PBL for Engagement and Empowerment in the Mathematics Classroom

Well, to everyone who has ever read my blog, I owe a debt of gratitude. I have been away for a while thinking of many different ways to go with this. I am focusing a great deal right now on my graduate studies, which are leading me in various paths. However, I’ve recently found out that I was granted a sabbatical this coming fall and will be (hopefully) finishing up my course work, which will allow me to begin my dissertation process.

What has started me on the path of writing again is, interestingly enough, questions of pedagogy and technology. For a long time now I have attempted to create a website that connected with this blog, the work I do with professional development and consulting with schools and teachers, my work with Problem-Based Learning, and my research interests. However, I think my problem right now is that all of those things are so varied that I am not in a good place to have some focus quite yet.

So in preparing for my work at my most favorite summer conference again this June, I am once again directing myself towards questions of technology. [Don't worry, fans of my pedagogical musings, I will soon get into that idea again..]

This past October, I went to a one-day seminar run by a large computer company, which is named after a fruit. As a huge national organization, it was pretty clear that they were a smoothly run ship, which had it’s order that it usually followed. As I checked in, I was handed a beautiful hard-cover notebook and pen, with their fruit emblem on the cover and a great handheld device that was mine to use for the day. I was truly impressed. I went in and sat down for the introductory address with the other teachers and technology directors from the surrounding schools in the area, who, I’m sure, were also impressed with the cool gadgetry we had been handed, excited about what we might learn. As we were introduced to the regional director of educational service for our area, a short graphic was projected on the screen in front of us. He gave us directions on how to turn on our handheld devices and put in our earphones in order to watch a short video on our handhelds. We were then look back up at him when we were finished. Everyone was very excited to make use of these tools and followed our directions carefully.

As I watched my short video full of interviews with teachers and children raving about their experiences with technology in the classroom, and how the 21st century skills needed for our global world would best be taught with these tools, I suddently looked around me at the room. I found it very odd that I was at a conference, sitting in a conference room, with about 50-60 people who were all watching an introductory video on their individual handhelds, when there was a person at the front of the room, who just as well could be telling us all of this as a group. I wondered why this Regional Director was allowing us to view this video in this manner as opposed to say, projecting it up on the screen and having us all watch it together - share the experience as a community. In fact, as I thought about this, I had a hard time focusing on the rest of the video. What brought me back to the reality of the situation was my peers finishing up watching theirs on the handhelds. In many ways, I felt like a pawn simply following the instructions, as I sometimes feel when standing on a grocery store line when there’s a video screen to keep me entertained while I’m waiting for the cashier to ring up my items next.

Technology, and more specifically the newer mobile technology that has most recently been established as mainstream, has truly sparked, or maybe resparked, my interest in the pedagogical question of local vs. global engagement of students in the classroom. What I mean by this is having students work individually (local engagement) with individual technology [read paper or electronic] or having students work in groups or collectively (global engagement). In terms of electronic technology, and specfiically the dynamicism of the computer screen, many educators feel that “largeness” of the bigscreen (e.g. an Interactive Whiteboard) is much more effective a learning tool then for example a handheld device or a laptop computer. I was able to have a very exciting “debate” about this last summer with a group of teachers at a conference, and many people have strong feelings about this. However, many feel that it is extremely dependent on the learning needs of the group.

My plan right now is to do a great deal of research and perhaps write something up on this - perhaps even specficially with respect to girls’ learning. I’m not quite sure, but the question truly interests me, mostly because I am quite averse to spending large amounts of money on technology that is not used in the most effective of ways in the classroom. It is very upsetting to me when schools put expectations on teachers to use technology, but there is little direction on the effectiveness of its use in the teaching. It is my hope to give some guidance in that direction. We’ll see if that can happen.

While I was holding a workshop this summer I got into a discussion with some teachers about the term “leading questions”. In most constructivist classrooms the term “leading” questions is actually looked upon favorably - as a good thing that allows students to often discover ideas on their own while the teacher leads them to the correct ideas. This is a very positive thing when you think about it, as opposed to the more traditional classroom where most main concepts are merely directly stated as fact and told to the student without any higher-order thinking involved or practiced. Although some students need that every once in a while (and that’s another blog entry altogether..), I believe it is the goal of mathematics education for students to learn those higher-order thinking and self-regulated learning skills. So in general, if we make the assumption that those are skills that a teacher wants to encourage in the classroom, how does one go about doing that without simply spoon-feeding students information?

