Breaking the Legacy of Mathematics Struggles

Here are some thoughts on what a student who has historically "struggled at mathematics" should be looking for in a course that will help them improve and have a chance to succeed in mathematics longer term:

• A balanced focus on the "three pillars of mathematics":
• Concepts, 
• Procedures, and
• Facts
• The opportunity to practice...every day!
• A chance to develop an appreciation for the applications of math...everywhere
• A teacher they can trust and who is driven to help them succeed
• Opportunities for periodic successes
• Opportunities to have some fun in class
• Resource(s) to go to for help when they need it (possibly online)
• Opportunities to recover from mistakes (or, more likely, bad decisions)
• Support outside of school - from family and friends

I am sure there is more but these are my thoughts for this evening.

Printing Hell - IT Policy Run Amok

There is has been a dark cloud hanging over my school for the last year or so. Generally, the school is a democratic, trusting place where students and staff are treated with respect and where problems are addressed collectively, with all parties having a chance to take part in collaboratively developing workable solutions.

However, our  IT department, along with the support of senior leadership (not exactly sure who), decided that they would unilaterally enforce a misguided printing policy on all students and staff with the presumed goal of saving money. This policy was rolled out in an e-mail with little initial fanfare or reaction.

Although dictatorial, authoritarian, despotic, tyrannical, autocratic, undemocratic and arbitrary, the policy at first seemed mostly harmless, albeit irritating:

• Limitations on the size of print jobs - less than 20 pages for staff (even less for students)
• No multiple copy printing (same file cannot be reprinted for extended time period)
• Trimester total page limits (70 pages) for students (fortunately, not for staff)
• Cumbersome interface for submitting and approving color printing (and page limits for all)

With the passage of time, the policy has grown increasingly annoying for both practical and philosophical reasons. After a year, it has become an infected, festering sore eating away at my (and others) soul (and, no, I am not being melodramatic). Here are just a few of the many reasons why:

• When printing a test I occasionally (despite my best efforts at online proofreading) make a mistake on a page...that I usually see while walking back from the printer. I would like to fix the mistake and reprint the single page. But, no go, there is a timer preventing me from printing a file with same name again (even if only a page) within specified period. So, I am forced to rename the file in order to print the problem page.
• The IT infrastructure is fragile at best, with significant downtime and clunky rollouts of new technology. For example, recent work on a web server resulted in multiple system outages during the primetime of student and staff use (7-11 pm). Why this rollout could not wait until the summer (when school was not in session) is a mystery - since IT is reluctant to share their super secret (and convoluted) strategic thinking with the lowly rank and file staff. I mention this because having an organization that has shown limited success in running their own business tell me how to run mine (vis-a-vis the printing policy) is just flat out annoying...and it pisses me off each and every day.
• Students are being impacted in their ability to complete the tasks they are being asked to do. For example, one teacher recently asked students to complete eight past exams in preparation for an upcoming end of trimester exam. The teacher did not print off 50 copies of each past exam (approx. 15 pages each) but some students wanted hardcopies to do their work. Well, it is the end of the semester and their printing quotas are exhausted. You get the picture. (NOTE: This is a perfect example of where a less printing intensive approach could have been taken.).

But, what really gets me, is that we are an international school focused on trust, collaboration and most recently "international mindedness". We have very creative, bright students and staff who would willingly work together to come up with more internally motivated approaches to saving money and resources associated with printing, if given the chance. Unilaterally implementing a misguided policy on printing which negatively impacts the entire organization is contrary to everything the school stands for and works against the principles that we try to install in our students each day.

One final anecdote related to this issue. A student recently won election to the Executive Council of the student government on the platform of "fixing the printer policy". I wish him the best and will do everything in my power to help him succeed in making his election promise a reality. This is an issue we should be working together to address not one that is going to be solved via bureaucratic mandates.

Disclaimer: The point of view put forth in this blog post does not reflect that of my school's IT department or the school's administrative leadership.

On the Love of Learning?

I love math and all things related. I find articles discussing applications of math in science, economics, medicine, you name it, interesting. I love reading about famous mathematicians and scientists and the breakthroughs they have made. I love math jokes. I pretty much love all things related to math, science and technology. I also love my wife, teaching, biking, food, photography, philosophy, reading, computers...and lots of other stuff. What I mean by "love" is that I always find exploring new ideas interesting, I always want to do and learn more, not just while I am at school "working", but all the time. Honestly, I don't really differentiate between "work" and "non-work", it is all one big continuum. It is impossible to just turn off one and turn on the other...they merge together for me.

