A Mathematician's Lament and Math Wars - Here We Go Again...

I found A Mathematician's Lament again, and realizing I had never finished reading the essay, I did. Plenty of LOL moments reading it.

The first big LOL moment was the part about the difference quotient on page 15. I definitely do that in my Honors Algebra 2, and the students hate it. I don't think it's necessarily a bad thing, but maybe I could present it in a different light so that it's not just "another boring math problem to solve because you'll eventually see it in calculus."

Then, there was the description of trig as "masturbatory definitional runaround." My first thought was that's an interesting word choice. Why would that comparison be made to trig? I then looked up the definition of the word, which also means excessively self-absorbed or self-indulgent. I guess that makes sense considering a person would be quite self absorbed when he's, well, never mind.... I just would love to teach that word to someone and have them use it in an English paper.

Simplicio exemplifies all of the same concerns I have. Do we put students at a disadvantage if they don't have basic arithmetic and algebra skills? And are we just creating an excuse for students instead of pushing them to actually learn how to do it? I get that calculators and computers can do a lot of this stuff now, but isn't an understanding of how the processes behind them work important?

Then, again starts the butting heads of educational philosophies on Twitter.

This Paul Bruno article and this one from AJC showed up in my feed on Friday and Saturday, promoting research that teacher-directed instruction is more effective than student-centered instruction, especially for students with math difficulty.

My response: No shit Sherlock! If most standardized tests and traditional classroom tests test a narrow set of skills, then of course explicitly teaching the students how to do what's on the test is going to result in higher scores.

The question becomes: what do you sacrifice for higher scores? Are we trying to train monkeys, or are we trying to teach students to love learning, think, and grow. Do we want students to learn math or learn to pass math tests? Do we want bored students playing (or more often not playing or playing poorly) the school game or do we want students conjecturing, discussing, arguing, productively struggling, and truly engaging in meaningful learning? Furthermore, I think we need to ask do these students really have "math difficulty," or are they just not good at taking tests?

I get a lot of complaints in my classes for "not going over" or "not reviewing" things enough because I do more student-centered activities. When I would try to do games or activities in the past (which I reduced significantly over the past few years in lieu of tasks), I would often be asked "Can't we just do a worksheet?" When summative assessments comprise the entire grade for my students, can I really blame them? My scores are sometimes lower than I would want, but the "aha" moments that happen in class are so powerful.

I really feel shitty when I make my students cry and when they feel bad over a test. Even though I've emphasized how mistakes help us grow and that I don't think anyone is or ever should think of themselves at stupid, I think the number at the top of the paper speaks much more loudly than I can. If I could find ways to affirm the things that they are good it and push them to keep learning, improving, and expending true effort, things would be a lot better.

Yes, I want my students to feel successful. But I want them to be able to more than pass a test that I've prepared them for. How many students go into a shell as soon as they see something that "we haven't learned" or "we didn't go over?" when they probably could do it. That's what I want my students to able to do: to actually APPLY and TRANSFER their knowledge. And I'm not sure you can do that if class is just example, example, worksheet, repeat, and then "study guide" with "problems just like the ones on the test." Sounds more like training monkeys to me.

So what are the options? Different assessments? I can do it more with my Algebra 2 Honors course since I'm the only one that teaches that, but in order to make more headway, I would need to convince more people, especially ones I teach common courses with. Plus, it's the regular students that would need it the most. My assessments would need to reflect growth and continuous improvement rather than a narrow set of skills.

And then the final question: if I change the way I assess and if I change my definition of what being good at math looks like, am I making it better for students, or am I just cheating?

Book Review: "Strength in Numbers" by Ilana Horn

I just finished the book, and want to offer some reflections. The book is more content specific application of how to implement complex instruction. I highly recommend reading it, and am looking forward to seeing how implementing these strategies works in my classes. I'll start by writing in the form of NSRF's 4 A's protocol, and then wrap up at the end.

Assumptions:

• Administration will support this model. It can be integrated into "the system."
• There are many ways to demonstrate mathematical "smartness."
• Students will cooperate with this system.