The idea of handing out leading questions is a very good one and guiding students through either partner or group discussion, exploration and practice, with teacher or student summary of findings in the classroom is an excellent model. This happens in many classrooms around our country to varying degrees of success in student learning. Many teachers who use a problem-based approach tend to use the “guiding” or “leading” question model a great deal in order for students to gain more ownership or the coursework and material, as this is a major part of the pedagogical practice and goals. However, I’ve never been a fan of the word “leading” question. It somehow brings to mind walking a dog. I picture taking a student by the neck on a “lead” and dragging them to the correct answer or point that I, as the teacher, am trying to make. This goes directly against the focus of PBL in general, where the students are guiding the discussion and their ideas are taking the lead. Maybe my analogy is a little overboard, but I’m trying to make a point here - sorry.

So in this workshop, as I often do when I’m public speaking, I struggled to find the right word to describe what I was trying to say. I wanted to describe pushing the students in the right direction. We all know the goal of the problem or where it’s going, where we hope the conversation will lead to the next problem or topic in the curriculum. However, we don’t want to lead them there. Hopefully, if the curriculum is well-written the students have the tools that they need to move forward and with the cooperation of each other, they should be able to follow the path laid out for them. I came up with the visual analogy of those water games where you push the little button and the water is pushed up and little hoops have to get whooshed around onto hooks or into buckets and you win by getting all of the balls or hoops in. Do you know what I mean? In those games the player pushing the button is the force behind the water which is the impetus for the motion the causes you to win. In a PBL classroom, the teacher is the force behind the students who are the impetus for the learning, so it is really the teamwork between the two that causes the success.

After thinking this through, I thought of the word “thrust” instead of “lead”. I guess if you think of a question as possibly thrusting the motion of learning out there and then letting the students go with the information and background knowledge they have and see where it goes. Of course, you may have to continue to thrust those questions at them, just like you would have to in that water game in order to win, but that’s half the fun of the game isn’t it?

I was asked recently by a wise young teacher about the idea of relegating so much of the practice of problem solving in pbl to “homework” time that is outside of class. In many teachers’ mind this takes the observation of the practice of problem solving out of the classroom. I’d love to discuss this a bit here, as I think it might clarify a lot about the procedures of learning problem solving through pbl that I believe have not bee articulated (at least that I haven’t really found articulated in any meaningful way in research articles).

When a student is assigned a problem that they have never seen before and take it home and grapple with it, what do we expect to see the next day? What were they supposed to do that evening? These expectations need to be clear to the student in order for pbl to be successful in terms of learning on many levels. First, a student must be able to dig into their “toolbox” or “toolkit” as many math teachers call it and find remnants and memories that strike a chord with that problem. Others might call this “recall of prior knowledge” or “making connections” or “transfer of skills”, whatever you want to call it, it is something that we want to make happen where a student can find a piece of a problem or solution that have seen before and connect it with a new problem that may not know the solution to in the present. Not only do we want them to connect the idea of a solution, but we want them to connect the idea of the conept as well. Now, do we expect this all to happen in one night of grappling with a new problem for a ninth grade geometry student? Most likely not, that would be practically impossible for a student of average caliber to come up with some mathematically significant new material on their own in one evening. However, if the foundational pieces are laid for them in a way with problems that are stepping stones and arrows pointing in the right direction, it does seem that with the right motivation, they might be able to put down enough recalled information to have a substantial amount of ideas to have productive discussion. This allows the classroom the next day to be the microcosm of the problem-solving real-world of the(substitute appropriate career path here…engineer, business manager, etc.) So in reality, although some of the independent problem solving happens during out-of-class time, a great deal of both the independent and group problem solving practice happens right there in class during the discussion and resolution of the problem the next day.

When a student is presenting the work they have come up with from the previous night’s attempts at a problem, she is presenting the problem solving that happened outside of class. However, even within that presentation she is problem solving. Perhaps she realized her error and because of a comment a classmate made, or an insight shared, or even a prodding by the instructor, she has self-corrected and changed her presentation entirely. This risk-taking and on-her-feet thinking is practice in problem-solving in and of itself. The rest of the class is practicing problem solving because they are learning what they did incorrectly in their recall, how to find the right ways to tell their classmate what she did incorrectly, and also how to be fair and reasonable in their comments to others. The group dynamic in this situation is very important.

