Category Archives: teaching philosophy

After Individualized Learning, What Comes Next?

This was my classroom in the latter part of the last block of the day.

I should point out that I usually have students seated closer together in groups. Conversation happens more organically in that configuration. I gave a quiz where I didn't want to set hard time limits. As each finished, I nudged them to work on their own on a PearDeck assignment.

This is what it looks like when everyone is working at their own pace. Each student with a single screen, each solving problems and answering questions.

I like that I can drift from student to student and either ask or answer questions when the time seems right. I can see each student's answers on the online teacher dashboard. I can decide which conversations I need to have. Students can also decide if they need to have conversations with me. I involved myself in student learning with surgical precision.

Some claim this is the future of learning in schools.

For me, the silence in the room today was unsatisfying. No sharing of ideas. No excitement shared between friends. Nothing that might compel a student to contemplate the other living, breathing beings in the room.

I don't do this every day, so I know this isn't how it will always be. Whenever I do this type of lesson, I know that the students are better off when they get what they need. I know it is good for them. My thinking always go to the next step. What will we do when we are back together in a big group, or at least in groups larger than one?

I asked the students the following question:

We have all worked independently today. What is the best way to use our time when we are back together?

Their answers gave me the direction I needed to think about the next steps:

  • Either going over answers so people who haven't put the answers in will know what to do and what the correct answer is.
  • Maybe review the questions people were confused with or got wrong a lot. This would help to review what we did.
  • Go over some of the answers together or answer some of the difficult problems.
  • I don't know.
  • To check where the majority of us got stuck and had trouble, and discuss how to figure out those problems. That, and to possibly discuss new concepts that we have yet to master.
  • Going through the questions that could be tricky and solve challenging problems together
  • I think it is good to have a mini lesson to quickly teach what we are learning and to go over, but also save time for students to work independently.
  • Give us a few examples at the beginning of class so we don't forget what we learned, and then learn new things.
  • To quickly go through and review all the topics we've learnt about polynomials.
  • Learn something new

Knowing what to go over is certainly where the online tools are helpful - they make incorrect answers or misconceptions stand out.

I know that the students prefer the social aspects of the classroom. Whatever our next step is, it should involve coming together and acknowledging and appreciating we are in a room of people learning together.

We need to make sure that we are social when being social is productive to learning.

We need to make sure students have time to learn and think on their own.

We need to make sure students can also learn what they need to know in the hands of an experienced guide.

All of these are crucial. Any one channel we use loses its effectiveness to learning when it becomes routine. I think this is especially the case when that routine involves staring at an entity that can't talk or laugh back.

Building Up or Breaking Down: Creating States-n-Plates

In piecing together States-n-Plates, I wanted to learn more about React, a web application library created by Facebook. In the process, I found myself finding parallels in how I go about learning anything.

Before I describe the details, I'll give a (hopefully brief) description of what React does.

An HTML page normally consists of HTML tags that tell the browser what to display, with other rules that also describe how the HTML should look. A page created through React reframes that page as a series of components that each serve a different function. The image components need to have the ability to be dragged onto the targets. The targets need to be able to accept a dragged image, and need to be able to indicate whether the image dragged onto the target corresponds with the correct plate. The scoreboard needs to know how many states have been correctly matched at any given time. The components also have the ability to respond when a user clicks, types, or drags other components into them.

In a well designed React application, each component uses information from the component that contains it in order to behave (or, um, react) as the application is used. Building the pathways for how this information flows from one component into another is deliberately designed so that each component can act independently from another.

When I first started working on creating States and Plates, I started with a fully formed webpage that looked much like the final product above. I followed the React documents that then suggested breaking the page down into components, one by one. I did this without really understanding in detail what I was doing, but was able to get the components to each have the appearance of the original web page, which felt like real progress. Eventually, my progress was halted when I reached the limits of what I understood. I needed help.

It was at this point when I picked up a book on React and started working through the basics. I began to understand better what the guiding philosophies of React were - the design decisions, the behaviors that one component had in response to another, and how to think through an application the React way. This was where it was helpful to read the perspectives of some people much more experienced then me - I understood the vocabulary they used and could make the connections I needed to make progress.

With some of the basics figured out, I rethought the application from scratch. Rather than starting with the webpage as a whole, I started creating components and making sure each one worked as expected before moving on.

