When The Activity Isn't Enough

I love the learn by playing nature of activities like Marbleslides.  In fact, I just visited a classroom yesterday where kids were digging in.  It was interesting to watch as students engaged in this environment.  It was fascinating to try to understanding their thinking.

If we walked into 100 classrooms where students were learning about graphing lines in slope-intercept form, we'd find more than our fair share of lessons where some sort of direct instruction is happening.  We'd likely hear academic vocabulary, see a formula for finding slope and probably even a general equation like y = mx + b.

I'm not against those things.  However, I'm for giving students an experience that can be precisely described by knowing those things. Activities like Marbleslides do this.

The activity isn't enough.

Here are four different students who are all engaged in the same activity.  Consider the following questions:

What do you notice?
What questions would you ask this student?
What could you have offered this student prior to starting this activity?


Student 1

Student 2

Student 3

Student 4

Here's what I see.

Student 1 is WAGging like crazy.  These are just random guesses. No adjusting or learning from feedback.  If this student achieves success, it'd be like a blind squirrel finding an acorn.

Student 2 is an answer chaser.  I mean literally, look at the guesses.  Once this student sees which part of the equation to adjust and the line moving in the right direction, the adjustments are incremental.

Student 3 is a strategic thinker.  Slope? Nah, don't need it.  y-intercept? Yeah, that's the stuff.  Let's trap the answer and close in on it.

Student 4 is engaged, believe it or not.  This student is paralyzed by options.  Just waiting for the correct answer to pop into the brain.


So, how do you respond to each student?

Pretend I'm Not Here

Yesterday we worked on this pattern. 

By the end of the period, we had two different rules.

n + n + 5     or      (n - 2) + (n - 2) + 9

Today we had to decide whether or not these two rules were equivalent.  We had a brief discussion about the different ways students could make their argument:  numerically, visually, symbolically or verbally.  I asked each student to choose a method they preferred and spend a few minutes constructing an argument.  The plan was to then have them pass their journal around the group and have their partners help them make their arguments more convincing.  

As I circled around the classroom, I noticed the work of a particular student who doesn't yet have the confidence I believe will eventually show up.  I stopped and asked him about his work. 

Me: So, tell me about what you have going on here?

Student:  ...

Me:  What type of argument are you trying to make here?

Student: Numbers. 

Me:  Ok, so what numbers are you choosing?

Student:  I chose 55.

Me:  Does it work for both rules?

Student:  Yes. 

Me:  Now that I'm sitting here with you and hear you explain, I can totally understand what you're trying to say.  

Me:  Let me ask you something:  Do you think that if you ripped this page out of your journal and left it for me to read after class, I'd be able to understand your argument?

Student:  No, I don't think so. 

Me:  Can you treat this as a rough draft and try to convince me as if I wasn't here?

Student:  Yes. 

Me:  Ok, great.  I'll come back and check in a bit. 

After a second pass around the class, I come back to this:

I asked if I could have his permission to take a picture of both and show it to the class.  We'd keep it secret if he wanted, I assured him.  When I projected the first iteration, other students tried to explain his thinking.  When I showed the work of the "second student", we all agreed it was much easier to follow the thinking.  Then I said, "This is the same kid."

Class:  "Wait, WHAT?!  

The coolest part of this was that when I wouldn't say the name of the student, many of his classmates said, "It's obvious Mr. Cox.  Look at him."

He was beaming. 

It Was a Simple Question...

...until it wasn't.

Features of Functions is the unit and the problem focused on the following graph.

Typical questions like:

1. What is f(2)?
2. For what values, if any, does f(x) = 3? 
3. What is the x-intercept?
4. etc.

 Groups were working well together and I asked them to write their agreed upon answers on their easels.  As we looked around the room, everyone agreed until we got to #6. 

      6.  On what intervals is f(x) increasing?

Naturally, everyone said the function increased on the interval [-4, 6]. Everyone except Group 7.  They're always contrary like this.  Probable just stirring the pot a little.  Just pat them on the head and move on.  

Except J says, "Mr. Cox, I stand by my answer."

"Wait, what? Do you know who I... ahem, tell me more."

"Well, since the function starts at -4, it's not increasing yet.  And since it ends at 6, it stops increasing."