So, do I have a problem relegating so much problem solving to students’ work alone for “homework” - no, not really. It does take a lot of coaxing and talking to them about the expectations of the teacher and changing their idea of success on homework. Being explicit about what you value on their nightly efforts is very important and having a grading rubric that does that for them helps - as I do. However, I do believe that having a pbl curriculum sends the message to students that the major goal of the mathematics classroom is that they should be learning to be problem solvers, and that that should be happening everywhere, all the time.

Hi everyone! Just returned from a week of discussing PBL at a conference with two wonderful groups of people, and other individuals along the way. I met lots of great math teachers who all had me thinking and justifying what I do almost continuously 24/7 for the whole week. It was exhausting, but exciting to discuss teaching so intensely for so long with so many. The best part about talking about PBL with people who are new to it is the fresh vision that I get; the new perspective on what I take for granted in so many ways. I want to thank everyone so much for all of the great input and the questions that you all asked during the week.

One of the most important discussions I had with my group was after we had just watched a video of my class for about 20 or so minutes. I had put forth the caveat that it was by no means the perfect class and there was one student that had struggled for a while with one problem about similar triangles. I could tell that it was making my participants very uncomfortable that I had let the student go on, and perhaps it was making other students confused. This is an excellent opportunity for us to discuss classroom management - or maybe a better term for it is “learning mangement”. In other words, when do the “needs of the many outweigh the needs of the few, or the one” (does anyone get that Star Trek reference)? When does the individual student’s learning become more (or less) important than that clarity for the group or vice versa? When is there a point of diminishing returns for the whole group?

Part of the philosophy of PBL is based in social constructivist learning theory where the class will co-construct the knowledge together, even with students at different learning abilities and levels all discussing problems. It is feasible that with consistency and the values of such a learning environment repeated to them that students become acclimated to this environment over the school year and begin to value it as well. This has been my experience at least. However, at certain times when a student is taking “too much” time and others might not jump in to help co-construct, it does leave a moment up to the teacher to decide to either help with the construction, give direct instruction (i.e. perhaps go against the pedagogical ideology of the classroom), redirect the construction to others in the room, or even be patient and allow the organic process even more time.

This was a very difficult thing for people who had not yet seen this type of learning in practice. Although all teachers have heard the adage to what ten seconds before you react to students comments or wait for an answer, the fact is we are all uncomfortable with silence or discomfort in the classroom. We are all uncomfortable with disorganization and potential chaos. However, it is clear that there is great value in allowing a student or a group of student to grapple with a problem, as they practice problem-solving skills. I think it will be important for me in the future to have better sets of skills to give teachers in order for them to aptly deal with, and be ready for, this discomfort.

However, the question of time committed to individual students needs as opposed to the groups needs, I feel is something that you can ask in any instructional approach - even direct instruction where students are allowed to ask questions (probably the only instructional approach where that question doesn’t apply is a lecture where student don’t respond at all). In any classroom, the instructor needs to think about organizing the time committed to dealing with students on an individual basis vis-a-vis dealing with the classroom as a whole - this is no different in PBL. The biggest difference may be that a teacher following this instructional approach may feel more comfortable allowing the student the freedom to grapple with the problem on their own for longer, which honestly is probably a good thing, if you have the time in your class period, of course.

Hi everyone, I just wanted to let you all know that I updated the url for my homepage for my students. My homepage can now be found at

community.emmawillard.org/Math/Schettino/index.htm

It will make it easier for people to access my Motivational problems which work well with most Precalculus or Advanced Algebra courses. This summer I also plan to do this for my Geometry Motivational problems as well. If you follow the links on my Class Calendar, you can also find the link in my Algebraic Geometry course to the pdf files that are the curriculum and that my colleagues and I have written that I reference a great deal in my entries.

More to come soon…

A square pyramid is a pyramid with a square base and four triangular lateral faces. The slant height is the distance from the vertex of the pyramid along a lateral face to the midpoint of a base edge. If the slant height is 10 and an edge of the square is 12, what is the altitude of this pyramid?

This is an example of a problem that probably is closest to what a traditional textbook reading would do for a student on a typical night’s homework. We do assume that students have had past experience with three-dimensional solids in middle school curriculum and have heard the term ’square pyramid’ before, although this is their first encounter with it in our book (it’s also defined in the reference section of our book too). The problem comes complete with a few other definitions just in case they aren’t familiar with slant height as well. However, although the definitions are in the problem, it can often seem that students don’t use their logic or problem solving skills to really understand what these new terms mean in the problem.

