Introducing logarithms

Having not taught logarithms for about 10 years, I looked through several blog posts detailing how to best introduce this topic to students, given how much less experience they've had with logs versus other types of functions. The main idea that I wanted to reinforce for students is that logs are defined as an inverse of exponential functions and to continue connecting them back to features of those functions, which they have had much more experience manipulating, graphing, and applying. As usual, Mimi had a lesson that resonated with me. I liked her formulation of a log equation as a form of a question.

From her blog:

But I wanted to add more emphasis on its derivation as an inverse. So I stole liberally from the CME Algebra 2 lesson on introducing logarithms. We started with a discussion of one-to-one functions (I was surprised to hear that they had never heard of this term before) to know if we could even go into the land of inverses. Then, once we had established that exponential functions were indeed one-to-one and did a quick summary of inverse functions, with which they were already familiar, I had them create a table of value for the function $y=2^{x}$ and then create a table for the inverse function, which we were going to call $L_{2}(x)$. We then started playing around with this function... What would it look like when graphed? Why? Is it also one-to-one? Why or why not? Can we evaluate it for certain values of x? Are there any restrictions on its domain or range? Why?

I really liked how this lesson went. It seemed to build a good deal of intuition for logarithmic functions that I felt had been missing for me when I'd taught it in the past. There was a feeling of exploring a new function, but not one that had been plopped down from the sky and was completely mysterious. There was definitely enough newness to keep it interesting, but not so much that it was overwhelming. I liked how the CME lesson didn't even call the function a logarithm, just L. The less bogging down in new terminology, the better, I say.

I showed students how to evaluate logarithms of any base using technology (both Wolfram-Alpha and Desmos are friendly to different bases) and then gave them the base change formula so that they could solve application problems now rather than waiting until we had developed more of the properties and derived the formula. We'll derive it soon, but I think that it will be more motivating now that they know why it's useful and I wanted them to be able to apply logarithms on day 1 without fending off tons of properties coming at them from left and right.

I think that it was also helpful that while we were working with exponential functions and equations, I would routinely throw in questions that required figuring out the value of the exponent, which students were forced to solve via guess and check and every time, I would mention that soon, we will be able to find these values more precisely with logs. Again, I like the idea of removing the mystery and foreignness of a new concept as much as possible... previewing and embedding it within more familiar concepts makes it less scary and connects it to prior knowledge when it is finally encountered.

Here's my actual handout:

Day 5 - Logarithms

I do regret that I didn't do a project in this unit, but we've been doing tons of projects and at this point, content coverage needs to take a bit of a priority. Also, assigning projects requires me to grade the previously assigned projects and well, yeah, let's just say that I'm a teensy bit behind.

Update:

Just saw that Henry Picciotto wrote a recent blog post explaining his approach to teaching logarithms as super-scientific notation. I would like to rework my lesson plan to include both approaches for next time.

SBG: Assessing Mathematical Practices

In an earlier post, I wrote about the challenges of giving meaningful feedback without using grades as motivation. But now, I am thinking about the challenges when grades are part of the picture. Over the summer, we put together a set of mathematical practices (a mix of aspects unique to our school, Common Core, Park School's Mathematical Habits of Mind, and work done by Avery (his post on Mathematical Habits is here), who teaches 5th and 6th grade Math at our school). The five main categories are

• Investigate, Explore, and Play
• Represent
• Reason
• Communicate
• Growth Mindset

Within each one are four sub-categories and four "levels:" Emerging, Developing, Strong, and Leading. 

Mathematical Practices 14-15

This template formed the backbone of my feedback this year. Every chapter has its own content objectives, but the practices continue the entire year and are consistently being used by all of the Upper School math teachers.

In order for this to be useful to students, most assignments had a content component, which was assessed separately, and a practices component. No assignment included all of the practices, but each one included at least a few. I would let students know in advance which practices I would be assessing with a particular assignment/project. Sometimes, there would be a self-assessment component first ("highlight the level you think you have demonstrated for each practice assessed on this assignment and give evidence for your conclusion."). They would get back a rubric with the appropriate cells highlighted, along with comments and suggestions for improvement.

