Tuesday, May 3, 2016

Teacher Preappreciation

Yesterday's post was feedback from a group of students that I want to do better for. That applies to this group of preservice teachers as well, but today's post is just going to be some bon mots from these guys about teaching.

I am such a teacher fan boy; I consider myself so fortunate to spend so much of my time working with people who are dedicating themselves to others. Hopefully this post shows that this extends even to before they're in the classroom. I encourage you to click on the link to their blogs and poke around a bit. I hope you have the time to read these - it's like getting a dozen semesters of elementary math ed. And you'll be able to see why I am going to miss these guys.

Kalyn gets us started: "Getting together a small group of soon-to-be teachers with the goal of having students understand and even like math more than what typical stereotypes say, creates a strong classroom. During the semester we had the opportunity to work with different people and learn from each other. We didn’t always agree on what we thought the answer was or on a particular way to solve a problem, but that was kind of the point of the class. We learned to express our ideas, and hear others ideas, and LEARN from each other. The ideas that many of my colleagues had were great and it is said that two works better than one, and I am pretty sure that 13 works even better than 2." 
     One exemplar: "Half of It: The last writing I want to submit is based on a problem that I explored. The problem was simple in nature, but contained a lot of strategy and mathematical thinking. Breaking down a problem and writing about it was something that I never did before this class, and I felt that I learned a lot from it. http://kalynjoy.weebly.com/blog/half-of-it"

Dayna: "One of the main ideas i learned from this class is to let the kids make discoveries about the content. I am now a firm believer in that students learn a lot when working with new things on their own. this also goes with being able to work with others. We demonstrated this idea in class everyday and i found myself learning so many new ideas and methods from everyone in our class. I also learned about time and how it is going to take every student a different amount of time to do their math problems."
     One exemplar: CGI story types. http://daynaball.weebly.com/mth223/through-the-eyes-of-a-teacher, "I choose this blog because i think it is a great take away from a class that confused me. I think it is important to know these strategies and making sense of them so i can help my students make sense of them in the future."

Dana: "The big idea I will take away from this class is that we need to delve into the student's thinking.  It is not enough for students to get the right answer on math problems, they need to be able to think mathematically and explain their answer.  I never thought about asking a student for their thinking before this class, and now I cannot imagine teaching math without do so"
    One exemplar on Sorting Geometery. http://danamspielberger.weebly.com/blog/the-unasked-question "For this one, I liked the challenge of writing a math blog post.  On all of my other posts, I think I tried to stay way from talking about anything that had something to do with math, but for this one I was specifically challenged to do the opposite.  This allowed me to explore math problems and pick one that I thought was interesting.  This post is one of my best because I was able to talk about something interesting to me while still being challenged."

Sarah: "This course has made me re-evaluate teaching and see the impact of looking deeper into mathematical concepts. Even the concepts that we consider simple and elementary level contains patterns and characteristics that we never evaluated as a child. An example would be double-digit subtraction. If a student asked me exactly why do we borrow, would I be able to give them a complete answer? When I think about it, I was never taught why and I honestly never questioned it. Math is so full of “just do it” and any questions are often answered with “because you just do.” How can you teach children to do it without fully understanding it yourself? This class has made me explore concepts deeper so that I can answer those difficult questions."
     One exemplar, Teaching with a Purpose: https://sarahacoutts.wordpress.com/2016/03/19/teaching-for-a-purpose/, "because I was the most passionate about this writing. I wrote it shortly after my mission trip over spring break and I reflected on my time in Dallas while also connecting it to the kind of teacher I want to be."

