Monday, December 15, 2014


Today, my 7th graders worked on a great activity from 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.

Monday, November 24, 2014

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.

Wednesday, October 22, 2014

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?

Friday, October 17, 2014

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 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.


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?

Monday, September 8, 2014

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?

Thursday, September 4, 2014

Digital workflow

In my previous post, I blogged about starting to use Google classroom with my 10th grade students this year. So far, I'm having not the easiest time with figuring out exactly how I want to use it. When you post to your classroom stream, you have the option of posting as an announcement or an assignment. Every time you post an assignment, Google classroom makes a folder with that assignment name and automatically drops work that students turn in into that folder on your Drive. However, students seem to be having a hard time using the "turn in" feature. For the assignment due today, they were supposed to make a spreadsheet of data (using a coin flipping simulator to flip coins and record the number of heads) as well as answer questions about their data. Apparently, Google classroom doesn't appreciate you trying to turn in two things for one assignment. I was told that there's an option to turn in one document and then add another, but I haven't tried this yet. Also, what to do for students who really want to answer the questions by hand?

This brings me to my second question... in the past, I have had students keep a very organized binder with sections. Not quite an Interactive Notebook, but something that could serve as a reference and be easily searchable and reflected upon. I had a great system for managing student workflow and feedback. Now that I'm using Google classroom, I'm trying to figure if I should go completely digital and have students create organized folders in Drive and scan/take pictures of their work or if I should try to maintain a hybrid system of some kind. My goal is for students to be able to compile and reflect on a portfolio of their work, as well as be able to turn in and receive back work in an organized manner. I'm a big fan of systems and right now, everyone is turning things in completely willy nilly and it's driving me crazy. I feel like I have this small window right now while the year is still in formation mode and students are eager to please to create an organized and simple system that makes sense for different types of Math work.

So please, clue me in to your wise ways. If your workflow system has a digital component, I would love to hear about it. How flexible are you? What's your take on helping students be organized? What do you do that you like or don't like? If you went all digital, how did it go? Tell me all of your things!!

Thursday, August 28, 2014

First day of class and starting my stats unit

Tomorrow is my first day teaching at my new school. It's 2 am, I've finally finished working out my lesson plans and I'm way too excited for sleep. I've had a chance to meet most of my students on a pre-term camping trip on Monday and Tuesday, but tomorrow is the first time that we're going to be doing math together. So. Excited. Yay math.

In the interest of greasing the wheels of interdisciplinary work, we decided to start the year for all 10th graders with a statistics unit that will dovetail with work they will be doing in their Biology classes to design experiments and test null hypotheses using chi-square. All sounds awesome, except for the part where I haven't ever taught statistics before and haven't even looked at it since I took it in college myself a long time ago in a galaxy far, far away. Aack. I pestered just about all the stats teachers on twitter with questions, so hopefully, my plan for tomorrow isn't completely wrong and ridiculous. It's mostly adapted from The Pit and the Pendulum unit in IMP: Year 1, but amended to include more technology and be done in one day rather than several.

I have 45 minutes with each 10th grade class. As I usually do on the first day of school, I'm going to project a seating chart of groups and directions for doing a quick, individual writing activity so that I have time to go around, get names, and do administrative start-of-class stuff. This year, I'm going to be trying Google Classroom, so I'm going to have them join the class that I've already set up and complete the first assignment on there, which has a few questions for them to answer. My four questions are:

o   Describe a class that you have really enjoyed – what was awesome about it?
o   Describe a class that you did not enjoy – what was difficult or unpleasant about it?
o   What are some questions that you have about the class?

o   What are some questions that you have about me?

I'm going to collate the results and use them to structure the next few classes. I think that I originally got this idea from @delta_dc, who blogged about it here. I used it last year and I really liked how it transformed the description of the class and its procedures from a passive droning on by me to a more active engagement of coming from student questions.

Once this is done, I'm going to have students pair up within their groups and gather data on their pulse rate. They will record the data in a shared Google spreadsheet and play around with the charts available in there to analyze it. They will discuss in groups, then I will call on specific groups to share out their thoughts. The main discussion topics that I want brought up are:

o   How can we represent this data?
o   What might be interesting to find?
o   How do we expect the shape of the frequency distribution to look? Why?
o   What other variables might be distributed in a similar way? What variables do you think might be distributed differently?

o   Why do we care about shapes of frequency distributions?