R chimes in, "Ok, J.  I see your point and I'd be willing to say the interval is (-4, 6] because at -4 it hasn't started increasing, but at 6 it's been increasing and then stops.  Maybe we include one and not the other."

This took us on an interesting discussion about what we really mean by rate of change, increase and decrease; how our interpretation is influenced by our left-to-right reading convention; and how many points we actually need to identify a rate of change.  We talked about instantaneous rate of change and how you can actually have a "slope" using one point.  

But I still have questions.  J was looking at the endpoints of the functions as if they were a relative maximum and minimum.  They aren't included in the increase interval because the rate of change is actually 0 at those points.  Was he correct to think this now?  Was this simply a really good wrong answer?  Should he be considered correct on the argument alone?  

What say you?

Well, since you asked...

We've been looking at the volume of prisms, cylinders and cones this week.  I had students working on a project where they had to build one of each with equal heights and widths/diameters.  The idea is to explore the volume of each and see how the eventual formulas will relate to one another.

Then, Jacob traces can on his paper and cuts out the circle.  He cuts a radius and begins rolling the paper (as if he's cut out different sized sectors) to make different cones.  He comes up and says, "Mr. Cox,  I think the cone that is almost flat has the highest volume because the tighter I roll the paper, the less stuff I can fit in it."

Me:  What if the circle is flat?  What's the volume then?

J: There isn't any volume.

Me:  So then when does the cone go from 'flat' to having the most possible volume?

J:  ...

Me: ...

J: What do you mean?

Me:  Maybe there's some kind of sweet spot where the volume gets bigger then starts to get smaller.

J:  Let me think about that.

At this point, I was with Jacob.  I didn't really know what the volume did as the cone changed.  But we were both interested.

The next day, Jacob comes in and says, "Mr. Cox, I thought about what you were saying and I think you're right, there has to be some kind of sweet spot."

So, we sit down and go to work.

I'm thinking about how to model this thing and Jacob enlisted the help of a friend to gather data.  They're cutting sectors from a circle and making cones.  Jacob has dibs on 30, 60, 90, ... degrees and Armando has 15, 45, 75, ...

Our first bit of trouble came when Jacob said, "I can find the radius of the cone's base, but I'm having trouble getting the height because of this..."

Wish these rulers came with a bubble level. 

But, we figured out that the Pythagorean Theorem was a nice work around.

Now, does our data match the model?

It took a while, and thanks to CalcDave for cleaning things up, but this is a pretty cool function.

 Desmos graph is here.


We're estimating the maximum to be about 66 degrees.  And because my calculus is a little rusty, I'm thankful for the folks at WolframAlpha.

This particular function is using a circle with radius = 3.1. 

The function is a little dense at this point, but Jacob was dialed in as we talked about it.  The idea that these crazy expressions really just amounted to V_cone = ⅓πr²h blew him away.

Full Circle

It was one of those moments when I was trying to explain something to them and they ended up explaining something to me.

We're in the middle of a unit on volume and exploring prisms, cylinders and cones.  I was inspired by James Tanton's ability to explain things by getting at their essence. As if to say, "we can call a cylinder a 'cylinder' but it's just a prism made of circles--or a cone can be called a 'cone' but is it really any different than a pyramid?"

It was one of those, sitting around a campfire moments.  We're using stacks of paper and stacks of CDs to demonstrate why calculating the base area is critical because the rest of the solid is just like a stack of that area and no matter where we slice the solid, we get the same shape--over and over again.

Then comes the question about the cone.

The base is a circle but when you slice it, you get a...circle?  Wait, but it's a different circle.  Waitaminit. What about a pyramid?  Triangle base and when you slice it, you get a triangle.  But a different one.

Are the triangles related?

"They're similar.  Hey wait, this is a dilation."

And the tip of the pyramid is the center of dilation.

We did dilations in Unit 1.  This was a callback I didn't anticipate:  A pyramid is like a 3D representation of a dilation.

Thanks, kids.  I'd never thought of it that way before.

When It Can't Be Wrecked

We're getting some mileage out of this lately.  Today, I have a new problem to add to the pile of those that foster the process of hypothesis wrecking.

I posed the question with a rubric.

Can a unit fraction always be written as the sum of two unique unit fractions?

