Category: Lessons

Teaching the “Distance Formula” with PBL

As I write curriculum, I am constantly scouring the Internet for ideas and ways to improve my own work, as we all do.  I was just on the NCTM resources page the other day at their “Reasoning and Sense Making Task Library” and found this description of a task called “As the Crow Flies”:

“The distance formula is often presented as a “rule” for students to memorize. This task is designed to help students develop an understanding of the meaning of the formula.”

OK, wait – shouldn’t this just say, we shouldn’t be presenting the distance formula as a rule for students to memorize?  Instead we should be teaching it for understanding from the conceptual level and allowing students to realize the connection between the Pythagorean idea of distance and how it allows a student to find the distance between two points?  Why should we have a specific task designed to create the understanding after learning the formula when the formula is actually secondary?

There is a series of questions in the problems I have written/edited that allow students to come to this realization on their own.

First, a few basic Pythagorean Theorem problems to practice the format, remind themselves of simplifying radicals, Pythagorean Triples, etc.  Second, some coordinate plane review such as:

  1. Given A=(5,-3) and B=(0.6). Find the coordinates of a point C that makes angle ACB a right angle.

This is really an interesting discussion question for many reasons.  First many students have trouble understanding where the right angle is supposed to be. If they incorrectly read that the angle that should be right is ABC, then they are picturing a different right angle (and also doing a harder problem that we’ll get to soon!) but if they are reading ACB, it’s still an interesting question because there is more than one answer.

Students can sometimes visualize where the right angle can be (even both of the points) but may not be able to get the coordinates.  This discussion is important however because in order to come up with the distance formula later in general (with the x’s, y’s and subscripts – whoa, way confusing!) they need to realize what’s so special about that vertex’s coordinates. So if there is a student who is confused I usually ask the student presenting this problem, “Can you describe the way you found the coordinates for C?” Their answer usually goes something like this: “You just take the x of the one it’s below and y of the one it’s next to.” and other kids are either totally on board, or totally confused.  So then they need to make it a little more mathemtical so every else is on board.  Other kids often chime in with words like horizontal and vertical, x-coordinate and y-coordinate.  This is a really fun, useful and fruitful mathematical discussion in my experience.

We can then move to a problem like this:

2. Find the length of the hypotenuse of a right triangle ABC, where A = (1,2) and B = (5, 7). Give your answer is simplest radical form.

This is generally a problem that is given to students individually to grapple with for homework or in class in groups at the board.  After doing the one discussed above, they at least are prepared to find the vertex of the right triangle and see where it should be.

It’s honestly rare that a student can’t even draw the diagram – especially if they can make the connection with the previous problem. (Connection is one of the four pillars of the PBL Classroom).  One of the things that is often difficult for students is the idea of subtraction of the coordinates.  The can easily count the units to get the sizes of the legs in order to do the Pythagorean Theorem, but in order to generalize, for a later purpose…sorry, don’t want to steal the thunder…subtraction would be an interesting alternate solution method if someone comes up with it – and they usually do.

At this point if someone does come up with it, I usually do ask why can you subtract the coordinates like that to get the lengths of the sides and (you guessed it) there was an earlier problem that had student finding distance on a number line, so, many kids make that connection.

So finally we get to this, maybe a couple of problems later:

Again, students are asked to use their prior knowledge and contemplate a way that they might be able to describe of finding a way to express the distance between two points in a plane.  This is after discussing notation, discussing how to visualize that distance, discussing subscripts, and discussing the purposes (in other problems) of why we might actually need to find the distances between two points. Because the Pythagorean Theorem squares the lengths of the sides (BC and AC) I’ve never had a kid get all upset about the fact that we don’t put the absolute value signs around the difference for the sides – we’re gonna square it anyway, so who cares if it’s negative?  Kids usually say, “if it’s negative, let’s just subtract the other way and it’s be positive.” We just get right to the point that all we are finding is the hypotenuse of a right triangle which has been the Pythagorean Theorem all along.