Amber: "Students have fixed mindsets when they shut down with math. They might be convinced that they are not ‘math people,’ be afraid of failure, or have outside sources telling them that math is unattainable. Students often shut down, and many associate math with torturous fact-memorizing and stressful timed tests. It is my role as the teacher to discourage this type of mindset in students. This means being flexible, accepting and encouraging mistakes, and modeling how fun math can be. This is especially important with people of color of women-all marginalized groups in math. I am all about equity and equal opportunities for all my students, and I can indirectly influence their future success by promoting math as a non-threatening, creative endeavor worth doing. There are no ‘math-people.’ There are just people and math, and everyone should find value in math."
     One exemplar, Whole Number Sense. http://ambergerrits.weebly.com/blog/february-07th-2016, "this one is so different from my other posts. It is an in-depth analysis that I honestly put a lot of work and time into (probably at least 4 hours of work). Like you mentioned in your comment, after writing about the whole number concepts, I have internalized them even more. There is more to write about with respect to instruction, but that is a whole other post. I edited it slightly-I proof read it and added spaces between paragraphs for clarity."

Ally: "Children are crazy creatures. They're just tiny little adults, but you know really small. Then, something I've learned from this class is that although they're small, they're mighty. They are SO smart. When we went to teach them I had this preconceived idea that they wouldn't understand some of the math that we were trying to teach them. For example, the last time we went to teach them, so on Friday. We had a set ratio, but the children needed to get the amount of sticky notes that used that ratio. At first, I had a hard time trying to figure out how that was going to work, let alone a child that is half my age trying to figure it out. I walked into that class room Friday morning having very little faith in them. This I understand is awful and I shouldn't think this way, but if I couldn't figure it out, how could they? Then, one group by one group they were understanding... WHAT?! I was amazed. This was fantastic."
     One exemplar, Flip for Math. http://allyboomsma.weebly.com/blog/i-flip-for-math-or-do-i "My last post I believe is one of my best because I was creative, and full of detail about my person gymnast life. This was the first blog post we did and I was happy that you enjoyed it too." 

Chris: "When we began our first counting circle on the first day of class, I was immediately nervous because I was afraid of being put on the spot. However, I learned quickly that this class, like the counting circle was open for mistakes, and was accepting of all kinds of input. This is encouraging for someone who is naturally introverted. Being able to come to class and worry about the important topics and not about the atmosphere of the classroom seems like a small aspect, but it makes a huge difference in my learning. It has also opened my eyes to the way I want my future classrooms to feel. This class has truly helped me foster an open mind about making mistakes, and using them to grow."
     One exemplar: Letter to a Concerned Parent. http://stromelemntarymathblog.weebly.com/blog/making-sense-of-new-methods "I worked hard to write a piece that combined daily life for me into my passion to teach and what we have learned in class thus far. I think this writing accomplishes taking the class to the next level. I did my best to take a scenario that I think teachers today are facing and came up with a solution that applied both the mathematical strategies and other teaching strategies I adopted from the class. I think this paper is decently written, and shows my understanding of what we have learned in class in a unique way."

Danielle: "Throughout this semester we have discussed several content areas that intimidate me, mostly because of my experiences learning them in elementary school. When we covered an area of concern for me I always learned new methods for approaching and solving the problems. Coming away from this class I feel much more comfortable and confident in my teaching abilities because I feel more comfortable and confident in my understanding and ability to solve the problems."
     One exemplar, Relearning How To Multiply. https://minsterd.wordpress.com/2016/01/27/relearning-how-to-multiply/ "I believe that this is an exemplar because I was able to reflect upon my previous knowledge, what I learned in our class discussions, and how I want to teach in the future to write on a subject that I struggled the most on and showed several ways someone can use to solve a problem."

Kathleen: "What I've learned about teaching is how to take a student's thought and write it down. Being able to record what someone is thinking is very hard. I want to know what they're thinking. I don't want to assume anything. As a teacher I need to be able to express what they're thinking. I never want them to feel cheated, or like I'm not understanding them."
     One exemplar, Math and Gym, http://mathleen.weebly.com/blog/lesson-idea-math-and-gym "an actual lesson I've done and put together for another class here at GV and I like the way it incorporates gym and math."

