Geometry – Page 3 – A Portrait of a Math Teacher as an Aging Man

Borrowing from the MTBoS

I’m guessing that most of you reading this are familiar with the awkward acronym for the Math Twitter Blog-o–Sphere – one of the joys of tapping into this community is that they are remarkably generous about sharing ideas and resources. Today in our Geometry classes (I teach only one of the five sections we have at our school) we used an activity written by Kate Nowak (@k8nowak on twitter). It is an activity based in GeoGebra and allows the student to explore the ratio between lengths of legs in a right triangle. You can find the document we used here . I modified (very slightly) the document that Kate originally posted here. Next time I use it I will tweak it a bit. I have only twelve students in my class and chose not to explicitly team them up. They talked with their neighbors as they are usually encouraged to. However, the directions either need to be tweaked so that team references are excluded or I need to clearly team them up. I am also debating question 7. A number of students did not make the explicit leap from using the ratio they found on page 1 and using it here. I don’t necessarily want to give away too much but I may add a little prompt that they should consider the work that they have already completed. We set up a google spreadsheet and in the next couple of days I will refer to this repeatedly to show that different students working on different triangles were arriving at the same ratio. We make a big explicit deal about scale factors between similar figures. I do not think we spent enough time pointing out that scale factors within figures will also match up for similar figures. I will definitely make this more of a point of emphasis next time through my text.

I cannot thank Kate enough for sharing this activity. My students worked well and I am convinced that they will have a more solid grasp of trig ratios moving forward. As I plan out the rest of the unit I am also going to be borrowing from Sam Shah’s latest post about trig. You can find that over here.

Man – the benefits my students are reaping from people that they will never meet – such as Kate Nowak, Jennifer Silverman, John Golden, Jed Butler, Sam Shah, Pamela Wilson, Meg Craig, and so many more – is just remarkable.

Show but Don’t Tell

So in Geometry today we began to study the ‘special’ right triangles and I had an idea last night that I wanted to try. I handed each of my students two pieces of paper, a ruler, and a protractor. On the first page I asked them to draw an isosceles right triangle on each side and asked them to have the legs of their triangles be different lengths. I polled the students and had them tell me one of their leg lengths. I then asked them to find the length of the hypotenuse and tell me what number they get when dividing the hypotenuse by the leg length. I, of course, got a variety of answers all of which hovered around 1.4. Some students used the Pythagorean theorem and gave me decimal approximations. Some used the Pythagorean theorem and gave me radical answers. Some measured the hypotenuse with their rulers. I asked them why these answers seemed so close to each other – I specifically avoided the word similar here. Luckily, one of my students told me that all the isosceles right triangles were similar to each other. I pushed back a bit and asked what that had to do with ratios within one triangle. We usually discuss similarity ratios between triangles. The explanations from the students were not as concise as I hoped but we all seemed comfortable that rations within a triangle will be the same when looking at two triangles (or in this case 12) that are similar to each other. Since a few students used radicals we had the exact ratio in front of us and a quick solution using algebra confirmed that the ratio was the square root of 2. Success!

Next up I asked them to draw two equilateral triangles and construct an altitude. Now I asked for the ratio between the altitude and a side length. These answers all hovered around 0.87. We were running out of time now so I did a little more telling than I wanted to but we saw the ratio for the three sides of this new right triangle were 1 : square root of 3 : 2

I have to say I was pleased with their persistence, with their measuring/equation solving, and with the idea that we could see these ratios without simply giving them formulas to try and remember. I may be an incurable optimist, but it feels to me that these ratios will be easier to remember at this point. Now I need to have the discipline to avoid using the words for the trig ratios for at least a few days. I am going to steal ideas from Kate Nowak (here is her trig blog post) and Jennifer Wilson (you can find her trig wisdom here) as I attempt to shepherd my Geometry students through the tangle of right triangle trig. I feel that we had a good start today!