I will also try to work in some terminology related to normal distributions, measures of central tendency, samples and populations, and connect this content back to work that they did with probabilities last year. Depending on time, I will ask groups to debrief what makes for good groupwork and good class discussions either today or the next time that we meet because I really want to make sure that there is at least 5 minutes at the end of class for students to do an individual reflection, which they will also share with me through the Google classroom site.

The individual reflection will contain the following questions (students can choose to focus on one, two, or all of these):

o   Something that I found interesting today
o   Something that I found confusing today

o   Something that I’m wondering about that is related to what we did today

This plan seems to contain all of the elements of a first day lesson that are important to me:

  • something to start the class immediately to set the norm that we will start every class period with work
  • a chance for students to share their prior experiences with me, which will help me plan better
  • activities that use individual, pair, group, and class structures so that we can start setting norms for all of these and practice moving from one configuration to another
  • learning of content that's central to the course
  • a few different forms of technology, with which I definitely want students building familiarity
  • end of class reflection
Excited to see how it goes and welcome and would appreciate any feedback!

Actual handout for group investigation:

UPDATE: Woot! Everything went better than expected! Kids were engaged and I got lots of interesting questions through the start of class Google survey. I also spontaneously decided right at the start of the group activity to do a "participation quiz" described by @samjshah, just not as a quiz, but as a way to give feedback to each group. I didn't categorize behaviors and comments/questions as positive or negative, just tried to record what I heard and saw objectively. It gave us a nice jumping off point to discuss group norms.

I did learn that when students complete a Google form that's part of an assignment, Google classroom doesn't count that as having turned in the assignment and continues to tell students that this assignment is due.

Sunday, August 10, 2014

Integrated Curricula - the Good, the Bad, and the Ugly

In working out the details of the Upper School Math curriculum a few weeks ago, we came to a consensus that things would be much more interesting, relevant, and connected if we created an integrated sequence of course that combined the curricula found in geometry, algebra 2, and pre-calculus classes, along with an infusion of statistics, probability theory, programming, and fun math things that are not usually found in standard high school math classes.

After high-fiving ourselves for a bit, we realized that going down this route answers a few questions, but creates about a zillion new ones. Such as, where do we place new students who have already taken one or more of the standard high school Math courses? 

Do we teach all students the same integrated content or do we create "regular" and "accelerated" versions? If we teach the same content to everyone, how do we allow for sufficient support or challenge for students who are either struggling or need more? How do we match up our crazy courses to the standards expected by the University of California board for accreditation? If a student transfers from our school to another one, how will the new school be able to place her into their standard system? How can we convince the rest of the school community that these courses are going to be awesome and worth the hassle?

Apparently, integrated math courses have a reputation as being fluffier or geared towards struggling students (news to me), so we will also have to communicate clearly to everyone why we are advocating for this and how they will benefit and challenge students.

The other issue that is quickly coming to light and making it obvious to me why more schools aren't pursuing this option is how demanding it is of teachers' time and knowledge. Despite my education and teaching experience, I'll be the first to tell you that my knowledge of statistics and programming is pretty minimal. Trying to write either of those two things into the curriculum is a major headache for me... I can lecture on some terminology and have students do examples and exercises, but I don't have the depth of knowledge to write a really awesome activity or project that will be rich and student-centered and exploratory and all the good stuff that I want to be doing in my classroom. Teaching integrated courses requires the teacher to have a much wider net of knowledge and the fluency to weave that knowledge into their teaching. This is really, really hard. 

I feel like non-traditional curricula have more room to go farther in either direction... they can be really, really awesome and exciting and students can learn a ton. But if done badly, they can leave students (and families) frustrated and with large gaps in their knowledge. There's less of a ceiling, but also less of a floor. And that scares me a lot.

People who teach integrated classes: any advice?

Sunday, July 13, 2014

Designing a curriculum: essential questions for Geometry & Algebra 2

I started working at my new school last week and it's been full-speed, non-stop all the way since day 1. I feel like I'm pedaling like mad down one of the steep slopes that are so popular around here and my brakes are broken.