Pros: very specific and detailed feedback, it was very clear to students how highly the practices were being valued, they understood them better as they continued to self-assess and get feedback on them over time, they began incorporating the language of the practices in their overall reflections on the course and in their work for the class, and they demonstrated progress and growth over time

Cons: there are so many sub-categories and so much detail that it took a while before students were really clear as to what each one meant, it took an almost unreasonably long period of time for me to do each assessment and justify each rating, compiling all of this in a non-formulaic way for a final semester grade was a Herculean effort that I don't know that I can ever undertake again, and the sheer volume of feedback that this resulted in for students, families, and advisors was overwhelming and therefore not practical in the long run

The main change that I think we will make for next year is to eliminate the sub-categories. They can be there in the background if we want to make reference to specific aspects of each practice, but always including each one is just too much. I would also like to build in more time for students to revise their work and make improvements on a project they've gotten back rather than waiting for the next project or paper in order to improve. I'm currently in discussions with other 10th grade teachers to use the last week of the school year as a time for students to put together a portfolio that will include one paper or project from Math, History, English, Science, and World Language from earlier in the year, but revised and improved to incorporate the feedback and learning that has taken place since then. I would love for revision and iteration to be a regular part of the learning cycle in all of our math classes and for feedback to be a step along the way, not the end.

I also need more regular ways to give feedback to students on practices that are not always assessed on projects, such as ones having to do with their growth mindset and collaboration and contribution towards class. There is so much already to plan, assess, and give feedback on that this one definitely slips through the cracks. But I keep reminding myself that if I want something to be a vital part of the class and for students to make progress on it, I need to regularly assess it, give feedback on it, provide explicit instruction on how to improve it, and ample opportunity to revise and iterate and apply it again and again. It makes sense to me that quality is way better than quantity here. Decreasing the number of practices, but assessing them more often and with depth, clear feedback, and explicit instruction and mentoring of students to move them along the spectrum is much better than spreading myself thin.

Collaboration

Today, my 7th graders worked on a great activity from nrich.maths.org that combined practice with the distributive property (ostensibly, the content we are learning) with some very important aspects of groupwork that I wanted to highlight and discuss. Thanks to @Veganmathbeagle for tweeting it out a few days ago.

The activity provides 16 cards in which there are 4 sets of 4 equivalent expressions. The four members of a group start out with 4 random cards and the task ends when every member of the group has 4 equivalent cards. Key rules: no talking or non-verbal communication of any sort AND you cannot take a card from anyone else, only give one of your cards to someone. Each member of the group must have at least 2 cards at any time. If there is an extra person in a group, he or she acts as an observer to the process and takes notes on the ways in which the group members helped each other.

The expressions in the activity - the link above has them in an easy, printable version

This was challenging for my students both from a content perspective and due to the emphasis on collaboration. It was amazing to watch how well some groups gelled and how others were brought to a standstill by a disengaged student.

Comments from my students (roughly paraphrased) when I asked them to reflect on what made this task hard:

"If one person wasn't trying, the whole group got stuck."

"You couldn't do the work for anyone else."

"Some of them were hard and I just wanted to do the easy ones that I knew I could get and leave the hard ones for someone else. But sometimes, everyone left the hard ones for someone else and there was no someone else."

"It made me do more work than I usually do because my group was depending on me."

These are real issues that happen in groups, but are often concealed because other members do pick up the slack. They are really hard to solve in most situations because we do want students discussing and creating a single group product, which means that students who choose to do the bare minimum often can do so. Of course, I do try to build in individual accountability into group tasks, asking a random member of the group to explain the group's work or asking an individual follow-up question that each person must answer on their own. I have done "group quizzes" in order to give feedback to students on their collaborative skills. But this was definitely the most aware and open that I've ever seen my students in discussing the disparity in the level of effort that often takes place when working in groups. I'm hoping that in future tasks, we can refer back to this activity and students will have a better sense of their need to work with more parity and engagement. If you know of any other activities or ways to improve individual accountability in group tasks, please do share.

Some ways that I modified the activity: half-way through, I allowed students to use scratch paper. This reduced the cognitive load a great deal and allowed them to work more productively. In one class that was really struggling, I allowed the groups to talk to each other for a few minutes at the end. Different groups may need more or less of the restrictions in order to create the right level of challenge.

Sequences and Series and Differentiation

Things are moving right along in my 10th grade classes. We wrapped up the Stats unit with some really fun individual research projects in which students created a question about our school community that they wanted to answer, collected data, and performed either chi square or z-tests to answer their questions. I was really, really happy with the level of work students put into their projects and how much ownership they took over their learning.

Here is a picture of the summary slides I asked them to create to summarize their research questions and conclusions. It was really nice to be able to display the results of our labors to the school community.

We started working with sequences and series. This is a relatively short unit and I am pretty happy with the unit projects, which were due last week. Students needed to create their own visual pattern, write recursive and closed form rules for the pattern and its differences and sums, and try to prove one of their formulas using induction. That last part proved really hard for just about everyone. Maybe it's because I haven't really taught proof by induction before, but it was just a painful slog for everyone involved. I have no idea how to teach it in a constructivist fashion as it seems so far removed from the way that most students would approach a proof.