Delighted

A quick post here – I want to share something delightful that a few Geometry students did this morning. We had our last test of the winter term today and here is one of the last questions:

Prove that the points A (x, y), B ( x + 1, y + 3), C (x + 4, y + 5), and D (x + 3, y + 2) are the vertices of the parallelogram ABCD. Prove this is true by one of the following two methods:

By showing that one pair of opposite sides are congruent and parallel.

By showing that both pairs of opposite sides are parallel to each other.

So, I was hoping that the majority of my students would take the quick and easy option of calculating slopes rather than messing with distances. I also hoped that the coordinates having variables in them would make them slow down, be careful, and remember a touch of algebra. I grade page by page and I have graded three of the papers with this problem on it. One student said ‘We can let x and y be 0 so the coordinates are (0,0), (1,3), (4,5), and (3,2)’ I love this thinking. She avoided the worry of dealing with the variables here. It’s a little slippery to determine just how clearly she was thinking here. She might have just been dodging a bullet. One student said ‘I will first transform this parallelogram by the vector <-x, -y> and then we will have the coordinates A’ (0,0), B’ (1,3), C’ (4,5), and D’ (3,2)’ Now, it is ABUNDANTLY clear that he knew exactly what he was doing. I’m so delighted by this that I felt I should share.

This and my great AP Stats classes today made for a pretty terrific day!

Fighting for Understanding versus Doing

A pretty interesting conversation unfolded in Geometry this morning. We are getting ready to explore similarity, so I gave the kiddos a quick assignment on solving proportional equations with one variable. This was meant to be pure review. When we started talking about these problems I, of course, heard talk about cross-multiplying, cross products, and even heard one student exclaim something about the old keep-change-flip idea. I decided to stand firm and talk about why we were able to do what we do with these proportional equations. We started simply with the equation like $frac{x}{5}=frac{3}{7}$ One of my students was taking a vocal lead in discussing cross products and I asked her what equation to write next. She told me to write $7x=15$ . I agreed that this was correct and most of my students recognized what she was doing. I then asked them to pause and wrote the following equation $frac{x}{35}=frac{6}{35}$ I asked everyone what they thought the value of x had to be in this situation. They all seemed to agree pretty quickly that x must be 6. So I got them to agree that an equation with one fraction on each side AND the same denominator demanded that the numerators would be equal. They all seemed to think I was making too big a deal out of this. I then asked them if I could do the following to $frac{x}{5}=frac{3}{7}$ . I asked if I could multiply the right hand side by $frac{5}{5}$ while multiplying the left hand side by $frac{7}{7}$ . One student protested that I need to do the same thing to each side of the equation. I, of course, agreed with her but I asked her to look more closely at what I was doing. She agreed that I was doing the same thing even though it looked different. Most of my students still seemed to think that I was making too big a deal out of this. Next came the payoff. I picked the following problem from the homework: $frac{x}{4}+5=frac{x}{5}+4$ I pointed out that our cross product idea was not really a comfortable fit here. My KCF student quickly suggested that we clear the fractions out of the problem by multiplying by 20. I agreed that this would certainly work but asked if I could try something different. So I wrote the following equation: $left ( frac{5}{5}left ( frac{x}{4}+5 right ) right )=left ( frac{4}{4}left ( frac{x}{5}+4 right ) right )$ I was immediately met with resistance. I begged for patience and made them a promise. I told them that I would carefully explain why I was doing what I was doing and that if they unanimously decided that they did not like this approach, then I would cease and desist. I pointed out that we were, again, doing the same thing to each side even though it looked different. I made an argument that multiplying by smaller numbers decreased my chances of arithmetic mistakes and I pointed out that this technique made the common denominator for the problem obvious. The equation became $frac{5x}{20}+5=frac{4x}{20}+4$ . I saw some signs of visible relief as they saw that this was now a pretty easy equation to process. Combining like terms gave us $frac{x}{20}=-1$ and a conclusion that $x=-20$ . I then solved the problem the more standard way by multiplying everything by 20 to begin with. I felt like it was a bit of a triumph when they voted that this new technique did not need to be banned from our vocabulary. I know that this is not revolutionary, but I certainly think that I made some strides here. My students are well-trained in mechanics and they know what works. I want to have serious conversations about the ideas behind why these techniques work.