Things I Agree with:

• "Learning is not the same as achievement" (p. 12). My biggest fear (okay I'll be honest, it's an unfortunate reality) is that this is the truth. Do the grades of my students necessarily match up with their understanding? Or is it their ability to memorize what they need to know for a test? Is it their ability to cheat and get the right answer? Some students' grades are higher than their level of understanding; some are lower. I try to be as fair as possible in remedying this, but sometimes there's more doubt than you can give students the benefit of. This is why I strongly consider a greater implementation of SBG. Moreso, I want students to focus on their learning and not their grade. As Jo Boaler says in this video, math should be a "learning subject" and not a "performance subject."

• Establishing classroom norms and posting them: I think the norms of "Take turns, listen to others' ideas, disagree with ideas, not people, be respectful, helping is not the same as giving answers, confusion is part of learning, and say your 'becauses,'" (p. 28) create a really great structure for engaging all students and creating a good classroom culture.

• "Start with challenging stuff, not easy stuff" (p. 39). I saw a clinical student of mine successfully do this, when I was really skeptical of how it would work out. The engagement was at a very high level, and the students put in really great effort. I was afraid that they would quickly disengage because they wouldn't know what they were doing. Now, three students were extremely disengaged, but I wonder if this could have been helped with more structure and the use of norms in place. This is going to be another substantial shift of my practice next year.

• Peer observing: positives and questions. I love this. I love "questions" instead of criticisms. It is so much more nonjudgemental and I think helps the teacher reflect without feeling threatened. I also agree that starting with a positive is a good idea. Sometimes observers are so concerned about helping people improve, they miss the opportunity to set a good tone for discussion and reflection. I also think about this in giving my students feedback. Do I start with a positive? Do I ask them questions to help them reflect on their learning and improve, or do I judge too much? (Full disclosure: I judge students too much, and I want to make a conscious effort to change this.) 

Things I Argue with:

• "Mathematics is not hierarchical." I can understand how ideas can be connected, but simply speaking, people don't learn to run before they learn to walk. A lot of math is based on structure and repeated reasoning, and while I can see the need to apply ideas to different places, in terms of math for the sake of math (which is still important!), you need to build on your prior knowledge. How are you supposed to add two rational expressions if you don't understand how to add two fractions?

• I might be missing the author's intention, but I wonder if the author is implying that although quick and accurate calculation is a type of mathematical "smartness," other types are also just as acceptable. I've gotten off the quick part. But, I think it is heresy to say accuracy is not important. Accuracy makes sure that the building doesn't fall down and that a patient doesn't overdose. Accuracy ensures that stockholders don't lose their life savings. While procedural skills are not everything, they are still important. Furthermore, some students will need them in future courses. It is not fair to these students to handicap them in future courses because they cannot carry out important procedures. We still need to require deep understanding of the math behind the procedures, but kids do still need the skills. Although it appears sarcastic and exaggerated, I think this blogger brings up a very important concern in this entry and this statement he makes in the comments: "It's all part of the 'multiple ways to be smart' or 'assigning competence' process of CI. So Juan can't add, but he can explain the group's solution. Sally can't multiply, but she can draw the poster. And so on." I'm sorry, but Juan MUST be able to add and Sally MUST be able to multiply. Being able to explain a solution does not absolve a student from being able to do the problem correctly. Skills are still important. 

• "Watch your pace" (p. 90). If you can get your entire department on board (and maybe entire country????), this sounds great. Reality doesn't sound so great. In reality, the Common Core and the PARCC now say that students will be tested on this list of standards at the end of the year. Furthermore, my livelihood might depend on this. How can a student do something he has never been exposed to if we haven't covered it? Secondly, math courses in the United States are mainly hierarchical. If a teacher in a future course expects students to know things from a previous course that they don't know, then the students are at a huge disadvantage. Finally, the "we shouldn't move on until the students are ready" might be pedagogically sound, but it also takes responsibility off the student if the expectation is "I don't need to put in any extra effort, because the teacher will just go over it again." How does that foster students' sense of ownership and responsibility? (Separate post about this issue to follow later.) In theory, I really do "agree," but I have a hard time getting it to work within the reality of "the system."