I generally have students write a journal entry about this amazing revelation for them at the beginning of the year and voila!  It’s right there for them, in their journal for the whole year – no memorization needed.  They understand the concept, know how to use it and actually love the idea because now they can just see a right triangle every time they need a distance.  It’s how so many of my students say that have “never learned the distance formla” – they just use the Pythagorean theorem to find distance.  I love it.

End of Term Reflections

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Phew…exams given…check…exams graded…check…comments written…check…kids on bus…check.  Now I can relax.  Oh wait, don’t I leave tomorrow to drive to my sister’s for Thanksgiving?

Such is the life of a teacher, no?  Just when you think you are on “vacation” there’s always something else to do.  I had an exam on Saturday then worked the rest of Saturday and Sunday finishing up that grading and writing my comments that were due this morning at 9 am.  But wait, I told some people I would write a blogpost about what my classroom is like, so I really wanted to do that too.  That’s OK though, I think it’s important for me to reflect back on this fall term – what worked and what didn’t for my classes.

I have three sections of geometry this year that I teach with PBL and a calculus class that I would say is something of a hybrid because we do have a textbook (as an AP class I needed to do what the other teachers were doing), but I do many problems throughout the lessons.

In my geometry classes, the student have iPads on which they have GeoGebra, Desmos and Notability where they have a pdf of their text (the problems we use) and where they do all of their homework digitally.  My class period for that course alternate between small group discussions in the Innovation Classroom in the library on Mondays and Thursdays and whole class discussions with student presentations of partial solutions (a la Jo Boaler or Harkness) on Tuesdays and Fridays. (We meet four times a week 3 45-minute periods and 1 70-minute period.)  Because my curriculum is a whole-curriculum PBL model, we spend most of the time discussing the attempts that the students made at the problems from the night before.  However, in class the discussion centers around seeing what the prior knowledge was that the presenter brought to the problem and making sure they understood what the question was asking.

classroom-shot1
Whole Class Discussion in regular classroom

 

geom-class-2
Small Group Discussions in Innovation Lab

If this didn’t happen we end up hearing from others that can add to the discussion by asking clarifying questions or connecting the question to another problem we have done (see Student Analysis of Contribution sheet).

One of the things that I had noticed this fall in the whole class discussion was that the students were focusing more on if the student doing the presentation was right immediately as opposed to the quality or attributes of the solution method.  There was little curiosity about how they arrived at their solution, the process of problem solving or the process of using their prior knowledge.  Unfortunately, it took me a while to figure this pattern out and I felt that it had also weeded itself into the small group discussion as well.

One day in the small group discussions, it became clear to me that the students were just looking for the one student who had the “right” answer and they thought they were “done” with the question.  This spurred a huge conversation about what they were supposed to be doing in the conversation as a whole.  I felt totally irresponsible in my teaching and that I had not done a good enough job in describing to them the types of conversations they were supposed to be having.

This raised so many questions for me:

  1. How did I fail to communicate what the objectives of discussing the problems was to the students?
  2. Why is this class so different from classes in the past (even my current period 7 class)?
  3. How can I change this now at this point in the year?
  4. How can I stress the importance of valuing the multiple perspectives again when they didn’t hear it the first time?

In my experience, sometimes when students are moving forward with the fixed mindset of getting to the right answer and moving on, it is very difficult to change that to a more inquiry-valued mindset that allows them to see how understanding a problem or method from a different view (graphical vs algebraic for example) will actually be helpful for them.

My plan right now is to start the winter term with an interesting problem next Tuesday.

“A circular table is pushed into the corner of the room so that it touches both walls. A mark is made on the table that is exactly 18 inches from one wall and 25 inches from the other.  What is the radius of the table?”

table-picture-problem

I have done this problem for many years with students and I have found the it works best when they are in groups.  I usually give them the whole period to discuss it and I also give them this Problem Solving Framework that I adapted from Robert Kaplinsky’s wonderful one from his website.  I am hoping to have a discussion before they do this problem about listening to each other’s ideas in order to maximize their productivity time in class together.  We’ll see how it goes.

