A Few Problems of My Own

Some of my favorite assessment problems from this semester in College Algebra. Each problem is labeled with the standards it mainly addresses. The main function families for our course are:

• Q - quadratics
• P - polynomials degree 3 and higher
• R - rational functions
• E - exponential
• L - logarithmic

And then there's a category of function concepts and special functions.

For each function family, there's four repeating standards:

1. Basics - vocabulary, characteristics
2. Representation - being able to move amongst table, graph and equation
3. Symbolic - traditional algebraic skills, solving and simplifying.
4. Context - solving in application. Being able to mathematize situations.

I like having a structure, as students getting used to the Standards Based Grading is one of the big hurdles in the course. That's why all the problems are labeled with standards, too.

These problems are different than the ones I use in class or for homework, with more scaffolding, and often reminiscent of things we've tried in groups. They also choose problems on which they're ready to be assessed. They can write up longer responses at home to turn in on problems that they do not do in class.

This first one's not mine. They read a Glen Waddell post on the Exeter method, so I thought it was a nice connection to put on an assessment.  The element of the cup really made clear which students understood what the height function was saying.

We did a fun activity in class on sound frequencies that you can hear at different ages (due to hearing loss/damage) in class when we were talking about decibels. Frequency is not logarithmic, of course, but I love using a topic as arcane as logarithms to make sense of something they've heard of (get it?) as much as decibels.

One of their favorite application problems in the semester was using exponentials to model weekend movie grosses, which are roughly exponential decay. But of course the sum of geometric sequences is also a transformation of an exponential. This problem unfortunately highlighted careful reading or the lack of it, which is not super-useful, so that some people tried an A*k^x model. But it did show the people who were making sense of that not working, and the people who blindly accepted the results. Desmos reliers (as opposed to Desmos-only-if-my-calculator-can't-do-it) did better on this.

This problem was another great one for function notation. The difference between evaluation and solving was crystal clear. And it was such a nice pattern for the people who correctly interpreted the problem that there was a good payoff.

The top problem here is also not mine, of course, coming from WODB.ca. Well, originally. The task of making a function to look like it is a great representation prompt. Students who made sense of marble slides were really able to strut their stuff.

 It was the second and third problems that were really interesting to me here. They were open middle-like in the variety of different methods people used to solve them. Recursive rules, which we never used in class, really, excellent table use, some regression... such a nice mix.

I liked it enough that I wrote a follow up for the last SBAR opportunity. (AKA the final.)

One thing that we never addressed directly that we got to the last time I taught this course is that the sum of a certain degree polynomial sequence gives the next degree. Much like the differences give the next lower degree. I LOVE that structure.

For the special function types they only had to demonstrate one of the four kind of standards that they did for the main function families. Both of these offered a lot of opportunities for sense making. In particular the normal distribution question highlighted whether people understood mean and standard deviation as descriptors.

If you have feedback on writing assessment questions for SBARs, I'd love to hear it. Whether it's modifying these or a whole different direction. Here's all my assessments from the semester in a Google folder, if that could be of help or interest.

PS. Thanks Ann and #MTBoS30 - I've now blogged more in May than I did all January to April.

Civil War

My son Xavier is a dedicated comic book fellow. A sophomore in high school, it has been almost a dozen years of near obsession. But unlike most of his ilk, he is open to either DC or Marvel. He does tend to be more into one or another at any particular time.

Why are we so tribal? In a society that can look down on comics (less and less, though, it seems) I always wondered why pick one and denounce the other? I like Hulk and Batman, deal with it.

Anyhow, this weekend is definitely about Mother's Day Civil War.

Aside: my spouse Karen is kind of anti-Mother's Day. She both worries for people for whom the holiday is painful (of which there are many) and is downright angry that what was originally a day where Ann Reeves Jarvis and Julia Ward Howe (separately) encourage peace work and prayer. After the actual Civil War it was important to them to never have such a rending again.

But we have our own civil war in math teaching. Even called the math wars for extra inflammatory effect. It's not like the US Civil War, with serious different goals setting opponents against each other. Both sides of the math wars know that they are fighting for what's best for students. I don't say this to suggest a CNN-like, two sides to this issue, so let's pretend it's Coke and Pepsi. I am far off to one side of the math wars. But I know that the great majority of people with whom I disagree are people of good intentions. I have serious differences with anyone who fights the math wars with an 'ends justify the means' style, though, and I am naturally suspicious of anyone with financial interests in the war's outcome.

The best paper I know on the subject is from Alan Schoenfeld: The Math Wars. He gives a thorough history, and begins our current conflict around the release of the first and second NCTM Standards. A key-to-me paragraph:

"The Standards, buttressed by NCTM’s call for “mathematics for all” and the equity agenda in Everybody Counts, clearly sat in the education-for- democratic-equality and education-for-social-mobility camps. In contrast, whatever the intention may have been, the reality was that the traditional curriculum was a vehicle for social efficiency and the perpetuation of privilege. Statistically speaking, the rich stayed rich and the poor got disenfranchised. There is a long history of data indicating that race and socioeconomic status correlate with mathematics performance, with drop-out rates, and with economic opportunity (Kozol, 1992; National Action Committee for Minorities in Engineering, 1997; NSF, 2000). Thus the Standards could be seen as a threat to the current social order."

