Why PBL? Part One

Why do our children have to complete four PBLs in a two month period, separate from their ‘regular’ schoolwork? Why can’t their ‘regular’ schoolwork be taught in such a way that they learn and can draw parallels to their world outside of school?

Not that the content should match their lives, but the way they learn that content; the way they organize and make it a part of who they are in school should have some relevance to how they organize and deal with the stuff outside of school.

These two parts of their lives should mesh, not be two such disparate worlds that they cannot be reconciled.

Here is one solution:

Making PBL Disappear: Why PBL? Part Two

What do you mean, it “…can’t be solved?!?”

I came across an Algebra I review problem the other day on Classworks. The challenge was to solve a quadratic using the Quadratic Formula. One of the answer choices was “can’t be solved.” Which I did not notice.

I was working with three students who did not understand what to do. Once I wrote out the quadratic formula, (actually, all I had to write was the negative b plus or minus!) they began to remember. One boy immediately told the other two how to find a, b, and c. That required a discussion about standard form, so we had to do a little rearranging of the problem given on the screen. Once we got the formula equal to zero, the second student plugged the numbers into the correct places! The third began offering solutions to various parts. I thought we were doing pretty good! Until we came up with a negative under the radical.

Like the music in Jaws… Dum, de dum, dum… They looked at me, dumbstruck.

“What do we do, Ms. Maxcy?”

I asked them if they had learned about imaginary numbers. (Of course they hadn’t – yet. This was only Algebra I! But sometimes I forget which level I am teaching… Which is another story altogether!!!)

Still not checking the given answer choices, I blithely proceeded to give them a brief ‘reminder’ lesson on real and imaginary numbers. They continued to look at me blankly.

As I magically (to them) unraveled the answer as 2 plus/minus 2i sqrt 11 divided by 3, they stared at me. Then they stared at their answer choices. They looked back at me.

“It’s not there, Ms. Maxcy.”

At this point, admit it, we teachers think, “it’s got to be there, that’s the right answer; why is it not there? Gosh, did I do it wrong?” And then we doublecheck our answer. And then it hit me. This was Algebra I. We don’t teach imaginary numbers. Yet. It was then that I finally looked at the answer choices…

The correct answer was there, but it wasn’t the correct answer at all!

Right there in front of me, there was the answer that the students were supposed to choose: choice “D) Can’t be solved.”

Right there in front of me, there was the answer that the students were supposed to choose: choice “D) Can’t be solved.” This is a terrible choice! It’s not the right answer! It’s not a good answer! Okay, so we don’t teach them imaginary numbers in Algebra I, why don’t we just list the result with a negative under the radical as the answer?!?

The kids get used to seeing the beast (negative radical) and we teach them how to simplify in Algebra II or geometry, depending on your school system. But, please, NOT “can’t be solved”!

That is just setting them up for trouble ahead! Lay the foundations, don’t build a wall that will have to be torn down later. Please!

Rant finished. Thank you for listening.

Murder Mystery Solved with Trig!

Dateline: April 14, 2016

The murder of Maria, whose body was conveniently found at right angles to Leg Streets A and B, has been solved! Investigators found the weapon across the river, apparently thrown there by her assailant while he was running down Leg Street A in an attempt to escape. A quick thinking officer (who had majored in math at the police academy) was able to calculate an angle measure for the angle made by the throw from the perp and the street leading to the victim.  Another savvy investigator was able to determine the distance from the suspect to the location of the attack.

With the mathematical evidence in hand, investigators were able to triangulate a conviction. Math teachers everywhere weighed in, saying it has the proportionate ability to change the way investigators do business!

Dimensions of the prisoner’s defense will be released at a later date. Film at eleven.

Okay, so I don’t really have film (we forgot to assign the job of reporter!) What I do have are a room full of kids who can now set up the proper proportions for trig problems!

Here’s how the crime went down:

Scene 1: Before the murder, I handed six students a few props:

Each student had to use the prop to arrange themselves into a triangle. The other students watching were, um, helping. (that’s what they called it!)

