Trying to find math inside everything else

Archive for the ‘assessment’ Category

Making It Stick

After going to Anna Vance’s session on Make It Stick, I implemented some of the ideas she presented and thought I found reasonable success with them. However, as I hadn’t read the book, it was a little half-hearted and could be improved. When I was looking through my library’s education e-book collection and saw it there (amidst a sea of worthless looking books, save Other People’s Children, which I also checked out) I decided to pick it up.

A few things stuck out to me, some of which I tweeted, but the one that I keep thinking about is the Leitner System, which they describe thusly:

This struck me for a few reasons. First, I love the idea that the “flashcards” don’t have to be what we typically think of as flash cards, but rather representations of anything we need to practice. Second, it’s a system that is learner-led, so if I can get my young mathematicians onto the system, they can run it themselves. (And extend it to other parts of their lives.)

So my thought became thus: how can I weave this system into my classroom? Here’s my thoughts. I’d love some feedback.

  1. Create a system of boxes (folders? tabs?) – I’m envisioning four in a set – for each student.
  2. At the end of each lesson, have the class write on (an) index card(s) something from that lesson that they think they should know. (This practice of summarizing their learning is also mentioned in Make It Stick.) It could be a knowledge fact (the definition of a polygon), a skill (solving a linear equation), or something broader (what are some ways systems of linear inequalities are applied?). If it is a skill or broad question, it should not have a specific example. (So they shouldn’t have a card that has them solving 3x + 2 = 8 every time they see it.) Then put those cards in box 1.
  3. Their standing HW is to practice whatever is in Box 1 every day. If it says something like “Solve an equation,” they need to generate their own equation, then solve it. (Generation is also mentioned by Make It Stick as a way to increase stickiness.) When they get it right, move it down a box. When box 2 is full, practice those the next session, and so on.
  4. On Fridays, give some time in class for students to practice, especially their box 2 or 3, if they didn’t have the time to do that at home. Then give the usual quiz.
  5. After taking a quiz, they should then reflect on what they did and didn’t know, and if there is something they didn’t know that isn’t on one of their cards, make a card for it right then and put it in box 1.
  6. To qualify for a quiz retake, all the topics for a quiz need to be on cards in Box 3 or 4. Otherwise, they need to study more before they can retake. (This would mostly be an honor system, as nothing stops them from just putting the cards in there.)

Does that sound feasible? What needs improvement?

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assessment, schooling, theory

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Whole Class Test

I teach an SAT Math Prep this year, which has been an interesting challenge. We basically started off with lessons on all the different content in the exam, then had a long section on tactics (which can be framed as test-taking tactics but I noticed are often just tactics for solving problems in general, which was nice). But we reached the end of those, and the (in-school) SAT is a month away. The obvious thing to do is to just keep doing practice exams, but that can get a bit boring, for both me and the students. Plus, the class that meets Tues/Thurs hasn’t had very many graded assessments this marking period, so I needed to give them something.

I had decided that grading them on correctness in a practice SAT is not appropriate. I had told them this before, and they knew their grades on their assignments were more for things like how they applied the tactic we were learning. But last class they walked in and I gave them a Part 3 exam (the non-calculator part) and told them it would be graded – but there would be a plot twist. For right now, just take it individually, except this half of the room should start from the back and go forward. Oh, and you get 5 fewer minutes than normal.

While they were working, I went around on my whiteboards and put up the numbers 1 through 20 well spread out, and an ABCD for 1-15. (I wish I had taken pictures!) This started to get them suspicious. When time was up, I told them my grading scheme: it was out of 5 pts, and they lost a point for every question they got wrong. So if you got 15 right, that’s a 0. But! They had the remaining 20 minutes of class to work together and figure out what the right answers should be. And if anyone got less than 15, the whole class lost a point – forcing them all to work together. (With limits, of course – they won’t be penalized for that kid who went to the bathroom for 15 minutes during this, for example.)

A suggestion I made to them was to go around and make votes for their answer for each question. A clear consensus might mean that that is the right answer. However! Don’t be afraid to put your answer down even if everyone else’s is different. I’ve seen questions where only one person got it right. I told them they need to convince each other of what the right answer is.

Let me tell you, I heard so many great conversations as they and I went around the room. Because it’s the SAT, no one gets them all right, so everyone is being pushed to make a convincing argument that their answer is right. Students who weren’t sure got explanations from others. It was delightful!

About halfway, I noticed a clear consensus for about 15 of the 20 questions, but the middle 5 were really quite split. So I lead the class in sharing out their reasoning for some of those questions – never saying what the right answer was, but again letting them convince each other.

It was a nice collaborative effort – I highly recommend it.

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assessment, math

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Vietnamese Age and the SAT

Dwight Eisenhower was born on October 14, 1890, and died on March 28, 1969. What was his age, in years, at the time of his death?