Things I Aspire to:

• I aspire to make my classroom a place of positive interdependence, where students can take ownership of their learning, and I don't get in their way.
• I aspire to have all students, even the reluctant learners and low-performers, to be engaged and to always put in their best effort towards understanding and continuous improvement.
• I aspire to implement classroom policies that support math as a "learning subject" rather than a "performance subject."
• I aspire to make sure that even while working under a Complex Instruction framework, students get the skills they need to be successful in future courses if they need them.

All in all, I think people would gain a lot from reading this book. The ideas are really helpful to find and design engaging lessons and create a good classroom environment, especially as many schools are using the Charlotte Danielson framework for teacher evaluation. Some of my lessons don't go well because they need more structure, and I think some exceptionally good structures are provided in this book. Furthermore, the book is very helpful in trying to find ways to work with a wide range of abilities of students. Our school is dropping low-level classes next year, so almost all students will be together.

I am left with two big questions after reading this book:

1) Will this really work? Or, even after assigning competence and explicitly stating that all students need to be ready to answer, but even a partial answer can be helpful, will kids continue to insist that "I don't know" when I'm asking them a question that they have no excuse not to answer, and the class is just sitting there waiting because I want to make sure I'm using appropriate wait time and still holding the student accountable for answering but the rest of the class is getting fidgety because most of them already know the answer?

2) Ideally, there would be no "honors" classes either. How do you use this structure to differentiate up and teach different levels within the same class, especially if honors-level students need extra skills for their future courses (e.g. "honors algebra 2 skills needed for calculus but not necessarily needed for the algebra 2 kid going to art school.)? 

Tests: To Time or Not to Time?

This will be the first in a series of issues that have been on my mind lately. Since this article popped up on my twitter feed today, I felt it would be a good issue to start with, as I think it leads to some of my other internal struggles with my beliefs about math teaching and about teaching in general. I am going to to present these in dramatic fashion, as a dialogue between Skeptical Zach and Ideal Zach. (Are these really the names I want? I'm not sure but that's what I'm going with for now.)

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Ideal Zach: What an interesting article about math anxiety. This suggestion of eliminating timed tests is especially interesting.

Skeptical Zach: It's especially stupid, that's what it is! Don't you want our students to be fluent? Timed tests are important. They let you know whether or not a student really gets it.

IZ: See, but that's the issue. The anxiety that is generated from timed tests might actually hide how well the student is getting it.

SZ: Well, they better get used to it. A lot of tests are timed. College professors are especially strict about not letting students finish tests. We have to prepare our kids for the future!

IZ: I'm sure that's not universally true. Even if it is, we want to make sure we do our best so that they will be prepared for the future.

SZ: Well, I'll tell you something else. If the students actually studied for the test, they shouldn't have any trouble finishing it. We always try to be fair in taking the test ourselves and then multiplying the amount of time in order to see if it's a fair amount of time. Plus, you know that some students who don't finish will try to come back later after they've gone and looked up the answer.

IZ: There you go blaming the students again. But now I have another question. If the kids can really easily go back and look up the answer, isn't that more of a problem with what types of questions we are asking them on our tests?

SZ: I don't know. I think that might be a good discussion for another day.

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A final note on this, two years ago I was a total dick about not letting students finish tests, especially with my honors classes. A few of the tests probably were really pushing it on the length, too. When I was working on some of the lessons in Jo Boaler's class last summer, I felt like the biggest asshole ever. I likely have caused a lot of math anxiety in several people.

This last year I had a change of heart. I shortened a lot of my tests, and if extra time was needed, I allowed it. Sometimes, I would put the condition on it that I would ask them an extra question to make sure they really knew the material and were not just going and looking up the answer. The change in students' dispositions towards class and towards math in general were monumental. In many ways, I was very happy with this past year.

A problem though (and maybe I'm just impatient) was that I felt sometimes that the students were taking a lot longer than they should. I'm sure for the most part they "studied," but I don't think most of them did any homework outside of class. (Our school doesn't grade homework, so it's been a real struggle, but again, these last two things I mentioned are worthy of their own posts, which I'll get to later this summer.) In other words, my biggest worry about not timing tests is that it lets students off the hook for not preparing as well as they should. Again, though, I think another issue to explore further on its own.

All about the right answer?

Still have tons of draft blog posts I haven't finished, including ones reflecting on "math wars," and some of my internal moral dilemmas related to that.