Back at it again tomorrow!

PS – Thanks to David Wees and Zach Coverstone for valuable assistance in learning some LaTex for this post. I hope it looks right when I hit publish

Working on the Holiday

So, one of the oddities about teaching at an independent school is that days off that are taken fro granted most places are seen as prime days for campus visits here. So, we were in session today. No need to feel sad about this though as we have plenty of vacation time as well. Just another Monday.

But…it wasn’t. It was a terrific teaching day and I want to make sure that I make note of it even if only for my own pleasure.

1st Bell – Geometry. I was looking forward to returning their excellent tests from last Friday but first I wanted to dip into our new unit. I passed out paper, rulers, and protractors and gave them a simple task. Draw some polygons. I insisted on not defining what a polygon was and I did not reveal why they had protractors. I tasked each of them with drawing six polygons and I saw some pretty great stuff. Complex, crisply drawn concave polygons. Some students stuck to the middle of the road and drew squares, triangles, etc. Then the fun began. I started asking for definitions of polygons and I framed the question this way: Explain to my 11 year old son what a polygon is. Brainstorming began. I heard about the need for line segments as sides, I heard about the limitation that there had to be at least three sides, I heard conversations about polygons that pointed inside versus those that did not. Someone offered up the word concave and I pressed for a definition. Everyone seemed happy about the inside-ness of some points on a concave polygon. I pressed them not to use the word point for where the line segments met and they offered up vertex as a better alternative. We decided that each vertex needed two (and only two) line segments and one boy suggested that polygons that were not concave (we agreed to convex quickly) should have angles larger than 90 degrees. He backed off of that but I will definitely revisit this idea soon. When he tossed out that idea he was greeted with references to equilateral triangles. Well played. Then the highlight of the morning came for me. I drew a figure on the board that was made of line segments, that had two line segments at each vertex but it was clearly not a polygon. The reason why is that my figure was not closed. One of my students used that exact language and I pressed, again with my 11 year old son in mind, what we meant by closed. Miranda said ‘Imagine it has water in the boundaries. If it’s closed, the water can’t get out.’ I thought that this was a lovely image. I then closed my crazy drawing but blocked off access to some regions while doing so. I was quickly told that the water needed to be able to get everywhere. I hope that this image stays with my students the way that it is staying with me. As we wrapped up class in a blur of vocabulary about quadrilaterals one of my students said to her neighbor, ‘What a great way to start the week. We got to sit and draw.’ I’ll count this one as a success.

Bells Three and Six – AP Statistics. My senior heavy Stats class did not come back from winter break with much of a sense of urgency. I did not want to just launch right into a new chapter on the heels of the disappointing chapter test we had on Friday. My Computer Science colleague had recently shared with me information about Sicherman Dice which are two six-sided dice that are not standard dice but their sum has the same probability distribution as the sum of two standard dice. I presented my students with a challenge. Describe two six-sided dice that replicate the probability distribution of two standard die. No other directions really. I have a rudimentary handout I gave them and you can grab it here. I fielded questions as they chatted about this problem with their neighbors. Can the die have negative numbers? Can the die have fractions? Can the die have zero? I kept replying in the reluctantly affirmative and checked on their progress. Most of them had a pretty logical attack where they would transform one standard die in a certain direction, say subtract two from every face and then transform the other one in the opposite direction. Not very sophisticated, but it was nice algebraic logic. One student was working on fractions trying to balance combinations of 1/3 and 2/3 so that she would always get integer answers. Overall, it was the most focused energy from this group that I have seen in two weeks. I hope that this is an omen for our next unit. They were pretty surprised by the reveal and I am curious, in retrospect, that they never asked if any die could have repeated values as this is necessary for the Sicherman Dice to work their magic.