By the end, I felt comfortable thinking about my application both from a bird's eye view and on an individual component level. I needed to have the experience of breaking the idea down into individual pieces and seeing how they interacted with each other to produce the whole. I needed to take time seeing what rules guided the function of one component in order to understand the whole. If I had started by reading the documentation as my step one, I would not have had the context that the big picture view of the application yielded for when I actually did so in my learning. Both views were important, and neither view was sufficient on its own to lead to full understanding or transfer.

We need to give our students opportunities to have both views of the content we teach. Insisting that student mastery of the basics is a necessary gatekeeper to higher levels of thought misses opportunities to understand the context of that basic knowledge. Student exploration of concepts through Desmos or Geogebra or problem solving is a great way to engage with the standards of mathematical practice, but without discussion, review of underlying concepts, or (gasp!) direct instruction where needed, opportunities for growth might be limited.

Let's make sure, as a team, that we are attacking this problem from both ends.

Zack Miller and his PBL/PBL Merger

I've been skeptical about Project Based Learning (PBL) for a while, despite many people trying to sell me on it as being a step up from more traditional instruction. I like the concept, but I have too many reservations with veto-power to keep me from jumping in completely. These are my main issues:

  • While PBL provides a rich environment for learning, it doesn't work as an effective means for assessment.
  • The idea presumes that every mathematical topic has a problem that forms a solid context for a project. In the recent past, I've made a commitment to my students to not force a context on them if it doesn't actually fit, and to tell them outright when I've artificially created one. There are many rich applications for many topics, but this is not generally the case.
  • I have never seen a project based rubric that I really like, even (and especially) ones that I've created myself. Something ends up being incentivized unintentionally, or students focus on the wrong elements despite my best efforts to prevent this. This is my fault though, not PBL.

All of that said, I'm really enjoying Zach Miller's ongoing series on combining Project Based Learning with elements of Problem Based Learning, which I also like, but also with reservations.

His posts are pushing my thinking a bit, and I'm liking how it's getting me to adjust some plans for this semester. Check out his blog when you can.

More of What Matters

Just over a year ago, my first daughter was born. Like most newborns, she couldn't hold her head up. Yesterday I watched her pick up her pacifier from a table and plop it in her mouth.

It made me realize that I've witnessed the entire progression of skill development that made that happen. She developed an awareness of the muscles needed to grip, lift, and rotate. She learned to use visual feedback to move her stubby fingers to the right location in her field of view, close them, and then move the pacifier to her mouth, which she could not see. This learning was all hers - nobody had to tell her what to do, or why she needed to go through the steps. My daughter's skills will continue to develop as she grows and encounters the great variety of people and places in this beautiful world we all inhabit together. I've made more of an effort this school year to be present for these moments, and it has been among the best decisions of my life.

There have been times in my teaching career when I've thought about how nice it would be to be able to start completely from scratch with students. With no misconceptions, bad habits, or fear of failing like the time before, maybe I could help them be more successful. I quickly stop myself. First of all, my own instruction is not perfect, and in all likelihood, the students would still discover shortcuts that might work in my class, but be detrimental to their success in the next. Second, and more importantly, our students are not computers to be programmed. The experiences they bring to our classrooms is what makes this job what it is. As rewarding as it might be for us as teachers to see students make all of the progress we want for them, teaching is not about us.

Our students have all walked their brains through a unique path that passes through our classrooms, and that path certainly doesn't end within them. The knowledge and skills that our students take from our classrooms are also not under our control. Our students' lives will hopefully be a long series of meaningful moments of discovery and excitement, and the reality is that those moments that will last are likely not the ones that transpire in our classrooms. That said, the time we all have together is precious and always decreasing.

As we start a new year today, let us remind ourselves to work to make the important things matter. This takes work because of the noise of distractions that surrounds us. Build opportunities to grow instead of dig in to old habits. Make a deliberate effort to express gratitude to the people that you are with. Recognize the value in being at this place, at this time. We will not always be successful, but accepting this is part of becoming more successful on the next attempt.

Be well, everyone.

Acting Upon One's Beliefs

Suppose you know in your heart that what you believe is right, and others disagree.

If you are in control, do you charge on ahead and act upon your beliefs?

Do you frame this as merely a communication problem, or is it something bigger?

What is your responsibility to...

  • ...those who agree with you?
  • ...those who disagree with you?
  • ...those who are impacted most by what happens next?

How do your answers to these questions change if you are not the one in control?

How do your answers to these questions change if you are in the minority?

These questions have bounced around in my head since last week, and not just in relation to the U.S. election. It bothers me a lot that I can't answer my own questions with any certainty right now.

