Jonathan Lind
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I often leave experiences like this feeling guilty, like I’m not doing enough. Looking back over this course, I’m going to change my mindset by saying that I’m on the right track. I am doing many of the things outlined in this course, and I’m interested in trying some of the things I haven’t implemented yet. This experience has given me some motivation to get through the rest of this school year and some excitement about implementing some new practices for next year.

Trust between students and teachers is super important, and I’m glad you mentioned it here. A lot of students come to my class pretty damaged by their earlier experiences, so it often takes awhile to get them to believe that we all want the same thing: them to understand math better!
Readdressing informally is great, but there’s a bit culture of “that’s not fair” at my current school. I feel like if I tried this, I would get complaints that all students didn’t receive exactly the same opportunities or problems as each other. Maybe I just need to do it and deal with whatever happens.

Great illustration of how to use time. I often feel guilty about not doing the right thing all the time. Sometimes, I resort to direct instruction; I don’t like to teach that way, and I know I can do better, but sometimes that’s all I’ve got. I have found that more practice on my part designing and implementing problem solving lessons, the easier it is to plan them. I’m also building up a repertoire of lessons and techniques I feel good about, which also makes it easier.
I think the colorcoding at the end of this video was a great illustration of time use; also the classroom clock is something I think about a great deal.

I agree with some of the apprehension expressed above. I don’t think I can handle this with three preps, without a welldesigned set of curricular materials (and a teaching team that’s really into it).
There are baby steps that I think I already incorporate, though. One example: our geometry team came up with right triangle trig as one of the most important takeaways from the class, and we decided that every unit following the trig unit would incorporate trigonometry in the problem solving tasks.

I love the idea of spiraling, but I would need some kind of road map to start it for each particular course. And then I would need my school’s assessment policies to catch up with what all the research says ðŸ™‚

Introducing matrices this week
Attention: Using transition diagrams to create or show a natural need for matrices. It’s a class of curious students, and this is a brand new format for presenting info, so I think it will graph their attention.
Generation: Students will turn the diagram into an organized table, which we’ll start referring to as a “matrix”
Emotion: I think when we can get students to see that something they have experience with (tables) is already the new content we’re going to talk about, especially if they can come up with it themselves, that creates an emotional connection.
We’ll see about spacing…

This is a really boring example, but in a recent coordinate geometry unit, I had a plan to do a
First lesson: problem solving/ reasoning lesson with a variety of problems involving lines, circles, and quadrilaterals on the coordinate plane
Next lesson: partitioning line segments in ratios, which is one of the trickier skills in the unit, and the last new skill the students learn.
From some individual formative work during the first lesson, it was clear that the majority still needed some practice on basic skills like distance, slope, etc, so going to the second lesson would probably have been a disaster.
Pivot lesson: students who needed it worked on practice problems from DeltaMath or Khan Academy, with my support or in groups. Students who didn’t worked on a Desmos activity to apply some of the skills they new to unusual situations. The entire class then moved on to partitioning together. It was a needed break from the rushed pace, and got us all back on track together.

Noticing some parallels between our classroom setups, Jon cards on the boards, students standing, etc. Fun to see that in another classroom.
I find the use of anticipation, when I can find the time, to be pretty powerful, particularly in cases like you mentioned where not all of the strategies you thought of are used, so it’s a good time to step in and share different ways of doing things.
I feel like the during moves get easier with practice. They’re a little daunting at first, but they become just what you do every day.

Anticipating a problem that becomes much more manageable with a transition matrix.
 This reply was modified 4 months, 1 week ago by Jonathan Lind.

So, this week 11th graders will be working on basic matrix operations. Future goals would be representing and solving systems of equations, and eventually working with pretty complicated probability systems using matrices. Thinking about these future goals helps me deal with questions about why we do matrices at all: to do a lot of calculations efficiently.

I’m introducing matrices to my 11th grade IB class. We don’t teach matrices at all earlier in the curriculum, so I thought it would be interesting to think about a learning progression for matrices.

Introducing radian measure next week in Geometry, and thinking this could be an opportunity to call back to some earlier understandings of pi. We can get out the string and see how many radii it takes to get around a circle. We can talk about why we use degrees, and why it doesn’t make great mathematical sense. That’s all I’m coming up with right now…

I had a frustration recently with my 11th grade Higher Level IB class the other day that brought to mind the idea of memorization vs automaticity. It was basically solving complicated algebraic expressions (involving radicals, pi, fractions, etc), and they couldn’t see the connections between all the algebra that had gotten them to that point. You don’t get to HL math without being a traditionally “good” algebra student, and probably even someone who enjoys algebraic manipulation. My students had so much experience and success solving algebraic equations with whole numbers, but they just weren’t making the connection between, say, 3x=6 and 5pi/sqrt2=6.
They kept asking for help, and I just kept on writing simpler algebra problems on the board. Eventually, we all came to an AHA moment, but it felt to me as if I was in front of a room of students who had memorized, rather than really understood, algebraic manipulation.

