What a week: Ending classes, field trips to Funtown and Boston, and Flex Friday celebration. That last thing was today.

Flex Friday is this thing we do at Baxter where kids work on long-term projects. They have to submit a proposal that goes through this pretty rigorous approval process. But then they get to work on their project. All year. Every Friday.

The concept is based on Google’s 20% time idea, where Google employees get to spend 20% of their time working on their own projects. Actual or perceived at Google makes no difference at Baxter. We have it. And at the end of the year, they get to show us all what they’ve accomplished with their 20%.

So, today, our students presented and exhibited their results. They shared what they have built or created. They talked about their struggles and successes. They focused on what they had learned (time management, organization, and communication being the big three). And then when I get home, this note from our principal is in my inbox:

Dear Baxter students,
I am so proud of you. Days like today reveal just how much you have accomplished this year. You are so passionate and articulate with adults and with each other. You are professionals, in dress and demeanor. You are honest when you share your successes and when you talk about your failures, and all of you seem to know that both are part of learning. I love it when you are innovative and your projects this year sought to reach the boundaries of what you know, what any of us know: what we can do with a mechanism or ingredient or measuring device; what we can make with paint or pencil or editing program; what we can build in the lab, on the street, or in cyberspace; what we can grow; how to bring compassion to a community of innovators. You are curious and creative and you are already changing the shape of the world around you. Thank you for changing mine.


And thank you, Michele, for making Baxter Academy the place where I want to be.

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A Visit from the Charter Commission

Baxter Academy is a charter school in Maine. We are independent, meaning that we are not “owned” by some education company and we are not “aligned” to any particular program that already exists. We create our curriculum, our courses, and our materials. We answer to our students & their parents, to each other, to our Board of Directors, and to the Charter Commission.

Each year, at the end of the year, we get a visit from the Charter Commission. Well, it’s happened for the past two years – I imagine that they intend to continue this practice. The visit is kind of like a cross between an IRS audit and a NEASC accreditation visit. This year the Commission decided to take about 45 minutes in the middle of the day to split up and have conversations with different groups: students, teachers, parents, and community members. At other times they met with administration and the Board. At the end of the day, they gave some feedback to our Head of School:

They reported excited students and satisfied parents. They shared with us, at the end of the day, that teachers reported a strong sense of community, that Flex Friday is great, that special ed students are thriving, but that we need more time to communicate with parents around proficiency based diplomas. Teachers also liked working in start up mode, even though it is hard. Students liked best: the community, being able to connect inside school learning with outside the school, their control of their own learning, choosing their own curriculum, that teachers are guides, that they can challenge teachers respectfully and get answers. They report very little bullying and a culture of kids sticking up for one another.  If they could change one thing it would be to have more space, to add sports, more balanced m/f ratio, more clarity around grading.

“You can tell after even 2 minutes in the building that this is a great school environment.”

Clearly there is more work to do, but getting good feedback at the end of the year makes the rest of the hard work all worth it.

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Why Standardize Normal Distributions

The new trimester started on Monday and I’m teaching a class called “Designing Experiments and Studies.” It’s a statistics class, so we’re starting with a bit about normal distributions. Most of the students in the class are juniors, but they’ve had very little instruction in statistics. They didn’t get it from me last year, so any knowledge that they might have is probably from middle school.

Today, I posed this question:

baby weights

And then I gave them some time to work it out. Here’s what happened in the class discussions (a bit condensed – the actual discussions took about 15 minutes in each class):

S1: The boy would weigh more compared to other boys because the boy is 0.25 pounds away from being one standard deviation above the mean, while the girl is 0.5 pounds away from being one standard deviation above the mean. Since the boy is closer to being one standard deviation above the mean, the boy weighs more, compared to other boys.

S2: But, 0.25 lbs for the boys is not really comparable to 0.5 lbs for girls because the standard deviations are different. I agree that the boy weighs more, but it’s because the boy is about 92% of the way to being one standard deviation above the mean, while the girl is only 75% of the way to being one standard deviation above the mean.

S1: What does that matter?

S3: It’s like if you’re getting close to leveling up (I know this sounds really geeky), but if you’re 10 points away from leveling up on a 1000 point scale, you’re a lot closer than if you’re 10 points away from leveling up on a 15 point scale. Even though you’re still ten points away, you’re a lot closer on that 1000 point scale.

S4: But you’re comparing boys to boys and girls to girls. You’re not comparing boys to girls.