A short q&a followed:

Me: Okay leg A, are you opposite or adjacent to angle b?

Hapless Student holding Leg A sign: “I’m opposite, um, no, I’m right next to him (indicating student holding the angle b sign)! What does adjacent mean, again?”

We were able to sort out the definitions, and the students holding the leg signs got pretty good at determining whether they were “opposite” legs or “adjacent”  legs. A big moment came as students noticed that they could be opposite OR adjacent. More importantly, they were able to articulate WHY the status would change.

More importantly, they were able to articulate WHY the status would change.

Scene 2: The next six students were given the cards. This time, I stood back and let the first group help position the players. A little skirmish ensued as Leg A and Leg B were being positioned. After a brief discussion about whether or not leg locations could be interchangeable (did Leg A have to go in the same place as the first triangle?), it was decided that as long as a leg were placed on each side of the 90 degree angle, it didn’t matter what we called them.

The opposite and adjacent discussion began again. It was fun watching students correct these new players, or making them guess by giving them tantalizing clues!

(If you ever want to know what you look like teaching, give your students the reins. Mimicry is not dead!)

Scene 3: With everyone up to speed on definitions, the murder could now commence! Maria was positioned. Ryan was immediately suspect, as we put the crime scene tape in his hand and instructed him to escape a bit down the hall. The “weapon” was given another piece of crime scene tape and told to take off in the opposite direction. The “hypotenuse” was asked how far the “perp” had thrown the weapon. We stretched the crime scene tape from the suspect to the weapon location. It was at this moment that I heard several students say “Hey, we made a triangle.”

It was at this moment that I heard several students say “Hey, we made a triangle.”

(Scary, I know, right?)

After a bit more discussion, the students determined that we needed an angle and we needed the distance from the body to the suspect to set up a proportion to solve for the distance. Two students were dispatched with the piece of crime scene tape that had been held between the victim and the suspect (Leg A, for those of you following along). Twelve inch square floor tiles assisted in the crime scene measurement. I used my oversized protractor to come up with the angle measure, and we were ready to set up some proportions!

Back inside the room, our eager detectives checked their trig proportion info sheet (yes! They used their NOTES!) and settled on cosine, adjacent and hypotenuse. I stood back and watched them argue over who was going to set up the problem, exactly how to set it up, and how to enter the information into the calculator. Then I watched them convince one another which answer was correct.

Concrete to representational to modeling AND peer tutoring…I love it! I would say that a murder wasn’t the only thing that got solved today!

We’d like you to take over this class, six weeks into second semester….

Six weeks into second semester, I was asked to become the teacher for a group of Algebra II students.

“It isn’t that I don’t like math. Learning takes time in math, and I don’t always get the time it takes to really understand it.”*

How many more of our students feel this way, but instead of telling us with words, they distract, joke, sleep, or skip class:

…Math is such an interesting subject that can be “explored” in so many different ways, however, in school here I don’t really get to learn it to a point where I say yeah this is what I know, I fully understand it. We move on from topic to topic so quickly that the process of me creating links is interrupted and I practice only for the test in order to get high grades.

Taking Time Learning Math:A Student’s Perspective by Evan Weinberg

Would I want to come to my class?
This question haunts me. What are my kids seeing, feeling, thinking? Why does this kid come, but stay totally uninvolved? Why does this child talk, constantly, but about anything but math? Where did curiosity go? Is my class a class I would look forward to?

My personal enjoyment of math comes from the struggle with ideas and the satisfaction I get from my connection of and understanding of the relationships among those ideas. It’s like a huge puzzle that will take the rest of my lifetime to fully understand. The student’s comments in Evan Weinberg’s post resonated with what I see happening with my students. They are not learning math so much as preparing for a test about math.

They are not learning math so much as preparing for a test about math.

