Reflection Protocols

Reflecting on reflection ...

What would a reflection protocol look like for teachers? Assuming it's an independent activity, here are a few ideas.


Timeframe: today's class
Goal: General reflection
1. What surprised you about today's class? List 1 or more things that stand out in your memory.

Then, go about and pick one thing to focus on for the rest of this reflection.
2. What did the event reveal to you or remind you about your students?

3. What did the event reveal to you or remind you about your own priorities and values?

4. What will you do differently tomorrow or next week because of this event?




Timeframe: Math Class Unit (To be done towards the end of the unit, but before any summative assessments)
Goal: Developing Growth Mindset, encouraging sense-making

1. What is a concept that many of your students still struggle with at the end of the unit?

2. What fundamental misunderstandings do they have that might get in the way of developing this concept?

3. What experiences might help break through those misunderstandings?

4. Is there a different way to teach this concept that you haven't tried? A different perspective or application?




Timeframe: before tomorrow's lesson
Goal: preparing
1. What concepts will students need before they do this lesson?

2. What do you expect them to get wrong, and what non-standard ways might they find to answer the problem?

3. How will you go about encouraging sense-making or risk-taking or developing your classroom community during this lesson?

Then, after the lesson:
4. What worked and didn't work from your answer above?

My Number Theory Class :)

What do I want to teach my students next year?

I will have the most amazing class. About 8 - 10 kids, 7th and 8th graders, who are all avid math learners, who have excelled beyond their years. They ask good questions, constantly draw connections, love to tackle new problems, they are willing to work through frustration when problems take a long time to solve, and they tend to justify their reasoning. Many know how to write proofs. All have completed Algebra I, several have completed Algebra II and Precalculus. I have worked with most of them before, and I like them, and I think they like me. Yeah, insanely lucky.

So, what is there to teach them? Content-wise, plenty. The whole world of math is at their feet. My intension is to spend most of the year on Number Theory. Mostly because Number Theory is awesome, fundamental, and deep, and also because the textbook I have is awesome and readable and funny and engaging.

But in terms of enduring understandings, and in terms of real take-away lessons about academics and about math: what do I want to teach? I want to have a few ideas to start the year with, so I can set a direction for the year.

What about the art of questioning? Have they ever looked at a situation in math class and come up with their own questions? Probably not much. I've tried it a bit, and only gotten so far with them, and then let it go. That's something that mathematicians do that they have not yet internalized.

The problem is, I'm not used to teaching this skill. It is SO much easier to pose a question than to just present a situation and leave the questions to the kids. My teachers did not model this for me! And I left grad school at just about the point when I'd be expected to come up with my own questions.  I can do it (I mean, that's why I write such challenging tests and fun worksheets!), but to teach questioning, that would be new.

Hmm, what if I start with the sequence that came up on Twitter today ...

1, 11, 21, 1211, 111221, ...

It's a really annoying pattern, especially if you like math. You can run yourself ragged trying to find an answer, but (SPOILER ALERT) you just read it aloud: one 1; two 1's; one 2, one 1; etc. It's not really math. Blah.

But you can ask math questions about it. What is the largest digit that appears? Any number without a 1? What proportion of the digits are 1's in the long run? Does the pattern ever repeat? What about this variation:

1, 11, 21, 1112, 3112, 211213, 312213, ...

How are they different? How do the properties of the two sequences compare?

The point is, I can start with this annoying non-math sequence, and then ask the students to ask math questions about it. That might be a great place to start this journey together. We need to take time to ask more questions as we go, I need to remember to give them room to ask questions. And I need to give them situations that are worth asking questions about. Games. Stories. Pictures.

Okay, one goal set. What is next?

I can teach them revision. That is a skill that they will need for life, how to go back and make your work better, more approachable to an outsider, and more aesthetically pleasing to another mathematician. It's something you should do with all writing, and math at this level is writing. One of my favorite things about grad school was the way that people would not be satisfied with answering a question - they'd want a GOOD answer. A beautiful answer. I think these kids already have the sensibility - they will argue over whose method is the better one, which is the essence here. But I want to connect it to the way they write up their work. To pick a problem to go back and rewrite, and write it more clearly the second time. How do you get kids to value that work? I should talk with the English teachers about that, they must do it all the time.


Okay, two goals, not bad for one blog entry! Here I am asking the big questions. I'll come back and revise my thinking about it later in the summer.

Cheers!

The Reading Wars?

I just read an article in The Atlantic, "The Radical Case for Teaching Kids Stuff," by Natalie Wexler. The idea is that students should not learn reading skills in a content vacuum. Reading skills develop when students are given deep and meaningful content to read about, especially if that reading is part of a larger exploration of a subject.

The studies and anecdotes she cites suggest that students are interested in the complex world around them, and their interest in the world will propel them to be interested in school and become better readers. She talks about content like trees and birds and Mesopotamia and Greek myths. She does not say that these subjects must seem immediately "relevant" to the kids. The content is not "American Idol" or text messages or ice cream sandwiches. This is very different from a message that I receive as a math teacher, that kids must see how the math is relevant to them on a personal, immediate level or else they will not be interested in it. (BTW - I have never noticed that to be true in my own classes.)