Rubric
5: Precise proof that demonstrates all cases (abstract, general rule)
4: Reasonable argument that demonstrates some cases (numeric, gives examples)
3: Gut level or weak argument
2: Does not present an argument

1: No evidence of understanding

Students played around with a few unit fractions and after a few minutes we had a couple of them.

Shortly, we had a student come up with an hypothesis:

which was soon followed by another student example:

Uh-oh, that doesn't fit the pattern.

"Does this example wreck our hypothesis?"

This led to a nice conversation on whether this new example and our hypothesis can coexist.  It was interesting to see how many students initially thought the hypothesis was wrecked.

We tested a few more examples and shared results--all confirming our hypothesis.

Then I asked, "So where does this put us on the rubric?"

And a student asks, "What has to happen for a 4 to become a 5?"

In other words, when does a numeric (quantitative) argument become abstract _[1]?

Had to pause.  This one is worth it.  So we discussed simple example:

I quickly came up with the question and answers 1, 3, and 4.  At lunch I added 2, which really added to the conversation for 6th period.

Which answer provides the stronger argument?  Most saw 4 as the strongest and agreed 1 was the weakest.  But very few saw 2 on the same level as 4.  Then one student says, "I see that 2 and 4 are similar but 4 is just kinda strung out."

Yep, the kid has a feel for brute force vs. elegance.  Love it.

By the end, we agreed that 2 and 4 were more abstract and 3 was more quantitative. What about 1?

Well, 1 was what they would've considered a great answer a few months ago.

[1] This is what prompted my question about SMP 2 on Twitter. 

Dirty Triangles

I've been out for a couple of days--let's just say that I can think of better ways to drop 10 pounds--so, I'm in a really special frame of mind today.  While I was out, I left a few distance/rate/time problems for students to solve.  Upon my return, I was asking students about the problems and many students had similar responses.

S:  "This is easy, you just use the Dirt Triangle."

Me:  "The what?"

S:  "The Dirt Triangle."

Me:  "Hmm. I don't know what that is."

S:  "Look, Mr. Cox it's like this...

"...You cover up the one you're looking for and if the other two are next to each other, you multiply.  If one is above the other, you divide."

Me:  "Really? That's strange.  I never learned the Dirt Triangle. I learned...


The Turd Triangle

S₁: "No, that won't work. That's not what he_[1] told us."

S₂: "He said it didn't matter how we wrote it."

Me: "So which is it; does one work or are they the same?  Make your case and be ready to defend it."


Helping students develop a turd detector one day at a time.



[1] Students picked up the triangle in another class.  They said that the formulas were given early on and explained.  However, many were still missing problems so the triangle was introduced.  

Instead

You know what?

Instead of having to teach things like perpendicular bisectors and systems of equations, I just wish we could do things like this.

Hypothesis Wrecking and the Diagonal Problem

We've been doing more problems lately where students can gather data and look for patterns.  Today's installment is via the Diagonal Problem which I think I first saw via Kate.

I'm noticing that more kids are gaining confidence in looking for patterns, forming hypotheses and then seeing if they can make the hypothesis fail.  The phrase that seems to be gaining ground when it comes to hypothesis testing is "wreck it"-as in "Oh, you think you have a rule?  See if you can wreck it."

This diagonal problem is nice because a lot of students seem to zero in on special cases. For example, an n x n (or I just call them squares) rectangle has a diagonal that passes through n squares.  There have also been some nice attempts at nailing down rules for odd x odd and even x even rectangles.  We're finding that special cases don't lead us right to a general rule, but the information can be useful.  

I've put together a flow chart that seems to be helpful. 

Some students get caught in the Do research-->do you see a pattern?--> Do research loop others are making it to the hypothesis before being kicked back to research. All are having to come face to face with their impatience.  Some are owning it.  

There are a lot of mistakes being made.  There's some frustration.  There's arguing.  There's collaboration. 

There's learning. 

I Like Triangles

Last night, I asked if anyone could point me back to this fantastic animated factorization visualization. (h/t @calcdave)

Now, I'm kicking myself for not thinking to use this in the first weeks of the school year.  Talk about some Fake World math doing a number on pseudo engagement strategies.

I started the animation at the end of each period and walked out to greet students as they walked in.  Once everyone got settled, I walked back in the room and each time the dots would circle up, I'd yell, "PRIME!"

"Alright, I'm up 1-0. PRIME!, man I'm smoking you guys."

