty good learning model. That’s something that I have a hard time getting my students to do with book work.

Play Science 2

So, the sharing, the idea-making, the consensus-building, the authentic assessment are all good things. Obviously I am not simply advocating letting the students play around all day. But perhaps by using play, we can improve engagement and the students seem to more naturally fall into a more authentic problem-solving mindset. When I consider helping them draw out the learning, some thoughts come to mind.

First, it seems like during the whole process of exploration, design, construction, testing, revising, and demonstrating, there needs to be an abundance of contents-specific vocabulary. The marble didn’t “bonk into that block.” That block “applied a force” to the marble. Students don’t “figure out how big the shape is.” The “find the area” or “circumference” or “volume.”

Second, students don’t seem naturally inclined to take data or to keep records. In the past two weeks, it seems that students are avid experimenters and do a pretty good job of verbally analyzing the problems if the plan didn’t work. Practically NONE of them documented anything on paper. No sketches, no data, no records of updates. This is an important part of the problem-solving process that would have to be established as a norm.

Third, the activities have to be tiered. Video games are great at this. The entry point tends to be quite low. The first couple of levels are pretty manageable and then the intensity and difficulty pick up. People get locked into video games through that model and people get unlocked quite quickly once the game has been beaten. Both Table Timers and The Helium Balloon Problem worked with this model. Then entry point was low for both activities and it was easy enough to begin to approach the goal, but perfecting the design and executing the plan took much more care. Then, once they hit ten seconds, we’d challenge them to add five seconds to their timer.

Fourth, I think that the groups need to be expected to summarize and present their work to each other and to field questions from the class. Class norms should allow for questioning of each other’s work and students can learn a lot about their own design, but also about the content when they know that they are going to have questions coming from their peers. Also, it would seem like this would encourage more thoughtful designs, too. Besides this, idea sharing gives the students an opportunity to look at other designs, integrate specific vocabulary into more regular use, and get the students comfortable with collaborating.

 

Play Science 3

I don’t think that playing around is the answer to everything, but I know that in my own experiences, it seems to be the forgotten learning model and if I’ve learned anything these past two weeks, it’s that an environment that produces enthusiastic student participation shouldn’t be ignored.