One of my favorite courses in college was on the philosophy of John Dewey. We examined social issues, such as abortion, from the pragmatic viewpoint. This involved looking at what divided people on an issue, and what separated the activists from the majority. It turns out that on most issues the majority is not separated by much. Activists often have no room for compromise, however, because for activists the issue is about their way of life. For example, with abortions, it seems like we could all agree on condom distribution. But the activists will fight over it.

So what separates the activists from the majority in the math wars? I'd love to know what you see from your perspective. My current theory is that it's about threat and equity. On one side they see change as against them or wanting to take something away, and other the other side they see the status quo as oppressive to many.  That so many are fearful of math adds fuel to the fire.

The pragmatist appoach would be to try to get the non-activists together. We are all concerned with student learning, and have a lot to talk about. It always strikes me that anything that is shown to be effective for improving teaching boils down to having teachers talking about some kind of shared experience. This fits with my framework that teaching is local and human. When I hear Lani Horn talk about contextualized discussions of teaching, this is what I think of. When I hear Catherine Lewis talk about lesson study, this is what I think of. When I hear Jim Knight talk about instructional coaching, this is what I think of.

We haven't seen Civil War yet - and we are a strict no spoiler family, anyhow - but when we do, I'm guessing that a lot of the fighting will have been avoidable. Similarly in the math wars, beware of those who love the battle, and look for those who are loving the students.

Save our arguments for which is better, Superman v Batman or Civil War!

Guess Again

Graham Fletcher was tweeting from a Marian Small presentation this morning.

I think the theme was number sense activities. These line questions caught my interest so I wanted to make a GeoGebra version. This post is about how I made it. I'm always torn about these things because so much of the good math is in making them, but I'm mostly making them for students. But learning how to make in GeoGebra is often not something that is  engaging to a majority of them...

If you want to try it first, it's on GeoGebraTube.



















First thought: Pick a random value - maybe  1 to 100? - and a smaller value, then have students guess a higher value. (Basically Marian's second photo.) Should it just be multiples of 5? I liked the idea of multiples, so I randomly pick a scale from 1 to 5, then pick a value 1 to 30 for a central value. Pick a number lower and higher, and the setup is done.

Steve Phelps was playing with GeoGebra color the other day, so I added the unnecessary frill of different colors, but it makes it a little cooler to me. Replaying it, I realized it would definitely be better if different points were unknown, so the sketch randomly selects the low, medium or high point for guessing.

A GeoGebra fine point. If you have the values randomly determined directly, then anything that makes the sketch recompute changes the value. So I have variables set randomly, but just assign those values to game values when the New button is pressed using the SetValue command. The guesses count also uses the SetValue command: SetValue[numguess, numguess+1] everytime the guess value changes.

Then it's just bells & whistles using the condition to show object on the Advance tab. A random element of {1,2,3} is picked to hide a value, which shows if guess == value. Some different text

The "Within..." is probably the sketchiest part. I want some further information, but not perfect, so it randomly assigns a number up to two times how far away you are.

There's a bit of oddness from uploading to GeoGebraTube in terms of scaling, so I had to ask users to hit the button to start. There's a command GGBOnInit that might solve this, but I don't know how to use this yet.

Pretty fun to make, and reasonably fun to play. I don't have students to play it with at the moment, so if you get a chance, let me know how it goes.

And - as always - if you have something you'd like to see made, drop me a line!

Forceful Teaching

The problem with #MTBoS30, aside from that every day could be my last, is that is in May, which includes May 4, which necessitates a Star Wars post. Yes, every other #MTBoS30 post today could be technically wrong.

1) David Coffey, famous starter of mathmemes that invariably suck me in, started a SW mathematical practices meme. I can't remember which of these were his or mine.

2) Star Wars and math teaching.

There's Dan's famous explanation of Three Acts Lessons in terms of Star Wars. Or the twitter discussions of Obi Wan as a teacher.

Frankly, Harry Potter is a much better context for teaching discussions. Umbridge, Dumbledore, Defense against the Dark Arts...? Come on!

Obi Wan just does not have much time teaching. Basically he's the one who gets to make the pitch. "There's this amazing thing! You, too, can experience it! And you have the talent for it because of genetics!"

Yoda we can talk about. Even though he gets to teach one-on-one, and literally have the student move in with him...

If I'm talking about teaching, it starts with Cambourne's conditions of learning. (Also introduced to me by Dave Coffey. Great to have a colleague like that.)

 So Yoda...

• Engagement: Luke definitely sees himself as a force user, Jedi skills will further his purposes, and he trusts Yoda to some extent that the environment is safe. (Vision tree.)
• Immersion: not many Jedi's left, but Yoda seems to try to communicate Jedi culture.
• Responsibility: I don't know. It seems like Yoda sets the agenda.
• Employment: most of the teaching we say is Luke trying to use the force to do things.
• Approximation: Yoda is a pretty harsh critic. When Luke fails Yoda jumps straight to the idea that Luke can not do it, he is too old or impatient or like his father.
• Demonstration: Yoda does as needed. (X-Wing.)
• Expectations: Despite his negative feedback, Yoda definitely sees Luke as a potential Jedi.
• Response: see approximation.

All in all, Yoda does pretty well by this measure. Let's mark him exceeds expectations and slot him for a pay increase if there's a budget for that.

 

In The Force Awakens, Rey seems to be learning directly from the force. In a way, that's what we want math learners to do. But this contrasts with our content standards that cover  500 years or more of math history.  So we compromise, hoping for little bits of discovery on topics we choose. But maybe they can still learn the ways of math and become powerful in understanding.

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.

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.

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.