Bell Seven – AP Calculus BC

We’re just getting ready to start exploring the magic of Taylor Series. We took baby steps today reminding ourselves of the language of arithmetic and geometric sequences and series. I always think that this material is such fun to untangle. Tomorrow we’ll play with GeoGebra and I will try to tease out of them the key ideas about how to make a polygon behave like the sine function. Nothing much else to report here.

Success!

I have had a very active blogging week thinking about (and writing about) my Geometry class. I have three preparations this year, AP Statistics, AP Calculus BC, and Geometry. I’m not proud of it, but I know that my attention to each class varies at different times of the year. Iy’s not a simple matter of 33 1/3 % of my planning energy being spent on one class at any time. Do many of you go through this as well? By the way, how many preps do most folks have?

Anyways, I blogged in December about my discomfort with HW in Geometry and gathered some nice ideas. I blogged about my decisions about changing habits and it has felt like a raging success. Five to seven minutes at the beginning of class of students sharing their work with each other and correcting each other/reinforcing each other/ sharing their miseries, etc. It’s just been a really terrific week with them and I have let them know how much I appreciate their demeanor, their energy, their willingness to share with each other. Today we had a quiz (you can grab it from here) on Sections 6.1 – 6.3 of our text (you can grab that here) exploring centers of triangles. We’ve talked about perpendicular bisectors, altitudes, medians, and angle bisectors this week. We have played with GeoGebra and looked at how, in each case, all three segments have a common point where they coincide. We’ve talked about which ones could coincide outside the circle and those are not popular choices as the best center of the triangle. We had a great lab activity yesterday (you can grab that here) and it developed into an interesting debate where one group of students nominated the intersection of the angle bisectors as the best representation of the center of a triangle while the other three groups all felt that the intersection of the medians was best. As we had a healthy debate I found myself wishing that I had been clever enough to have physical triangles to manipulate. Next year, I want to be prepared with cardboard triangles of various types with these two candidates for center marked out. I dropped the ball on this one anticipating that everyone would feel best about the centroid. What really impressed me was that the group arguing for the angle bisectors had GeoGebra construct a circle that had this incenter as its center and showed that this circle touched all three sides. I was THRILLED that they thought of this argument.

So, this morning I felt confident as my cherubs asked their last few questions before the quiz and the results are in. I have 12 students in this class and 4 of them earned perfect scores with another 4 earning an A on the quiz. Their class average was 93%!!! I’m thrilled by this. I think that this is due to a number of factors.

In general, my students have had more energy this week in January than they did in the few weeks leading up to our winter break.
I believe that the HW strategy has made a positive difference.
I believe that the extensive use of GeoGebra in class is finally spreading to the home. I have overheard a number of students this week make reference to looking at GeoGebra while doing their HW this week. I am a firm believer in the power of these graphing programs and, for my Geometry students at least, I think that this is the best of the bunch.
I worked hard during break planning out this unit for me and for my Geometry team of two terrific colleagues. This thoughtfulness has paid off.

Oh yeah, one final thought. As a long-time Calculus teacher I have a strong preference for lines in the point-slope format. Every one of my students presented at least one of their line answers in this format. Woo-hoo!!!

On a Roll

Man, my Geometry students are on a roll right now. Today we went through our same new HW procedure again. I was quiet for the first 5 – 8 minutes of our 40 minute class while my students shared their HW with each other. They were asking each other good questions and catching each other’s mistakes. They are still a little shaky at times on their line equation writing skills and their line intersection skills, but the mistakes they are making are much more of the arithmetic and detail type rather than broad conceptual mistakes about what to do.