The other challenging part of this unit for me has been appropriate differentiation. For several students, writing rules and finding patterns seemed intuitive and they flew through classwork problems, while others have really struggled and I could tell they needed more support. Most of what we do in class is groupwork based, which has its advantages and disadvantages in terms of supporting struggling students. They can get help and work with peers, but they can also chill on the sidelines and rely on others to do most of the work. I do call on random group members to explain the group's work, but this isn't the same as actually doing the group's work. There is also a big discrepancy between students who are seeking me out for extra help outside of class and those who are avoiding me. Spoiler alert: it's not the students who really need the help who seek it out, for the most part. 

When I teach middle school students, I feel comfortable emailing home or just telling a student that they are required to work with me during lunch or before/after school. For high school students though, it feels overly babyish to do this. I want them to have independence and learn to reflect on their understanding and ask for help. Conferences were a great time for me to communicate to students what I would like to see them doing differently, but the challenge now is to find the time to follow up with individual students and remind them of the commitments they made in their conferences. It's a tough balance between giving them freedom to make their own choices and mistakes and also coaching them in how to learn from those choices and mistakes. One thing that I would like to do is to meet with each student one-on-one right after Thanksgiving break to discuss their progress. As always, finding the time to do this is a challenge.

Accountability without grades

We all want to teach for the love of learning and I bet lots of us wish that we didn't have to give grades. I firmly believe that grades should not be used for motivation, BUT, when done right, they are super useful as a way to clearly communicate what students have learned and where they need to put in more work.

In  my 10th grade classes, we are using standards based grading, and so far, it's supporting the goals that I have for my classes immensely because the grades are composed of both mastery of learning objectives and the "softer" practices we also want to foster in students. Grades are seen as individual pathways and ways to continue improving and to get more focused coaching from teachers on how to get there. 

In my 7th grade classes, for which I don't give grades, only narrative feedback, I am really struggling with how to focus students and have them work at getting better without the structure and clear communication imposed by a grading system. For example, I gave a quiz a few days ago and there were a number of students who didn't demonstrate mastery on a few topics. Today, I gave the quizzes back with lots of feedback. I also made a spreadsheet like this for each student, with an assessment of mastery on each topic and very specific comments as to what they should be working on:

Quizzes were given back and students were told to rework problems on the quiz that they got wrong, first asking their group for help if they were stuck and then me if the whole group was stuck on the same question. They were also given extra practice problems for each learning objective. The result was pretty crappy. They were not engaged with this at all. Instead of helping students or groups with questions on which they were stuck, as I imagined I would be doing, I spent my time policing a class of students who wanted to do anything in the world but the task at hand and putting out behavior fires.

My middle school classes do well with open tasks, interesting projects, games, and puzzles and I totally believe that we should have lots of those things in a Math class. But, I also believe that students need to be able to demonstrate understanding of course objectives. Without grades, I don't know how to build in accountability for doing the latter. If you're 12 and don't really care if you can't set up and solve percent problems, how do I make you care? Is a class supposed to be full of fun and rich activities at all times?

Stats wrap up

Gaaahh. I've been so busy with the new gig this past month that I've hardly had time for sleep and the occasional run, much less blogging or hanging out in the #MTBoS. I want to give a quick summary of the Statistics unit that I'm wrapping up with my 10th graders. It's been a really, really fun unit, in large part thanks to my awesome coworker @michaelpeller, who's graciously been allowing me to steal all of his sweet, sweet statistics projects and explaining stats things to me slowly and repeatedly.

Stats is hard sometimes

We spent the first three weeks working with one-variable statistics: measures of central tendency and spread and understanding normal distributions and standard deviations. I pulled a lot of activities and problems from the Interactive Math Program, a really great high school textbook series for integrated Math. This unit was so rich in applications and connected well to the probability theory students studied last year.

The first project that brought things together for students had them analyzing the massive international data sets available at gapminder.org/data. Students picked a particular set of data that was interesting to them (anything ranging from infant mortality to literacy rates to square kilometers of forest) and analyzed it for the world over a period of several years using statistical measures, as well as for the United States and another country (I asked them to use a country in which the foreign language they're studying is spoken so that they could report on their findings in their language class). Full directions for the project here. Great way to get students to be excited about means and z-scores! I was blown away by their projects - lots of students researched to learn more background about their question and to explain the differences in the world data vs. what was happening in the U.S. and the other country they analyzed. The presentations took an insane amount of time though, what with students going way over the recommended time frame, technology malfunctioning, and two fire alarms that happened on consecutive class days. Teachers who have students present their projects in class, any suggestions on structuring this better? Classmates were attentive and asked good questions (I had them give feedback to each other that was shared so that helped with engagement, I think), but it took a loooooong time. Given how little time I have this year to teach so much content, I was hyperventilating a bit at giving away a full week for presentations.