Modeling with Soap Bubbles

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I am so very lucky to have a guest teacher with me this year at my school.  Maria Hernandez (from the North Carolina School of Science and Math) is probably one of the most energetic and knowledgeable teachers, speakers and mathematicians you could ever find – and we got her for the whole year!  We are so excited.  I am working with her and she is so much fun to work with.  I have been teaching calculus with PBL for almost 20 years now and thought I had all the fun I could but no!  Maria is bringing modeling into my curriculum and I’m enjoying every minute of it.

As we started teaching optimization this week, Maria had this wonderful idea that she had done before where we want to find the shortest path that connects four houses.

picture-of-houses

I let the kids play with this for about 10 minutes and then did this wonderful demonstration with some liquid soap bubbles and glycerin.  We had two pieces of plastic and four screws that represented the houses.  As the kids watched, I dipped the plastic frame into the liquid and voila-file_000

Right away the students saw what they were looking for in the shortest path.  Now they had to come up with the function and do some calculus. As they talked and worked in groups, It was clear that using a variable or one that would help them create the right function was not as easy as they thought.  However,  I was requiring them to write up what they were doing and find a solution so they were working hard.

file_000-1

We have been doing a lot of writing in Calculus this fall so far and they are getting used to being deliberate about their words and articulating their ideas in mathematical ways.

Here is the outline of the work they did in class: Shortest Path Lab

and here is the rubric that I will be using to grade it.

rubric-for-lab-3-2

The engagement of students and the buzz of the classroom was enough to let me know that this type of problem was interesting enough to them – more than the traditional “fold up the sides of the box.”  The experience they had in conjecturing, viewing, writing the algebra and solving with calculus was a true modeling experience.

If you decide to do this problem or have done something like it before, please share – I’d love to do more like this.  I am very lucky to have a live-in PD person with me this year and am grateful every day for Maria!

 

What is “Low Ability” Anyway? Comparing a Point to “Room”

One of my big “beefs” at my school is the fact that we have three levels of tracking – count ’em, three.  There’s the honors track, that of course at a college prep school, most kids think they belong in.  There’s the regular track, that which is still pretty quick and difficult, and there’s the track that the kids who are sometimes, I would say, just not very motivated to learn math, or have less interest in math, or maybe come from a school with a less rigorous math program, are placed in.  These are the kids who probably all their lives have been told they are not “math people” and have been pigeon-holed as an “artist” or “writer” so they won’t actually need math when they get older.  This really, really irks me.  But I do it – I go along with a system that has been in place way before I got here.  I’m only one person – even though I cite Jo Boaler’s list of research showing why “tracking” in general is just a bad idea and hurtful – I know I can’t win.

Anyway, I really shouldn’t complain because the department has let me do my thing with the geometry curriculum and I have written a PBL curriculum for the three levels.  In my 201 book, I have created scaffolded problems that I think work really well with these “low-abiity” kids and often challenge them enough to make them realize how much ability they actually have.  We just started talking about dimension and we watch these clips from Flatland: The Movie, where Arthur T. Square meets the King of Pointland and then meets the King of Lineland.

We had a great conversation about why the King of Pointland keeps saying “Hello Me, Hello Me” and can’t really understand why there’s anyone else there.  We talked about why the King of Lineland doesn’t understand where the Square is because he only understands the directions of left and right.  One of the kids goes, “Is this kind of like what happened in the movie Interstellar? I think he went through a black hole and just appeared in the future or something.” Now, I hadn’t seen that movie but then another kid said, “Well, I’m not sure it was like that.” But then one of the other students says, “No, the King of Pointland is kind of like the kid in the movie Room.  Did you see that movie?”  I nodded in understanding and so did many of the others in class. The student went on, “In Room, the little boy grew up thinking that “Room” was his whole universe so that was all he understood, and that’s why the King of Pointland seems so nuts. That point with no dimensions is all he can understand – there’s no one else in the world.”  I was so blown away by that analogy.  She really had an understanding of the idea of the limitations of being alone in the universe of a point. I had never had a kid in a “regular” or “honors” class make a connection like that – but then again, Room just came out!