Oriana: "I chose a math major because I want to teach in elementary schools and show young learners what fun and amazing things we can do in math. Math consists of creativity, exploring, and not just one right answer every single time. I want to create an environment where math is fun and intriguing. Here at Grand Valley, as I go through different math classes, I feel as if am equipped with tools to share and create an environment that portrays the fun, and exciting side of mathematics."
     One exemplar, Learn the Facts. http://obenin.weebly.com/blog/learn-the-facts "I chose this one because I've tutored in elementary classrooms and helped children who struggle in math. One of the ideas that kids keep coming to me for is no knowing their facts. It's been hard for me before this class because I tried to explain repeated addition, but they still struggled because students didn't have them memorized. This topic really hit home and is applicable. I know methods to help students out."

Brittany: "For me that really hit home because as a student I HATED math. I was the quite student in the classroom that teachers liked to call on to hear me say something in class. I was terrified by that at all times but especially so during math class, because I didn't think I was any good at it and didn't want everyone in my class to know that. I would spend hours at home sitting trying to do my homework, and trying to get help from my parent, who didn't either understand what I was doing or didn't know how to explain it to me so I would understand. Now I'm not going to go into that right now because that is a whole other bag of worms that I won't open up today, but those are real issues that I had and my future students will have."
     One exemplar, Math as a Foreign Language. brittanykloe.weebly.com/blog/math-foreign-language "I picked this one because it really speaks to who I am and what I think."

Stephanie: "My favorite aspect of the class was the openness, and the acceptance of everyone. No one judged anyone for being wrong, or for not understand something no matter how easy the topic might have seemed. We were able to learn from our mistakes instead of being yelled at or put down for not being perfect all of the time."
     One exemplar, Why so negative? http://stephanieepetersen.weebly.com/blog/thoughts-can-be-deceiving was one of my best works because it was something that hit home with me, and I was able to relate with the topic really well. I am very passionate about rooting for the underdog, so actually read text proving my point on how we always give up on the underdog so easily was a fun read, and it made writing the blog very easy.

Heather's one exemplar is her course reflection, and she has a fitting conclusion: "Often I found myself questioning everything I had previously known about math and education. I know that the main purpose of this course was to give us future educators the chance to learn techniques and practice teaching math to young children, however I found that many of the things that we were taught I can use in my life. ... Above all else it was respect that made this class stand out in my mind. A true respect for my peers and knowing that I had their respect in return, as well as our respect for the professor and most importantly his respect for us as students and future educators. He truly took in our ideas, helped us develop them into something even better. It was this same respect that helped us blossom throughout the year, questioning more and becoming more confident in ourselves rather than seeking approval."

I am officially verklempt.

Monday, May 2, 2016

Student Takeout

OK, maybe more like student takeaway. 

The last writing assignment in College Algebra (of 6) is a bit different. The first 5 are them writing up a problem in depth, explaining thinking. I give them feedback, and they can revise or not based on their wishes. At the end of the semester the writing grade is half % out of 6 writings written, and half evaluation of three exemplars the student chooses on the 5 Cs. 

Five Cs: developed with David Coffey, based on NCTM's Communication Process standard.
  1. Clear (occasionally will find things not clear because of penmanship, accumulation of grammar/spelling or lack of paragraphs.)
  2. Coherent - there is a point and the writing addresses it.
  3. Complete - shows 2 hours-ish of work. Contains important relevant support towards the point of the writing.
  4. Content - the most relevant math is included, reasoning is shared and correctness of idea use and computation.
  5. Consolidated - the writing has a conclusion, synthesizing or summarizing or extending. (This is the hardest aspect for students to adopt.)
When working with Dave things should alliterate or rhyme.

The last writing was to be on these questions: What do you see as the big ideas of algebra? What did you learn about them? 

I'm under no illusion that these are objective since they are not anonymous, but they align more or less with my impressions.