Kids caught on really quick and started looking for the circled numbers.  In fact, it took many students a while to realize that the applet literally said "prime."

We started out by looking at the patterns and how each number was represented visually.

But, next I said, "You know what, I really like triangles. What is the smallest number that will give us a triangle?"

This one's easy. 

"Alright, what's the next number that will give us nothing but triangles?  Write your guesses on your easel."

Guesses were about 50-50 between 6 and 9. 

"Alright, what about the next one?"  

Still guesses were a little sporadic.  But by the time we got to 81, most students thought they figured out a pattern.  From 243 on, we were at about 100%. 

After about 10 minutes of doing this and discussing our results, I put up this slide.

There was a nice discussion on clarifying our question.  Three different student offerings illustrated the idea of First Idea; Best Idea. 

Student 1:  At what stage is each triangle?

Student 2:  How many triangles are in the green circle?

Student 3:  How many dots are there in each circle?  *Boom*

Now get to it and be prepared to justify your answer.  

The first student said there were 9, 27, 81 and 243 dots.

"Ok, great. So how did you do that?"

"Well, the green circle has 9 dots, then I multiplied by 3 to get the red. Multiplied by 3 again to get the blue and then by 3 again to get the black."

"Alright, so let's press on this idea a little."

I know what question I want to ask, but I just bit my tongue until a student speaks up.

"How can you be sure that there are 9 dots in the green circle?"  *There it is*

"I estimated."

Here's where it gets good. 

 From across the room, I hear, "It looks like there are 9 dots in the green circle, but we have to look past that."

Wait, what?

"Yeah, we can't trust the picture because the dots are too small.  We know there are 6,561 dots on the whole page and there are three black circles of dots.  We have to start there."

So, Dave, keep this link handy, I'm sure I'll be asking for it again next year.  

First Idea; Best Idea...

...and the Worst Idea

Creating a Culture of Questions was, by far, the most popular post on this blog until someone somewhere starting linking to the post on Exponent Rules.

I think a natural follow up to the Culture piece would be with regards to establishing a classroom culture where feedback is given and accepted.


The First Idea is the Best Idea and the Worst Idea

The first time students hear this, I usually get, "Gosh, that's mean."

But we discuss how the first person who puts forth an idea holds the best idea as there is nothing to which we can compare it.  But using the same logic, this idea should be the worst.  This assumes the flow of ideas that should follow.

I think this encourages two important things:

1.  "If I go first, it doesn't matter that my idea isn't fully formed."  This student has established a floor on which each other student can stand and/or build.

2.  "I can take someone's idea and help them make it better."  The real work is done by the first follower.  This student chips away at any imperfections and helps the first student refine her idea.  Subsequent students then follow suit.


What's this look like?

Yesterday, we trying to determine the equation between the points below and students wanted the y-intercept.

Students were using what they knew about slope to find other points and had to wrestle with the fact this particular line doesn't have a lattice point for a y-intercept.  Once we were finished, I asked students to write down any questions they had.

Student 1:  "I have a comment."

"Ok, what is it?"

Student 1: "No matter which points we choose, the slope simplifies to the same thing."

"Can you turn your observation into a question?"

Student 1: "Will that happen all the time?"

Now here is where it happens.

"I can misunderstand [Student 1]'s question, can we make this more precise?"

Student 2: "Will the slopes always simplify to the same thing?"

Student 3: "Will the slopes between two points always simplify to the same thing?"

"Are we only using two points?"

Student 4: "Will the slopes between three points always simplify to the same thing?"

Student 5:  "Will the slopes between any two pairs of points always simplify to the same thing?"

Student 6: "Are the slopes between any two pairs of points always equal?"

"Are we really talking about any 4 points here?"

Student 7: "Are the slopes between any two pairs of points on a line always equal?"

I'm Bringing Multiple Choice Back

So here's the idea:

One problem with multiple paths to solution.  Students connect as many skills as they can to the problem.  I listed eight possible skills two of which wouldn't necessarily apply to the problem. Students had to assess themselves on the skills they demonstrated. 

Question of the day:  "Mr. Cox, is it possible to use all of these skills?"

Answer to Question of the day:  "It's possible that some of the skills don't apply."

For this first iteration, I used the standard Ticket Problem. 

Below are samples of student work. 

As an exercise for the reader:

1) What are your thoughts on this process?  

