math – emergent math

Five steps to plan a problem based lesson

Posted by Geoff in math

Far from a simple undertaking, incorporating more inquiry in your class is a challenging process. You and your students may have to unlearn some of the tendencies you’ve built up over the years. So I hope you don’t take this “five steps” post as a flippant, “it’s so easy” post. The opposite is true: problem based learning can require more effort than your typical lesson planning. I’ve broken down my approach here.

Step 1: Find the problem. Modify if necessary.

Easy start, huh? Just go through every problem set in every resource on the World Wide Web and find a problem. Should take no more than two to three hundred hours. Or you could follow my approach.

I start with the standard, then I go searching for problems. That search begins with my own Common Core Curriculum Maps, where I find my trustiest repositories of problems from bloggers and organizations that have made their resources freely available. I’m so grateful for these caches of math tasks.

Another place I go searching is the problem set in the associated section in the textbook (a gasped hush falls over the audience). The application section in typical textbook problems sets are actually decent places to find the kernel of quality problems. Textbook problems typically take a bit of adaptation, but you can rest assured they’re at least aligned with the intended standard.

From *Necessary Conditions: Teaching Secondary Math with Academic Safety, Quality Tasks, and Effective Facilitation*

Step 2: Plan the outcomes and assessment

How are you going to assess students? And, if the problems are group tasks, how are you going to ensure that individual students know the material and had a chance to contribute. I have to go-tos:

For the groupwork, I use a rubric; one that assesses both the collaboration of the students and the content put on paper (or poster paper).

For students’ individual content knowledge, I give a few practice problems. Depending on how much time I expect after we share out our problem solutions, I may find three or four. Or, better yet, use David’s “Choose Your Own Problems” method.

Step 3: Plan the launch

How are you going to launch the problem? I always start with some sort of problem decoding routine. I want to make sure that every student understands the problem and have a few immediate next steps the moment we turn to groupwork.

I also want to consider students who might need a few extra moments to process the problem. Sure, I could be talking about students with IEPs, but even students without designated modifications may need extra time to process. I need extra time to process, often in writing, before I’m ready to engage. Consider how you’ll structure your problem launch so all students can engage.

The strategy I used most often was to give students the problem the day before. Their “homework” was to read through the problem and identify any of the following: unknown vocabulary, important information, ideas for solving, drawing the scenario, etc. This way the day of the problem based lesson was not the first time students see the problem. They can come prepared with ideas and questions.

Step 4: Prepare the scaffolding

Many problem based lessons require some sort of scaffold during or after the problem. Sometimes the scaffold may be a whole class lecture, other times it might be a small workshop. It’s also possible your scaffolding might be in the form of “hint cards.” These would be little hints to get kids unstuck as they progress through a problem. These hints could be in the form of questions:

Step 5: Identify students and skills so as to promote academic status

One of the (if not the) benefits of problem based learning is that complex tasks afford the opportunity to demonstrate ingenuity, creativity, and camaraderie in a way that a rote, teacher-centric lesson cannot. And while you’ll certainly find opportunities in the moment, it helps to plan ahead as well: what are the types of mathematical thinking may come up during the problem? Who are the specific students in your class that you’d like to offer a confidence boost to?

Most lessons plans are very task and agenda driven. I’d encourage you to bucket some space on your lesson plan template for assigning academic status. Think about students you haven’t connected with in a while. Examine your biases. I’ve attempted to do so on the lesson plan template from Necessary Conditions. You can use this template or not, but I do suggest having a place to capture this planning.

And “voila!“

There’s certainly so much more than goes into planning a problem based lesson, but hopefully this will give you an insight into my planning process. For some of those additional things, check out my selected blog posts and problem based learning pathway.

Math Mindset and Attitudes Survey

Posted by Geoff in math

The start of the school year offers a unique time in the academic calendar to obtain some baseline data on how your students view themselves as mathematicians and the discipline of math itself. Most beginning-of-year info sheets solicit information about students’ passions and/or guardians’ phone numbers. This year, I encourage you to ask some questions about math itself. Encourage kids to be honest: What do they like about the subject? What do they dislike? How do they (or do they) view themselves as mathematicians?

The following is a sample card from the Academic Safety Card Set based on Necessary Conditions.

I encourage a mix of free-write prompts and quantitate prompts. Here is a link to a Math Mindsets and Attitudes Survey I created and had a few schools implement.

This mix of free response and Agree / Disagree / Neutral/Not Sure gave us good, actionable data in the form of word jumbles and data around students’ mathematical identity.

You can access the PDF of the survey and facilitation guide from the companion website to my book, Necessary Conditions.

If you’d like the google form of the survey, send me a message and I can make a copy and share it. I’m also happy to help you break down and make sense of the data.

Specifics before Strategies

Posted by Geoff in assessment, math

In this blog post, we’ll explore how to get specific with math or non-math classroom issues before we develop strategies. We’ll also see an example of how to build a rubric from the ground up.

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“My kids just won’t work together.”

This (or something like it) is a common complaint I hear during professional development as I encourage teachers to facilitate rich tasks via small student groups. It’s an understandable pushback and also an unhelpful one. If you’re struggling with an issue in the classroom – whether it be groupwork, late work, or anything – the first step is to Get Specific.

Let’s review the opening statement with some questions and commentary:

“My kids…” Which kids? All of them? Some of them? Nelson Muntz? How many and which kids are we talking about? Is there a commonality between these students?