But I found a great illustration of the issue, and why it at least continues to produce that "cognitive dissonance" inside me about what I should be valuing in math class and in education in general that compelled me so much that I forced myself to write and publish this, even though I have other things I really need to get done.

So I'm grading this Honors Algebra 2 test. This is one of the high difficulty level problems. Traditional way of solving it (the way I taught it as well, and how the textbook examples in the books I use illustrate it) is to square both sides twice. With the algorithm on the test generator (which I should have worked out the problem to check the numbers, but was bad and didn't :/ ), this one actually turned into a particularly nasty, but still factorable quadratic at the end. In fact, most of these problems are doctored to have rational solutions. (Cue a debate about factoring and real-life situations not having nice solutions like this where the problems have been rigged in order to be factorable, but that's a story for another day.)

Anyways, here's a student with the problem worked out correctly by the standard way:

As I'm going through these, trying to get them done because quarter grades are due the beginning of this week, I come across this solution. First thing I notice is that the answer is wrong. OK, it's wrong, so I'm just going to quickly glance at it to figure out how I would rate it. I have to pause for a second, because I can't figure out exactly what's going on. Luckily, now I'm curious, and have to look at it more closely. And that's when it hits me. I'll let you look at it first to see if you can see it. She did her work on the right side of the page.

Here's a better view:

I re-work it this way myself, just to figure out what's going on. Then, I finally figure out that she took the square root at the end instead of squaring it. So in one way, it's wrong, and that's kind of a big conceptual gap. (Regardless, she didn't even bother to set the other factor equal to zero since she saw it was not going to result in a good solution.)

BUT THE IDEA IS F-ING BRILLIANT! How did she see the quadratic form that was already there? And, the numbers are so much easier to factor or put into the quadratic formula.

For whatever reason, I had never thought of approaching the problem this way. I immediately had to try it with a bunch of other problems. (Does it always work? Does it work if both radicals have two terms in it? etc...) All the other ones I tried, I could work it. It requires a really keen sense of structure and arranging and re-arranging the terms, but it works!

So even though there's a wrong answer, here's a student who has taken mathematical thinking and "Look for and make use of structure" to heart, has totally schooled her teacher, and has restored his faith in mathematics teaching when a week ago it was at a really low point.

But the answer's wrong...

So do I penalize her for not getting the right answer? (Maybe, also depends on what the rest of the equations she solved look like, too, I guess.). Do I tell her this is brilliant and embodies what "doing math" actually means? (Yes).

And do people see why I continually doubt and question everything about teaching mathematics, and teaching in general? If all I wanted was right answers, would this student have had the guts to try this? I do emphasize thinking and problem-solving over procedures and memorization, although the part I struggle with is if grading doesn't mesh really well with this.

I guess it's a matter of what do I want to worship? Is it right answers and grades? Or deep, creative, good mathematical thinking? I think I know what my heart says, but my head keeps getting in the way.

What are the students' "needs"?

This post addresses expectations. It also addresses many of my skepticisms about things I've seen in the How to Learn Math course and in reading Carol Dweck's Mindset. I think it comes down to expectations: expectations of ourselves and expectations of students.
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So one of the challenges that comes in teaching math is the wide range of students' skills and retention from previous courses (hell, even retention from the current course). Our school has done a lot of work with differentiation, assessments, learning targets, etc. One of the ideas in a lot of the differentiation literature is that students should not be assessed the same way. The standards should be the same, but not necessarily the level of the tasks. I think myself and most (maybe all?) of my colleagues are extremely skeptical of this idea.

Yet, the growth mindset comes from helping students grow rather than judging them. It seems to suggest further evidence against using grades. But, if you're in a situation where you have to give a grade, is it fair to give the same grade to two people when one performs way higher than the other. One person is exceeding a standard, one can do it consistently, and one only has a really basic, incomplete understanding of it. It's one thing to say they can meet a standard. It's another for them to have understood it at a level sufficient to be able to use it for future mathematics. I know "Fair Isn't Equal," but when you have external forces (parents and colleges) looking at these grade marks, the reality is it still is important, regardless of whether or not it should be.