S2: Yes, you actually do have to compare boys to girls, in the end, to know who weighs more for their own group.

Me: How did you figure out that the boy was 92% of the way to being one standard deviation above?

S2: Well, the boy is 2.75 lbs more than the mean weight and 2.75 / 3.0 is about .92. I did the same thing with the girl and got 75%.

At this point I showed them a table of z-scores, kind of like this one and we talked about percentiles. Looking at the table, they determined that the boy was at about the 82nd percentile, while the girl was at about the 77th percentile. Therefore, the boy weighed more, compared to other boys, than the girl weighed, compared to other girls.

I have two sections of this class, and this recreation of the conversation happened in both classes. I’m so happy when my students make sense of mathematics and reason through problems. I never had to tell them the formula to figure out a z-score, or why that might be useful or necessary. They came up with it.


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Order of Operations and Facebook

I received this text from my brother: Pam, who created the “Order of Operations” rule? Well, that’s a curious question. Why is he asking? I’d never thought about who “created” these rules. They just kind of made sense to me. Before I did any research, I thought for myself why these rules made sense to me. Here’s my response to my brother.

Suppose I needed to calculate 4 + 3 * 5. Without the order of operations, I would just make each calculation as I come to it. In this case 4 + 3 = 7 and 7 * 5 = 35. But, 4 + 3 * 5 is equivalent to 3 * 5 + 4, right? Because of the commutative property of addition, the order that we add numbers in doesn’t matter – we get the same result either way. So if that’s true that 4 + 3 * 5 = 3 * 5 + 4, then both calculations should give us the same result. But, if we don’t have conventions around this then the second expression, 3 * 5 + 4, would result in 19. Clearly that’s not equivalent to 35.

So why would we choose to multiply first instead of adding? We can think of the expression above (4 + 3 * 5) to mean that I am adding 4 onto 3 bunches of 5. I guess I think that it’s kind of implied that I would want to know what 3 bunches of 5 are before I add 4 onto that number. That’s why we would multiply before we would add.

There is a similar argument to be made with division. Suppose I come across 3 + 1/2. Without our order of operations, we might conclude that this was equivalent to 4/2 (or 2). But isn’t 3 + 1/2 the same as 3 and a half, or 7/2 (which is clearly not the same thing as 2)?

The PEMDAS mnemonic is a little misleading, too. It suggests that multiplication takes precedence over division and that addition takes precedence over subtraction. That’s not true. Multiplication and division are at the same level as are addition and subtraction. For example, if you want to calculate 6 * 3 / 2, you can first calculate 6 / 2 and then multiply that result by 3. Or, you can calculate 6 * 3 and then divide that result by 2. Either way, you end up with a result of 9. You can reason similarly with addition and subtraction.

Personally, I think that the real question is “Who invented parentheses?” I mean, to show grouping by using symbols is just genius. That would change the outcome of the original expression, right? (4 + 3) * 5 is very different from 4 + (3 * 5). At some point, mathematicians agreed that they didn’t need to write the parentheses around 3 * 5 (maybe for the reason that I stated above, maybe not), but that they would need to explicitly group (4 + 3) if that’s what they meant.

My Google search found this response to your question from Ask Dr Math.

Turns out there’s this Facebook post that asks you to calculate 6+1*0+2/2. Some people say the result is 7 and others say it is 1. Which is correct? That’s where the order of operations comes in. And that’s what was behind my brother’s question.


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A few of my favorite things

Following the lead of the good folks over at One Good Thing, I’m sharing a couple of fun tidbits from my life at school this week.

Today a student came to ask for help with a trig problem he had. He’s part of a team building a 3D printer and stand and he needed to figure out how long to make a brace of some kind. (I don’t really know exactly what his team is working on, but that’s the gist.) He had everything set up properly, but it wasn’t making any sense to him. Turns out, his calculator was set to radian mode. Yay that he recognized that something was wrong. Yay that he asked for help. Yay that he knew what he was doing.

One of our math classes this term is called Euclidean Geometry & Introduction to Logic. The teacher (not me) has been focusing on precise communication of reasoning. The other day I observed a student in my advisory ask for some peer feedback on a proof. The first student he asked had been out sick for two days, so he very kindly declined. Then next student, also a member of my advisory, gave very solid and constructive feedback about how the proof could be improved. I love it when they talk math with each other.