Class Notes and Workflow (On The Other Side of the Wall)

I've struggled in the past with the role of class notes. I wrote more than a year ago about my solution using Microsoft OneNote. Since moving out of China, I've realized just how far behind I am in just awareness of what Google Docs are capable of doing. My new school uses them extensively for all sorts of organizational and administrative purposes, not to mention applications in the classroom. I decided to upgrade my class notebook system this year to make better use of these tools. Now that we're approaching three months in, I'm feeling pretty happy about my system thus far.

I now make all my handouts on Google Docs. The bandwidth and lack of a Great Firewall make it a reliable way to have access to files both at school and at home, which means that I'm not dragging my computer back and forth anymore. There's something to be said for carrying a minimalist backpack, especially given the temperatures here. I relied on iCloud Drive last year which worked well enough, but the fact that I'm not worrying about files syncing between home and school is a clear change for the better. These files are titled U3D02 - CW - Title of Day's Lesson to signify 'Unit 3, Day 2' for ease of identifying files and their order. These are starting points for class activities, resources to use during class such as Desmos activities, videos, or other parts of what might be useful to students learning a given topic.

screen-shot-2016-10-31-at-8-16-35-pm

Each handout is shared with the class through Hapara teacher dashboard and Google Drive, and I give students read access on each file. Two students are randomly picked to be responsible for class notes. These two students make a copy of this handout during class, name it with the same title and unit/day designation, and then change CW (class work) to NB (notebook) file to indicate the purpose of this file.

I take notes during class using Notability and my Wacom tablet. It's easy to copy and paste images from the digital handout into the notes, and then annotate them as needed. I take photos of student work with my phone and use Airdrop to get them to my classroom laptop. At the end of the class, I paste images of the notes I take during class into the relevant part of the notes. The two students are responsible for solving problems from the class handout and from homework, taking pictures, and putting them into the notes file on Google Docs. Links to these files are then shared on the course website with the rest of the class.

My class handouts are still printed on A5 paper as an analog backup, and quizzes are usually still on paper as well. I still insist on students doing problems by hand since that's ultimately how they will be assessed. The computer is there for access to Desmos, Geogebra, and the digital handout.

The most satisfying part of all of this is that students are being remarkably proactive about asking for materials to be shared, letting me know when they think something should be added to a handout, or adding it themselves when they have editing access to the file. There is also a flow of suggestions and comments to the students that are responsible for each day's lesson.

It's pretty amazing what is possible when a major world power isn't disrupting the technology you want to use in the classroom (or for whatever) on a regular basis.

Scaling up SBG for the New Year

In my new school, the mean size of my classes has doubled. The maximum size is now 22 students, a fact about which I am not complaining. I've missed the ease of getting students to interact with simple proximity as the major factor.

I have also been given the freedom to continue with the standards based grading system that I've used over the past four years. The reality of needing to adapt my systems of assessment to these larger sizes has required me to reflect upon which aspects of my system need to be scaled, and what (if anything) needs to change.

The end result of that reflection has identified these three elements that need to remain in my system:

  • Students need to be assessed frequently through quizzes relating to one to two standards maximum.
  • These quizzes need to be graded and returned within the class period to ensure a short feedback cycle.
  • There must still be a tie between work done preparing for a reassessment and signing up for one.

Including the first element requires planning ahead. If quizzes are going to take up fifteen to twenty minutes of a class block, the rest of the block needs to be appropriately planned to ensure a balance between activities that respond to student learning needs, encourage reinforcement of old concepts, and allow interaction with new material. The second element dictates that those activities need to provide me time to grade the quizzes and enter them as standards grades before returning them to students. The third happens a bit later in the cycle as students act on their individualized needs to reassess on individual standards.

The major realization this year has been a refined need for standards that can be assessed within a twenty minute block. In the past, I've believed that a quiz that hits one or two aspects of the topic is good enough, and that an end of unit assessment will allow complete assessment on the whole topic. Now I see that a standard that has needs to have one component assessed on a quiz, and another component assessed on a test, really should be broken up into multiple standards. This has also meant that single standard quizzes are the way to go. I gave one quiz this week that tested a previously assessed standard, and then also assessed two new ones. Given how frantic I was in assessing mastery levels on three standards, I won't be doing that again.

The other part of this first element is the importance of writing efficiently targeted assessment questions. I need students to arrive at a right answer by applying their knowledge, not by accident or application of an algorithm. I need mistakes to be evidence of misunderstanding, not management of computational complexity. In short, I need assessment questions that assess what they are designed to assess. That takes time, but with my simplified schedule this year, I'm finding the time to do this important work.