But then, if you expect students to still do their work without any grade to work for, then it has to be "engaging." Even if they are getting grades, the work still has to be engaging, anyway. Any good teacher understands the importance of engagement. However, I think there is a great danger in engagement for the sake of engagement. I don't believe you can sacrifice content. If you do, then I think you're mortgaging their success in future courses so that they don't feel like a failure now.

So I think a lot of the skepticism boils down to this question of what is important: "Is it what they learn, or how they feel about it?"

Because regardless of whether or not it's compatible with students' interests, they have to learn how to write an equation of a line correctly. Plus, anytime you try to engage the student with a context, you run the risk of dis-engaging them further because now it's a word problem.

And even if you "explain what you're doing" or "correct and explain the mistakes you made," I still think it's very important to be to actually do the problem correctly. I agree with the process being more important, but getting the correct answer (and Attending to Precision) is still part of the process. Part of the constructive dialogue and feedback probably still needs to be "This still isn't correct, and you still need to work on it more." I'm not sure that you can say someone is proficient at something if they still are not getting a correct answer.

I also find it a little perplexing the dueling messages we get. We're expected to align our curriculum to Common Core, which assumes that everyone learns the same things the same year (and the fact that whether or not they can demonstrate this on a test will determine teachers' livelihoods.) However, we're also supposed to be using differentiated instruction, which operates on the premise that everyone learns differently and at their own pace.

The Problem with Grades

Going to do this kind of backwards to make this easier to read. I'll include more background info at the end, if it helps set a better context. But, I really want the heart of what I have to say to stand out.

My other internal debates to follow in this string of posts:

- Process/Problem Solving/Analyzing Mistakes vs. Precision (Getting a correct answer)
- Responding to students' needs vs. "Enabling"
- Responding to students' needs vs. Curriculum content
- Depth and Problem Solving vs. Curriculum content
- Keeping students from "gaming the system"
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In a nutshell, this post is about me wanting to stop grading students. At very least, to stop grading using averages. I think it's done so much damage to myself, my students, and my classroom over the past couple years. I think it's caused some students to hate math.

I've followed a lot of Joe Bower's work on classrooms without grading over the past few months. I think it's really good stuff. I think my two biggest problems with grades are the negative feelings/pressures they put on students and that they give students an acceptable place to stop learning or improving.

I think my ideal system would be using some kind of feedback portal, such as this one, and then sitting down with students at the end of the semester where they and I decide their final grade. Bascially, everything would be both "formative" and "summative." Anything can be used to provide continual evidence of learning, yet nothing counts as an individual grade. The feedback is the important thing.

Maybe though, this post is more about my fears of doing this:

1) How many students will see this as a reason not to have to do anything? There a lot of students in my school who have played the system for years.  Regardless of the feedback they get, how motivated will students be with very few immediate consequences for not doing the work?

2) I've talked about this, and it does not seem to be a philosophy that other teachers would believe in. Many believe standards-based grading, or not grading at all, would hurt students because colleges still only look at GPA. I've complained about teachers who are too easy on their students, and I worry that although it would end up being far from the truth, I would get lumped into this category. A lot of teachers (and I admit to being one of them quite a bit) say that students have gotten lazier because we allow them to re-test, don't grade homework, and are very soft with deadlines. I worry about this problem getting much bigger without grades.

3) I know that for now, I would still have to a give a final grade. I love the idea of the students being a big part of determining their own grade. But, with parents and students who still worship GPA, what's to stop students from (or parents pushing their students to) argue for a grade that is much higher than what reflects their actual learning? If students know that they can get by without doing the work, then why wouldn't they slack off?

4) Students already play games between teachers. "I didn't do my homework because I had a summative that was due in another class." While that offends me, it's only a rational thought. Some teachers also count "reading quizzes" as summative assessments, or just count normal assignments as summative just to "make" students do them, even if it's questionable whether they are assessing competence vs. compliance. (I'll still keep an open mind that I could be in the wrong on this one. Maybe I'm too committed to following the rules, and I should be breaking this rule if it would help my students? But I still ultimately believe that not grading homework is the correct way to go, even though it's much more difficult.) If I'm the only teacher who uses this system, how often will my work be pre-empted by a "summative" in another class?

