## The Nature of Variables for Students vs. Programmers

Dan Meyer has provoked us again with this post questioning the meaning of variables in programming compared with how they exist in the minds of our students.

I previously wrote about something I tried at the beginning of last year with my students that probed this question a bit. My contention then was that writing expressions is something that occurs with students only in math class world, and that it is an inherently non-interactive process. The spirit of what variables do is something with which students have familiarity. It’s the abstraction of the mathematical representation that pushes that familiarity away from them.

I’m going to use a different expression problem since the one in Dan’s post doesn’t do it for me.

### Dan estimates that around 3/4 of any group of people drink soda.

Students could each click on the people go through the process of figuring out how many in each group drink soda according to Dan’s estimate, and would record the number in each group. The third group serves to construct a bit of controversy for discussion purposes. In doing this four times, students are presumably going through a similar process each time.

Mathematics serves to create structure for this repetition, but on its own, is not necessarily in the realm of what our students would do to manage this repetition. Programming provides a way to bridge this gap using the same idea of variables that exists in the mathematical realm, and here is where the value sits for this discussion.

In the post I mentioned previously, I said that I briefly showed students how to type expressions into a spreadsheet and play around with inputs and outputs so that they match concrete values. In a non 1:1 laptop classroom, I might start with this:

A calculation links the outputs to the inputs in each of these tables. Students have concrete values sitting in front of them, so they will notice that each of these tables must be making the wrong calculations, even though they each have one correct value. Here, we have the computer making the same calculation each time, but these calculations do not work in each case. This is the wrong model to match our data. The computer is doing exactly what we are telling it to do, but the model is wrong.

How do we fix this, class? Obviously we use a different computational model. I might have students decide in a group what calculation I need to do to correctly reproduce the values from the exercise, and elicit those suggestions from them.

Once we establish this correct model, this calculation we are making is common to every set of data. We can show that this calculation makes an interesting prediction of 7.5 people liking soda in the group of 10. We can use this calculation to predict how many people in a group of 28 drink soda (and in a 1:1 classroom, I’d have them go through this entire programming process themselves.)

I might now generate a table hundreds of entries long and ask whether there is a better way to represent the set of all possible answers to this question. The table will work, but it is tedious. We need a better way. How do we do this? Here is where variables come in.

Programmers use variables because they want to build a program that produces a correct output for every possible input that might be used to solve a given problem or design. Mathematicians also want to have the same level of universality, and have a syntax and structure that allows for efficient communication of that universality. Computers are really good at calculating. The human brain is really good at managing the abstraction of designing those calculations. This, ultimately, is what we want students to be able to do, but they often get lost in both the design stage and the calculation stage, especially because these get divorced from the actual problem students are trying to solve.

If we can have students spend more time in the design stage and get feedback on whether their calculations are correct, that’s the sweet spot for making the jump to using mathematical variables.

## Standards Based Grading, Year Two (Year-In-Review)

This was my second year using standards based grading with my classes. I wrote last year about how my first iteration went, and made some adjustments this year.

## What did I do?

• I continued using my 1-5 standard scale and scale rubric that I developed last year. This is also described in the post above.
• As I wrote about in a previous post, I created an online reassessment organization tool that made it easier to have students sign up and organize their reassessments.
• The new requirement for students signing up for reassessments involved credits, which students earned through doing homework, seeing me for tutoring
• I included a number of projects that were assessed as project standards using the same 1-5 scale. the rubric for this scale was given to students along with the project description. Each project, like the regular course learning standards, could be resubmitted and reassessed after getting feedback and revising.

## What worked:

• My rate of reassessment was substantially better in the second semester. I tweeted out this graph of my reassessments over the course of the semester: There was a huge rush at the end of the semester to reassess – that was nothing new – but the rate was significantly more consistent throughout. The volume of reassessments was substantially higher. There were also fewer students than in the first semester that did not take advantage of reassessment opportunities. Certain students did make up a large proportion of the total set of reassessments, but this was nowhere near as skewed a distribution as in the first semester.
• Students took advantage of the project standards to revise and resubmit their work. I gave a living proof project that required students to make a video in which they went through a geometric proof and explained the steps. Many students responded to my feedback about mathematical correctness, quality of their video, and re-recorded their video to receive a higher grade.
• Student attitude about SBG was positive at the end of the year. Students knew that they could do to improve their grade. While I did have blank questions on some unit assessments, students seemed to be more likely to try and solve questions more frequently than in the past. This is purely a qualitative observation, so take that for what it is.