2) How did each student do?

Let me know in the comments. 

Student A

Student B

Student C

Student D

When Perseverance Pays Off

Our high schools are committed to taking the integrated path with the first three courses and since the middle schools will be teaching these courses, I'm part of the team building the units.  I've been piloting the curriculum by the folks from Utah and, for the most part, I like it.  I'm particularly enjoying the learning cycle they employ:  Develop Understanding, Solidify Understanding and Practice Understanding, mostly because it's pretty easy to discuss with the majority of teachers.

This task was at the beginning of a learning cycle.

Source: Mathematics Vision Project

I had a student, H,  come to me before class and say, "Mr. Cox, I spent like three hours on problem 3 last night.  I couldn't quite get it."

During class, students worked with their groups and started presenting solutions.  As I approach H's group, she gives a high-five to the student next to her.

Me: "What's that about?"

H: "We figured it out!  I get it now."  Then she shows me her solution.

"Feels nice, huh?"

"Yeah, I think I'm gonna cry."

Me too, H.  Me too.

Adventures in Pedagogy: Four Zero

Aidan was having trouble with subtracting numbers that required renaming renamingborrowing, umm, taking away more than you have. He was simply taking the bottom smaller digit from the larger regardless of which one was the subtrahend or minuend on top or bottom.

He picked up some sort of rule along the way and was obviously misusing it. So he decided to get a little creative.

Take a look.


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CCSS 8: Unit Building

At the end of February, representatives from grade levels K-8 spent two days unpacking the CCSS and clustering them into units of study. My previous experience with unpacking standards became a process of identifying "essential" standards which assumed the existence of non-essential standards.  Those standards that didn't make the cut were ultimately ignored in favor of those that were most heavily tested essential.
We have the same provider leading these new sessions, so I was a little worried we would end up looking for content to cut rather than incorporate.  So far, that hasn't been the case.  Obviously, there are certain topics that will require more focus (eg. linear relationships as opposed to exponents) but the goal has been to see how and where these supporting standards fit with the focus standards.
Our team has come up with the following units of study.  We haven't reached the point where I can discuss specific activities/tasks, but I'd like some feedback on the pedagogy that motivated the clustering and sequencing.

Transformational Geometry

Use the coordinate plane to discuss transformations, congruence and similarity.  Use dilations as an application of the ratios/proportions work done in grades 6 and 7.  Use a graph as a tool to describe proportional relationships.

Data Analysis

Use bivariate data to create scatter plots which can then be the jumping off point for informal line of best fit (where the line may have an initial value other than zero) and an introduction for a future defining of function/non-function.

Linear Relationships

Graphing, graphing and more graphing.  Take the informal line-of-best-fit and formalize the definition.  Allow math to be it's own context.  Graph systems of equations and look for common point (read: solution to system).

Equations

Use the work done in graphing systems to motivate more abstract symbolic manipulation required for solving linear equations.

Exponents

This was a tough one.  Expressions with integer exponents and scientific notation seemed like an island unto themselves.  We are still working on finding a place that fits nicely for these ideas.

Functions

Use the informal introduction to functions and formally define a function.  Look at linear and non-linear functions.  Compare functions using different representations (ie. graph vs. table vs. equation vs. verbal).

Pythagorean Theorem

I think this one speaks for itself.

3D Geometry

Problem solving involving the volume of cones, cylinders and spheres.
We are trying to move from the concrete/informal to the abstract/formal while allowing students to explore these ideas while creating their own formal definitions.  I'm particularly interested in the sequence that runs from Data Analysis to Functions (note: Exponents look to be a unit that can be dropped in and our school calendar lends itself to having that unit kick off the second semester) as it may receive the most push-back from our high school colleagues.  Traditional textbooks usually go the route of
Functions-->Equations-->Graphing-->Applications so we're going to have to have solid rationale.
No one pushes back better than you all.  I'm counting on that.