Getting Kids to Drive the Learning

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It doesn’t always work this way, but it would be awesome if it did.  When PBL is perfect or ideal, the students are the ones who make the natural connections or at least see the need or motivation for the problems that we are doing.  Yeah, some of them are just really interesting problems and the get pulled in by their own curiosity, but as all math teachers know, we have a responsibility to make sure that students learn a certain amount of topics, it is quite that simple.  If students from my geometry class are going into an algebra II class with trigonometry the next year where their teacher will expect them to know certain topics, I better do my job and make sure they have learned it.

So how do I, as a PBL teacher, foster the values for the problem-based learning that I have while at the same being true to the curriculum that I know I have a responsibility to?  This is probably one of the biggest dilemmas I face on a daily basis.  Where’s the balance between the time that I can spend allowing the students to struggle, explore, enjoy, move through difficulty, etc. – all that stuff that I know is good for them – while at the same being sure that that darn “coverage” is also happening?

So here’s a little story – I have a colleague sitting in on my classes just to see how I teach – because he is interested in creating an atmosphere like I have in my classes in his.  We have just introduced and worked on problems relating to the tangent function in right triangle trigonometry in the past week and now it was time to introduce inverse tangent.  I do this with a problem from our curriculum that hopefully allows students to realize that the tangent function only is useful when you know the angle.FullSizeRender (3)

So as students realize they can’t get the angle from their calculator nor can they get it exactly from the measurement on their protractor (students had values ranging from 35 to 38 degrees when we compared), one of the students in my class says, “Ms. Schettino, wouldn’t it be great if there was a way to undo the tangent?” and the other kids are kind of interested in what she said. She continues, “Yeah, like if the calculator could just give us the angle if we put in the slope.  That’s what we want.”  I stood there in amazement because that was exactly what I wanted someone to crave or see the need for.  It was one of those “holy crap, this is working” moments where you can see that the kids are taking over the learning.  I turned to the kids and just said, “yeah, that would be awesome, wouldn’t it?  Why don’t you keep working on the next problem?” and that had them try to figure out what the inverse tangent button did on their calculator.  They ended up pressing this magical button and taking inverse tangent of 0.75 (without telling them why they were using 0.75 from the previous problem) to see if they could recognize the connection between what they had just done and what they were doing.

At the end of the class, the colleague who was observing came up to me and said, “How did you do that?” and I said, “What do you mean?” and he said, “How did you get the kids to want to learn about inverse tangent? I mean they just fell right into the thing you wanted them to learn about.  That was crazy.”  I really had to think about that.  I didn’t feel like I did anything honestly, the kids did it all.  I mean what made them all of a sudden care about getting the angle?  Why were they invested?   It doesn’t always happen in my classroom that’s for sure.  This is not a perfect science – there’s no recipe for it to work – take a great curriculum, interested kids, an open, respectful learning environment and mix well?

I do think however that a huge part of it is the culture that has been created throughout the year and the investment that they have made in their ownership and authorship in their own learning. We have valued their ideas so much that they have come to realize that it is their ideas and not mine that can end up driving the learning – and yes, I do end up feeling a little guilty because I do have a plan.  I do have something that I want them to learn, but somehow have created enough interest, excitement and curiosity that they feel like they did it.  It is pretty crazy.