When I gave this problem last week, the student that put it on the board did it incorrectly because she assume that the slant height was actually the lateral edge (the length of the congruent sides of the isosceles triangle that is the lateral face of the pyramid). So even though the defintion of the slant height was right there in the problem, why was this student unable to transfer this definition to the problem? Well, there could be many reasons. When students read a problem for the first time, I theorize that new material takes time to become a part of their toolkit, or set of skills that they would use. So in this case, even though the definition is inthe problem, and possibly she even understood the definition, transfering that to the problem situation was difficult since the new idea of slant height was “too new”. She actually new to use the pythagorean theorem, just used it with the wrong triangle.

I also theorize that it is easier for students to visualize the right triangle that is formed by the lateral edge, the altitude and the square base because the lateral edge is physically drawn in (in a diagram that students generally draw) already. We are going to change that for next year and actually draw in the slant height in a diagram in the text.

There are a number of problems in our curriculum where a great deal of assumptions are made about students reading the definitions and applying them to the problem at hand. This result is more positive on specific problems when students have the motivation to complete the problems on their own. The kids who are finding the curriculum instrisically challenging and interesting don’t have a problem realizing that applying the definition is an expectation. I’ve also found that the more you expect from them on a regular basis, the more they begin to fulfill that expectation too.

OK, well maybe that title isn’t really what this problem is about, but many teachers might think this problem is too hard for a lot of geometry students. Initially, we were hesitant to include it in our discussion of circles, since we don’t discuss the all-to-familiar unit circle really at all in our geometry course discusison of trigonometry. But this problem is really more of a discussion of the special right triangle (45-45-90) or if students choose the prior method of the intersection of the line and circle (see previous blog entry). This is a really interesting problem with different solutions, so it’s fun to see how different students approach it.

The vertices of a square with sides parallel to the coordinate axes, lie on a circle centered at the origin with radius 5. Find the vertices of the square.

What have we discussed up until now? The students have formalized the idea of the equation of a circle centered at the origin as x^2+y^2=r^2 (from the Pythagorean Theorem) so they should be able to come up with the equation of the circle. They also know the terms of the radius, diameter, coordinate axes and show be able to set up the picture.

When I did this problem in class, many students started by drawing rectangles inscribed in the circle with dimensions of 6 x 8, realizing that the diagonal would be 10. This was a great start. They realized that somewhere in between the 6 x8 and 8 x 6 rectangle was the square they were looking for. They were then able to actually draw the diagram with that square in the circle. They could then approximate the coordinates which helped a lot in the class discussion. I felt very proud of students who were able to come to class with even that done. However, other students got even farther.

This is where I wish I were able to draw some diagrams on my blog. I need to do a little more research with Edublogs to see if this is possible. Most students then drew the diameter as the hypotenuse of an isosceles right triangle and either used the Pythagorean theorem to solve for the side, or remembered the ratio of the 45-45-90 triangle, and simply divided 10 by sqrt(2). Once they realized the side of the triangle would be 10/sqrt(2), they just needed to transfer that information to the coordinate plane. This is where a good diagram was important.

Since the origin was at the center of the circle, and the radius was half the diameter, most students could easily extrapolate the fact that you would simply divide the side of the isosceles triangle in half to obtain the coordinates. In other words, you’d just go over 5/sqrt(2) and up 5/sqrt(2) to get to the vertex in the first quadrant and do the same for the other quadrants (taking into account the negatives for left, right, up and down).

This was an amazing problem for discussion since another student did the problem by finding the intersection of the line y=x and the circle x^2+y^2=25. This student had been able to independently transfer the skill of substituting the line equation into the circle equation from a previous problem. This was the minority of the class however, but still someone was else to do it. I was impressed. If you’d like to try this problem it was #8 on p.50 at my curriculum

Here’s a great problem relating parallelograms and area -

A parallelogram has 10-inch and 18-inch sides and an area of 144 square inches.
a. How far apart are the 18-inch sides?
b. How far apart are the 10-inch sides?
c. What are the angles of the parallelogram?
d. How long are the diagonals?

OK, what’s the pre-existing knowledge or discussion that has happened before the problem is presented to the students? So far, the students have discussed the fact that the area of a parallelogram is simply the base times the height (or altitude, perpendicular distance between the parallel sides), but have not done a huge number of repetitive area problems, I’d say not even two or three of them. They’ve done a dissection discussion problem a few pages back about how to cut off a right triangle and move it over and the parallelogram basically becomes a rectangle, and that’s why it’s the same area formula.