(A) 77
(B) 78
(C) 79
(D) 80

When my boyfriend went to his grandmother’s funeral, he found himself confused about exactly how old she was. Was she 93 or 94? He heard different people say different things. Eventually he figured it out. In Vietnam (and apparently in other places in East Asia), when you are born, you are 1. The next year, you are 2. And this ticks over at the beginning of the solar year, not on your birthday. So I, born on December 22, would, 374 days after my birth, have been considered to be 3 years old using this reckoning. (It might be more accurate to say that I’m in my 3rd year – being alive during 1985, 1986, and 1987 at that point.)

Earlier this week, in my SAT Problem Solving class, we encountered the problem at the top of this post. The correct answer, according to the book, is (B) 78. But according to the Vietnamese reckoning, he’d be 80, and the answer would be (D).

Before my boyfriend went to that funeral, I wouldn’t have even looked at this question twice. I had never heard of another way of determining age. And I’m willing to bet the people who wrote this question haven’t, either.

It’s a small example of the way tests can be biased, and how having more diverse voices in the process could help avoid this kind of mistake.

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assessment, social justice

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Quick but Comprehensive Feedback

So my portfolio idea was working out well, but I was getting overwhelmed with the written feedback. It took so long to write that sometimes my hand felt like it would fall off! I needed a new strategy. Luckily, David Wees had one for me, so I thought I’d share it with you all, since it’s worked really well.

Instead of writing all the feedback, as I go through and check an assignment and finding something I want to comment on, if I think it might be a common mistake, I type it up on a word document on my computer, numbered. Then I just put the circled number on the page itself. When I’m done I have a comprehensive list of feedback that I print out and attach to each assignment. Now every student knows both the common errors and has specific feedback on what they need to fix.

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assessment

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SoP Portfolios

When we learned about planning back in grad school, we were told that if something is important, you need to assess it; if you don’t assess it, then it’s not important. When the Common Core Standards came out, most math teachers were very excited about the Standards of Practice. The problem was, of course, that the Standards of Practice are hard to assess. So most standardized tests that use the Common Core don’t assess them, which of course means they don’t get implemented in the same way as the content standards. The Standards of Practice are important to me (though I frame them as the Mathematical Habits of Mind), and that means I need to find a way to assess them. But I’ve never had a really good way to test them before – I was always kind of making it up as I went along. Now, though, I think I’ve hit on something now that really works.

After seeing Ashli’s video about not putting grades on papers, I stopped doing it this year – but having the grades still be there in the online gradebook wasn’t quite what I wanted to do, and it became very hard to keep track things, especially because classwork and homework were what I used to measure the Standards of Practice. This semester I gave written feedback on all the assignments that I’ve given but recorded no grades – not even in my own gradebook – the only thing I kept track of was if something was incomplete or missing.

If we decide that every assignment is a formative assessment, we can’t possibly grade it as students are learning the material. So instead each assignment is like a first draft (or second or third) and students can read the feedback that I gave and make changes in order to improve their work. Come the end of the marking period (or eventually the end of the semester) students create a portfolio of their work. They don’t need to include everything that they’ve done but rather a representative sample that shows that they apply the Standards of Practice/Habits of Mind as they work.

The portfolio has a cover sheet (shown below) that that asks them to reflect on what habits they have used in their mathematical work this semester.

They have to find evidence of their own habits in their work and write a few sentences citing that evidence. I gave suggestions of which assignments might be easier to find evidence of those habits in. And they only had to include work that they cited as evidence as part of the portfolio.

To get us started we read through the rubrics that I created for the habits and created posters of what those habits might look like when doing assignments. We have them hanging on the walls of my classroom – that way I can referr to them easily when something comes up (such as when we worked on Des-man, I tried to emphasize the tinkering nature of the process).

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The first portfolios coming in have been graded and some of them were stellar and others need some work, but it was the first time and they are not really used to this whole reflective idea. But I have noticed that most of my students have started to use that vocabulary more and have become more aware of the kind of things that they are expected of them in the long-term, not just immediate math facts. I think if I start this from the beginning next year it’ll create a really great culture of thinking using the habits and the standards of practice.

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assessment

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End of Marking Period Grading

How did I let myself get so far behind? Maybe I did it on purpose – I was so annoyed with the focus on grades that I wanted to let it sit back and push toward focusing on the content. Maybe it was my minor addiction to Civilization V. Maybe I just give out too many assignments.

I stayed until 6 today, skipping out on happy hour, to try to whittle down the pile before bringing it home, because marking period grades are due on Tuesday. However, because I was also doing notebook checks, I prioritized those and didn’t do much whittling.

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When a student saw how much I had to grade, he said, “Wow, that’s so much. But I can see why you’re behind – you actually have to read everything and make comments and check if it’s right, not like some other teachers.”

