## What Does the Desmos for Probability Look Like?

Desmos, the online graphing calculator, activity builder, and general favorite of the MTBoS does phenomenal work.

I found myself wondering over the past few days about the statistics and probability world and how there isn't a Desmos parallel in that realm for easy experimentation. You can put an algebraic expression in Desmos and graph it in a few keystrokes. You can solve a problem algebraically, and then graph to see if you are correct. There are multiple ways to confirm graphically or numerically what you have solved algebraically, or any other permutation of the three.

It's also easy to put a bunch of marbles in a jar, pick one, replace, and repeat, though this becomes tedious beyond a few trials. Such a small data set often isn't enough to really see long term patterns, particularly in cases where you are trying to test whether the theoretical probability you have calculated is correct or not. For subtle cases that involve replacement versus no replacement, the differences between the theoretical probabilities of events are small if there are enough marbles in the jar.

Creating a simulation and running it half a million times is possible in a spreadsheet or a number of computer languages, but the barrier to entry there is not trivial. I've written simulations myself of various problems and usually make predictions for what I think is going to happen. I then will usually work to find the theoretical probability by hand.

So what would this sort of probability playground look like? There are some examples out there already. Here's one from CPM for small numbers of trials. I haven't done an exhaustive search, but I haven't seen anything that truly allows full experimentation at the level I'm hoping to achieve. Here are some ideas for what I would love to see exist:

• Natural language definitions for sources of possible outcomes. By this, I mean being able to define outcomes verbally. This might mean "rain" and "no rain", with the assumption that having only two labels means these events are complementary. This might mean we define numbers of items for each possible outcome, or simply enter the probability of each as a decimal. The key thing is that I do not want to require labeling events as A or B, and throwing notation around. Let's see if we can make this as visual and easy to explore
• Ease of setting up conditional outcomes for compound events.. If event A (I know, I'm breaking the previous rule here) happens, only B and C are possible, and event D is only possible if event A does not occur.
• Sinks that easily allow for large numbers of trials. I might want to have a single trial be generated a million times - tell me the proportion of all of the different outcomes. Make it easy for me to count up instances of binomial probability and see how many times, out of ten, I get three or more successes. Tell me when I'm not looking at all of the possibilities. For example, give me some visual indication that when I'm picking two marbles from a jar, that if I only have both red or both blue in my possible outcomes, I'm missing outcomes in which there is one of each.
• Make it easy to tap into existing complex data sets for exploration purposes. Include some data sets that are timely and relevant. The US election comes to mind.

I realize also that this is a tall order, but I've seen how far the Desmos team has explored the algebraic/numerical space. Now that they have expanded into the Geometry space through their beta, I wonder if they (or someone else for that matter) has something like this probability exploration tool on their roadmap.

## Building Arguments with Probability and the Clips App

I don't like projects for assessment. I do like in class projects for the purposes of fostering discussion and other forms of interactions. I decided to put together something fun to build time into the unit while students developed their skills in applying binomial probability. From student feedback, they actually said it was fun, so this wasn't just hopeful thinking (this time). This also had the added value of giving students a change to work on Common Core mathematical practice standard 3: Construct viable arguments and critique the reasoning of others.

I gave pairs of groups of students a statement. The center paragraph was the same for both - a statement about probabilities. The paragraphs preceding and following that were different - conflicting contexts for each statement. Here's an example.

I ended up writing four sets of situations to make sure that each class had at least two groups working on the same probability statement, but different arguments.

I asked students to do calculations and write a 100 word abstract stating their argument. After learning that the Clips app, recently released by Apple, made for a really easy way for students creatively describe and document their thinking, I also asked students to create a two minute video documenting the situation and their argument. You can see a selection of the video results below.

Students were really challenged to search for the calculations and results that supported their arguments. Some reported that they felt dishonest doing so.

You can check out all four sets of scenarios and the rubric I used here. The students said that working in teams and working through this task was enjoyable and actually reinforced their understanding of how to use binomial probability. As with a previous unit, this project was graded for completion, not for a grade, a fact I stated up front. So far, the students haven't actually said this was a problem for them, and the quality of what they produced didn't seem to suffer much.

### "I'm just going to teach it again from the basics."

This approach makes some assumptions:

• Students that didn't understand the topic on a first exposure will benefit from just seeing the topic be developed again.
• Students that did understand the first time will get confirmation of what they remember.
• Colleagues that taught this in the past didn't necessarily cover everything, so this ensures students see a complete presentation of the topic.

All of these are assumptions that serve a teacher-centered classroom model. No teacher wants to be to blame when a student forgets an essential component of knowledge for a given topic, I get that. I have a hard time seeing the presentation of a complete topic as anything other than a checklist of items for a teacher to present.