I have a little trouble seeing the final stage from the example: isn’t distribution a whole extra topic? If the goal is multiplication, can we define the mastery stage as a different concept?
EditTalked to my wife who teaches elementary, and watched the following video involving donuts, and this make a whole lot more sense to me now ðŸ™‚
Anyway, we’re working on circle theorems in Geometry, and introducing cyclic quadrilaterals. We’ll be working towards the understanding that opposite angles must be supplementary because they open onto arcs that make up the whole circle. Students have already worked with inscribed angles and know those relationships, so I don’t expect this to be a long journey, but it’s what came to mind.
1. I have no idea how quadrilaterals in circles relate to our prior learning. I wouldn’t be able to find the angles in an inscribed quadrilateral unless I know information that doesn’t relate to circles.
2. I know that there must be something going on with these quads, because we studied inscribed angles and all of these angles are inscribed.
3. I see that opposite angles open onto arcs that make up the entire circle, which is 360, and inscribed angles are half of their arcs, so the sum of opposite angles must be 180. I can find the other angle when given one.
4. I recognize inscribed quadrilaterals and automatically apply this rule to problems involving them.
5. I can find inscribed quadrilaterals and use them as a tool in solving more complex problems of angles and arcs in circles.
 This reply was modified 4 months, 3 weeks ago by Jonathan Lind.

Here’s a problem from a recent assessment where I used some student work to start a discussion about problem solving. The students both had similar strategies involving using subtraction to find irregular area on the coordinate plane. We can discuss efficiency (left student did a lot of extra work), connecting different representations (both students used labels pretty well), and areas where they could add strength to their solutions (confirming the answer in a different way).
I put together student work like this after assessments rather than going through the problems myself or providing answers myself, so that students have models for effective strategies. It’s a pretty effective way to show students expectations, while also showing them that they can do this, and don’t necessarily need a teacher to tell them how all the time.
 This reply was modified 4 months, 3 weeks ago by Jonathan Lind.

Awesome visual. It’s great that more and more of these are becoming available, as it isn’t super easy for the “average” teacher like me to throw these together ourselves. Let’s all keep sharing these!
Here’s a visual colorbased proof of the law of cosines that I use every year with at least one of my high school classes:

Multiple representations is something that’s actually on the rubrics we use to assess in math class. We encourage visual, algebraic, numeric, and verbal representations in the process of problem solving, and especially in the act of communicating reasoning. Students aren’t always used to doing this, but when they get into it, we see some innovative, novel ways of representing problems, and of connecting those representations together. Being able to make these connections makes the students better communicators, reasoners, and problem solvers in general.

I’m working on a geometry lesson for the coming week on ratios of line segments, with the ultimate goal of introducing the idea of calculating them using weighted averages. This is a good exercise to get my thinking around it! I’m always reminding myself to preplan on what to look for while students are solving problems, and this helped with that.
here’s my template:
 This reply was modified 5 months, 1 week ago by Jonathan Lind.

The hero’s journey makes sense as a path we hope our students take through math class. Productive struggle is super important, but it’s SO difficult to get students, especially students with 8+ years of being told how to do things, to accept the idea that NOT knowing how to do things is where real learning happens. I’m really happy to see a lot of elementary teachers in these workshops, and am excited to start getting more and more students who have had this kind of instruction in prior math classes.
I do have a wondering of my own for Jon and Kyle: I’ve been working on implementing the steps from Peter Liljedhal’s Building Thinking Classrooms this year (it’s all about the Canadians in my math world this year :). His research seems to encourage us to get them going on solving a thinking task as quickly as possible, while the steps presented here have a great deal of preamble and teacherdirected discussion before the students are “set loose” to solve the problem. I am sure that these ideas can both live in the same world (they do in my classroom), but I wonder if either of you could share your thoughts on this. (I’m attending a short workshop with him later this week, and will ask him as well)
 This reply was modified 5 months, 1 week ago by Jonathan Lind.

WODB is a goto for me if I need a starter. Two truths and a lie is awesome, and having students make their own can take an entire lesson. WYR is a new site to me, and I wish they had more for high school. Maybe I’ll look into putting some together; it’s a great structure to start a productive discussion.

My before and after of a typical goat on a rope problem is below; first round I left the general question in, but then realized that it wasn’t really necessary and not having the question would really open up the exploration and questioning phase.
When an 11th grade IB class was presented with this problem and asked to notice and wonder, they (having probably seen similar questions before) came up with the classic questions that could be asked here, but also came up with much more interesting questions, and we decided to answer one of those instead:
How long would the rope have to be so that the goat could reach the entire field if it was tied to the opposite corner.
This also gave students an opportunity to estimate the length of the rope after they asked for and were given the dimensions of the field.
IB questions are often very scaffolded, which is great if you’re taking a test, but pretty lame for classroom explorations. This exercise helped the class have a much richer discussion than we would have solving the original problem.