My last post was about my excitement over using the Numbas web site to create and generate the quizzes. A major bottleneck in grading these quizzes quickly in the past has been not necessarily having answers to the questions I give. Numbas allows me to program and display calculated answers based on the randomized values used to generate the questions.

Numbas has a feature that allows students to take the exam entirely online and enter their answers to be graded automatically. In this situation, I have students pass in their work as well. While I like the speed this offers, that advantage primarily exists in cases where students answer questions correctly. If they make mistakes, I look at the written work and figure out what went wrong, and individual values require that I recalculate along the way. This isn't a huge problem, but it brings into question the need for individualized values which are (as far as I know right now) the only option for the fully online assessment. The option I like more is the printed worksheet theme that allows generation of printable quizzes. I make four versions and pass these out, and then there are only four sets of answers to have to compare student work against.

With the answers, I can grade the quizzes and give feedback where needed on wrong answers in no more than ten or fifteen minutes total. This time is divided into short intervals throughout the class block while students are working individually. The lesson and class activities need to be designed to provide this time so I can focus on grading.

The third element is still under development, but my credit system from previous years is going to make an appearance. Construction is still underway on that one. Please pardon the dust.


P.S:

If you're an ed-tech company that wants to impress me, make it easy for me to (a) generate different versions of good assessment questions with answers, (b) distribute those questions to students, (c) capture the student thinking and writing that goes with that question so that I can adjust my instruction accordingly, and (d) make it super easy to share that thinking in different ways.

That step of capturing student work is the roughest element of the UX experience of the four. At this time, nothing beats looking at a student's paper for evidence of their thinking, and then deciding what comes next based on experience. Snapping a picture with a phone is the best I've got right now. Please don't bring up using tablets and a stylus. We aren't there yet.

Right now there are solutions that hit two or three, but I'm greedy. Let me know if you know about a tool that might be what I'm looking for.

Context and Learning Names

I wrote yesterday about my decision to try learning names of my students on the first day.

As of the middle of week two, I've learned the names of every student within each class with few exceptions. In some of the bigger groups, I mix one or two names that start with the same first letter, but I correct myself pretty quickly. I've come to recognize some individual traits that make each student unique within the group, and am feeling comfortable building on my knowledge of their names to find out more about who they are.

In the hallways, in line for lunch, and walking around campus, I struggle. Outside of the classroom, I lack the context of those names that I can usually lean back upon to remember them. With the students all mixed up together, including with students that I don't have in my classes, it takes longer to put a name with the face. As I develop an understanding of the students beyond names, this struggle will go away.

The analogy to learning in any classroom context stands on its own, so I won't ruin it with more commentary.

IMG_3451

Generality vs. Specificity

We want our students to have problem solving methods that are general enough to work in any situation. If we assign a series of exercises that are too similar to each other, it becomes easy for students to lock onto the wrong pattern, or to use a 'trick' that works just frequently enough to seem worth the effort to learn it.

One thing I tried this year was to prompt students to make themselves aware of the spectrum from generality to specificity. What works for solving specifically this question? What general ideas apply to answering all of the problems on the page?

I used my randomized question generator to help create problems that worked this way. Here's an example:

Screen Shot 2016-07-01 at 11.05.02 PM

I only started a deliberate effort to prompt these conversations at the middle of the second semester. I wish I was doing it all year.

Hacking the 100-point Scale - Part 4: Playing with Neural Networks

First, a review of where we've been in the series:

  • The 100 point scale suffers from issues related to its historical use and difficulties of communicating what it means.
  • It might be beneficial to have a solid link between the 100 point scale (since it likely isn't going anywhere) and the idea of achievement levels. This does not need to be rigidly defined as 90 - 100 = A, 80-89 = B, and so on.
  • I asked for you to help me collect some data. I gave you a made up rubric with three categories, and three descriptors for each, and asked you to categorize these as achievement levels 1 - 4. Thank you to everyone who participated!

This brings us to today's post, where I try to bring these ideas together.

In case you only have time for a quick overview, here's the tl;dr:

I fed the rubric scores you all sent me after the previous post to train a neural network. I then used that neural network to grade all possible rubric scores and generate achievement levels of 1, 2, 3, or 4.

Scroll down to the image to see the results.

Now to the meat of the matter.

