## Monday, December 11, 2017

### PBL in Algebra Day 4

Here is a link to my previous posts about using QFT to start the project as well as links to my posts about the first two days of the unit.

### Day 4: Writing a Draft Claim

This was our 4th day of the statistics unit and our project. At this point in the unit, students had many opportunities to interact with the data, so I thought it was a good time for students to start making the claim for their project.

I had two big ah-ha moments as I was planning for this lesson.

First, since student would need to make a claim from the data we collected, I realized they needed to understand how to represent variables. Mainly, when do we use a scatter plot as opposed to a box plot or histogram (or pie chart, for that matter). Thus, I added a learning goal that I have not taught before: students will be able to identify variables as quantitative or categorical.

This went WAY better than I anticipated. We talked about several examples as a class, then I gave students a list of some of the variables from the survey and had them identify the variables as qualitative or categorical. This was measurable #5 and was assessed individually. Students could revise work that had a low grades, though very students needed to revise.

This helped a lot over the next couple lessons when students were looking for statistical evidence to support their claim. As students asked for a particular graph, I could ask them if they were representing a quantitative or categorical variable. And this helped us decide which type of graph wen needed (and we had rich conversations).

The second ah-ha moment happened the Friday before this lesson. My student teacher and I were talking about the project and our plans when I realized that we hadn't done the project ourselves! I decided that we should go through the project 'as students' to better inform our next steps and planning. Our project was an Instagram story--thanks Diarra Gueye for putting together the project (I don't have Instagram and wouldn't know how to make a story on there!).

I'm super glad we did the project 'as students' because I realized that we made our claim (homework is unfair) AFTER looking at the data. Hence, when I planned the Day 4 lesson, I included time for students to look through the data to help them think about the claim they would be making.  I also made a bunch of graphs (scatter plots, histograms, box plots and pie charts) from the data so they could get an idea of what was available.  Engaging in the project as students helped me see where they would get stuck and it also helped me think about what I need to do to support them through the tough parts of the project.

I'm also glad I structured the lessons so that Monday's goal was a draft claim. Monday afternoon, I looked over the draft claims and gave students feedback so that on Tuesday they could finalize their claim and start to think about evidence.

Here are a few examples of claims groups made. The biggest errors were: posing questions not making a claim and correlation and causation. Overall, however, I think they made a good start on their claim.

As has been the trend with this project, engagement was super. Students had rich discussions, asked tough questions and called me or the student teacher over when they needed information to move forward. The challenging part, as has been the trend, was doing 8 different things with 8 different groups.

I realize the power of direct instruction is that it is much easier to tell 28 students one thing and expect them to "meet me halfway". I have the intellectual authority and it is students 'job' to engage in the content as I deliver it to them.

PBL flips that script. Each student (or group) is working on something authentic, meaningful and important to them; however, that means that not every group is doing the same thing, at the same time. Thus, students call me when they have a need and I fill that need with the information I have. Within the clear parameters of my project, students have the authority to decide when they need what information to move forward. Thus giving them more of the authority and the power in the lesson. And this creates the engagement.

More to come!

## Wednesday, December 6, 2017

### PBL in Algebra Days 2 and 3

Here is a link to my previous post about our launch and QFT to start the project.

### Days 2 and 3: Histograms and Boxplots

After the project launch and QFT on Monday, we jumped into the 'content' on Tuesday. The content goal of day 2:

c  I can create a dot plot or box plot given data.
c  I can describe the shapecenterspread, and outliers of data given a histogram, dot plot, or box plot.

c  I can compare the shapecenterspread, and outliers of two sets of data given histogram, dot plot, or box plot.

I didn't expect students to master all this in one day! But this lesson introduced univariate statistics being represented using a histogram or dotplot. On Tuesday a period is 90 minutes and I planned to have about 40 minutes of content (traditional) and 40 minutes of project time.

I also put students into groups on Day 2. At the start of the lesson, I gave students a survey asking a few questions. I had planned to give the survey on Monday, then make groups after they left. However, we ran out of time on Monday for the survey. I'm fortunate to have a student teacher, so while she gave the histogram notes, I looked at the survey results and used the student responses to build groups.

The survey was 4 questions:

I used the broad categories from the QFT for the 1st and 2nd choice project questions in the survey.