## What needs work:

• Students hoarded their reassessment credits. This is part of the reason the reassessment rush was so severe at the end of the semester. Students didn’t want to use their credits until they were sure they were ready, which meant that a number were unused by the end of the year. Even by the end of the year, more than a quarter of credits that had been earned weren’t used for reassessments. <p\> I don’t know if this means I need to make them expire, or that I need to be more aggressive in pursuing students to use the credits that they earned. I’m wrestling a lot with this as I reflect this summer.
• I need to improve the system for assessing during the class period. I had students sign up for reassessments knowing that the last 15 – 20 minutes of the class period would be available for it, but not many took advantage of this. Some preferred to do this before or after school, but some students couldn’t reassess then because of transportation issues. I don’t want to unfairly advantage those who live near the school by the system.
• I need to continue to improve my workflow for selecting and assigning reassessments. There is still some inefficiency in the time between seeing what students are assessing on and selecting a set of questions. I think part of this can be improved by asking students to report their current grade for a given standard when signing up. Some students want to demonstrate basic proficiency, while others are shooting for a 4 or 5, requiring questions that are a bit higher level. I also might combine my reassessment sign up web application and the quiz application so that I’m not switching between two browser windows in the process.
• Students want to be able to sign up to meet with me to review a specific standard, not just be assessed on it. If students know specifically what they want to go over, and want some one-on-one time on it since they know that works well for them, I’m all for making that happen. This is an easy change to my current system.
• Students should be able to provide feedback to me on how things are going for them. I want to create a simple system that lets students rate their comprehension on a scale of 1 – 5 for each class period. This lets students assess me and my teaching on a similar scale to what I use to assess them, and might yield good information to help me know how to plan for the next class.

I’ve had some great conversations with colleagues about the ways that standards based grading has changed my teaching for the better. I’m looking forward to continuing to refine my model next year. The hard part is deciding exactly what refinements to make. That’s what summer reflection and conversations with other teachers is all about, so let’s keep that going, folks.

## tl;dr A Project Proposal:

I’d like to see expert ‘knowers’ in different fields each record a 2-4 minute video (uploaded to YouTube) in which they respond to one of the following prompts:

• Describe a situation in which a simple change to what you knew made something that was previously impossible, possible.
• Describe a moment when you had to unlearn what was known so that you could construct new ideas.
• What misconception in your field did you need to overcome in yourself to become successful?

I think that teachers model knowledge creation by devoting time to exploring it in their classes. I think we can show them that this process isn’t just something you do until you’ve made it – it is a way of life, especially for the most successful people in the world. I think a peek behind the curtain would be an exciting and meaningful way for students to see how the most knowledgeable in our society got that way.

## Long form:

One thing we do as teachers that makes students roll their eyes in response is this frequent follow up to a final answer: How do you know?

This is a testament to our commitment to being unsatisfied with an answer being merely right or wrong. We are intensely committed to understanding and emphasizing process as teachers because that’s where we add the most value. Process knowledge is valuable. An engineering company can release detailed manufacturing plans of a product design and know they will remain profitable because their value is often stored within the process of building the product, not the design itself. This is, as I understand it, much if the power of companies dealing in open source technologies.

In a field like ours, however, students often get a warped sense of the value of process. They don’t hear experts talking about their process of learning to be experts, which inevitably involves a lot of failure, learning, unlearning, and re-learning. In some of the most rapidly changing fields – medicine, technology, science for example – it is knowledge itself that is changing.

An important element of the IB program is the course in Theory of Knowledge (abbreviated TOK). In this course, students explore the nature of knowledge, how it represents truth, how truth may be relative, and other concepts crucial to understanding what it means to ‘know’ something to be true. From what I have heard from experienced IB educators, it can be a really satisfying course for both teachers and students. Elements of TOK are included as essential parts of all of the core courses that students take.

I can certainly find lots of specific ways to bring these concepts up in mathematics and science. Creating definitions and exploring the consequences of those definitions is fundamental to mathematics. Newton ‘knew’ that space was relative, but time was absolute. Einstein reasoned through a different set of rules that neither was absolute. These people, however, are characters in the world of science. Their processes of arriving at what they knew to be true don’t get much airtime.

What if we could get experts in fields talking about their process of knowing what they know? What if students could see these practitioners themselves describing how they struggled with unlearning what they previously believed to be absolutely true? I see only good things coming of this.

What do you think? Any takers?