# What Challenges Do Math Students Face?

For a college student in a math class (I’m thinking of a first year course, but this could apply to students in more advanced courses) what are the distinct challenges that they face (that they might not expect)?

• Contextual Knowledge – Most students will have seen lots of mathematics by the time they reach college and will have developed lots of ideas and practices related to doing math. However, they usually only know this in a specific context and often haven’t been really challenged to apply these ideas in new scenarios (i.e. non-textbook) and in places where they might not work the same way. Thus they don’t always have the ability to recognize what ideas are relevant in a situation or the ability to check if the applied method actually works. The most common instance of this is ‘forgetting’ basic algebra when it is encountered in a calculus class.
• Fragile Knowledge – Related to Contextual Knowledge, but involves applying advanced mathematical processes in a slightly different way, that might test a different aspect of the mathematics. In this case, although they may ‘know’ the process, they don’t know it well enough to actually use it widely.  The result is often panic and reverting to some primitive (and often wrong) mathematics.   An example of this would be working with rational expressions, and asking questions that are related to questions about whole number fractions, and the student ends up doing things like (12)/(16) = (12)/(16) = 1/3.
• Variable Overload – Students can handle a certain number of variables in an expression, but when that number is exceeded, they can’t make sense of it anymore.  This also extends to the use of nonstandard variable names and Greek (and other non-Latin) symbols. This can be made worse in contexts where the number of variables used is almost always 2 (e.g. x,y), and something with an extra variable comes along. The weirdest instance of this is confusion when facing standard formulas, like V =  π r2 h for the volume of a cylinder in a class like Calculus where you’ve worked primarily with (x,y,t). The students may not even know what to do with π, like is it a variable or what?
• Abstraction – This is related to variable overload as college math instructors, being math people, are often quick to go to the abstract case; you’d think that we count: 1, 2, 3, n. Students will have been exposed to abstraction and will have some specific relation to it (usually hate), but it will be fragile and in very specific contexts (sometimes limited to abstract versions of well-understood concrete versions) , and usually won’t be something they do on their own or are comfortable with.  In most college math classes, abstraction comes early (and often) and it expected that students can connect the specific details of examples, etc. with the general theory.

The big issue is whether the responsibility to deal with these challenges belongs to the instructor or to the student. As with most classroom issues, it probably belongs to both with the instructor responsible for course structure that helps students become aware of these challenges and provide opportunities and feedback to help them make adjustments according to these challenges.  The students would have the responsibility to self-assess and make adjustments and accommodations based on their own level of difficulty with the challenges. Both would have the responsibility of not ignoring these challenges, but to face them head on throughout the course.

# Office Hours

Some thoughts about office hours with probably no real or useful conclusion.

I’ve been thinking about office hours lately. I don’t really have a point to it all, so I’ll just go with questions and partial answers.  Enjoy!

Should one have scheduled physical office hours? It is convenient for both teacher and students to hold online office hours, but there are always somethings that are best (and only?) done in person. So unless the course is entirely online with no local students, then one should at least have physical office hours.  And it is also convenient for all for office hours to be ‘By Appointment only’ but there is a certain need and security in having at least one scheduled hour per week. In general, students don’t come to office hours enough, but occasionally they need to be able to see you in person, and so yes, have a scheduled physical office hour.

Is it a good or a bad sign if lots of students come to your office hours? It is a good sign in that it shows that you are an approachable teacher, and that students are taking some initiative to be successful in you class. It is a bad sign in that it shows that some students are not being reached during the regular class, that you’ve maybe crossed the line from instructor to friend, and that students aren’t doing much on their own to figure stuff out and are depending on you. It’s probably bad overall, if lots of the same students come all the time and they either need you to basically explain everything again, or they spend most of the time just shooting the breeze.