However, I can count on one hand the number of students who I have run across in the last seven years of teaching who truly "love" learning. I get the impression that many just want to succeed in school, they want to get good grades, they want to get into a good college, they want to make their parents happy...but seldom do they want to be at school to truly learn. There are exceptions, but most want to get out of school as quickly as possible (on a given day, at the end of the year, when they finally graduate). They are always looking for the next best thing...other than school. There are not many who truly seem to "love learning".

I am always looking for ways to make headway in unraveling this dilemma. Is it possible for the mass of high school students to truly love learning or is it a point of view that develops with time...like a fine wine?

Lifelong Learning & Summer Break

I just finished a rambling dinner conversation with my wife on whether and how students (and adults) like to learn. Earlier today, I wrote an e-mail to the parents of a number of my students letting them know about an online course offered by Stanford this summer, "How to Learn Math". While I was drafting the e-mail I kept thinking: "Most high school student are not going to want to take a class involving math over the summer. Will any of them (students or parents) actually see this course through to completion?"

This thought took me down numerous meandering paths of thought:

• Lifelong learning is meant to be interesting, fun and rewarding. Why do some students find high school the antithesis of this?
• Why are students fixated on getting out of school and doing anything but lifelong learning in the summer?
• What do I mean by lifelong learning? Does this learning necessarily occur in a physical building or during an online course? If not, when does it occur?
• Do students really shutdown for the summer or do they eventually get the urge to start structured learning before school restarts in the fall? Are they ready for school in the fall - even if they may not say it to a friend or an adult out loud?
• What would students do to learn if left to their own devices - without the formal structure that school provides? What would motivate them to push forward?
• With the Internet, there are many educational options available. I have shared numerous, interesting (to me) online course options for the summer with students in my upper level classes. Will even one student actually be sufficiently motivated to invest the time to complete an online course? Pessimistically, I am guessing not.
• What do students actually do with the 2+ months of "free" time during the summer? Is it all about sleeping and various forms of mental and physical stimulation?
• Do students still read books over the summer months?

OK, I know, there are not many tangible action items here, but it does provide some food for thought. I am genuinely curious about this topic and would love to put some more structure around understanding the role of summer break (or lack thereof) and how it relates to lifelong learning.

Accreditation - What Value Does It Provide?

Every school goes through a periodic accreditation process - often multiple, duplicative processes. I have been part of accreditation processes at schools in Montana, Japan and now Spain. This year my current school is completing the International Baccalaureate (IB) Diploma Program accreditation process as well as initiating the Middle States Association of School accreditation process for grades PK-12.

I do believe it is important that schools are accredited by a third party, but my issue/concern is that most staff who get involved in the process quickly become more focused on "completing the paperwork" associated with the process than with critically assessing the actual day-to-day functioning of the school and developing a detailed action plan focused on making the school better. Most action plans that get developed are put in place to meet the paperwork requirements of accreditation. They are then quickly forgotten...until the next accreditation cycle begins 5-7 years later. It is no wonder staff members are reluctant to take the repetitive accreditation processes seriously or to invest their limited time in taking part in accreditation teams and committees.

I am part of the accreditation team and process this year at our school. My goal is to try to keep a positive attitude and focus on completing the self-assessment critically and developing an action plan for the mathematics focus area that can actually be implemented and tracked over the next 5-7 years. Although I will be long gone, and the action plan may be quickly forgotten, I hope to do my part to critically evaluate how our school is doing in helping our students become better in mathematics.

Wish me luck!

Your Relationship with Students: Caring & Empathy in Teaching Math

Helping students learn math can be complex, difficult and requires a long-term commitment. It also requires that a "relationship" of trust be built with your students. Today I had about 20 different students in my room before and after school (as early as 7:30 am and as late as 5:00 pm). Interestingly, none of them were actually my current students. Many were students from past classes I had taught or students that I didn't even know. They had one thing in common...they all had a test in IB Math Studies coming up sometime in the next week.