So, in my class, most students got part a, pretty quickly, just divinding 144 by 18 and getting 8. The other type of previous problem the students have done is being given the area of a triangle and all the sides, using each side in turn to find each of the three altitudes, so they area used to turning the triangle around to find each of the heights. So now we are seeing if they can transfer that method to a parallelogram in this problem. This took getting all students to the board in my class.

Let me back up for a second. We also had covered problems that had discussed both the tangent and sine functions for right triangle trigonometery (which are introduced before cosine in our curriuculum). Hence, the students were desperately trying to find the height to the side of 10 using a right triangle, which could not work, of course, without being able to break up the side of 18. Some of them guessed lucky making a 6-8-10 triangle, which did get them the right answer, but they could not justify it - no beans in my classroom.

So, we finally had to get them to draw the parallelogram with the 10 as the base, then students finally realized they could use the area once again. This was the a-ha moment and it spread like wildfire. Thank God. Once we got through this point, they were able to find the angles with the trig no problem.

However, finding the diagonals took some manipulating of right triangles with the heights. The shorter one was easier with the right triangle within the parallelogram. This created a right triangle with legs of 8 and 12 and the hypotenuse was the diagonal. Putting a right triangle outisde the parallelogram made a bigger right triangle with legs of 18 and 8 and that hypotenuse was the longer diagonal. Since I expected so many students to not have totally completed this problem, but only have attempted it, I made time for this to be done at the board in pairs during classtime. It was time well spent, and many students wrote this problem up in their journals - very well, I might add.

If you’d like to try this problem in class it was p. 47 #2.

OK, so there’s a problem that goes like this:

Sketch the circle whose equation is x^2+y^2=100. Using the same system of coordinate axes, graph the line x+3y=10, which should intersect the circle twice - at A =(10,0) and at another point B in the second quadrant. Estimate the coordinates of B. Now use algebra to find them exactly. Segment AB is called a chord of a circle.

OK, what have the kids seen before this? They have been introduced the equations of circles that are centered at the origin. Simply, we did this by asking them to plot points that were say 5 units away from the origin and we discussed from the standpoint of the pythagorean theorem. This happened about 3 pages prior to this question. We’ve also done a few problems about circumcenters and the smallest circle that fit around a right triangle. However, this is the first circle of a radius 10, I believe. This is also the first time, they are asked to graph a line and a circle on the same coordinate axes.

This is also the first time they are being asked to approximate the point of intersection. The point does end up being a lattice point (what we refer to as a lattice point is a point that has integer valued coordinates), so the students probably will be able to come to class with the answer, which happens to be(-8,6) if they did a decent job of graphing it well.

What they have not done before is algebraically substituting in an expression for y. So this is the interesting part - to see if anyone thinks to do this. Most student probably won’t. In the past, many student remember solving systems of equations in algebra class, but they get confused when they see the quadratic terms. Once prodded in the right direction to ’substitute’ however, they realize that the linear equation is easy to solve for x, and see what they can do. This quadratic is actually not that hard to solve, pretty simple factoring involved and no quadratic formula, so it’s’ a nice first attempt at this type of problem. I will report back with what happens in my class tomorrow.

This problem also introduces the terminology of a chord of a circle. I generally like to have the students come up with a definition at this point so that we can formally have one to move forward in the problems as well. Overall, lots of good stuff in this problem, mostly generated by the students, even the substitution idea sometimes.

If you’d like to try this problem it is # 4 on p.49 at my website. You can get pdf files for the entire curriculum there. Have a look.

So I have this new idea for a direction for this blog and I’m hoping that it might take off. I’ve archived all the old “transition” entries hoping that those can still hang out and be categorized in case I need to refer to them in some future entry. But what I’m hoping to turn this blog into now is something of an organizational place for the categorization of our curriculum at my school. Since our geometry curriculum went PBL, I’ve been looking for a way to categorize the problems. Every summer though my colleagues and I rewrite and edit the problems so categorizing them by problem number and page actually doesn’t really work.

but then I thought, a blog would really be awesome because what it does it create this hypertext categorization that will let us connect the problems by categorizes and tags. I’m going to try it and see what happens. We can always go in and edit and and later on see if it makes a lot of sense or not. The only thing that will be problematic is that there seems to be no equation editor to speak of. I need to write to someone at edublogs and see if there is one somewhere that can be accessed. I’ll see what I can do about that.

Wish me luck.