Some responses moving towards my goals:
  • "I liked this class because it was the first class I have had where the students were running the math class, and the professor was there to guide us, not tell us that we were wrong. Our ability to voice our opinions was very nice..."
  • "Now, I feel like these concepts are things that I will actually be able to apply because I understand WHY they are done." 
  • "Logs have always been tricky to me, really tricky. What we did in class allowed me to break them down and really understand the rules and properties. " (Introduced them without the name. I call them  pixies because - pxe - reverses exp.)
  • "However, I really appreciate how easy going the class was, there are not grades to define us and SBARS did not have to be straight forward answers (in fact, they were not supposed to be at all), they helped us explain why we got the answers we did and that was a big factor for me. " 
  • "This course was more focused on learning how math works, not just memorizing and regurgitating. I really appreciated this because it made math much more interesting to me. I learned how mathematical properties worked, why and how. The way that the course was graded I believed was great. Some people, including myself are just not good test takers. The SBARs were great, you could work on your own time to show understanding as well as ask questions if you could not figure something out, For me this is a much more effective way of learning."
  • "It was nice to actually figure out where answers to problems came from, and how equations made graphs." 
 Some responses point out where to grow:
  • "I am never somebody whose been good at math and this new way of learning was definitely a bit of a difficult transition for me. Learning why we do problems the way we do was something I struggled to wrap my head around, considering I often didn't know how to do some of the problems in the first place."
  • "Generally as a student, explaining things are not how I learn, so to have to explain things was a new type of learning for me."
  •  For a College Algebra class, it was challenging and interesting!
  • "I still struggle with explaining why just because sometimes i don't really know what to say, other than it just is what it is. As a student i learn with direction and this type of class was less of  direction and more free flowing which is okay. Just hard to get used to."
  • "This class was actually really difficult for me, personally. I have never been good at math, so I was kind of doomed from the start."
  • "it has been a little challenging for me to understand what my grade in the class has been throughout the semester. I prefer to know my overall grade in a course so that I can prioritize my workload and be as efficient with my time as possible. "
And a warning to you K-12 colleagues: "I took up to Calculus in high school but I never once was asked to explain how i found the answer." I'm sure this is not true, but it was the student's perception.

To sum up, I think I will also use a student's words. This seems to capture both sides. "This class was quite difficult for me because it was a new style of education. I am a person who likes things black and white, which has allowed me to succeeded in math classes in the past. In most math classes once you find the answer, the problem is considered done, and you move on. However, this course has challenged me to think deeper into what I am actually doing in order to reach an answer. I think this type of thinking is more realistic as to what math really is."

What I am asking the students to do is real to them, but I need to find more ways to support them and build culture in 28 meetings.

Sunday, May 1, 2016

Blogging to Blog

OK. I've been thinking about Anne's #MTBoS30 challenge and I'm in. It was ParkStar's post that was the tipping point somehow. But probably just for today.

I'm in the middle of grading finals, grades due Tuesday. Noon, though hopefully it won't come to that. Meant to have them done last Friday, but I've been taking every distracted path possible. It just doesn't seem as relevant. Grading with no feedback, as only .1% ever come back to see this stuff. A minority even want to see their grade breakdown. The learning is done, so why am I still here?

In particular my college algebra class that has been such a challenge. They took my lovingly crafted questions and gave them short shrift. They're showing me all the ways in which they did not learn what I wanted them to learn... and I oscillate between blaming them (which, thankfully, the #globalmath on Math Trauma by Kasi Allen helped me spot) and blaming myself.

But I know the blame cause is shared. I have to create the conditions for learning that I influence, but they have to engage and be responsible, too. And I know that I'm a better teacher for those struggles.

The struggle now is to find evidence of understanding for the standards, to fight for them to get the best grade possible. I wish we could still talk about it - so much of what they did bears discussion. I'm annoyed by people who did nothing to the last minute (although I was one of those students and am now one of those teachers; not doing nothing, but...) at the same time as I'm so happy that they didn't give up. Even if it was for a grade, and not for learning.

Here are my current learning targets for college algebra. Lots of ideas for revision, so maybe that's grist for some 2 ≤ day ≤ 30.

Friday, March 18, 2016

Block Market

How much is that number worth? It's all about location, location, location.

This is another story of impulsive teaching. I'm not recommending that, but we got to a good place, so I want to tell someone.