Today we were concentrating on medians and they seem convinced that the medians should always intersect inside the circle. Last night’s HW (which you can find here along with all our other HW assignments) was on Section 6.2 and they were finding medians and their point of intersection. I also asked them to find the perimeter of an original triangle and the triangle made by connecting the midpoints of the original. This allowed us to do a little noticing and wondering in class together. Everyone seemed pretty convinced that the perimeter of the interior triangle should always be half the perimeter of the ‘parent’ triangle. We displayed this on GeoGebra and I asked them to pick a certain type of triangle that they wanted to explore. One student suggested that a right triangle would be fun so we moved out vertices to the origin, the x-axis, and the y-axis. The interior triangle was still half the size and now the noticing began. They noticed that the smaller triangle not only was also a right triangle but that its acute angles seemed to be the same as the acute angles of the original. They noticed that the four right triangles formed inside were probably all congruent. They noticed that the centroid of the original triangle was also the centroid of the smaller triangle. Then Tara asked about yesterday’s peek at angle bisectors and whether they would meet where the medians met. I asked if there might be a special triangle they could think of where this is true and Miranda guessed that our favorite right triangle, the 5 – 12 – 13 triangle might be special enough. Sadly, it was not, but I was happy to hear a quick guess at this familiar old friend. Then Julia suggested that an equilateral triangle might fit the bill. I worried about how to manipulate our given GeoGebra sketch to match up and she cleverly told me to start a new screen with a regular polygon. Class concluded by seeing that GeoGebra was confirming that Julia’s guess was correct. What a great 40 minutes! I also made a point of telling them that they were on a roll and I hope it carries over to our GeoGebra lab day tomorrow. This is called Chapter Six GeoGebra activity in the dropbox file I had the link to above.

I should have dwelled a little more on my second stats class yesterday. I was really pleased with the three different formulas that those four groups generated. I was especially intrigued by the group that decided that the minimum number in their sample plus the maximum number in their sample should be a good estimator for the true max in the population. I discussed this idea with my morning stats class and we had a pretty vigorous debate over how appropriate this was. Playing with our TI and drawing random samples of 5 from a group of 342 (kind of like the German Tank problem!) convinced them that this technique actually turns out to be pretty accurate.

It’s easy to actively blog when it’s fun to relate what’s happening in class. I hope I can keep up a reasonable pace, if not daily, for the year.

Beginning New Habits AND a Fun Activity

This morning in Geometry I started by not talking for the first five minutes while my students shared their HW with each other. They talked about their answers, they puzzled over why/where they differed and they talked about using GeoGebra on their own to explore the intersection of perpendicular bisectors of triangles. I was SO delighted I almost wanted to call the rest of the day off. I did not, though and I’m glad I stuck it out.

We looked together at GeoGebra, reviewed (again) how to find the intersection of lines, we let GeoGebra confirm that we were right. We remembered from yesterday that these lines coincide on the hypotenuse for a right triangle, in the interior for an acute triangle, and outside of an obtuse triangle. After playing with another GeoGebra sketch we all agreed that this behavior made this point of coincidence a pretty poor candidate for the center of a triangle. I pointed out that one of our students had suggested – on his way out of class yesterday – that we should concentrate on vertices rather than midpoints of sides. Again, we let GeoGebra take over and looked at a compromise by constructing a line through a vertex AND through the midpoint of a side. I named this for them as the median. I also displayed that these medians seem to ALWAYS intersect in the interior of a triangle and I named this point for them as the centroid. We all agreed that this name was ‘center-y’ enough. As time ran out, at the suggestion of another student, we asked GeoGebra to construct angle bisectors. It does so, but draws an exterior line as well. They did not complain when I erased them, but I want to examine what is really happening there. It felt a little too much like I was waving away a distraction. We saw that these angle bisectors intersect in the interior as well – setting up a great debate for tomorrow about which center is the center-est. Just thrilled with how they hung together during the intro time and during the quick GeoGebra exploring. Need to commit to both HW review time tomorrow and to revisiting the blur of activity on GeoGebra. I am planning on a lab day for Thursday so that they can manipulate these ideas themselves.