This seems particularly relevant to me to be because of a tweet from a week ago, saying that in math you need to learn the basic ideas first, and apply them afterwards, just like you need to learn to read first, and then read to learn afterwards. She disagreed with that on the reading side.

I wonder - does she have a view on math education? If it is in line with her view on reading education, she would believe in "content-rich math." But how would that be interpreted? Is the content a series of multiplication facts followed by the distributive property? Or is "rich content" a deep understanding of relationships in math, and the relationship between math and a broader understanding of the world from an analytical perspective?

I wonder - this is a bit of a call for back to basics -- schools should teach stuff! If this idea catches on (and the article I read made it seem like a reasonable one - but I have not seen the other side yet), and schools move away from standards and into a back to basics approach, will math get caught in that tide yet again? Because I see her call for a deep and rich, contextualized approach to learning science, humanities and reading. And that I what I want for math teaching as well. But I worry that the tag line -- "schools should teach stuff" -- will be misinterpreted as a call for dry, repetitive, meaningless math.

Well, we've been down that road before, and we are getting better at fighting our way out of it.

Oops! My bad.

Earlier today, I wrote up a whole post about how Twitter is an echo chamber. I gave an example: there was this one tweet that I have agreed with and half disagreed with. But there were, like, 15 replies, and they were all just saying, "Yes! Totally! Could not agree more!" and I was a bit worried as I put in my "Yes, but..." reply.

But then I went back to add the actual tweet to show my point. (Show, not tell, Mr. Ratner!) And I looked at the whole big list of replies. I was a little surprised by what I saw. There were three that agreed completely. There were about ten that partially agreed, adding their own nuance and perspective. It was really a great, if choppy, conversation. (Each point was so short, with these tiny tweets -- grr, twitter!!).

So, what was it that made me think everyone was being a yes-man and blowing meaningless sunshine at each other? Why did I get that impression when it was clearly not there?

Now on Twitter

I'm brand new to Twitter, alongside being new to blogging. Wow, what a world. It is so easy to fill your time with blabber. I mean, it's not like everything I say is deep and meaningful. But really, this?

I posted a question and asked for some advice. By morning, my notifications had about 15 tweets listed. Yay! Then I read them. Two people gave advice. Many retweeted or liked my question. Some liked the replies. But for three screens worth of responses, only two people said anything. What's that about?

And why does everyone agree so vehemently with each other? Posts with replies are usually full of "Oh my god, yes!" "That is SO TRUE!" etc. And then there are so many of them, the list goes on. Do you really read through all those yesses? Maybe there is some interesting insight hidden in reply 13 out of 24, but you'd need to read through all 24 to even know. Ugh.

And I am having a really hard time getting my thoughts into one little tweet! I include nuance, and nuance takes space. Sure, I know about the six-word biography. "For sale: Baby shoes, never worn." Yes, some genius can write six words and express a whole world of emotion and thought. But I need a full-on paragraph, sometimes even two, to say anything worth saying. When those characters turn red and twitter says, "You can talk, but now you've talked too much, this part needs to go!" I feel like I must have done something wrong. (Yes, reply to your own post, post two in a row, I get it. But the red highlighter still freaks me out.) Will I end up using that crazy short millennial speak? U goin to Ss hous 2night? Will I just start leaving out "that" and "the"? How will this impact me over the years?

And yet, this combo of blogging and tweeting is interesting. I realized when I signed up for twitter that I'd need a place to write actual thoughts, hence the blog. Lots of people seem to do the same. I write out my thought in this blog as if a) they matter, b) people care, and c) it will be read one day. Then, I go to twitter and reply to others' posts as if a) I know what I'm talking about and b) they will listen to me. And I see others replying to my tweets, so I know for sure that people are reading them. Wow, that's is heady stuff, a big ego-changer.

Close encounters

Met someone in a coffee shop today. She noticed the quote on my shirt: "Don't let school get in the way of your education." I mentioned that I was trying to write up a statistics course that let the kids do some good digging and exploration on their own. She gave me some really good ideas: first, I should actually do the research project that I'm going to ask the students to do, and show them my work, and be honest about my difficulties. They will be more willing to admit their own difficulties when they see mine. Not a bad ideas!

We got to talking - she is also a middle school math teacher. But the most interesting thing she told me was not about school at all, it was about helping kids in find their way in the community. The story went like this:

I walked by a group of three girls sitting out in front of the high school this morning. I passed by two hours later and they were still there. I asked them, "What are you doing here? You've just been sitting here for two hours. Do any of you have a job?" They said no. So I walked them over to CVS and I made sure they all got jobs. They were asking, "Who are you anyway?" But I just said, "No, doesn't matter who I am. You need a job. And this is not free - there is one more thing you need to do. You need to put half of your paycheck in a savings account."

Wow. I hope that is a true story. But either way, it's sure an inspirational one!