So at least my hard work is being noticed.

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assessment

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A Boss Fight?

One of the things about arranging your grading system like a game, as well as being a math game aficionado, is that it is pretty easy to combine the two. While yes, students can take quizzes or write essays to gain levels, they can also beat me in a math game. Of course, I’m not easy to beat, so winning against me would really show some mastery. (I do, though, allow them to gang up on me when the game is more than 2 players.)

The only students that really challenge me are the ones that hang out in my room at lunch, even though I’ve offered the challenge to everyone. And it’s cute because when they do lose they get even more determined, often because they may lose by a very small margin. (This is occasionally by design.)

The only game I’ve lost so far is Blokus, where the two Kevins beat me (but my score was still above the 4th player). As a reward, I gave them a level in Visualizer, as I figured that was the most applicable skill to winning the game. Planning ahead and visualizing paths in your mind is a useful skill. That same skill is the reward if they beat me in Ricochet Robots. In that game a team of Jane and Kevin tied me, so I still gave them reward, but they didn’t win.

It’s interesting trying to match games with skills. For example, the reward for winning at 24 is Tinkerer (since you need to play with numbers and try different things to succeed). It’s easy for games I made myself: if they can win at the Factor Draft (an upcoming post, I swear), they are a master of factoring. I have considered giving some points, not quite mastery, if they win against their classmates or my co-teacher, but to be a master, you gotta beat the final boss.

I’d love to have a bigger collection of games that I can use as assessment of skills, not just algebraic skills but the Standards of Practice as well. Any suggestions?

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assessment, games

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Rubrics for Standards

So my grading experiment has been going on for a month now, and so far I think it’s going well. But I was pretty stressed about getting it up and running, because a lot of the work was front-loaded. The thing I was particularly working to get done was my mega-rubric. I wanted to make a rubric that showed what exactly students needed to prove they understand to move up a level in a particular learning goal.

So here’s what I made (I call it the SPELS Book to go along with the students’ SPELS sheet):

I started by making the proficient categories, and for the first 8 (The Habits of Mind/Standards of Practice) it was pretty easy to scale them down to Novice, and then to add an additional high-level habit to become masters.

I was stuck, though, on the more Skill-Based Standards. I had all the things I wanted the students to show in each category, but how do I denote if they “sometimes” show me they can graph a linear equation? If I was doing quizzes all the time, like in the past, I could say something like “70% correct shows Apprentice levels.” But I wasn’t, and it seemed like a nightmare to keep track of across varying assignments.

So instead, my co-teacher had the idea that, if each topic had 4 sub-skills that I wanted them to know, we could rank them from easiest to hardest and just have that be the levels. So my system inadvertently became a binary SBG system, but still with the SBG and Level Up shell. Now if a student shows they understand a sub-skill, they level up. If they don’t, I write a comment on their assignment giving advice on what they should do in the future. What remains to be seen is how much they take me up on that advice. We’ll see.

Also, I’d LOVE any feedback you have on the rubric, and how I can improve it. Thanks!

Downloads

SPELS Book (pdf)

Updated Student Character Sheet (pdf)

Updated Student Character Sheet (pages)

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assessment

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Habits of Mind, Standards of Practice

For the past three years, I’ve loosely organized my classroom around the Mathematical Habits of Mind which I first read about in grad school at Bard. I would give the students a survey to determine which habits are their strengths and which are their weaknesses, group them so each group have many strengths, and go from there. Last year I even used the habits as the names of some of my learning goals in my grading.

As I was planning for this year and the transition to the Common Core, I was thinking about how to assess and promote the Standards of Practice. And I realized that they are very similar to what I was already doing with the Habits of Mind. In fact, having a habit of mind would often lead to performing a certain practice! In that way, the SoP are actually the benchmarks by which I can determine if the habits of mind are being used.

Let me demonstrate:

Students should be pattern sniffers. This one is fairly straight-forward. SoP7 demands that students look for and make use of structure. What else is structure but patterns? Those patterns are the very fabric of what we explore when we do math, and discovering them is what leads to even greater conclusions.

Students should be experimenters. The article mentions that students should try large or small numbers, vary parameters, record results, etc. But now think about SoP1 – Make Sense of Problems and Persevere in Solving Them. How else do you do that except by experimenting? Especially if we are talking about a real problem and not just an exercise, mathematicians make things concrete and try out things to they can find patterns and make conjectures. It’s only after they have done that that they can move forward with solving a problem. And if they are stuck…they try something else! Experimenting is the best way to persevere.

Students should be describers. There are many ways mathematicians describe what they do, but one of the most is to Attend to Precision (as evidenced in things like the Peanut Butter & Jelly activity, depending on how you do it.) Students should practice saying what they mean in a way that is understandable to everyone listening. Precision is important for a good describer so that everyone listening or reading thinks the same thing. How else to properly share your mathematical thinking?