How can I get more students to come to my office hours? It’s probably not really a problem of getting more students, but of getting the right students to come to office hours. For that I recommend making sure that students know of your office hours regularly, know why they should come, and know what to expect when the do come. You should also make the office hours effective, meaning don’t be too friendly, address the concerns or questions, make sure their concerns or questions are resolved to some satisfaction, and let them move on. If you do have specific students that you think should come, the direct approach of a note on homework or test, or an email, inviting (requiring) them to come by is best. It can also help if you refer to productive things that happened during office hours while in class, like ‘As one of your classmates, brought up during office hours …’.

How can I get fewer students to come to my office hours (when they have other, more appropriate, opportunities)? This relates to teaching a large-lecture (140+ students) with GTA support (1 per group of 35ish). In this case, I want students to first meet with his or her GTA and only come to me with bigger issues or as recommended by the GTA. For this, I’d publish and announce the GTA office hours early and often, and make sure that the GTAs are showing up. I’d make it clear that the GTA is the first contact and that I won’t get involved until they’ve been tried. I’d minimize my office hours, maybe limiting them to 1 scheduled hour, with additional contact through email or a discussion board. Overall, I’d push information to the web, and individual tutoring to the GTAs. I’d probably pull back a bit from being too approachable, making office contact more formal and not so inviting.  I might also replace office hours with group sessions (with GTA support). (If it’s a smaller class, and you just don’t want any students coming by, just be a jerk, be non-helpful, be unavailable, say mean things; few will come, none will come back)

How many office hours should I have per week? Some schools have rules, we don’t, except you do need to post something about office hours, and try to have at least one scheduled per week. The number and schedule depends on the class and the students (and you). I like to schedule 2 per week, on the days before class, with extras available by appointment. This also requires using some web based means of either announcements, emails, or a discussion board, and using the small spots before and after class to resolve quick technical questions. Have a small number of scheduled hours can also work well if you otherwise have an open-door policy, i.e. if they stop by, with a short question, and my door’s open, they can ask; if it is a more involved visit, I’ll ask them to come to office hours or schedule something.

What e-resources do you recommend for office hour substitutes? Good old email is fine for individual simple yes/no questions.  For classwide information, I use the announcement capability in our LMS.  I’ve also used Piazza as it allows both corporate and individual messages, peer-to-peer responses, and supports TeX. I have a colleague who uses Facebook Chat (or whatever it’s called), and the discussion board in Google (or on an LMS). I’ve thought about using Google Hangout or Skype with a whiteboard.  With Google Hangout, the session could be recorded for later. The best answer might depend on what your students have easy access to and are willing to use.  It also depends on whether you want synchronous  or asynchronous communication, individual or group, or the ability to save (and edit) responses for later. I don’t think the right (or wrong) choice will really make a dramatic difference in student participation, but might make a difference in the instructor’s enthusiasm and participation.

# What’s Good for Prospective Math Teachers is Good for All

Taking a discussion of what’s good for prospective math teachers (explanation, representation, translation, and conversation of/about math) and seeing that’s it’s really good for all math students.

As part of our conversation with Dr. Ball (see the previous post), the question came up about how we should be preparing future math teachers. Because we wanted to consider teachers at all levels (K-12) and didn’t really want to get into specific mathematical topics, Dr. Ball presented 3 ideas supported by research and one that was a current interest. I’ll go through the ideas below, but what struck me as I thought about the ideas was that they could and should be part of every math class because they were key in what it means to be a mathematician, and thus good for all students.

The ideas (with my clarifications, comments and examples) (educators = prospective math teachers in math classes):

Being able to explain mathematical ideas accurately – For educators this means explaining concepts, practices, etc. in both a variety of ways and at different levels, and doing so with mathematical accuracy; this involves more than just mathematical knowledge but the basis does come from mathematics. For the rest of the students, as we desire them to not just be able to do the mathematics, but also they should be able to  understand the concepts behind the processes, and there’s probably no better way to develop and assess this type of learning is to explain it, especially using different methods. It would be very natural to include activities involving these type of explanations by all students in most of our math classes.