In my AP Stats I tried out the German Tank problem using resources found here at the Stats Monkey site. My two classes dealt with this in pretty different ways. My smaller class (12 students today) worked in 3 groups of four. I made a mistake in responding to one of the first ideas I heard. One group decided to invoke the empirical rule and guessed that the # of tanks was their sample mean plus three standard deviations. I responded positively to them but this simply steered the other two groups into following this lead. In my other class I was smarter and quieter. Here I had four groups of four. One group invoked the empirical rule but they also pooled their three samples together. One group used the inverse normal function on their calculator seeking a point where the area was 0.999. One group added their sample minimum to their sample maximum guessing that they should be (roughly) equidistant from the extremes. The final group doubled their median guessing it should be halfway to the max. I was thrilled with the level of discussion and the variety of responses. A great step forward from yesterday’s disappointment where they largely ignored my Radiolab assignment.

I’ll count this day in the victory column for sure.

First Day Back – A Tale of Triumph and Sadness

My first period class (we call them Bells here, rather than Periods) is my Geometry class. I started by sharing with the the NYTimes story about ‘The Interview’ and was pleased that they quickly attacked this as a system of equation. I had a secret plot for starting with this problem. We are getting ready to explore triangle bisectors of various sorts. I started out with this question for my students: ‘What does it mean to call a point a center for an object?’ Luckily, this prompted a quick recollection of centers of circles along with a nice attempt at remembering a sound definition for a circle. I then asked them to consider what would be the center of a square. One of my students, a freshman named Matthew, quickly proposed that the intersection of the diagonals would be his point of choice. I opened GeoGebra, drew a rather random square and tested Matthew’s idea. We saw that this was in fact equidistant from the vertices. I then asked about distance to the sides. This required a quick conversation to remind them of what we mean when we talk about distance from a point to a line or to a line segment. We quickly came to an agreement that the perpendicular distance was what we wanted. GeoGebra confirmed that this ‘center’ was equidistant from the sides of the square, but I pretended to be troubled that this second equal distance was not equal to the first equal distance. My students quickly overruled me and were comfy with this point as the center. Next, I asked what the center of a triangle might be. I had three students each volunteer and ordered pair as a vertex of the triangle. It turned out that they formed a right triangle. We agreed the idea of perpendicular bisectors (which we had JUST looked at for the square!) was the way to go. Some quick GeoGebra showed that these lines coincided at the midpoint of the hypotenuse. I was pleased that this raised questions. No one jumped to the conclusion that this would always happen and a student named Tara quickly guessed that this was happening due to the original triangle being a right one. I then moved one of the vertices so that the triangle was acute and, happily, we noticed two things together. First, the perpendicular bisectors still coincided as I moved a vertex. Second, they coincided inside the triangle. Matthew then asked to see what happened with an obtuse triangle and we saw the point of coincidence migrate outside the boundaries of the triangle. It was great to notice that they still met at a point, but the idea of a ‘center’ being outside the triangle did not make anyone happy. Matthew observed that this point did not feel very ‘centery’ to him. Awesome stuff. Finally, since we had GeoGebra to confirm our work, it did not seem that intimidating to go ahead and find the coordinates of the point of intersection for these lines. My secret plot of having them think about systems of equations at the beginning of class paid off. Overall, a wonderful way to start the new year. Tomorrow, I’ll try my idea of HW review at the beginning of class and see how that feels.