In my preservice elementary course, we were headed towards decimals, passing through place value, so it was time for the base 10 blocks. A wise elementary teacher taught me that new manipulatives should always start with play time. (If you can't, tell them when they will be able to play. Chris's other lesson was to use each new manipulative as a chance for the students to tell you the rules about using them. Pro tip.) The wooden Base 10 blocks we have are particularly good for building.  But playtime always ends with: 'so what did you notice about these?'

They found the 10 fit into the next one pattern, and noticed irregularity in these old, hand-cut materials.

I love when manipulatives are used for a purpose or a problem rather than a set of exercises. So I asked how many blocks were in each tub, in terms of the small cubes as the unit.   3510, 4873, 4508, 4508, 3377. Hey! That's not very fair. Ooh, I have an idea: what if each group gave the next group half of their blocks? 4191, 4691, 4003, 4508, 3443. Is that any better? Some say yes, some no.  Let's give half again! 4441, 4500, 3741, 4256, 3817. Still disagreement about what's happening.

Okay. Let's settle this like mathematicians. Make a display of the data that proves your point. We collected one more round of give half away.

 I was really impressed at the diversity of displays by happenstance. It made for a great discussion of results. As we often do, I asked for each group to get feedback from the other students: one specific thing you like about their work, and one thing that would make it stronger.
 I missed one of the graphs, but here are the other three.
The first graph shown, people liked how it made the visual comparison of the round by round numbers. Convinced people that the numbers each round were getting closer.

This display charted each group's round by round count compared to the mean.

The classic lineplot follows each group's total round by round. People agreed that this showed convergence to the mean the most strongly.

But while people were making their graphs, I noticed something, and had groups record their total for each block on the back board. What's the problem?

Our individual block distribution is out of wack. Two groups don't even have enough to compose the next unit! We had to make some trades to get a better balance. We could play...


We went table by table with people proposing trades. The whole class decided if a trade was fair. It was the most fun I've ever seen composing and decomposing by place value. Trading was heavy and fast paced. But occasionally we had to stop to check fairness. We even had one crazy three way trade. There was lots of interesting reasoning about the quantities and how they got out of whack even while the totals converged. Final count - not bad.

The idea of social relevance in math class has always been an interest. My colleague Georgi Klein made great use of Marilyn Frankenstein's algebra work. And we almost got a chance to hire Mathew Felton who looks at the political aspect of math learning. So I closed with an observation that with so many math problems about maximizing or candy, it might be nice to address big issues, and disparity is something that's going to be an issue. I got a little preachy, really. But it felt like a good day, with some real values in our place value.


Transitioning to decimals, after work with a fixed unit, we traditionally do something like the top part of this next activity. (Probably originated with Jan Shroyer.) It starts the idea of shifting the unit for different situations. Pretty effective. This time around I added the problems at the bottom as puzzles. They were very interesting for the students to think about, and seemed to push consolidation of their decimal strategies. It really requires a lot of reunitizing. I'd love to know how middle school students thought about them. Each group made up a puzzle of their own to swap, and that also seemed beneficial.

I'd love to hear your thoughts about political values in math class, block market trading for place value, or the representation puzzles.

Sunday, February 28, 2016

Exploring the #MTBoS

My elementary preservice teachers (PST) are exploring professional development this week. The first assignment was to do a webinar or our local conference, Math in Action. Global Math was Problem Strings with Pam Harris (awesome) and Christopher Danielson is keynoting Math in Action, so fortuitous timing, say no more. The second assignment is to find a blogpost to recommend to teachers, so I thought I would pass these along. The list of leads I gave them is:

Apologies for any exclusions. These are all people who's work has come up so far in class. What elementary blogs would you add?

Their recommendations:
Dana -  https://mathmindsblog.wordpress.com/2016/02/11/rhombus-vs-diamond/
Summary: This blog post is about a class of students looking at four different shapes, and trying to find the odd one out.
Review: I thought this blog was great because instead of simply telling the students the names of the different shapes, the teacher let them think and reason for themselves; she allowed them to come up with and defend their answer by themselves.