So, the obvious question was, "Why were they coming to me for help?" There are many possible reasons but here are a few of the most likely (from my perspective):

• They trust me and are comfortable coming to me for help
• Their friends (who know me) told them I would help them and encouraged them to visit me
• They know I have confidence in them and want to see them succeed
• They want to understand the math and pass their class
• I am patient and explain mathematics in a way and at a pace that matches their learning style
• I give them an opportunity to practice and confirm/deepen their understanding
• I make the time working on mathematics enjoyable and fun - with a sense of humor
• And their course teacher is not readily available to provide help

I am sure there are other reasons (for example, desperation) that these student come to me, but I think this list is a pretty good start.

MORAL  OF THE STORY: Make yourself overtly available to work with ANY students on math (don't miss the opportunity). Show the students respect and that you care. Empathize with their situation and adjust to meet their needs. Make the visit to work with you productive and enjoyable. If you can do these things, over the long term, you will build a reputation with students for being a teacher who they can trust and a teacher who truly has their best interests at heart. This is something that takes (a lot of) time, and once established, every effort should be made to keep the relationship and reputation firmly in place. Best of all, it makes teaching math that much more rewarding!

IB Math SL & HL: Problems vs. Practice

Most of my students come to me in grade 11 for their first year of IB mathematics (either SL or HL). One of the big challenges I face is transitioning students from "practice" to "problems". I define practice as repeating the same skills over and over until you get good - for example, learning how to complete the square. "Problems", on the other hand, are new questions that require a student to combine multiple mathematical concept in new ways to work towards an answer.

Many of my students have been raised on "practice" but have limited exposure to "problems". This is a serious "problem" for them when they begin the IB mathematics curriculum. My assessments are almost entirely made up of problems, not additional practice. Early on it is not uncommon for a student to approach me after a test and say something along the lines of, "It's not fair, I haven't seen that problem before." Needless to say, I have little sympathy and proceed to explain the difference between "practice" and a "problem".

Eventually, they understand that it is going to be a real "problem" for them if they do not learn how to solve actual mathematical "problems". They come around, but it takes time and "practice". In the end, they eventually get a sense of what real mathematics is all about.

The Importance of Practice: Concepts, Facts & Procedures

Here is another article, Practice Makes Perfect, from Daniel Willingham that I found interesting. I highlighted a few points in the link provided. Some notable quotes:

It is difficult to overstate the value of practice. For a new skill to become automatic or for new knowledge to become long-lasting, sustained practice, beyond the point of mastery, is necessary.

Practice until you are perfect and you will be perfect only briefly. What's necessary is sustained practice. By sustained practice I mean regular, ongoing review or use of the target material (e.g., regularly using new calculating skills to solve increasingly more complex math problems, reflecting on recently-learned historical material as one studies a subsequent history unit, taking regular quizzes or tests that draw on material learned earlier in the year). This kind of practice past the point of mastery is necessary to meet any of these three important goals of instruction: acquiring facts and knowledge, learning skills, or becoming an expert.

Our ability to think would be limited indeed if there were not ways to overcome the space constraint of working memory. One of the more important mechanisms is the development of automaticity. When cognitive processes (e.g., reading, writing grammatically, reading a map, identifying the dependent variable in a science experiment, using simple mathematical procedures) become automatic, they demand very little space in working memory, they occur rapidly, and they often occur without conscious effort.

In mathematics, it seems clear that there is a mandate to develop automaticity in certain base facts and procedures. Failure to develop automaticity in their early years of mathematics is one of the primary roadblocks for students I teach in grades 10-12 and much of the time spent in remediation is focused on developing skills that should have been made "automatic" in earlier years.

There is much more in the original article and these points tie back perfectly to the early Willingham article I shared titled, Is It True That Some People Just Can't Learn Math.

Stop Saying "Balance is Key" When Discussing Technology

I have added Lisa Nielsen's blog to my RSS reader list...recommended. Here is a recent post on "balance" as it relates to using technology (for education & life in general).

The next time you find yourself afraid that the "overuse" of technology is a harbinger of the end of civilization as we know it, calm down. It is. But, behind those screens you might very well be surprised to find people who pursue their passion in a way that makes them most productive.

Stop Saying "Balance" is Key When Discussing Technology

Sunday Morning Moodle

Everyone has their own routine on Sunday morning. For me, it is time to update my teaching plan for the next school week, and work on assignments and online activities in Moodle - and today write a post for my blog. As the school year draws to a close, this morning I reflected on the work that I have put into Moodle this year for my IB Math SL Year 1 course.