After students finished the content portion of the class (notes and then partner talk and class discussion to practice describing and analyzing histograms), they were given data from one of my classes (data I pulled from the survey mentioned in this post) and asked students to make a histogram of the data and then analyze the histogram.

 example of the type of graph of made using the class data

This is a huge improvement in how I've taught this in the past. We've always done notes, but then we look at random histograms that I pull of the internet for practice. I LOVE that we were looking at the data.

Students created the histogram in their project groups. While this was part of the project work time, this work didn't directly relate to their project--though it could. And it wasn't a group grade; while students worked in their groups and could support each other, they were individually accountable for turning in this work.  The reason behind this is I want to make sure I can assess students individually for content as well as assess students as a group for the project.

What I loved is that students were so engaged during this work time. I think the engagement happened for a few reasons:
• they used their class data, not random data from the internet.
• they had choice: they chose the data they wanted to use, they could also choose where to sit with their group.
• they could choose when to turn the histogram in. It was due at the end of the block, but students could also turn it in the next class for full credit if they felt what they finished at the end of the block wasn't high quality. About 25% of the students wanted to keep it to finish and turn in the next class because they wanted to create a high quality product.
The other thing I loved about this lesson is how the open-ended structure of the group work created a need. For example, as students were working on their histograms in small groups, the inevitable problem came up of which bin to put data that falls on a border of a bin. If the bins goes from 10 to 15, where would a data point 10 go? In the previous bin or the 10 to 15 bin? I LOVED that students would call me over because there was a need and I would fulfill that need.

In previous years, I would just share that info with the class and 2/3 of the class would not hear me, etc.

### Day 3: Boxplots

Day 3 looked a lot like Day 2 except the learning goal was boxplots instead of histograms.

c  I can create a dot plot or box plot given data.
c  I can describe the shapecenterspread, and outliers of data given a histogram, dot plot, or box plot.

c  I can compare the shapecenterspread, and outliers of two sets of data given histogram, dot plot, or box plot.

We did notes on boxplots, then students described boxplots I made from our data with partners. Finally students got into groups and created a boxplot using data from our class.

 example of the type of graph of made using the class data

### Reflection

There are definitely improvements I want to make to this unit next year. I don't think what I did is perfect, though it is better than how I taught this unit in previous years. I'm looking forward to continuing this process--and I'm thinking a lot about how to incorporate more PBL into my curriculum. My only problem is that I think algebra, or at least how algebra is taught at my school, isn't a good fit for my PBL.

More to come!

## Tuesday, December 5, 2017

### PBL in Algebra Day 1

This year I've decided to teach statistics using a PBL structure.

In sheltered algebra the project focus (driving question) is on using statistics help the public come to an alternative understanding about the impact of immigration on our country. That project starts in January, though I am doing some language production work now. I have a few posts on this work. Here is the first and second post on this work.

In mainstream algebra, we are also doing our statistics unit using PBL Here the project focus (driving question) is on how can we use statistics to tell a story about...the about is open-ended. They could talk about their particular class, both my algebra classes (I teach two sections of mainstream algebra) or it could be about the 9th graders in algebra at our school.

The other algebra teachers and I created a survey which we gave to all our students a few weeks ago. We are then sharing this data with students and using the data from their survey to teach stats.

c  I can determine if a statistical relationship is best described by a linear model or nonlinear model.
c  I can use scatter plots of data to create lines of best fit (by hand or using technology).
c  I can use lines of best fit to make predictions.
c  I can describe the form, direction, strength, and outliers of an association.
c  I can interpret the slope and y-intercept in a statistical situation.
c  I can create a dot plot or box plot given data.
c  I can describe the shape, center, spread, and outliers of data given a histogram, dot plot, or box plot.

c  I can compare the shape, center, spread, and outliers of two sets of data given histogram, dot plot, or box plot.
c  I can interpret a given correlation coefficient from a scatter plot.
c  I can determine and explain when it is best to use median or mean to describe the center of data.
c  I can distinguish between correlation and causation (association is not causation) because there might be a lurking variable.

I'm on my 5th lesson. I'll share a snapshot of what we did each day for content and for the project.

### Day 1: QFT

We launched the project by looking at a starting statistic from their data. I showed students the following slide:

I used the Talking Points structure to get students thinking about these statistics.  I wish I hadn't included the max and min data on the slide. I think it got confusing for students to see the minimum and median were both 0 (at least at this point in the unit).