The current situation of ‘learn how to do this; learn how to do that’ mentality is slowwwwly changing over to ‘understand why this is so; why does this relationship work’ exploration. It will need a shift in how we teach, letting kids struggle and connect ideas (we must facilitate this exploration, but not down some tightly designed path), and changing our view of grades and mastery. I can’t say I don’t have the answer- I am working on an answer that works for me and for my students. And I’m sure I am not the only one teacher who has found the path that is taking them closer to the ideal.

This post grew out of my response to Evan’s column. His response,

“I completely agree that this is a shift, and it is ongoing. Clearly, despite the changes I’ve made to the way I teach, students still get the sense that the test is the important part, which means there is still a great deal of improvement yet to be made!”

*Taking Time Learning Math:A Student’s Perspective by Evan Weinberg

“…maths is really not about formulas, it’s about finding patterns in things, and how they influence people. And that’s a big part of art…”

This statement from STEAM: How art can help math students really struck a chord for me:

“The big point of it is that maths is really not about formulas, it’s about finding patterns in things, and how they influence people. And that’s a big part of art,”Year 12 student Maxim Adams

I am currently teaching geometry (among other maths!) and we are in the sweet zone of shapes: triangles lead to parallelograms which lead from 2D to 3D to the idea of 4D (think twisted paper*). All of this incorporates the effects of angles on shape which affects (and effects) area. Some shapes are more ‘beautiful’ than others. Some shapes and forms lead us to wanting to know more…

I use paper folding to illustrate these shapes and connections. Patty paper is very light and a great beginning medium to illustrate the connections between shape and angle, how changing the angles changes the shape. Origami fascinates me, and I use it to fascinate my geometry students, and to lead them into exploring the relationships found there. The zig zag origami is complex and beautiful, and it takes a bit of time to teach, but as my students learn to fold, we are talking about math. Even their mistakes, when they have to go back and refold, teach them about the shapes and connections. And as STEAM director Melissa Silk says,

“I think it needs to be embedded from a very early age…When I’m teaching design, I’ll try to bring in a mathematical element. It’s worth students trying to make those connections without having to find it arbitrarily later in life.”

These ‘connections’ she speaks of are not innate. Students see without understanding. Teaching reveals and makes obvious; children begin to see math in everything in the world around them. It will allow them to make much deeper connections, more fluid connections, throughout their education and beyond.

Art is not math, it’s not boring. Folding paper, for many math students, isn’t ‘doing math’. While they may struggle to fold straight lines, to make inverse folds, to make it look ‘right’, they are creating something beautiful, and if I can teach them nothing else, I can show them math can be beautiful!
The idea, after introducing them to this complex looking, yet simple, folding technique, is to have them think about what it will take to replicate angles, curves and other shapes with their paper. Then they have to apply some design sense: how many folds? How long does the fold need to be? What direction do they need to fold? What angles will produce a curve? The folds are repetitions of a fold- what is the proportional relationship? In this image, there is a curve. Each fold is a point on that curve. It can be mapped on grid paper, it can be written as a formula. There is a connection!

Don’t be afraid to start with complex looking folds. It’s so powerful for students to struggle with something that seems beyond their abilities. Success is completely intrinsic and will lead to a student who is more confident. This feeling of success is habit forming. It is a primary building block in creating a student who perseveres!

(Check out 7 more art and math connections here.)

We can calculate the area of this beautiful 3D paper construction, we can create more of them by following the ‘formula’ or we can dramatically change or even destroy an image by changing the formula! This creative process stretches the imaginations of my students. The ‘what if’s are endless.

Many of my students will continue with this exercise beyond class, through drawings, foldings, and cut paper. There is a great simple foldable here that actually incorporates cuts to create a 3D effect. It’s a great way to jumpstart paper folding and cutting shapes to students that are not accustomed to paper-folding activities! I want my students to make graphic organizers that are beautiful to look at. Notes as art… What a concept!

Notes as Art… What a concept!

As teachers, we have authority…

As teachers, we have the authority to hold our students accountable for their learning. We do these children no favors by feeding them the answers or rewarding them for less than stellar work.

As someone with authority, we are charged also to show our students how to take on that authority in their own life and toward their own learning. Giving students choice and voice is part of that transference of authority. But:

Only those with authority can convey authority to others.