Integrated Teaching

For the past six years, our middle school science and social studies departments have kinda been given the short end of the stick.  The minutes given to math and ELA doubled at the expense of these two departments.  I don't know what the teachers were worried about, though, because only the 8th graders are tested and the test only covered, I don't know, three years worth of standards.  (And besides, Jason says teaching science in CA is easy.)
This year we decided to give some relief by reducing the minutes to math and ELA and adding a semester of integrated math/science and a semester of integrated ELA/social studies to each student's schedule.
My assignment this year is to teach one section of 7th grade math and four sections of the integrated class.  It's been tough because we are really creating this class as we go.
So far, this is what I've learned:

1. Students will do exactly what you tell them to do.  


2. Students have trouble breaking large (essential) questions down into smaller (guiding questions) questions. 


3. Students think "try harder" is a plan. 


4. 
   Polya and the guys who invented this were life coaches.   
   
   


5. Establishing protocols is essential. 


6. Scaffolding doesn't just refer to content. 


7. Changing "Why?" into "Tell me more about that" is magic. 


8. "Common knowledge" isn't so common.  


9. We need to do a better job of helping students distinguish between opinion and argument. 


10. After 16 years of teaching, I still get excited when students realize that thinking is better than memorizing. 


11. Any list worth reading stops at 10.  

On Problem Solving

Phil Daro:

The American teacher looks at a problem they're going to use in a lesson and asks themselves, "how can I teach my kids to get the answer to this problem?"  The Japanese teacher asks, "What's the mathematics they're supposed to learn from working on this problem? How can I get them to learn that mathematics?"

If you want better answers, ask better questions.

Common Opener

I really like where we are going here.  Our video production class has been broadcasting daily announcements live for a while and now we are really starting to take advantage of this time to establish some campus habits of mind. Recently, we started a series of "Common Opener" segments airing on Tuesdays and Thursdays focused on Flexible Thinking.  Math was up first and we used the Number Talk as the vehicle.

Basically, we are asking students to deconstruct and reconstruct numbers by looking for ways to find the answer that may be different than what they're used to.  We want them to leave their comfort zones just a bit.

Once the announcements are finished, teachers send their "exit slips" up to the office (or a student aide will pick them up) and I end up with a delivery of ~500 index cards/half slips of paper with student responses.  I quickly go through them and look for patterns.  It's been interesting to say the least.  Many students have started to look for new ways to work with numbers while a large majority still stick with the standard algorithms, but I'm optimistic that students will become more comfortable with the uncomfortable.

Hopefully, this acts as a vehicle to further the student/student, student/teacher and teacher/teacher conversations about learning on our campus.  What really encourages me is the opportunity we have to show how big ideas show up in all over the place.  Once the other content areas have a chance to interpret Flexible Thinking, we may see some real aha moments with our students.

You can find the announcements here.

Hypothetically...

What if we could design a project-based class for middle school students that integrated math and science-- that provided context for math and gave instructional minutes back to science?  What if this class capitalized on the natural correlation between the Practices For K-12 Science Classroom and the Standards for Mathematical Practice.  What if students were asked to design investigations, used modeling to move from the concrete to the abstract and then presented their findings for peer review?  What if they used technology to take snapshots of their learning along the way and kept a running journal of the process?

What if...

The Comma

If [this], then [that].  We talk a lot about [this] and [that] in the math classroom. Teacher supplies [this], student responds with [that]. They even have names:  hypothesis and conclusion.  But, what about the comma?   All the power of the entire process is summed up with a tiny little "," that is all too often ignored.

No more.  It's time to give the comma a voice.

We were getting ready to add rational expressions and I wanted my students do have a workable rule for adding fractions with like and unlike denominators.  My goal was to develop the idea that when adding fractions with unlike denominators:

A lot of students don't see this very clearly.  They do what they do to jump from [this] to [that].  And most of the problems end up looking a lot like

with no real understanding of what's taken place between the hypothesis and conclusion.  And up to this point, no one has really cared because Johnny was able to find the correct answer on multiple choice scavenger hunts with a great deal of accuracy added fractions like a champ in 6th and 7th grade. However, when Johnny gets to algebra, and sees

for the first time, you'd think he's never worked with fractions before.


Time to talk about the comma.  It was actually a pretty simple adjustment to a simple question, but the conversations it generated made all the difference in the world.

This quickly became

See, the beauty here is that the process became the outcome.  The numbers become the variables and we get a good grip on how one-third plus two-sevenths becomes thirteen-twentyfirsts.  The abstract isn't so abstract and the easy part is swapping out the 1, 2, 3, and 7 for a, b, c, and d.

Mission accomplished.  Now, lets hope they remember it tomorrow.