Rubric design is not easy. It takes quite a bit of careful thought to decide on descriptors, point values and much of the time we don't have a team of experts on the payroll to do this for us.

On the other hand, we're asked to make judgements on students all the time. These judgements are difficult and subjective at times. Mathematical tools like averages help reduce the workload, but they do this at the expense of reducing the information available.

The data you all gave me was the result of educational judgment, and that judgement comes from what you prioritize. In the final step of my Desmos activity, I asked what you typically use to relate a rubric score to a numerical grade. Here are some of the responses.

From @aknauft:

I need to see a consistent pattern of top rubric scores before I assign the top numerical grade. Similarly, if the student does *not* have a consistent set of low rubric scores, I will *not* give them the low numerical grade.
Here specifically, I was looking for:
3 scores of 1 --> skill level 1
2 scores of 2 or 1 score of 3 --> skill level 2 or more
2 scores of 3 --> skill level 3 or more
3 scores of 3 --> skill level 4

From Will:

Sum 'points'
3 or 4 points= 1
5 or 6 points = 2
7 points= 3
8 or 9 points = 4

From Clara:

1 is 60-70
2 is 70-80
3 is 80-90
4 is 90-100
However, 4 is not achievable based on your image.
Also to finely split each point into 10 gradients feels too subjective.
Equivalency to 100 (proportion) would leave everyone except those scoring 3 on the 4 or scale, failing.

Participant Paul also shared some helpful percentages that directly relate the 1 - 4 scale to percentages, perhaps off of his school's grading policy. I'd love to know more. Dennis (on the previous post) commented that multi-component analysis should be done to set the relative weights of the different categories. I agree with his point that this is important and that it can easily be done in a spreadsheet. The difficulty is setting the weights.

The experience of assigning grades using percentages is a time saver, and is easy because of its historical use. Generating the scales as the contributors above did is helpful for relating how a student did on a task to their level. My suggestion is that the percentages we use for achievement levels should be an output of the rubric design process, not an input. In other words, we've got it all backwards.

I used the data you all gave me and fed it into a neural network. This is a way of teaching a computer to make decisions based on a set of example data. I wanted the network to understand how you all thought a particular set of rubric scores would relate to achievement level, and then see how the network would then score a different set of rubric scores.

Based solely on the six example grades I asked you to give, here are the achievement levels the neural network spit out:

ml-rubric-output

I was impressed with how the network scored with the twenty one (out of 27 possible permutations) that you didn't score. It might not be perfect, and you might not agree with every one. The amazing part of this process, however, is that any results you disagree with could be tagged with the score you prefer, and then the network could retrain on that additional training data. You (or a department of teachers) could go through this process and train your own rubric fairly quickly.

I was also curious about the sums of the scores that led to a given achievement level. This is after all what we usually do with these rubrics and record in the grade book. I graphed the rounded results in Desmos. Achievement level is on the vertical axis, and sum is on the horizontal.

One thing that struck me is the fuzziness around certain sum values. A score of 6, for example, leads to a 1, 2, or a 3. I thought there might be some clear sum values that might serve as good thresholds for the different levels, but this isn't the case. This means that simply taking the percentage of points earned and scaling into the ten point ranges for A, B, C, and D removes some important information about what a student actually did on the rubric.

A better way to translate these rubric scores might be to simply give numerical grades that indicate the levels, and communicate the levels that way as part of the score in the grade book. "A score of 75 indicates the student was a level 2."

Where do we go from here? I'm not sure. I'm not advocating that a computer do our grading for us. Along the lines of many of my posts here, I think the computer can help alleviate some of the busy work and increase our efficiency. We're the ones saying what's important. I did another data set where I went through the same process, but acted like the third category was less important than the other two. Here's the result of using that modified training data:

ml-rubric-output-modified

It's interesting how this changed the results, but I haven't dug into them very deeply.

I just know that something needs to change. I had students come to me after final exam grades were put in last week (which, by the way, were raw percentage grades) and being confused by what their grades meant. The floor for failing grades is a 50, and some students interpreted this to mean that they started with a 50, and then any additional points they earned were added on to that grade. I use the 50 as a floor, meaning that a 30% raw score is listed as a 50% in the final exam grade. We need to improve our communication, and there's a lot of work to do if the scale isn't going away.

I'm interested in the idea of a page that would let you train any rubric of any size through a series of clicks. What thoughts do you have at the end of this exploration?


Technical Details:

I used the Javascript implementation of a neural network here to do the training. The visualizations were all made using the Raphael JS library.