Dayna - This is a great lesson to combine English and math and get students excited to learn. My response to the lesson is that I love that as a teacher you get to see the students thought process when they are working on this problem. http://marilynburnsmathblog.com/wordpress/chrysanthemum-an-oldie-but-goodie/

Kalyn - https://tjzager.wordpress.com/2015/06/06/comparisons-a-little-bit-more-older/
This post talked about how comparison problem are everywhere, even outside of school. Although they can be difficult at first, the lightbulb goes off and the problem makes sense! I really enjoyed this post and the person example that it gives of her daughter and the conversation that they had about math, but also about life. A lot of good stuff here!

Ally - http://exit10a.blogspot.com/2015/12/22-30-50-100.html
This is a great blog. It's about how is he works with "Alex", going through counting. It was a great read.

Amber - For my blog post, I chose to look at some more of Graham Fletcher's stuff. And although we aren't learning about volume right now, I thought this post was a great representation, which shows real life problem solving. It offers that children are robbed when force fed uninteresting story problems from a text book, and Graham offers an interesting 3 ACT problem as a substitute. My reaction to this concise, yet powerful read is that I would like to try a problem like the one he brings up. I bet students would be very interested in it. http://gfletchy.com/2016/02/14/im-placing-a-hit-on-the-pseudo-context/

Sarah - https://mathmindsblog.wordpress.com/2016/02/05/obsessed-with-counting-collections/ Summary: Second graders explore sorting by counting by 2's, 5's, and 10's. A similar activity is conducted with first grade students. One students has difficulty counting when there is a leftover present. Review: This post really made me think deeply and question the use of ten frames. Kristin Gray does an excellent job of questioning the thinking of students and that is something that I, personally, need some work on. 

Chris - http://followinglearning.blogspot.com/2016/02/art-for-maths-sake.html
Summary: This blogpost is about an "artsy-mathy" activity he did with students involving creating trees out of factor trees.
Review: I absolutely loved this post because it addressed an issue with "artsy-mathy" activities, which is that they tend to be less artsy than an art lesson and less math than a math lesson. I enjoyed how he addressed the issue by making more out of the project and creating an exhibit where the students could teach to younger students.

Orina - http://marilynburnsmathblog.com/wordpress/four-strikes-and-youre-out/ I thought this game was really interesting and fun because students have to use number sense to try to win. A lot of students are familiar with hangman and it's a math way to play a fun game.She comments at the end about how this game is competitive and can be cooperative also. Playing games can bring more fun to the classrooms, but she comments that we must make sure there are some competitive and some that are about cooperation.

Kathleen - http://followinglearning.blogspot.fr/2016/02/quadrilateral-sets-lesson.html
this teacher was trying to get her students to understand grouping and have then work together to see what made the shapes different and what the noticed in general. a lot of then saw that there are many triangles in the rectangles.

Heather - http://gfletchy.com/2016/02/14/im-placing-a-hit-on-the-pseudo-context/ I really enjoyed this blog post because it addressed the question we all ask ourselves, when will I ever use this? I like the way he addresses practical examples and being able to take those practical examples and use those for practice problems not "mind numbing" problems. (John says: "be sure to see Joe's follow up.")

Stephanie - http://exit10a.blogspot.com/2016/02/a-post-about-counting-circles.html
Summary: Counting circles can serve as more than one purpose of just counting; it helps you practice standards, recognize patterns, etc.
Review: I never knew you could do a counting circle in some many different ways; the more questions you ask your students about it, the more they will think about it, and deeply understand the material better.

I admire their taste in posts! 

The other thing is how much I want to thank these bloggers. By sharing your classroom you are having a profound effect on other teachers - and on the future. It takes time and vulnerability for you to write, and I want to thank you for it.

Saturday, February 20, 2016

Teacher's Block

I am struggling with my college algebra class.

There is the classic misalignment of what they think math is and what I want them to be able to do. The makeup of the class is almost entirely people who are done with math after this course. In our department, we're trying to separate this from precalc, making a new course for people moving on in math and this course will be for people finishing their math. The goal is to make this course more conceptual, and prepare students for use of mathematics in other subjects. Previously this course had so many skill objectives that teachers were put into coverage mode. Tenure track faculty teach it occasionally, full time affiliate instructors sometimes and most frequently adjuncts.