Yup, I live here now

We're in year 2 of a new Upper School and the challenge is to rethink the traditional Math curriculum so that it promotes engagement, interdisciplinary connections, design thinking, and deep content learning without compromising students' ability to do well on the SATs, AP exams, and you know, their future life in a potentially traditional world that may frown upon our hippie ways.

After an amazing day working with Denise Pope, a professor at Stanford's School of Ed and writer of the book "Doing School,"who basically schooled us in the purpose of education, I felt ready to tackle some curriculum writing. She reviewed some of the ideas from backwards design from Wiggins: don't start with activities, but with the desired results. Then, figure out how you will know whether the desired results have been achieved. Then create activities that lead to the desired learning. Her guidelines for selecting enduring understanding goals were:

  • Represents a big idea with enduring value beyond the classroom
  • Resides at the heart of the discipline
  • Geared to students' interests and developmental needs
  • Open-ended, complex, provocative
  • Fits with school standards/expectations

So, I'm putting together my list of "essential questions" and "enduring understandings" for the integrated Geometry/Algebra 2 class that most students at my school will be taking during 9th and 10th grade. Please, please, please give me all of your feedback and criticisms and suggestions for improvement either here or on twitter (@borschtwithanna). I hope that this can be a fruitful springboard for discussion and curriculum framing.

  • How is a system of definitions, postulates, axioms, and theorems created and made cohesive? What does it mean for something to be true?
  • How can objects be transformed? Which of their characteristics stay constant and which change and why? How can we measure and describe what changes? How does this relate to concepts of proportionality and similarity?
  • How are angles on a plane related to each other? How do we know when two lines are parallel or perpendicular to each other?
  • How can we do algebra on a coordinate plane?
  • What is congruency? How do we know when figures are congruent to each other? Why might that be useful to know?
  • How can you determine whether a triangle is isosceles or equilateral? Why might that be useful to know?
  • What are some special relationships for right triangles? How do they relate to polygons? Why might this be useful to know?
  • What can we measure about figures? How do we measure these qualities? How do measurements change when we change dimensions?
  • What relationships are formed when lines intersect circles? 

Algebra 2:
  • What are the underlying principles of solving equations? How do we solve specific kinds of equations: linear, absolute value, quadratic, exponential, radical, and rational?
  • How are equations and inequalities related to each other? How are inequalities represented graphically? How are inequalities solved?
  • What are functions? How can they be represented and what do they represent? How can they be combined or reversed, algebraically and graphically? Why might any of this be useful?
  • What are the key characteristics of the following specific functions: linear, absolute value, quadratic, exponential, logarithmic, radical, and rational? How can we represent situations using specific functions?
  • What are asymptotes? How are they related to graphs and to equations of functions?
  • How are functions transformed? What is the relationship between the equation of a function and its graph?
  • How do functions model data? How do you know if the model is accurate?
  • How can multiple constraints be represented with systems? How can systems be solved?
  • How can expressions be simplified? How can expressions be combined or operated upon? How can expressions be factored? Why might any of this be useful?
  • What are different ways to categorize numbers? How are the different categories of numbers related to each other? How can different categories of numbers be represented graphically? How can we perform operations on different categories of numbers?
  • What are polynomials? What operations can be performed on them and how? How can they be represented graphically? How are their roots related to their graphs and equations? How can their end behavior be described and related to their graphs and equations? Why might any of this be useful?
  • What are properties of exponents and logarithms? Why are they true? How are exponents related to radicals?
  • How can sequences and series be described and evaluated? How are recursive and explicit formulas different from and related to each other? How are arithmetic and geometric sequences different from each other? How do we know whether a series diverges or converges? 
  • What are the measures of variation and how are they computed? Why might they be useful?
  • What are the measures of central tendency and how are they computed? Why might they be useful?
  • How can probabilities of events be determined? How are theoretical and experimental probabilities different from each other? 

* I should add that I haven't forgotten about mathematical practices, habits of mind, etc. We are planning to assess on a common set of these that will be the same across all of the math courses. Just breaking that up into a separate post.