We must first model that which we wish to convey.

How do you handle your authority? Do you give it away by allowing students to misbehave? Do you hold students accountable for missed work? For making up the work missed when they are absent? Do you hold them responsible for deadlines?

Children know instinctively who is in charge. They see our weaknesses and exploit them. They respect authority– well used, not that which is authoritarian!

Over time, after days and days of class and expectation, they will begin to reflect that which you give them. You may not see it, but others will.

My son was always rebelling against my authority, at home. But when speaking with others, I was told of his joy, his politeness, his willingness to help other adults and friends with the chores he did not willingly do at home.

Model authority. Pass this torch onto your students by expecting them to respect your authority, and you will help the next generation to understand what it means, and how to take up its mantle responsibly.

I know that giving students choices in learning works precisely because we are passing on authority!

Could this be why formal “college seat time” is going away, and has already begun being replaced by things like certificate training, MOOCs, the rise of information available on the Internet, and why a resurgence in apprenticeships is already on the horizon (it just looks different, because entrepreneurs are the new apprentices)!

The title is a quote from a treatise on Tim Urban’s blog  about Elon Musk. Urban is trying to understand Musk’s success by examining how he thinks, and how Musk’s thinking affects his choices. Read the full 4-part series here.

As a teacher, and as someone who reads widely on any subject I feel I need to know more about, the title quote makes powerful sense. Author Urban calls this ‘first principles’:

‘A scientist gathers together only what he or she knows to be true—the first principles—and uses those as the puzzle pieces with which to construct a conclusion.’ *

For me, first principles is something I’ve always practiced. I just didn’t call it that. I love to learn, to read, to gather information, and test it out against what I know. I learned early not to take what others say without checking and confirming- maybe because as a kid I was rather rebellious (my dad says I have to learn everything the hard way!), maybe because my naïveté allowed me to be made to feel stupid, and I do not like to feel stupid! As I read further into the analysis of Musk, I found a companion to my own thought processes, albeit worded differently than I would have ever thought:

‘ Musk sees… his brain software as the most important product he owns—and since there aren’t companies out there designing brain software, he designed his own, beta tests it every day, and makes constant updates. That’s why he’s so outrageously effective, why he can disrupt multiple huge industries at once, why he can learn so quickly, strategize so cleverly, and visualize the future so clearly.’ *

Urban’s statement about this awakens my “spidey sense”** as a teacher. All of us have hardware (our physical sense) and software (our brains). As a teacher, I feel strongly that this is the goal of my interaction with my kids: to teach them how to learn quickly, to strategize cleverly, and to visualize the future clearly. This is where success, innovation, and fulfillment (yes- doing that which we are so passionate about) must spring from. Our world’s future is at stake if we don’t bring out in our children these abilities.

‘When your childhood attempts at understanding are met with “Because I said so,” and you absorb the implicit message “Your own reasoning capability is sh*t, don’t even try, just follow these rules… ,” you grow up with little confidence in your own reasoning process. When you’re never forced to build your own reasoning pathways, you’re able to skip the hard process of digging deep to discover your own values and the sometimes painful experience of testing those values in the real world and learning you want to adjust them—and so you grow up a total reasoning amateur.’*

Teaching our children the value of ‘first principles’is critical in education. It’s what our children lack: the need to know for themselves, and not rely on what they are told by others: what ‘conventional wisdom’ says must be true.