This past week I’ve been working on probability with a class of 11th grade IB year 1 class. I started with a textbook 3circle Venn diagram problem about speaking additional languages, with a list of information, and then parts ae asking about number of elements in certain sets and probabilities.
Anticipation: Before showing the problem, I asked my students whether they thought a person speaking, for example, French in addition to English might be independent or not from a person speaking Spanish in addition to English. This was a good opportunity to discuss and address the common misconceptions about independence, and really work on our understanding of the concept as it relates to probability.
I did not find a good tie in for estimation, but I was just now thinking that I could have changed the languages to make it more relevant to my situation (an international school in the Persian Gulf region), and then asked them to estimate how many people in a random group of 60 students they think might speak these languages. Maybe next time this would be a good opportunity for engaging them in the problem. I also forgot to do a notice and wonder, but could have easily added that in when they were first presented with the clues.
I removed some of the clues about the number of students in intersections, and removed the instructions to draw a Venn diagram, and gave them an abbreviated set of clues. Their job was to figure out what information to ask for so that they could find out how many total students were in the survey. This is my favorite part of all of this, because you get to really see students thinking through things and working together to figure things out. They eventually got to Venns, and asked the necessary questions, and were able to move on to figuring out whether speaking these languages were independent of each other.
I need to like tattoo on the back of my hand that this makes math class more fun for everyone so that I remember to do it more often. It’s usually pretty easy to modify problems if I can remember to think about it!

This lesson was a great reminder. I’m going to look through some of my lessons this week and see where I can withhold information, and add some notice/wonder, questioning, and estimation to them. Sometimes kids don’t realize they need a piece of information and they start doing the problem anyway, and the moment they realize and ask for that info is always one of the highlights of my day.

I’ve been working on implementing problem based lessons for most of the time I’ve been teaching in the classroom, and I’m still working on it. Some thoughts:
1. I struggled with the energy and preparation I need as a teacher to present the task, monitor student work, and consolidate efficiently and meaningfully. This has gotten better with experience, but it’s still a struggle. Peter Liljedahl’s work in Building Thinking Classrooms has recently helped a great deal.
2. When I started, I thought that every single lesson had to be like this. It wore me out, and it wore my students out, and there was no time for consolidation. We had a lot of fun, but it felt very disorganized. There’s still a place for kids doing problem sets in class, and there are still appropriate times for direct instruction. As mentioned in this discussion, we don’t need to throw everything out; this is just another tool to use. For me, it has become the foundation of my classroom practice, but it took awhile to get there.
3. When I was first introduced to 3act tasks in particular, I didn’t totally buy into the structure (especially the beginning with the notice/wonder, the estimation, etc), but was pretty impressed with how it worked in practice. I don’t do it for every lesson, even lessons that are based around one task, but it’s part of the practice in my classroom. To start out, I found a few tasks that were right for my class, and implemented them in a sort of “by the book” way. The results gave me enough confidence in what I was doing to continue, and eventually develop problem based lessons in a way that was manageable for me. Give it a shot if you haven’t already!
 This reply was modified 6 months ago by Jonathan Lind.

I agree with a lot of what’s said here, but what first came to mind for me (particularly as a HS teacher) was:
Good: have been told they’re good at math for a long time, and have generally positive emotions associated with math class; want to keep doing the things they’ve been doing that have given them those positive “rewards” First to finish, 100% correct, good mimics of teacher methods
Bad: have been told they’re bad at for a long time, and have generally negative emotions associated with math class; have significant (and understandable) barriers to trying
Both: Want a comfortable safe environment; want to be recognized

1. A time when they helped someone else
2. A time when they were helped by someone else
3. A time where they didn’t want class to be over.

Isn’t it great to be able to work in groups again?
This sounds engaging. Using yourself (and your dog) is a great way of making some connection to the material. I’ve never used data generation to get into trig functions, but it sounds like a nice way to make them see how the functions work.

I also have trouble imagining a place for manipulatives in my higher level high school classes. I teach geometry, too, and have found some areas where manipulatives can be helpful: talking about angle side relationships in triangles, it helps to have a few rulers or even different lengths of sticks around to test out relationships; actually moving shapes around to explore congruence and transformations; and like Kerri said 3D models for volume and surface area. It’s pretty tricky as the math gets more abstract, though, and I’ve found that most students who might gravitate towards using manipulatives will end up drawing a representation of those anyway, which is a pretty good problem solving strategy at the higher levels in itself.