Being able to use a variety of representations of mathematical ideas appropriately –  For educators this is the essential skill to address the diversity of learning styles in a typical class; it forms the basis for the Rule of 4 (or 5) (i.e. symbolic, in spoken and written words, in tables, and in graphics) and the focus in most curriculum on developing student fluency with data in these variety of forms. For the rest of the students, being able to convert ideas from one form to another is a necessary skill for learning in different teaching environments, is useful for making sense of new ideas, and is valuable for ones who will work in applied fields with non-mathematicians. Presentations in math classes should already be using a variety of representations and it would not take much to add opportunities for students to work at expressing ideas in different ways.

Being able to translate mathematical ideas from one context to another –  This is more about the nature of mathematics as it focuses on learning the essential nature of a mathematical idea (sometimes called the invariances)  and then seeing how that idea is expressed in different areas of mathematics.  For the educators, the focus might be on ideas that are more prevalent in K-12 mathematics, like proportionality and equivalence, that also are present in higher mathematics. But, it also would extend to looking at how ideas in the higher math classes exist as extensions of K-12 topics, and how the ideas of translation, equivalence, etc. are directly manipulated to develop new ideas or new areas of mathematics and to develop deeper understanding of those ideas. For the rest of the students, this is an essential part of mathematics and should be a part of every math course (and not just some sort of capstone experience). All students should see all of mathematics as both a distinct set of areas, but, more importantly, as a strongly connected set of ideas, with higher level ideas held in common. This is (or should be) part of the instruction in each course, and is probably part of the student work, but, due to it’s primacy in mathematics, should be increased in all classes.

Being able to have an accurate, comprehensible conversation about mathematics – For all this is just the ability to express ideas correctly. So often we only ask students to write mathematics and often accept some pretty incoherent scribbles as ‘justification’. For the educators, this would just be the baseline of being able to speak mathematics and to make good logical arguments. For the rest of the students, we ‘talk’ to our students all the time, and this would just ask them to be able to do the same. Including student oral communication in a class can be challenging both in scheduling and assessing, but could be done more informally as part of other assessments, like having a student orally present the results of a problem, or using online resources to record themselves, and then use some peer evaluation for assessment.

Overall any efforts to modify math courses with a focus on improving the preparation of future math teachers, would inherently improve the quality of the course for all math majors.

# Applied Mathematics of Teaching

A new aspect of math teacher development of using a mathematical perspective to look at pedagogy.

Recently Dr. Deborah Ball (Univ. of Michigan), came to our campus to give a presentation and some of us were invited to meet with her earlier to discuss the education of prospective math teachers. This post and the next (and maybe others?) are related to ideas that came up in that conversation. I didn’t take detailed notes of who said what, so it is best to assume that anything that is awesome belongs to Dr. Ball and anything less so belongs to me.

First a little background. When there is talk about teacher development it is usually broken down into various aspects depending on perspective, source, etc., and usually some diagram similar to this shows up:

(source http://tpack.org)

The ideas of Content Knowledge (CK) and Pedagogical Knowledge (PK) are fairly easy to understand.  Pedagogical Content Knowledge is a newer idea, and, as the diagram indicates, is a blend of PK and CK.  Usually this blend is in the form of taking ideas from PK (e.g. how students learn and effective classroom practices) and looks at what those say about teaching in the particular content area.  It usually addresses questions like “How should we teach topic X?”, and so is usually taught from the education (PK) perspective.