5) I teach one of my courses with two other teachers. There are expectations for consistency within teams. I'm not sure how I could do this while the other teachers use a traditional grading system.

6) How will this fly in the community? We have two cultures within our school: one of a Chicago school, and one of the suburban schools around us. Most of the suburban schools around us are very traditional, and have been able to stay that way because of their upper-middle class clientele. They haven't had as many problems with low test scores because they don't have much poverty in their schools. The status quo works, so there's not much need to fix it. I'm sure most of the area around me favors traditional schooling. The parents who are most involved in the school fall in the mindset of the suburban schools around us. If I ever had the need or opportunity to move schools, and were to interview for a position or work at one of these schools, would I be accepted there with this kind of philosophy?

7) Is it a manageable system? (Although it certainly seems more manageable than trying to be strict and manage everything for the re-dos. And seems to have more integrity than just spoon-feeding students the test so that they don't have to do re-dos or just giving students high grades that they don't deserve to limit the re-dos and keep everybody off my back.)

8) What about students who should fail? They shouldn't pass on to the next course without the required knowledge. How does this look in a system without grades? What about athletic eligibility? Every week I have to submit a list of failing students? What does this look like without grades?

As you can see, though I think I want to do this, I see many obstacles. More background info about these thoughts and why I feel this way is below if you're still interested.
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One of the main topics so far of the "How to Learn Math" course has been the role of mistakes: the way they cause new synapses to grow, how making mistakes is the most useful thing to be doing, how they are the key to innovation, and so on.

And I do try to give this message to students in my classroom. Mistakes are great because we all can learn from them.

So I say this, and as I'm going through this course, everything keeps screaming at me, "Zach, you are a big f-ing hypocrite." What kind of mixed messages had I been sending to students?

You see, in the past few years our school has gone to only counting summative assessments in grades. One of the ways I tried to convince students to do homework was to tell them that it's good to make mistakes on your homework. If you're going to make mistakes, you want to find that out on your homework where it doesn't hurt you, instead of finding out on the test where it does hurt you.

Because it's one thing for a student to hear me say that you learn from mistakes. It's another thing for the student to see the "50" on her test and try not to cry, because she actually did do the homework well and worked hard on it. She thought she knew it, too. And now she's freaking out because she has to take the time to put in the work to re-do the test, all while trying to keep up with the current material in class and balancing her other courses and her sport.

We also put a maximum grade on re-tests. It seemed like a good idea at the time, but I don't think it worked the way we wanted it to. The only people it seemed to punish were mostly the people who actually were re-testing because they cared about learning it. It hasn't done anything to stop students who are intent on failing the first time and being a chronic re-tester from doing so.

What this has led to over the past couple years has been students trying to work the system - trying to do as little math as possible while getting the highest possible grade. Most days, I walk into class, and on a good day, maybe three or four students have practiced outside of class to an acceptable level. This seems to be the consensus among many teachers, that students just don't do homework anymore because they don't have to.

The first game students would play was don't do anything, bomb the test, do the remediation work, memorize the old test, and then do the re-test (since many re-test problems are similar to the original one.) Then we started requiring homework to be done, contracts to be signed, etc., but students still only paid lip-service to them and were still playing the "bomb the first one, memorize it for the re-test game." I know I could have been stricter at times, but I also don't need someone asking me why over half of my students are failing, which might possibly be the case.

It's been really frustrating, and I really want students to start actually trying to learn the math and not trying to pass a test or accumulate enough points (to pass, for a B, etc.).

Dante's Mathematical Inferno

This is my first reflection on the "How to Learn Math" free online course that I'm taking, and also kind of a "brain dump" on my struggle with some very common controversial current education issues.

This course has really affected me, even emotionally at a couple of points, and is giving me a stronger push to try to implement some really drastic changes in my classroom.

The bad thing though, is that it's made me wonder what level of "Math Teacher Hell" I deserve to be condemned to, after some of the "torturous injustices" I've subjected my students to in the past.

I'm going to break my internal debates into a few separate posts, just to keep things organized and to keep posts from getting too long.

Hopefully with this confessional, I can do some "penance" this year and eventually have my "math teacher sins" forgiven.

Actually, I'm hoping it will help my students learn more.