The course started off on the wrong foot. Two problems that have been a smash in the past went awry. On the house painter problem they were unable to convince themselves of the answer. And on the fair pay problem a misconception was shared and caught on so that it became insoluble. When they came back, one student had a nice intuitive solution that he could not convince the class - even with my help - that it was correct. Finally someone asked me point blank if his answer was right. "Yes, but one of our goals is that you are able to decide for yourself."

Ay ay ay.

By nature, they are reticent to talk to each other. Despite my urging repeatedly, and sharing how math is best learned through discussion. Many don't engage in activities, they're waiting for me to tell them how, many are not doing the homework, and absenteeism is about 20%. Standards based grading has been a tougher sell than usual because this class, as a group, wants the math that was.

I've removed a lot of choice, I've been doing more demonstrations and spending time on the teacher half of gradual release of responsibility, and I'm super explicit about what problems show which standards on assessments. They still won't talk, and many won't engage in in-class inquiry. They will make up something rather than ask about a question they don't understand on an assessment.

Ay ay ay.

The other day I saw this image on Twitter, but stupidly didn't catch the source. Simon Gregg remembered - it was David Wees!)
Our College Algebra picks up with quadratics, but a lot of the work I do with students visualizing patterns is for linear content. (We did do the growth problems, though.) Doing some other work I realized that the students were not understanding symbolic representations as generalizing number patterns. (There are even such quadratic examples on my own blog!) They had been getting by on regression rather than representation. I though this problem would be a great introduction to this kind of generalizing.  It did seems to be helping connections form. I wanted to extend this to cubic or higher, so I built this pattern.

First we discussed what was going on, what they noticed and what they wondered about. Very few students wondered about how many cubes for the next building. More assumed that the next building was to be built with exactly that many orange blocks.  That's very different than my thinking, and emblematic of how difficult it is for me to anticipate how this class will respond to prompts.

I built a very scaffolded worksheet. I used to make stuff like this all the time, but have been moving away from it. But sometimes students need supports.

Another adjustment I'm trying to make is, instead of roaming the room to eavesdrop and do formative assessment,  to roam and ask questions, try to encourage table conversation, and hover over students doing work for other classes or just sitting. I feel a little awkward promoting engagement by (what feels like) intimidation, but students need supports sometimes.

One of the more successful areas of class so far has been the math writing. They have six assignments over the course of the semester, they can count towards SBARs, they can revise for their final exemplars. Several people are writing about this problem for their current writing. That tells me this was at a good place for them, and I'm happy to see some of the sense making.

Meg writes:

The strength of the temptation to give in, to teach them how they think they'd like to be taught is way stronger than I would have guessed. But so is my stubbornness to not give in. What I am trying to be wary of is to keep my stubbornness from stopping me giving the support that students need and teaching the students in front of me, rather than some fantasy class.

Sunday, January 24, 2016

My Favorite (Teaching): Improvisation

Sign in data. Most of the variety is size.
This is one of those things that's both a strength and a weakness of my teaching. I have a lot of ideas about things to try, but that is not especially professional. When people talk about the profession of teaching, we always seem to compare to doctors. You don't want your doctor making up treatments on the fly. Research! That's on what doctors should base treatment. Maybe it's okay because the stakes are less high - one topic in a math class vs. your health and well being, or because in teaching we are the researchers, too. Or maybe medical doctors are not the right comparison.

Regardless, I like to improvise! This is the story of two of those moments, in the same class period.

The characters, preservice elementary math teachers; the content, learning quadrilaterals with a focus on reasoning with properties; the setting, they've gone from describing quadrilaterals to thinking about their properties. Day 1 was spent describing quadrilaterals on geoboards to make, and thinking about a variety of different possibilities that still fit a type. They took home geoboard dot paper to make their own quadrilaterals, one of 11 types. (For us: square, rectangle, rhombus, parallelogram, isosceles trapezoid, right trapezoid, trapezoid, kite, chevron, convex, concave.) In general my teaching here is guided by the Van Hiele levels, in particular activities that give students reasons to transition from visual to analysis, from analysis to informal reasoning and then informal to formal, depending on the level. This is K-5 focused, so we don't push at formal reasoning too hardly.