Open ended, open middle

At last! I found the phrase! So often people talk about "open-ended" problems when they are talking about math puzzlers with exactly one solution. That's just not right. I adopted the language "problem-solving problems" but that's just awkward. But finally, some genius came up with the right phrase, the one I've been looking for all these years: Open Middle. That's it!

In my math-ed classes in grad school, they touted the benefit of open-ended problems. Like this one:

James needs to mow his lawn. The lawn is a rectangle, 12 feet by 16 feet. He needs to start and end in the top left corner, and he doesn't want to push the mower any longer than necessary to get the job done. The mower mows a path 14 inches wide. Show one path that James could take to mow his lawn that involves as little overlap as possible. 

I can imaging students having different good answers to this problem, with 3 or 4 "best paths" emerging in the end. Students might argue about whether or not turning the mower around a corner is more work, and some might talk about the need to overlap the path a little to make sure you got the edges fully. The problem is clearly stated, but no direction given for where to start, and multiple correct answers might come up.

Here is another problem that was also presented as "open-ended" in that same class.

A simplex lock is a lock with five buttons, numbered 1 - 5. You can set whatever combination you like, with the following constraints: 

• You need to have at least one button pushed in your combo, but you don't need to necessarily push all 5. 
• Each button can be pressed at most once. Once it is pressed, it stays down. 
• The order you press the numbers in matters. However, you can also press two or more numbers at the same time. So, 1 - 2 - 3 -4  is a different combo from 1 - 3 - 2 - 4, and it is different from 1 - 2/3 - 4 (where 2 and 3 get pressed together). But 1 - 2/3 - 4  is the same as 1 - 3/2 - 4. 

How many possible combinations are there? 

This is an interesting problem (to me at least). It is not obvious where to start, and I never learned a method for solving a combinatorics problem of this type before. I can imagine many possible ways to try to solve it - mostly breaking the solution space into "types" of combinations and then counting the combos in each type. For example, you might look at all the combinations with exactly 1 button, then those with exactly 2, then 3, etc. But different students might break it down along different lines, leading to rich discussions about what methods work best and what range of methods we could use.

But open-ended? Not quite. There is exactly one solution, one final number of possible combinations. In fact, this would be a great problem to bring up (if it weren't so complicated) when people tell you that math is always right or wrong, 2 + 2 equals 4 every time. Yes, it is true, math often has correct answers. But don't tell me math is straightforward. Here is a problem that has a single right answer, but finding it is anything but simple, and memorizing a procedure to solve it is just crazy. (Interesting conversation about that here.)

What happens when we talk about these as open-ended problems when they are not? Maybe we don't give actual open-ended problems a fair shake. Most of the actual open-ended problems I've come across involve modeling, perhaps with statistics, often making assumptions or estimations. This is such a valuable skill to have, and it leads to multiple "correct" answers. We need to be teaching more modeling problems, and let kids know that there is not always a single correct answer when they are using their math brains to do a math problem.

And I need to way to communicate with parents about problems that involve multiple paths, about the joy of problem-solving, about the creativity that comes about when you let your kids loose on a deep, complex (for them), problem with no clear path to the solution. "Open middle" describes it perfectly. The problem is stated clearly, and there is an answer to be found, but it is not at all prescribed how to get from here to there. You have options, and the process belongs to the the student, not to the teacher.

This is also a way to combat the idea that all math processes that come up in the classroom should be taught.

Because there are many that CAN be taught, and some classes teach them, but I don't think it does anyone any good. But that's for another blog post.

Games and Puzzles

Which fraction activity would you rather do? 

Who Am I? 

I am a fraction in reduced form. 

My denominator is 3 more than my numerator. 

My value is more than 0.5 and less than 0.7. 

My numerator is prime. 

Who am I? 

The Dice Game

Roll two dice. If you get doubles, roll again until you don't have doubles. Make a fraction with the smaller number over the larger. Then, your partner rolls the dice and makes a fraction with the two numbers the same way. The person with the larger fraction gets a point. 

I think a lot of people would go for the Dice Game over Who Am I?. Personally, I don't like how the Dice Game depends completely on chance. I prefer games that give me a chance to figure out how to win. But that's my view as an adult - I see a lot of kids get very excited by the roll of the dice! For all the teachers out there, which do you prefer? What about your students? 

I'm thinking about this question because of two students in my class last year. Neither was particularly motived most of the time, even though both were excellent mathematicians. But when I came up with some fun activity to do, I could usually grab the attention of at least one of them. Nope - scratch that. I could grab the attention of exactly one of them. 

One of them loved the dice game. He would come up with all sorts of creative (and correct) ways to explain to his partner why one of those fractions was larger, and did not want to stop playing. 

The other kid had no interest in the Dice Game. But let him loose on a "Who Am I?" puzzle, and he was working frantically away at it. He came up to ask me if his answer was correct, and then he was willing to walk me through a detailed explanation of his reasoning. 

It took me a long time to realize why I could not get both of them interested in the same activity, but it finally occurred to me. One of them loved games. The other loved puzzles. What's the difference -- what kind of kid likes one or the other? I don't have a handle on that yet. I'm just glad I used some of each this year, so both of those two students had a chance to feel like they were in their element once in a while.