#firstworldteacherproblems

For the past three weeks, we've been working with hypothesis testing, doing the classic M&M lab to introduce chi square testing. This part of the unit gave us great opportunity to do interdisciplinary work. Biology was also doing work with chi square testing so students got lots of practice in both math and science classes. When we moved on to tests of homogeneity and independence and z-tests, one of the History classes was studying social dimensions of race in the post colonial period in the Western Hemisphere so gave us some nice tables of data to analyze. Students did the statistical analysis in my class and got the context for what it means in their history class. Score. These kinds of easy areas of overlap are going to become harder to find when we move to more abstract units so I'm happy I could find some now.

Counting up observed values of M&Ms

Our final unit project is for students to design and conduct a research experiment on a question of interest to them and analyze their results using hypothesis testing. These are due next week and I'm very curious to see what results they generate.

The one complaint that students have had, and one that I really want to resolve, is that they feel like they don't conceptually understand all of the formulas. I did try to derive with them as many as we could, but some (like the formulas for sampling error) require more math than we currently have. This is something I've struggled with before... should we only be using and working with formulas/theorems that students are capable of deriving or at least being able to recreate the derivation of? I hate waving a magic wand and pulling formulas out of thin air (or at best, giving promises that one day, if they take more advanced statistics and probability theory, they too will be able to see behind the smoke and mirrors). Bleah. I love that students are dissatisfied with simply receiving formulas from on-high, and I want them to continue expecting rigor and proof in what we're doing. How do others make peace with this?

Formalizing and its challenges

I've really been feeling the tension recently between emphasizing creativity, different ways of thinking, innate mathematical processes that are genuinely student-driven and the type of formal math notation and expression that are needed in order for us to have a common language and to be able to demonstrate our understanding to people outside of our community.

This is the first year in a long time that I'm working with students (7th graders) whose almost entire math learning experience has been rich and validating of the importance of expressing their thoughts and ideas in ways that made sense to them. They have done a lot of open projects and pattern investigations. As a result, they are exceedingly curious and creative in their approaches. They are not into answer-getting, they listen to the ideas of others, and they demonstrate really cool insights and ways of thinking. Having said that, their notation and formalizing of thoughts is ghastly. Their work is just numbers and symbols all over the place, a very personal record that somewhat makes sense to the student writing it, but is incomprehensible to anyone else. Equal signs are placed willy nilly, variables are used with little rhyme or reason to sometimes mean one stage and sometimes mean the previous/next stage, the progression of thought skips blithely around the page in seemingly random directions.

So I'm in a position where I know that I need to teach some formalization of process, some common notation and standardization of the way that we communicate our thinking and show our work. But I want to do this in a way that doesn't destroy the freedom of thought that has been carefully cultivated by their previous teachers, their ownership of mathematics as personal expression. Every time I ask a student to show their work in the very specific, standard way, just like all the other round pegs, I feel a little bit like I'm crushing something wild and pure and free.

It's a math fairy in its natural habitat! Let it run wild and free!

Help me out, teachers of younger students. How do you help students channel their approaches without crushing their spirit? How do I know how much to push formal notation? Our high school does not have an Algebra 1 class so by the end of 8th grade, they are supposed to have learned the equivalent of a standard Algebra 1 class. In my previous school, formal and precise approaches were held in very high regard and students bought in and didn't question it. I received my yearly package of students, some of who maybe weren't so amazing at formalizing their thinking, but were definitely aware that this was a goal for which to strive and gave a reasonably good effort to make it happen. Not so here. I feel like I need to be fully confident and able to justify to these students (and their parents) that what I'm doing is for their best development as students of mathematics. And clearly, I have some doubts at the moment. 

If you teach middle school math, I'd love your thoughts and feedback. How do you get buy in to formalization? My approach so far has been to first let them tackle problems intuitively and then try to demonstrate how to convert that into a more formal way, but their response so far has been a bit of

I feel like I can create some need and urgency to communicate more clearly by having them read and edit each others' work, but that won't likely get them to writing in the standard ways that the rest of the math world shows their thinking. And how to approach formal ways of writing without narrowing their thinking and reducing ownership? Or is that a conflict that's inevitable and just part and parcel of continuing in one's studies as a student of mathematics?