My Introduction to Stats class was dealing with correlation vs causation this week. They were presented with these two variables: time in seconds spent draining a full bathtub, water depth in cm, and asked to identify the explanatory and response variables. Some students saw the draining time as the explanatory variable and others saw the size of the bathtub as the explanatory variable. The debate that ensued was engaging, animated, and enlightening. Plus, I was able to unleash the voice of a 9th grade girl who has been afraid to speak about math before that moment. Another student commented on her way out of class, “I’ve never had such an argument about bathtubs before!” I love it when we can respectfully disagree and have interesting conversations.

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Baxter Academy Shadow Day

Yesterday was a “Shadow Day” at Baxter Academy. That means that most of our students were off on a job shadow of their choosing. I’m anxious to hear about the shadows that they were able to arrange during the snowiest week of the winter, so far. I would have checked in today, but we have another snow day – the third this week.

Anyway, while our students were off doing their shadows, we had about 120 prospective students, interested in attending Baxter Academy next year, join us for a “simulated day.” The students were placed into 16 different groups, each led by a couple of current Baxter students through a day of classes that included a math class or two, a science class or two, humanities, and an elective or two.

I co-taught our modeling class with one of our science teachers. This is the introductory math & science class at Baxter. It’s technically two sections, but they are integrated and teamed up so that the two teachers are working with the same groups of students. Sometimes we meet separately, as a math class and a science class, and sometimes we meet together. I’ve written about the class before, and the kinds of modeling we have made them do.

But what do you do with a bunch of 8th graders who are are with you for only an hour? Introduce them to problem solving through with this TED talk by Randall Munroe. And then take a page from Dan Meyer’s Three Act problems – a page from your own back yard: Neptune*. A brief launch of the problem and off they went. Not every group was able to answer both parts of the question: How big is the Earth model and where is it located? But most groups were able to come up with a solution to at least one part.

The point of the day was to provide a realistic experience of what it’s like to be a Baxter student. We grouped them together with others they didn’t know before walking into the building. We asked them to collaborate to solve a problem they’d never seen before. We asked them to do math without giving them directions for a specific procedure to follow. We asked them to share their results in front of strangers. We gave them an authentic Baxter experience.

*For more information about the Maine Solar System Model, visit their website. It’s really a rather amazing trip along this remote section of US Route 1. I’ve done it – I’ve driven through the solar system.

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The Frozen Code – A Classroom Example

I could have kicked myself when I saw Dan Meyer’s post about Gelato Fiasco‘s Frozen Code. I mean, they are literally just blocks away from my school. I didn’t use Dan’s map to launch the problem (Gelato Fisaco only has two actual stores), but just dove into the main idea that he posed. Here’s the prompt.

I have a bunch of students, mostly 9th graders, in a class called Functions for Modeling. So today I gave them the challenge of finding a function rule for the Frozen Code. We started by assuming that the gelato that they purchase would cost $5.00. One class came up with three different rules:

IMG_1074The blue group originally had just the middle rule: P(T) = 0.05T + 3.40. Students in both classes were eager to point out that the rule only applies if the temperature drops below 32 degrees Fahrenheit, and they wanted to somehow make that clear in the function rule. So we started to add to the definition. Then someone pointed out that there would be a bottom limit to the discount, too. After all, Gelato Fiasco might be willing to give you gelato for free if you are that willing to venture out into the extreme cold and they are still open, but they are unlikely to pay you to come in. So we added the third bit about temperatures below -68 degrees Fahrenheit.

What I love about the above work is that before we got into all the piecewise stuff, I was able to ask them, “How do we know if these three function rules are equivalent?” They told me that they “generated the same results,” that they could “use algebra to change from one to another,” and that they “would all produce the same linear graph.” How cool is that? We were also able to discuss how one form easily told us the price (of a $5 gelato) when the temperature was 0 degree F and another form showed us all the calculations clearly.

But they weren’t satisfied. After all, every gelato purchase isn’t going to cost $5 – price is also a variable. So that’s how we got into functions that have two independent variables. I asked them to modify their rules to reflect this new information. Here’s what they did:


Once again I asked if these were equivalent and how they knew. We wondered what the graph would look like. Would it be flat or curvy?

We’ll check that out tomorrow.

[Update: Here are the graphs comparing the set price of $5 to the variable price model.]

gelato models

By the way, my students think this is a pretty good marketing strategy – you are fairly likely to get some kind of discount for buying gelato in the dead of winter, but not that big a discount, on average. Last Thursday would have been a great day, though. The temperature was -10 after the sun set.

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