What does a student do in this context? Why does the student that remembers everything have to sit through tasks that they demonstrably know how to complete? Why would we expect a student that struggled after a first exposure to benefit from seeing the same sequence of topics, but made "harder" by some arbitrary measure associated with course or grade?

I prefer the idea that we instead present students with a task that demands the knowledge and skills that are outcomes of the course. Tasks like open middle and 3-Act problems let us see where students are in the continuum of knowledge and problem solving. There are plenty of resources we can use to fill in the gaps for students where they exist - this is where online resources and activities shine. As teachers, we truly add value when we can build intellectual need for what we teach and foster discussion about interesting challenges and thought process. Most importantly, we can provide feedback that is focused and personal.

If I had to identify one fundamental change to my teaching philosophy over the last several years, it would be the acknowledgement that students are not blank slates. Assuming they are doesn't serve any of them well. Teaching compliance and patience to the strongest students is a pretty low level goal. Teaching what we say are the basics to those that never understood the basics in the first place disrespects these students as well.

Let's stop assuming we need to give our own overview of a topic. We aren't as good at it as we think we are. This only reinforces the idea that students are hungry and waiting for us to give them the knowledge they can't obtain any other way.

We must aim much higher than that.

## Scaling in Education

From today's New York Times article, The Broken Promises of Choice in New York City Schools.

"Ultimately, there just are not enough good schools to go around. And so it is a system in which some children win and others lose because of factors beyond their control — like where they live and how much money their families have."

The structures of education do not scale well. This is because good lessons, good classrooms, and good schools are all sourced from people, and people do not scale well. People cannot be copied. The human mind is exceedingly, beautifully complex - a fact that underlies the wonderful challenge of teaching. The talents, ideas, and experience of people that understand this reality are essential to making a school what it can be.

The work that must be done centers on building a culture that acknowledges and values the human basis of our profession. It takes energy and time from human beings to turn an empty room into a learning space. Budgeting for all of the costs of the inputs, financial or otherwise, is necessary to do this work.

Ideas scale easily because it costs virtually nothing to share them. Cultivating the relationships that are necessary to use those ideas to make opportunities for children needs to be our focus.

People matter. We should be skeptical of anyone that seeks to minimize this reality.

## Probability, Spreadsheets, and the Citizen Database

I've grown tired of the standard probability questions involving numbers of red, blue, and green marbles. Decks of cards are culturally biased and require a lot of background information to get in the game, as I wrote about a while ago. It seems that if there's any place where computational thinking should come into play, it's with probability and statistics. There are lots of open data sets out there, but few of them are (1) easy to parse for what a student might be looking for and (2) are in a form that allows students to easily make queries.

If you know of some that you've used successfully with classes, by all means let me know.

A couple of years ago, I built a web programming exercise to use to teach students about database queries. Spreadsheets are a lot more accessible though, so I re-wrote it to generate a giant spreadsheet of data for my Precalculus students to dig into as part of a unit on counting principle, probability, and statistics. I call it the Citizen Database, and you can access it here.

I wanted a set of data that could prompt all sorts of questions that could only be answered easily with a spreadsheet counting command. The citizens in the database can be described as follows:

• Each citizen belongs to one of twelve districts, numbered 1 - 12.
• Citizens are male or female.
• Citizens have their ages recorded in the database. Citizens 18 and below are considered minors. Citizens older than 18 and younger than 70 are adults. All citizens aged 70 and above are called seniors.
• Citizens each prefer one of the two sports teams: the Crusaders or the Orbiters.
• If a citizen is above the age of 18, they can vote for Mayor. There are two families that always run for mayor: the Crenshaw family and the Trymenaark family.
• Each citizen lives in either a home, apartment, villa, or mansion.
• A citizen above the age of 18 also uses some type of vehicle for transportation. They may rent a car, own a car, have a limousine, or take a helicopter.

I wrote another document showing how to do queries on a spreadsheet of data using some commands here. My students asked for some more help on creating queries using the COUNTIFS command on Google Sheets, so I also created the video below.

The fun thing has been seeing students acknowledge the fact that answering these questions would be a really poor use of the human brain, particularly given how quickly the computer comes up with an answer. One student went so far as to call this side-trip into spreadsheet usage "really actually useful", a comment which I decided only to appreciate.

Programming in Javascript, Python, Swift, whatever is great, but it takes a while to get to the point where you can do something that is actually impressive. Spreadsheets are an easy way in to computational thinking, and they are already installed on most student (and teacher) computers. We should be using them more frequently than we probably are in our practice.

If you are interested in how I generated the database, you can check out the code here at CodePen:

See the Pen CitizenDatabaseCreator by Evan Weinberg (@emwdx) on CodePen.