Everything Old is New Again…(or why teaching with PBL is so great)

So I heard that what everyone is saying about the new Star Wars Movie, The Force Awakens, is that “Everything Old is New Again” – go ahead google it, there are at least 5 or 6 blog posts or articles about how “BB-8 is the new R2D2” or “Jakku is the new Tattoine” or whatever.  I actually don’t have a problem with J.J. Abrams reusing old themes, character tropes or storylines because I think that really great stories are timeless and have meaning and lessons that surpass the movie that you are watching.  I still thought it was awesome.

This concept of everything old is new again really hit home to me today in my first period class when I was having the students do a classic problem that I probably first did in 1996 while I was under the tutelage of my own Yoda, Rick Parris (who I think wrote the problem, but if someone reading this knows differently, please let me know).  The problem goes like this:

Pat and Chris were out in their rowboat one day and Chris spied a water lily.  Knowing that Pat liked a mathematical challenge, Chris announced that, with the help of the plant, it was possible to calculate the depth of the water under the boat.  When pulled taut, directly over its root, the top of the plant was originally 10 inches above the water surface.  While Pat held the top of the plat, which remained rooted to the lake bottom, Chris gently rowed the boat five feet.  This forced Pat’s hand to the water surface.  Use this information to calculate the depth of the water.

What I usually do is have students get into groups and put them at the board and just let them go at it.  Today was no exception – the first day back from winter break and they were tired and not really into it.  At first they didn’t really know what to draw, how to go about making a diagram but slowly and surely they came up with some good pictures. Some of the common initial errors is not adjusting the units or mislabeling the lengths.  However, one of the toughest things for students to see eventually is that the length of the root is the depth of the water (let’s call it x) plus the ten inches outside of the water’s surface.  Most students end up solving this problem with the Pythagorean Theorem – I’ve been seeing it for almost 20 years done this way.  Although I never tire of the excitement they get in their eyes when they realize that the hypotenuse is x+10 and the leg is x.

However, since everything old is new again, today I had a student who actually is usually a rather quiet kid in class, not confused, just quiet, but in a group of three students he had put his diagram on a coordinate plane instead of just drawing a diagram like everyone else did.  This intrigued me.  He initially wrote an equation on the board like so:

y= 1/6 (x – 0)+10

and I came over and asked him about it.  He was telling me that he was trying to write the equation of one of the sides of the triangle and then I asked him how that was going to help to find the depth of the water.  He thought about that for a while and looked at his partners. They didn’t seem to have any ideas for him or were actually following why he was writing equations at all.  He immediately said something like, “Wait, I have another idea.” and proceeded to talk to his group about this:

Jacksons solution to Pat and Chris
Jackson’s Solution to the Pat & Chris Problem

He had realized from his diagram that the two sides of the triangle would be equal and that if we wrote the equation of the perpendicular bisector of the base of the isosceles triangle and found its y-intercept he would find the depth of the water.  He proceeded to find the midpoint of the base, then the slope of the base, took the opposite reciprocal and then evaluated the line at x=0 to find the y-intercept.  I was pretty impressed – I had never seen a student take this perspective on this problem before.

This made my whole day – I was really dreading going back to work after vacation and honestly, first period was the best class of the day when this wonderful, new method was shown to me and this great experience of this student’s persistence refreshed my hope and interest in this problem.  Perpendicular bisectors are the new Pythagorean Theorem!

One of the Original “Makers”

Apologies to any faithful readers out there – I have had a heck of a summer – way too much going on.  Usually during the summer, I keep up with my blog much more because I am doing such interesting readings and teaching conferences, etc. (although I’m running a conference for the first time in my life!) However, this summer I was dealing with one of my biggest losses – the passing of my father after his 8 year battle with breast cancer.  I thought I would honor him by writing a post talking about a problem that I wrote a few years ago, well actually a series of problems that utilized his work when teachers of algebra I asked me how I taught the concept of slope.  So dad, this one’s for you.