‘A command or a lesson or a word of wisdom that comes without any insight into the steps of logic it was built upon is feeding a kid a fish instead of teaching them to reason. And when that’s the way we’re brought up, we end up with a bucket of fish and no rod—a piece of installed software that we’ve learned how to use, but no ability to code anything ourselves.’*

My need for ‘first principles’ has been leading me to those ideas that will train my students for their future- one that will require them to code for themselves, going beyond ‘established’ wisdom’, carving out innovative solutions, and finding a future that enthralled and fascinates them. Without this, it becomes easier for a population to become manipulated by a leadership that does not have their best interests at heart. For more on this idea, check out How the Oil Industry Conquered Finance, Medicine, and Agriculture, by James Corbett. In it, Corbett references an essay by Frederick T. Gates, the man intimately connected to the origins of public schooling as we know it. The essay, The Country School of Tomorrow, Gates lays out his plan for education,

‘… we have limitless resources, and the people yield themselves with perfect docility to our molding hand… We shall not try to make these people or any of their children into philosophers or men of learning or of science. We are not to raise up from among them authors, orators, poets, or men of letters. We shall not search for embryo great artists, painters, musicians. Nor will we cherish even the humbler ambition to raise up from among them lawyers,
doctors, preachers, politicians, statesmen, of whom we now have ample supply.’*

From my view, I think we have chillingly succeeded (thanks to Rockefeller and his billions), in the first quoted section of this essay. I look with hope on what is happening as teachers everywhere have begun to break with tradition and seek to teach for understanding, and to teach students to think independently. Read a little further down Mr. Gates’- Frederick’s, not Bill’s- essay, to find the sweet hope and goal of education that  somehow became lost in the ‘monetization’ of public schooling,

‘…all that we shall try to do is just to create presently about these country homes an atmosphere and conditions such, that, if by chance a child of genius should spring up from the soil, that genius will surely bud and not be blighted.’

All of our children are budding geniuses- in their own time and in their own way. Finding and nurturing that bud is my ‘action plan’. I think I’ll continue upgrading my ‘software’ in pursuit of being the most effective teacher I can be, sharing what I find, in order to improve our education system in whatever way I am able. Excuse me while I go fill my ‘goal pool’ so that I can plan my ‘strategies’ to maximize my ‘experience’ and ‘feedback loop’ so that I can move my ‘goal attainment mechanism’ forward.

Let’s keep the conversation going!

*The Cook and The Chef: Musk’s Secret Sauce, T. Urban, Wait, Why Not

**No, I was never bitten by a radioactive spider. I am an intuitive woman (which some of you will no doubt see as a redundant phrase…)

‘One colleague suggested turning to the calculator and using the answers as an investigation. Why does the calculator give this answer? What rules is it following? Can you write a set of rules? What would the calculator say for this sum?’by mrbartonmaths

Student misconceptions are critical to planning successful lessons! In a recent series of posts (I think he is up to 11 posts now) mrbartonmaths explains how he uses diagnostic questions to delve into the misconceptions his students have about basic arithmetic.

The quote above came from #10 in the Insight of the Week series: order of operations. As mrbartonmaths explains,

‘…the misconceptions I think students hold are different to the ones they actually make, and I want to put this to the test on a larger scale.’

While I found all of the responses interesting, I was surprised by the number of responses that indicated students were trying to place parentheses (brackets*) into the problem where none existed! These students were trying to make sense of the problem using familiar notation. The only problem I saw is that students didn’t know the ‘rules’ of brackets!

My comment to mrbartonmaths:

Teach them the rules for brackets!

To return to the thought at the beginning of this little essay, ‘have them use the calculator to evaluate the rules the calculator is using’ would require students to identify and investigate their own misconceptions- an idea I find ultimately rewarding!

For a great interactive lesson- which includes some much needed awareness of when parentheses are needed- try this Make This Number game.

As a further step in ‘teachers as lifelong learners’, I love that mrbartonmaths has embarked on something he calls Guess the Misconception, an email poll he sends out weekly to those who are signed up. What misconceptions are you holding about your student’s misconceptions!?!

I am reminded of the time I asked my Algebra I student, who was having a lot of trouble solving basic algebra problems in one variable (3x + 7 = 13 for example), why he kept wanting to start with 3x first. I had spent time working with him on the ‘unwrapping’ idea, without success. He pointed out that he was dividing by 3 because it was first. Headsmack! (Me, not him!)

Don’t assume! (You remember what that does, right? Makes an a– out of u and me!)

Thank you, mrbartonmaths, for giving us a little bit more insight (and some great ideas) into best teaching practices!