Unfortunately, the feeling of triumph dissipated quickly. I have two Bells of AP Stats this year and I had asked these students to listen to a Radiolab Shorts episode called Are We Coins? and I gave them a handful of question prompts. I asked everyone to jot down some reaction notes and to bring these notes to class today. In my first class of 11 students I had three who showed clearly that they had listened to the episode. I had zero students with notes. I asked everyone to take out their notes and a number tried to fool me by having a notebook in front of them, but none of these had anything to do with my questions. In my next class of 16 students, three of them had notes and one or two others showed clear evidence of having paid attention to my request. Sigh…

I’ll dwell tonight on the Geometry kids instead and get ready to really dig into this idea of ‘center-ness’ for a triangle tomorrow. A couple already asked, on their way out of class, about using the vertices as anchors instead of midpoints. Should be fun tomorrow. Lots of noticing and wondering and a concession on my part to their need for HW reinforcement. Hoping for another great start to a day.

Thinking About Work and about Homework

I took my mom to the airport this morning which means I have one less reason to start thinking about work again. We start back on Jan 5 – as many of us do, I am sure – and I have a pile of papers to grade and some class planning to do. As I have written before, we are working with a new Geometry book that I wrote last summer. You can find a dropbox link to it here. I have also posted a folder of the HW assignments we’ve written for it here. I’ll be updating the HW folder regularly. I have been compiling a list of edits for the text and I’ll be doing the updating work this spring.

As I think about our next Geometry unit I have really been wrestling with the role of HW and how to incorporate it into my classroom routine. I’ve been mostly teaching AP classes the past few years and I have definitely adopted an attitude that HW is assigned but not graded. I’ll discuss it in class when someone asks a question, but I don’t plan around that activity. These questions from my students pop up on quiz days or on test review days but not too often otherwise. I am under no illusion that the students are doing all of what I ask from them, but their performances indicate that most of them are doing enough to get a grasp of the material. I believe that their understanding and mastery of the material would be deeper with more practice. I also believe that they would practice more diligently if I incorporated HW into my grading system somehow. What I have seen is that any way that I have tried to include this into my grading system encourages copying and shortcuts that are frustrating to me and reward bad behavior. This year, however, I am working with younger students again in the Geometry class. These students pretty clearly crave much more attention to HW than my older kids seem to want/need. Days when I post the last HW on my AppleTV and spend time directly addressing it are clearly appreciated. What I am wrestling with is the feeling that these days also lack some of the creative problem solving feel of days with more open-ended questions driving the conversation. Am I wrestling with the balance between what motivates me and what motivates my students? I know that there is a cliche that most of us become the teachers we had. I certainly think about some of the inspirational teachers I had and try to capture some of that magic. However, I also find myself thinking at least as much about the experiences that were NOT inspirational and try to avoid those scenarios. For me, many of those uninspiring days centered around 20 – 30 minute conversations about last night’s HW. Days where we seemed to be spinning our wheels. I know I was not the only student who had completed the HW from the night before and did not need to have most of those problems reviewed right after completion. I know that I have students who drift away from being engaged when we are going over problems they have already thought about and completed correctly and confidently. I am really struggling with how to best strike a balance between what some of my students crave and what I think is best for them. I think that our time together is best spent on open questions that push our understanding forward, problems that would be a real stretch for them on their own. I want to include those (periodically) on HW assignments and concentrate on THOSE problems when we are together. Given that a number of my students seem to crave the comfort of the routine of doing HW and then going over it together, I need to find comfortable ways to incorporate that without stalling other routines. One way I am thinking of – and I know that this is not original at all – is to have answers posted at the beginning of class and asking students to take the lead on any review questions that pop up. I can have a few kids up at once and cover this material more quickly and in a more student-centered way. This way, we can get to the more challenging questions together more quickly and I can respect the desire to have space for HW review. Since we are writing our own HW assignments, the students have no answer page to look at and this may be an important source of the difference in HW attention desired. My older kids have standard texts with answers in the back of the book for reassurance.

I’ll report back soon after we start again and I’ll write about how this works. In the meantime, as always, I welcome wisdom in the comments section or through my twitter @mrdardy