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.)

-----------------------------------------------------------------------------------------

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.

-----------------------------------------------------------------------------------------

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.