In 1986, my dad, Francesco (Frank) Schettino, was asked to work on the renovations for the centennial project for the Statue of Liberty.  He was a structural steel detailer (also known as a draftsman) but he was really good at his job.  Everywhere we went with my dad when I was younger, he would stop and comment about the way buildings were built or if the structure of some stairs, windows or door frames was out of wack.  He could tell you if something was going to fall down in 10 years, just by looking at it.  At his wake last week, one of the project managers from a steel construction company that he worked on jobs for told me that they would save the interesting, most challenging jobs for him because they knew he would love it and do it right.
photo (1)I remember sitting with my dad at his huge drafting desk and seeing the drawings of the spiral stairs in the Statue of Liberty.  He talked to me about the trigonometry and the geometry of the circles that were necessary for the widths that were regulated for the number of people that they needed to walk up and down the stairs.  This all blew my mind at the time – that he needed to consider all of this.  So to be able to write problems that introduce slope to students about this was just a bit simpler to me.

If you take a look at my motivational problems on slope and equations of lines I believe it’s numbers 2 and 3 that refer to his work (excuse the small typo).  Over the years I’ve meant to go back and edit these a number of times.  If you are someone who has taken my course at the Anja S. Greer Math, Science and Technology Conference at Exeter, you are probably familiar with this series of questions because we have discussed these at length and talked about how students have reacted to them (and how different adult teacher-students have as well).  We have assumed no prior knowledge of slope (especially the formula) or the terminology at all.

Some questions that have come up: (with both students and the teacher-students I’ve worked with)

1. What does a graphical representation of “stairs” mean to students?
2. What does “steeper” mean and what causes stairs to be steep?
3.  Why are we given the “average” horizontal run for the spiral stairs? Would another measurement be better?
4. Why does the problem ask for the rise/run ratios?  Is there a better way to measure steepness?
5. (from a teacher perspective) why introduce the term “slope” in #3? can we just keep calling it steepness?

These are such rich and interesting questions. The questions of scaffolding terminology and when and how to introduce concepts are always the most difficult.  Those we grapple with specifically for our own students.  I always err on the side of allowing them to keep calling it steepness as long as they want, but as soon as we need to start generalizing to the abstract idea of the equation of the line or coming up with how to calculate that “steepness” a common language of mathematics will be necessary.  This is also where I take a lesson from my dad in terms of my teaching.  His great parenting style was to listen to me and my sisters and see where we were at – how much did we know about a certain situation and how we were going to handle it.  If he felt like we knew what we were doing, he might wait and see how it turned out instead of jumping in and giving advice.  However, if he was really worried about what was going to happen, he wouldn’t hesitate to say something like “Well, I don’t know…”  His subtle concern but growing wisdom always let us know that there was something wrong in our logic but that he also trusted us to think things through – but we knew that he was always there to support and guide.  There’s definitely been a bit of his influence in my career and maybe now in yours too.

Spring has Sprung – and so has the French Garden!

So the spring term means two things for my Honors Geometry kids – the technology inquiry project and looking at the French Garden Problem.  So for those of you who are not familiar with both of those I’ll try to quickly fill you in while I talk about how they just happen to so coolly (is that an adverb?  if not I just made it up) overlapped this week.

My Spring Term Technology Inquiry Project is something I came up with three years ago when I really wanted a way to push my honors geometry students into thinking originally while at the same time assessing their knowledge of using technology.  I did a presentation last year at the Anja S. Greer Conference on Math, Science and Technology and the audience loved it.  Basically, I give students an inquiry question (one that I attribute to my good friend Tom Reardon) that they have to work on with technology and then they have to come up with their own inquiry question (which is, of course, the fun part) and explore that with technology and/or any other methods they wish.  I have received some pretty awesome projects in the past two years and I don’t think I am going to be disappointed this year either.