Students should be tinkerers. Okay, this one is my weakest connection, mostly because I did the other 7 first and these two were left. But maybe that’s mostly because I don’t think SoP5 is all that great. Being a tinkerer, however, is at the heart of mathematics itself. It is the question “What happens when I do this?” Using Tools Strategically is related in that it helps us lever that situation, helping us find out the answer so that we can move on to experimenting and conjecturing.

Students should be inventors. When we tinker and experiment, we discover interesting facts. But those facts remain nothing but interesting until the inventor comes up with a way to use them. Once a student notices a pattern about, saying, what happens whenever they multiply out two terms with the same base but different exponents, they can create a better, faster way of doing it. This is exactly what SoP8 asks.

Students should be visualizers. The article takes care to distinguish between visualizing things that are inherently visual (such as picturing your house) to visualizing a process by creating a visual analog that to process ideas and to clarify their meaning. This process is central to Modeling with Mathematics (SoP4). It is very difficult to model a process algebraically if you cannot see what is going on as variables change. To model, one must first visualize.

Students should be conjecturers. Students need to make conjectures not just from data but from a deeper understanding of the processes involved. SoP3 asks students to construct viable arguments (conjectures) and critique the reasoning of others. Notable, the habit of mind asks that students be able to critique their own reasoning, in order to push it further.

Students should be guessers. Of course, when we talk about guessing as math teachers, we really mean estimating. The difference between the two is a level of reasonableness. We always want to ask “What is too high? What is too low? Take a guess between.” Those guesses give use a great starting point for a problem. But how do you know what is too high? By Reasoning Abstractly and Quantitatively, SoP2. Building that number sense of a reasonable range strengthens our mathematical ability. We need to consider what units are involved and know what the numbers actually mean to do this.

What we do, or practice, as mathematicians is important, but what’s more important is how we go about things, and why. A common problem found in the math class is students not knowing where to begin. But if a student can develop these habits of mind, through practice, that should never be a problem.

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assessment, schooling, theory

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Level Up! +1 to Exponents, +2 to Equations

Previously on The Roots of the Equation: You All Have “A”s, You All Have “0”s, and Grade Out of 10? This One Goes to 11.

I like games. All kinds of games: video, board, tabletop, role playing. And so I often think about how games and teaching align. One thing (good) games really do well is provide a sense of progress (especially role-playing games). You start off with not many skills, but as you advance you build them up, learn new things, and can conquer tougher tasks. By the time you reach the end of the game, those things that were hard from the beginning ain’t nothing to you now.

Games don’t usually score you on every little thing that you do. What they do is take a more holistic view and then, at some point, say that you’ve done enough to go up a level. And I say, why can’t I grade that way?

Many people have lamented that the best grading system would have no grades, just feedback that students respond to to improve their learning. But grades are required from external factors: school districts, colleges, parents, principals. But maybe there’s a way around that.

Last time, I said grades should just be a sum of the levels of the learning goals. So now I’m picturing students having a “character sheet” that looks something like this.

I maybe have created that name just so I could tell students to take out their SPELS sheet.

I maybe have created that name just so I could tell students to take out their SPELS sheet.

Student Character Sheet 2

The N/A/J/P/M are my current grading system, Novice –> Apprentice –> Journeyman –> Proficient –> Master

At the beginning of the year we can do a pre-assessment to determine their “starting stats and skills.” Then as the year moves in, we do our work in class. But none of that worked is graded in the usual sense. We would write feedback on the assignment, giving areas for improvement, but the only time a grade is mentioned is when a standard improves. Even then, we don’t focus on what they are (“You now have a 3 in Exponent Rules”), but rather in how they’ve grown (“You gained one level in Exponent Rules!”). The former just highlights that they are not the best they could be. The latter highlights their constant growth and improving.

(Then, at the end, based on what I said in the last post, their grade is literally how many boxes are shaded on the sheet. Have 75 boxes shaded? That’s a 75.)

In order to do this effectively, what we really need to have are rubrics for each standard. That way we know what counts as evidence of a certain level in a standard across all assignments, so it doesn’t matter which assignment provides the evidence. The upside to this is that you do not need to then have a rubric for each assignment! You only need your standards rubrics, because that is all you are using. (The collection of these rubrics, then, in the hands of the students, are a road map to success.)

I’m pretty excited by this idea, and can’t wait to try it next year. This is my idea from the last two posts taken to the next level, with a clear focus on growth, and not deficit. We can’t get rid of grading, and I’m not 100% convinced that we should. But we can definitely minimize the damage that it does and use it to actually promote students’ learning. All we need to do is focus on how we always get better.

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assessment

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