I now have a complete course for year 1 (pre-Calculus) fully built out in Moodle. This includes all assignments, online activities and questions, and links to all supporting materials. The biggest chunk of work was related to building the question bank in Moodle to support the course. I just checked and my current question count is over 400 in the Math SL question bank. If I roughly estimate the average question takes me 15 minutes, then, conservatively, I have put in an extra 100 hours of work outside of school developing Moodle content - just for Math SL. And, I teach two other courses that require Moodle updates!

The good news is that (1) the Moodle content dramatically improves the ability of my students to retain and review content from the course, and (2) the work is now done at a level of quality that I am happy with - I won't have much new/redo work for this course next year.

So, looking back, it took a lot of time to get to this point. It has actually been a 4 year investment (with many 100s of hours of practice and improvement) since I started working with Moodle for my courses in Japan. But, I enjoy working with the technology, and I am now (finally) quite competent at using it.

Students Appreciate Organization & Planning

I have found that one of the things that students appreciate (and recognize) most is a well-planned and organized course. Here are some student survey responses to the question, "What are the best things about the course?":

• The course is phenomenally structured. It is clear and well organized with time to work in class that is hugely helpful. The learning is great!
• The organization and the fact that I know what is due the next day and the following month via Moodle.
• Everything is very organized and clear. There is a lot of things I can do to review and I never forget past lessons. I also have never had math so clear!
• I believe the best things are the organization of the topics and the structure of the class. The pre-made notes and the explanations are very convenient and helpful as well as the time that is usually left after the lecture is over.
• The course is difficult but extremely well planned and we've kept pace, giving me more confidence in how prepared we'll be. 
• Everything is always perfectly organized (I love the way Moodle is set up and the daily class notes) so that we can plan ahead and get feedback before, for example, turning in the first drafts of the IA.

Students expect teachers to have their act together - and they know it when you don't (trust me, I have heard them talk about some teachers who are clearly planning by the seat of their pants - not kind). They appreciate it when you do your job well and are the first to point out (to other teachers) when a teacher is dropping the ball.

Teachers really should take the time to get their courses superbly organized (at least one week in advance). Their students know the difference between well planned and not.

One Reason I Love Teaching Mathematics

One of the most satisfying parts of being a math teacher is to watch students begin the year unmotivated and "hating math" (an unfortunate reality during my time as a high school math teacher) and see them end the year motivated and proud of the work they have done, with a positive outlook and renewed excitement about their ability to succeed in future math classes.

Some teachers aspire to teach the upper level classes with the "best" students (make no mistake, I enjoy teaching these classes too). But, the most personal gratification for me comes from my work in my lowest level course - Integrated Algebra & Geometry (IAG). It is there that I think that I really have the greatest impact.

Today, I sent e-mails to the parents of two of my IAG students letting them know that their daughters had the two highest scores on the most recent test. Both of these students entered the year at the very bottom. Other teachers had commented that they were "challenging". I passed one of these students this morning in the hall and told her that she had gotten a 94% on the test. Another teacher walking in next to her actually said out load in a somewhat shocked voice, "Really?"

Over the year these two students (and others like them) have slowly started to have a little fun in math class and have begun to earn small victories. They managed to successfully complete their assignments, and they managed to submit work on-time and get full credit. They quickly learned that there was someone there to help them when they got stuck and to answer their e-mail at night as soon as they had a question. Every little bit of positive reinforcement they received stuck and further enhanced their self-confidence and desire to succeed in mathematics.

As the momentum of these small successes took hold it became even easier to keep it moving forward. The students kept working harder and harder, and wanted to experience more success (for themselves and, I like to think, a bit for me). They knew I cared about them and in return they wanted to show that they appreciated it and did not want to "let me down". This may sound a little self-centered, and it is only an hypothesis, but it rings true with my experience.

Yes, there were times I wanted to strangle each of them and there were also times that I was disappointed with their occasional lapses back into past, unproductive behaviors. But, at those times, I subtly (and usually with an edge of humor) let them know that I was disappointed. However, my actions always showed that I continued to care and that I still had confidence in their ability to succeed in my class and in mathematics.