The French Gardener Problem is famously used in my PBL courses at the MST Conference as well.  Everyone who has taken my course knows the fun and interesting conversations we have had about the many ways to solve it and the extensions that have been created by many of my friends – an ongoing conversation exists somewhere in the Blogosphere about the numerous solutions – In fact Tom sent me a link just last fall to a more technological solution at Chris Harrow’s blog. (We’re such geeks).  Great math people like Phillip Mallinson and Ron Lancaster have also been drawn in by the attractive guile of the The French Gardener Problem.  In this problem, the main question is what fraction of the area of the whole square is the octagon that is formed inside (what is the patio for the garden)?

So the other night, after we had worked on this question in class for a couple of days and the students had meet with me in order for me to approve their original inquiry question, a student stops by to discuss his question.  John starts off with, “I can’t think of anything really. What I had wanted to do, someone else already claimed.” (I’m not letting them do a question that someone else has already decided to look into.  So John sits in my study and thinks for a while. I told him that this part of the project was supposed to be the fun part.  I gave him some thoughts about extending some problems that he liked.  He said he had liked the French Garden Problem and thought it was really cool.  So I went back to some of my work and he started playing with GeoGebra.  Before I knew it he starts murmuring to himself, “Cool, cool….Cool! It’s an octagon too!”  I’m thinking to myself, what has he done now?  I go over to his computer and he’s created this diagram:

John's Original Inquiry Question
John’s Original Inquiry Question

I’m asking him, “What did you do? How did you get that?”  He says that he just started playing with the square and doing different things to it and ended up reflecting equilateral triangles into the square instead of connecting the vertices with the midpoints as in the original French Garden Problem.  Then he started seeing how much of the area this octagon was and it ended up that it was……you don’t think I’m going to tell you, do you?

Anyway, it just made my night, to see the difference in John when he came by and the by the time he left.  He was elated – like he had discovered the Pythagorean Theorem or something.  I just love this project and I would encourage anyone else to do the same thing.  Leave a comment if you end up doing it because I love to hear about any improvements I could make.

Succeeding at Helping Students to Fail?? Part 1: Meaning

Apologies faithful readers – those of you who know me well, know that I have been dealing with a great deal of personal issues and preparing for the summer PBL Math Teaching Summit, so I have taken a small hiatus from blogging for a while.  However, with that under control for now, I turn to reflecting on something that happened in class the other day and its relation to a great article I retweeted that was on TeachThought’s website the other day entitled Helping Students Fail.  I have been giving a lot of thought this year to the idea of Grit and Problem-Based Learning which has intrigued me for a while.  However, this article is one of the few I’ve seen that really speaks to some concrete steps that teachers can take to aid students on the journey of dealing with making mistakes and viewing them in a positive light.

I love the framework that the author gives here:

http://www.teachthought.com/teaching/the-role-of-failure-in-learning-helping-students-fail/
Helping Students Fail: A Framework by Terry Heick

Breaking the struggle into these four aspects of learning is very interesting to me (of course with respect to the PBL Classroom).  It dawned on me while reading this article that this is a continuous and completely ongoing process of learning to fail that happens.  It is so ubiquitous that the teacher and students are probably not even aware of it (or are so aware of it that that’s where the discomfort is emanating from).  It is so ubiquitous that I needed this framework for me to be able to even have it spelled out for me.

1. Meaning: In the PBL classroom, meaning is shaped everyday – the explicit separation between knowledge and performance is spelled out in discussion and the way students are asked to share their attempts at problems.  Jo Boaler might have spelled it out best in her paper desribing the Dance of Agency, where she explained the importance of sharing what she called “partial solutions.”  Using this language is really important to make sure that students don’t feel the need to have a complete solution when they present (because no matter how many times I say it, they still say, “Is it OK if it’s wrong/”)  In their mind, they feel their presentation is a performance.  However, the other day I had an interesting experience while students were presenting.  We were doing this problem in class and I had assigned two girls to present their ideas together:

A triangle has sides measure 9, 12 and 15 (what’s special about this triangle?).  Find the distances to the centroid from all three vertices.