The interesting part of the conversation with Dr. Ball was her idea of turning PCK around (and it thus might not really be best called PCK), by looking at what mathematical practices and ideas can say about particular PK in the mathematics classroom.  For example, given a specific curriculum, we would evaluate it in terms of whether or not it is correct mathematically. Or we could look at different arguments or ways of solving problems and determine which ones are valid mathematically. The goal of this approach is to encourage the math teachers to keep thinking mathematically in their classrooms and to keep the mathematical level of the pedagogy high. This approach would not be taken in isolation from the other perspectives and one can be mathematically correct and have it end up wrong in the students’ minds. For example, Dr. Ball mentioned a situation where properties of numbers were being defined and discussed and the point that zero is neither positive or negative came up. This is mathematically correct, but this fed into an early misconception that students had that zero is not a number and so they started skipping zero in various number line calculations.  So one would still need an awareness of the tradition PCK.

What I find exciting about this idea is that it provides a natural entry point for mathematicians into discussions of teacher development and a way to bring education materials into a traditional classroom. I can imagine asking students to evaluate arguments and curriculum materials for mathematical correctness and having that contribute to both their understanding of the mathematical topic and their understanding of student learning.

# Why they don’t know it.

Why students come across as not knowing something that we know they’ve been taught.

In a recent interview of Sir Micheal Atiyah (famous mathematian, etc.) he is quoted as saying

When someone tells me a general theorem I say that I want an example that is both simple and significant. It’s very easy to give simple examples that are not very interesting or interesting examples that are very difficult. If there isn’t a simple, interesting case, forget it.

And it reminded me of a conversation that I had a while back with a colleague about whether or not students in precalculus know the commutative, associative and distributive laws. My colleague said that student’s haven’t seen those laws, and so they don’t know them (and they should, and that’s what’s wrong with education, blah, blah, blah). I didn’t have the chance at the time, nor the desire to make a big deal out of it, and so I didn’t get to respond. But one advantage of a blog is that I can always go back and rewrite history, or at least post my response here and will tie it neatly to Atiyah’s quote.

My fundamental belief is that students appear to not know something that they have studied in the past, because either they don’t recognize it in the given context or it was not presented to them in a way that made it significant enough to remember.  The example of the first case that I come across most often in my discussions with instructors is in calculus when students have taken the derivative of a cubic polynomial and don’t know how to solve the resulting quadratic equation to find the critical points. In most cases, a quick reminder/review is all that is needed to get the students back on track. In this case, they know how to solve the problem when it is presented in the context of ‘Solving Equations’ but it doesn’t appear the same when it is ‘Find the Critical Points’. The example of the second is the point of the earlier discussion about the commutative, etc. laws. Students can and do use the laws, but without any recognition of what they are doing.  In our proofs course where we go through the field axioms carefully, I always have the students expand $(x+y)^2$ and denote all the axioms that they use. It usually takes about 20+ steps and we almost always miss at least one axiom.  (As a challenge to the reader, how many times, if any, are each of the distributive, associative and commutative laws used in expansion? (Answer below)).

A solution to these issues is connected to Atiyah’s quote. We need to raise the cognitive awareness of the things we teach, and this comes from increasing the variety and significance of contexts in which we see (and don’t see) the results. If we only see the laws in the context of normal arithmetic, where students have been using such things for years, then it shouldn’t surprise us that they don’t remember them.  I suspect that most people that studied math, didn’t really get those laws until they were introduced to some examples where they didn’t work (like matrix multiplication).   But just giving a new context is not enough. Advanced math students are taught that continuity does not imply differentiability, usually with an example like $f(x) = |x|$ for pointwise.  But for the general case, we often go through all the details of constructing the Weierstrass Function or something similar based on the Cantor Set.  This does provide a new context, but it is so complex and apparently isolated, that it doesn’t surprise me to see advanced students still thinking that continuity does imply differentiability. Later when the example(s) have been repeated (and maybe taught) does it finally sink in. In general, one example or counterexample is not enough to seal the deal. And, as a final note, I’d like to adopt the last part of Atiyah’s quote in that if we give simple but interesting examples, i.e. various and significance contexts, then maybe we should just not teach that topic until we can.