Every year this same process (which I would like to make more formal) repeats itself with many of my previously low performing students - of course, more with some than with others. I think the process is enabled by the following actions:

• Genuine care and interest in seeing students succeed in mathematics
• Unwavering confidence in their ability to succeed in mathematics
• High expectations of the quality of work from both students and myself
• Excitement about mathematics in general, and a willingness to share it with students
• Sense of humor and willingness to make math class fun, flexible and engaging - but serious too
• Many opportunities to practice, improve and succeed - repeated as needed
• Constant, timely (near real-time) feedback on performance and meeting (or not) expectations
• A chance to recover when mistakes are made - and they are!
• Availability to help - in class, throughout the school day, and online outside of school hours
• Constant reinforcement - both with the students and their families
• ...and others that I have not yet nailed down.

This is a fascinating topic that I intend to come back to in future posts.

More from Daniel Willingham

After getting a reminder (and some motivation) from the practical writing and insight of Daniel Willingham in "Is It True That Some People Just Can't Do Math" I went out searching for more. There are other great articles and videos by Willingham on his personal site Daniel Willingham.

Lost Article: Is It True That Some People Just Can't Do Math?

I spent about 30 minute tonight looking for an article from the past (I had forgotten the name) titled, "Is It True That Some People Just Can't Do Math?" by Daniel T. Willingham. Interestingly I found the article I was looking for embedded in a very recent article by Grant Wiggins titled, "Conceptual Understanding in Mathematics".

I first read the original article when it was published in 2010 but had lost track of it. During recent discussions at school there have been multiple lapses into the "concepts vs. facts vs. procedures" debate. I always felt that this article aligned with my personal thinking and experience on this topic and wanted to find it and share it with everyone. It is worth the read!

Is It True That Some People Just Can't Do Math?

Practice & Timely Feedback - Online Assignments in Moodle

Students appreciate timely feedback on their work and know that they need to practice to get better at math. Feedback not only allows them to know whether they understand something but it also motivates them to continue to improve. One of the ways I provide practice & timely feedback in my math classes is by using Moodle online quizzes.

The daily online activity focuses on content from the current topic but always includes review problems from past content. Review problems are randomly selected from over 400 existing problems (2-3 problems a day turns into a lot over time). I do this in both my Algebra and IB Math SL courses. An example question in Math SL is shown below:

Teacher View of a Moodle Question - Student cannot see correct answer.

As you can see, this student spent approximately 12 minutes working on this problem - with repeated attempts prior to determining the correct answer. Students can retake assignments multiple times until they get a perfect score. Not surprisingly, they spend a significant amount of time outside of class working on mathematics - and they truly enjoy it. They also appreciate the fact that they are getting constant reinforcement of concepts. There will be almost no additional review of material needed prior to our upcoming end-of-year assessment because they have been reviewing the material all year long! Feedback from students in a recent survey:

I also like the fact that in most tests there tend to be some review problems, which, again, forces us to go back and review previous material and have it always "fresh" in our minds (I wish this was done in more classes).

I really like how the quizzes are set up because even though it does take some time to complete them, they really do help and it serves as a sort of review and challenge. I also like how you can go back to the questions you missed and work on them. 

The downside...it takes time to develop the Moodle questions! If you don't enjoy technology then it's probably best not to start down this path. If you do enjoy technology and are interested I am always looking for collaborators to help extend my existing question banks and to try them out at other schools.

Common Core Suggestions for Intervention

I stumbled across the section "Supporting Students" while reading the Common Core (CC) - Appendix A on designing math courses to align with the CC. I was particularly intrigued by the Response to Intervention practices. Few of these suggestions are in place at my school...although we are looking for ways to put more support outside of the classroom in place gradually over the coming years.

---- from Appendix A of Common Core for Mathematics ----

Supporting Students

One of the hallmarks of the Common Core State Standards for Mathematics is the specification of content that all students must study in order to be college and career ready. This "college and career ready line" is a minimum for all students. However, this does not mean that all students should progress uniformly to that goal. Some students progress more slowly than others. These students will require additional support, and the following strategies, consistent with Response to Intervention practices, may be helpful:

• Creating a school-wide community of support for students;
• Providing students a "math support" class during the school day;
• After-school tutoring;
• Extended class time (or blocking of classes) in mathematics; and
  