I have an old set of quadrilateral cards that has a lot of visual confusion. Looks like a rhombus but is a parallelogram, etc. They've been good for me in the past, but the longer I teach, the more I want the students involved in the manufacture of math materials. So this time, they made the cards for homework. I made a couple of extra sets in case someone hadn't had a chance.

The first activity was the same as I've done in the past: Quadrilateral Concentration. Players shuffled up their cards (I had them put initials or a symbol on their own so they could get them back, but style would have been enough for most), and dealt them out into a grid. Turn over two cards and if there's a match - two quadrilaterals of the same type - you can pick them up. Most people knew the game already. The conversations were amazing. First question, "I turned over a square and a rectangle, can I pick them up?" The table ruled no, and I supported. The best arguments are over type, though. Either while playing, "hold on, I don't think that is a rhombus," or at the end... "Why don't these have matches?"

My fair quadrilateral
To summarize, I brought up to the document camera the ones that provoked the most discussion. We also used them to talk about variety again - as there were MANY congruent examples. And we discussed the most ambiguous case, which is probably now my favorite geoboard quadrilateral. People thought it was a kite, people thought it was a right trapezoid, people argued about the length of sides, people argued about the size of angles, people compared it to a square... loverly. It was especially nice because people kept cycling back to earlier claims, which seems to prove what I was suggesting about the power of our visual processing.

Another game I've played with the old cards is quadrilateral Go Fish.  We played with the same rules as concentration, using the most specific names possible. Suddenly it occurred to me that we could play concentration as we had, but switch the Go Fish rules to allow for more mathematical subtlety and strategy. Not everyone had played Go Fish, but enough had to make the rules explaining go smoothly in each group. To play a match, they had to be the most specific type. But when your opponent asked you for a type, you could give any that fit the characteristics. BAM! This was great, and almost instantly a better game than the original Go Fish. There was the start of some strategizing, where some people weren't asking for exactly what they wanted, and the conversations were spot on. This is a trapezoid, are you sure you don't have anything that fits, etc.

That's an improvisation that paid off. Better than what I had before.

The other idea on the spot was to go farther into combining properties. I wanted to make it natural to think about what if a quadrilateral was a this and a that. So spur of the moment, sidebar into a weird movie and TV discussion. I asked, what was an adjective that described a show or movie that you liked to watch. Then I shared how my wife's favorite genre was funny + scary. "Like Krampus?" (Side discussion on Krampus, which we recommend. But only one person had seen it, so...) I wrote down the 'equation' funny+scary=Ghostbusters. (Best example is probably Buffy, though.) Then they discussed at their table until each person had one to put on the board. I was worried about = abuse, so I did mention that what we're really doing is finding examples in the intersection.

And one table really got into trying to do adjective arithmetic. We talked about the examples & shows for a bit and then I transitioned to the purpose: what if we combine the quadrilateral types this way? Each table I wanted to come up with one quadrilateral equation. Got some good ones, and I shared about the role of conjecture in mathematics. To their list of four conjectures, I added some questions.

I connected this to the homework, which is to try the very challenging problem of a Venn diagram for all the quadrilateral types. We'll discuss those and the conjectures next class.

Passed it around again, and got much more
variety of property and orientation.

This improvisation was okay. Don't think I did much harm, it was a moment of high engagement, but not necessarily in mathematics. Well, it was mathematics, but not our quadrilateral content.  The disappointing thing is that the conversation about the shows - reasonably analytical - didn't carry over to the conversation about the quadrilaterals.

I'm okay with this, however, because even a bad result is going to happen sometimes. The same activity that is a gas burner with every class that has ever tried it can crash and burn. So the improvisation increases my store of supplies, keeps my interest, and gives me things to think about for student thinking.