The day before we had done a problem very similar to this with an equilateral triangle of sidelength 6 and the presenter had realized that he could connect this problem to the work we were doing with 30-60-90 triangles.  He then applied the Centroid Theorem which states that the centroid is 2/3 of the way from the vertex along the median.  So when the girls presented, they did this:

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They realized that the median from A was the hypotenuse of a right triangle and they could find its length with the Pythagorean Theorem. They then used the Centroid Theorem and found 2/3 of it. However, next, they did this:


FullSizeRender (2)It was great that they connected this problem to the previous day’s presentation where all of the distances were the same (I’m always asking them to look for connections). However, when I asked them the question of whether they expected those distances to all be equal, they had to think about that. We put the question out to the class and it started a great discussion about why sometimes they were the same and sometimes they weren’t. I won’t go into the whole solution here since the correct answer is not the point of this blogpost but what happened that evening is.

Later on that night, I received an email from one of the girls who was part of the presenting team. At the end of class, I had noticed that she seemed very quiet and I had asked her if she was confused about something else we were discussing towards the end of the class when the bell rang. She had said no and left class very awkwardly.

This is what she wrote to me:

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I had been working so hard to make students feel comfortable making mistakes that I wasn’t paying attention to who had made the mistakes and that they were actually comfortable making the mistakes and proud of making those mistakes and wanted credit for making those mistakes! I was dumbfounded. I just couldn’t believe it. My perception of (at least) this student’s ability to be comfortable with being wrong was so different than what her’s was. She was proud that her “mistake was a good one” and not just a “silly error” and I needed to give her the credit she deserved for taking a risk. I learned such a great lesson from this student on this day and I owe her so much (and don’t worry, I told her that in an email response)!

The separation between knowledge and performance has been made clear to at least some of my students and I am going to keep doing what I’m doing in the hope of getting this message to all of them.

What I learned over my NCTM break!: Part 1

Wow!  What an amazing three days I spent at the NCTM annual conference in New Orleans!  I can’t believe how much I learned (which actually never amazes me and always humbles me – one of the many reasons I love going to these conferences.)  I also hate leaving and knowing that I missed at least 20 sessions that conflicted with ones that I did go to, so now I’m catching up and trying to email the speakers that I didn’t quite get to see or get in contact with while there.

One of my major a-ha moments was in Gail Burrill’s session on logarithms.  You’d think that after 25 years of teaching that you’d understand how much you understand about logs right? Oh, no!  So she had us all have a very large number and we were doing an exercise where we had to put a post it note with that number (mine was 72, 753) on a scale of powers of 10 {10, 10^2, 10^3, 10^4, 10^5…}, her argument being that one of the main reasons to teach logs is to have a different scale for very large numbers.  So after all of these teachers did this, we analyzed where all of our numbers were on the scale – particularly between these numbers.  Since my number should’ve been between 10^4 and 10^5, I knew I put it in the right place – but oh no, I had it in the wrong place relative to the middle.  She asked us to calculate the middle of those two – 10^4.5 which was 31,622 and yes, I admit that’s very close to where I put my post-it.  I could blame the person who put their’s up first which said 75,289 and I just put mine by there’s but I won’t.  I just didn’t really think.  But I know this was a light bulb moment for many of the teachers in the room.  Students don’t really understand how a logarithm is an exponent in the first place and we were doing this exercise without even using the word “logarithm.”

Then we went down to the section below that was between 10^3 and 10^4 and checked some of those numbers.  They were very off too and Gail asked us what number we expected to be in the middle.  At this point, some of us pulled out our calculator (yes, I admit, I did) but some of the smartees in the room just said “3,162” and I finally got it.  By just dividing by 10 and looking at the scale in this nonlinear way, students would be able to make the connection between the algebraic properties of exponents and what a logarithm was.  I thought this was an amazing way to introduce logs.  Has anyone done this before?  Thanks so much Gail!!  I think I’m going to write a problem for my curriculum about this, it’s such an insightful experience.

More reflections to come – just can’t do it all at once – catching up on school work!