• Additional instruction during the summer

Watered-down courses which leave students uninspired to learn, unable to catch up to their peers and unready for success in post-secondary courses or for entry into many skilled professions upon graduation from high school are neither necessary nor desirable. The results of not providing students the necessary supports they need to succeed in high school are well-documented. Too often, after graduation, such students attempt to continue their education at 2- or 4-year post-secondary institutions only to find they must take remedial courses, spending time and money mastering high school level skills that they should have already acquired. This, in turn, has been documented to indicate a greater chance of these students not meeting their post-secondary goals, whether a certificate program, two- or four- year degree. As a result, in the workplace, many career pathways and advancement may be denied to them. To ensure students graduate fully prepared, those who enter high school under-prepared for high school mathematics courses must receive the support they need to get back on course and graduate ready for life after high school.

Furthermore, research shows that allowing low-achieving students to take low-level courses is not a recipe for academic success (Kifer, 1993). The research strongly suggests that the goal for districts should not be to stretch the high school mathematics standards over all four years. Rather, the goal should be to provide support so that all students can reach the college and career ready line by the end of the eleventh grade, ending their high school career with one of several high-quality mathematical courses that allows students the opportunity to deepen their understanding of the college- and career-ready standards.

Why Innovative Educators Should Look Down Upon "Look Up"

I saw the original "Look Up" video about a month ago.

There were messages that I could relate to but there were also some points that I felt the video missed about the value of social media. Lisa Nielsen hints at the flip side of the video in her post.

Why innovative educators should look down upon "Lookup"

A challenging issue that will definitely get more press as time passes. But in reality, the technology is here to stay, and we must figure out the best ways of using it and its future incarnations.

Facebook Focus: Rate of Change

Today, a student in my Algebra class was confused about what rate of change was and how to calculate the rate of change given the measurements for two quantities. The problem she had was:

Rather than respond directly to her e-mail I posted the following general response to our FB group. My response was:

This comment was also posted to my IB Math SL class that is about to begin an introduction of calculus. A very useful 10 minutes spent discussing a very important concept with two of my classes via FB. We'll see what type of student comments are generated, either in class tomorrow or later tonight online.

Using Facebook for (Math) Courses

I started using Facebook as a communications tool for my courses about 2 months ago - as an experiment. I had previously been using Moodle and e-mail (both of which I still use) as my primary means of online communications with students. Although both worked they were cumbersome to use and did not really motivate students to read the content or take part in online conversations about math and education. Hence the experiment with FB.

I setup a FB group for my IB math (SL & HL) classes and sent an invite to all class members. I had not used FB groups before. I found the setup and administration easy and students could join the group without having to become a FB "friend" (which I do not do with students).

I immediately realized that this experiment was going to be a hugh success since:

• Every class member already had a FB account and all joined the group within the day/week
• All class members could post and respond to messages with comments
• There was immediate feedback on who had viewed each post ("Seen by...")
• Class members (and I) could seamlessly post math-related content from any web page ("Share on FB") and any device (computer, mobile...)

My courses now use FB for many tasks, including:

• Posting course announcements
• Posting assignment and assessment information
• Posting reminders, hints, and other "immediate" content
• Students post questions on assignments - and help each other online. This is great because it (1) gets them to write about math, and (2) let's everyone in class read, or better yet, actively take part
• Students and I post math-related content, including articles, math jokes, links to useful sites...

It is a wonderful way to engage students in math outside the classroom and the possibilities for ways to use FB for the classroom are endless.

Some screenshots of a few examples are below.

Problem Posts & Discussion

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Test Summary

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Link to Article

Getting Started - Motivation

OK, to be honest, I have attempted to maintain a blog of my thoughts and experiences related to teaching math numerous times. I get started, write a few posts, then lose interest. This time I am going to take a slightly different tact. My goal is to post something, anything, at least once per day. Every day something interesting or thought provoking seems to occur. Maybe I try something new in the classroom or online that works or doesn't, or maybe a colleague does something special or frustrating. Whatever it is, it is typically worth my time to write about and reflect on. So here I go...

A quick introduction...

I am a high school math teacher at the American School of Madrid in Madrid, Spain. I have been teaching mathematics for 7 years and worked in the consulting industry for 20 years prior. I love teaching math and using technology to augment my teaching. I focus on IB mathematics courses, but have also taught Algebra, Geometry and remediation courses. I have taught in Montana, Japan and now Spain. I live in Montana during the summers.