——-

Solution: Distributive: 4, Commutative: 1+, Associative: 3+. (The + depends on whether you’ve defined the distributive law as left or right side only, and whether or not I accurately counted all the uses; associative is the hardest because we often don’t track the grouping of 3 or more terms in a sum or product)

# Turning ‘Cover’ into a Good Thing

Instead of thinking of covering in terms of material, think of it in terms of students.

Suppose you are teaching only one student, then you (hopefully) know how to listen to and watch the student and adapt the lessons to maximize his or her learning. In this case, if someone asked you what you covered, you’d define it in terms of what you know the student learned. If there was some topic that you discussed, but you know the student just didn’t get it, you probably wouldn’t claim that it was covered. This is because if in the future this student is asked about some topic you claimed to have covered, and acts like he or she had never seen it, then you’d just look foolish.

Now, suppose you are teaching 30+ students, and are using the best teaching techniques you have to maximize the learning. If someone asked you what you covered, you’d define it by what topics you covered, knowing full well that not every student got each and every topic. If later a student is shown to not know something you covered, you always have the excuse that ‘you did it in class’ or something similar. Being able to hide behind the ‘covered compact’ can mean that a teacher doesn’t really have to work at developing or improving his or her teaching so as to increase the quantity and quality of the learning that happens in his or her class.

As a new thought, consider the class as a collection of individuals, each having a profile of effective learning strategies.  For example, one student (or group of students) might learn best through direct instruction, which another might prefer hands-on activities. Now we can talk about what happened in a class over a semester by the topics, the variety of teaching methodologies that were used to teach each topic, and the results of the assessments, with the expectation that all the students’ learning strategies were covered for each topic. We might even want to report our results in terms of this breakdown, e.g. for Topic X, all students were covered, but for Topic Y, mostly students that learn through reading got the material, etc.  Or we might be able to list which topics we taught by what means.  Then when a student shows up in another class, the description of what they should know can be more detailed, and any weaknesses in his or her preparation can be traced back to the type of teaching done, not just whether or not it was ‘covered’.

This perspective brings up some new issues. For example, if one teaching style is highly effective for a majority of the class, but is totally ineffective for the rest of the class, is it proper to only use that style? What if the effective style for the rest of the class is bad for the majority, so that by any type of mixing of the two styles, the overall results (in total) are lower? Related to this is how much should students be expected to adapt to the teaching style(s) in a class? I don’t think there is an easy answer.  I’d probably err on the side of trying to reach all in some way rather than maximizing the overall result.  But I also think it is part of our job to help students learn to adapt, and so, at some point, I’d probably expect more of students in terms of being able to learn from a limited variety of styles.

# The Death Spiral of a Blog Post

How an idea goes from being a possible blog post to a black hole for time and energy.

A story of an idea: It started with a discussion with a colleague about evaluating teaching and my saying that I didn’t think that my department would agree on what was good teaching. And then I had the question, “Is there some form of teaching, that even if done in the most excellent manner, is not effective?” And then the spiral begins. This connects to an earlier unfulfilled idea about how that the measurement of effective teaching is really a question of the expected value (the effectiveness of the teaching for various student types against various distribution of student types) and, of course, the question of the time frame and actual measures for effective teaching. But then what teaching styles can I consider, and is it reasonable to look at the styles in isolation or at ones that probably nobody really uses. (For example, what about the stereotype of the professor that teaches the same class over and over from the same notes, and just reads them without any type of interaction with the students – I doubt anyone really does this anymore). Just in case you might think the spiral is settling down, it goes to the next level. I start thinking about how writing a blog post starting from a question is like the way one approaches mathematical modeling. One, ideally, starts with a question and then as the model develops, new questions and ideas start coming out, and it is easy to lose focus as one examines all the different possibilities. It is also challenging if you share the model with someone else as they will always have suggestions for other (reasonable) things to consider. And then it seems the blog post should be about that, but then it seems like the best idea is to write a post about the struggle to come up with a a focused post.  And so, here it is.