After students talking in small groups using the Talk Points structure, I introduced the project and the driving question (but I didn't say much else). My original driving question was "how can we use statistics to tell a story about social media, but that later morphed into "how can we use statistics to tell a story about...").

I used a QFT structure to get students thinking about what they wanted to explore in the data. To do this I gave each group a big piece of butcher paper. And gave the following instructions:

Students did this individually. Then, students changed closed ended questions to open ended questions.

Then students talked as a group about which questions they wanted to share with the class.

Then, each group shared their top 3 questions.  And, as a class, we started to group the questions into broad categories.

Each class ended up with the following categories:

One thing that amazed me about this process was the way the QFT structure gave students an opportunity to be smart in different ways--as we were trying to group the questions into categories, one student jumped up and shared her idea for grouping them--it turned out to be a great idea and we ended up using her idea. Many students were active that aren't' typically active in my class.

One thing that scared me about this lesson was the fact that I was giving up control--I was scared to give up control to the students (I'm being honest here), BUT I was pleased by the level of ownership students took for this project.

What scared me (giving students agency) is also what engaged them in the lesson. I realized that this lesson was an important moment in my teaching practice. And this lesson made me excited for the rest of this project!

More to come!

### The Stats Project Part –1

I've wanted to try PBL in math for a while now. Two things were holding me back. First, there is a lot to PBL and I wasn't ready as a teacher to tackle all that until this year (maybe...we'll see). Second, I don't think PBL is a natural fit for mathematics. It is hard to have authentic, public projects that align with some of the content I teach in algebra--solving for example.

However, last year I realized there was an opportunity for me to teach my Statistics Unit using PBL. Stats is real life! What a great opportunity to come up with an authentic, public project that aligns with the content.

In January 2018, I'm going to teach the Stats Unit via PBL for my Sheltered Algebra class. Sheltered algebra is for recent immigrants to the United States. The students have low English proficiency in reading, writing, speaking and listening. Some students speak conversational English, some speak no English. Most of the students are Latino, however, I also have students from China, Vietnam, Japan, the Philippines, and Palestine.

Thus, any blog posts about the Stats project are "pre unit". Here is a link to my first post on this project.

I expect this project to be messy. There is a lot I've thought about and planned for; but there is a lot of things that I know will catch me off guard and I cannot plan for those things. I will be sharing my experience (good, bad and ugly) on my blog.

### Language is Messy

One of the biggest parts of this PBL Unit on Immigration will be teaching EL students that language they need to discuss and make sense of the different representations of immigration statistics we will be looking at.

In anticipation of this, I have been showing students complex graphs in class and on homework and asking them things like "what does this graph tell you" or "what questions do you have for this graph".  Here are a few examples of the types of graphs we've looked at:

The reason I am doing this is so I can get a sense of what language they know and what language do I need to teach.

Some of the graphs have been class discussions and students just look at the graph and then discuss with a partner and then the class. Some have been on homework and I ask them to write a response.

One of the biggest things I've learned (so far) is that the graphs we usually look at in class are not the kind of graphs that are in the media. Accordingly, the language I usually teach in terms of how to describe graphs isn't useful.

For example, we talk a lot about y-intercept aka initial value in my class. And several students pointed out that the graph below does not have a y-intercept. I know that seems like a small thing, but for students that speaks little to no English, identifying the initial value from a graph is a great use of language and it is important to them as they make sense of a graph. I never taught them how to think about a graph with no initial value!

Also, very few graphs from the media are perfectly linear! So my usual sentence frame of "as the inputs increase by ___, the outputs increase/decrease by ___" isn't useful.

But this is good to know! I want to know where they struggle and what language supports I can put in place to help them.  I developed the following sentence frames to help students analyze the graphs we are currently looking at.  The big difference is how we describe linear v. non-linear graphs. This is something else we've gone over in class.

And students started using the sentence frames "the function is increasing for x values" and "the function is decreasing for x values" to describe graphs! Here is some of their work (good and bad).

### Formative Assessments

The point of all this is to say that sometimes a teacher needs to formatively assess students so the teacher can determine next steps. None of the assessing I did with these graphs was graded; that's not the purpose. I wouldn't know what language students need, and I wouldn't think through how to teach that language, if I hadn't done these activities.

Lots more to come on this project!

### My Dilemma in Grading Student PBL Work

There was this fear I had about PBL. I was afraid that the learning was not authentic because of the multiple opportunities students have to...