“…won’t…” What do you mean by “won’t”? They don’t want to? They don’t know how? What are they doing instead of working together?

“…work together.” What does it mean to “work together?” Does it mean to check each other’s work? Discuss a problem before they move on to the next? What structures have you provided to help them work together? What prompts, aside from “work together” have you provided to help students know what it means to work together?

I provide this example to demonstrate how many of us are vague with our comments, when what we need is specificity. We can’t get better when the issue is vague and nebulous.

Sticking with the “work together” situation… this is a common issue among students (and adults). I have two children who hate groupwork more than anything in the world. It’s natural to want to work individually. Some of us are even wired that way. I have a difficult time collaborating with peers far beyond “you do this, I’ll do that.” So let’s drill down.

What are the specific things you want students to do while working together?

Do you want students to offer words of encouragement?
Do you want students to check the answer with their peers before the move to the next problem?
Do you want everyone in the group to share an idea?
Do you want students to divvy up the work?

If any of these are the case, say so and don’t just say “work together.”

(***Rubric sense starts tingling***)

In fact, let’s create a small rubric for this.

Rubric Example

Let’s take the question around checking answers with their peers before moving to the next problem.

What is the behavior your want? Well, we just answered that in the previous sentence. Let’s make that the PROFICIENT column.

What is the current state of students? Let’s say students currently are working entirely individually such that they aren’t checking each other at all. That’s no good. Let’s put that in the far left column.

Now, what would be a stretch goal for students? What would it look like if students were really, really checking in on each other? Maybe: Checks with peers and makes sure everyone understands before proceeding.

And now to fill in the gap: what’s the midpoint between “doesn’t check” and “checks”? I’ll toss in the modifier “occasionally” but I’m guessing the more seasoned rubric developers may have better ideas. “Occasionally” isn’t terribly descriptive. Maybe we should be specific with language like “once or twice.” But that’s what I got and we’re in the middle of a PD session right now.

Now we have a small rubric on Corroborates Solutions With Peers which is an aspect of collaboration (not the whole thing). We can even use it throughout the year – every time we work on a problem set as a class. We can get better at it. We can improve the rubric as well (let’s go ahead and change it to “once or twice” and “all”).

Because we were able to get specific about the behavior we’re trying to assess, we can now communicate and scaffold towards it. You’ll be able to document with some reliability how many students are at what specific level of this specific aspect of collaboration. I’ll admit: I haven’t offered any strategies in this blog post to treat the initial issue around students working together. But once we have specifics – and rubrics are a great way to get specifics – we can start addressing the problem areas and celebrate the bright spots.

I’d encourage you to check out some of New Tech Networks rubrics around Collaboration, Agency, and Communication for other (better) exemplars.

To wrap up this with a meta-comment, I’m realizing more and more that I don’t often know what the second step is, but the first is to understand.

Your Student Portfolio System Begins Now

Posted by Geoff in assessment, math

As we transition back into School Mode, I’d like to offer a brief encouragement to use this school year to establish a system of student portfolios. If you’d like a “why” around this, I’ll point you to my Shadowcon Talk from a couple years ago.

If you’d prefer not to watch a video, here are the highlights:

Student portfolios allow students to demonstrate and realize their own growth over time (ok, just watch the first minute and a half of the video, up until “Damn, I’ve grown!”)
Rich tasks provide better data about what students know and can do than standardized test scores
Rich tasks better reflect our instruction and, as any follower of this blog or my twitter feed knows……

. @emergentmath "Assessment is at its best when it is ongoing and most difficult to distinguish from the teaching that is occurring."
—
Jenn Borgioli (@DataDiva) November 18, 2013

Provided you think of it and plan a bit before the school year starts, facilitating a portfolio system is not too difficult. Here’s what you need:

Six to ten rich problem-solving tasks
A calendar
A place to store student work
A tool to assess and/or have students self-reflect
A couple hours to collect the above items

Let’s take each one by one.

Six to ten rich problem solving tasks. In other words, Portfolio Problems. There are many places to find such tasks. I’ve started by asterisking problems in my Problem-Based Curriculum Maps that I think are worthy of a student portfolio. But I’m sure there are also excellent assessment items in your textbook. Yes, that’s right, your textbook.

If it helps, consider this scoring guide for Quality Tasks card for a quick check on whether or not a task is worthy. (From Necessary Conditions.)

A calendar. Put the tasks on the calendar now. Every 4-6 weeks block off a couple of days for a Portfolio Problem. You can change them later, but if they’re on the calendar, they’ll get deployed. If they aren’t, they won’t, as other seemingly more urgent business pops up. You can also build in twenty minutes of reflection and share-out time the following class period.

A place to store student work. Your options here are a file cabinet from an Army surplus store or Google Drive. (There are dozens of other options for physical work and digital work, but these are my go-to’s).

A tool to assess and/or have students self-reflect. After each problem you and/or your pupils will need to assess their work in the moment. Ideally, a you’d use a rubric with common indicators throughout the year. New Tech Network has Math rubrics (and a plethora of others, including Collaboration, Communication, and Agency) that work nicely. But feel free to use your own.

A couple hours to collect the above items. This is why we’re doing this now. Hopefully you have a couple hours of individual or departmental planning time built in to your in-service before the year starts. The most effective thing you can do with these precious hours is identify now – months in advance – the problems you’d like to serve as Portfolio Problems. Once you have those problems identified and on the calendar, there’s no stopping you.

Here is a related table from Necessary Conditions.

Problem Based Learning and supporting English Language Learners

Posted by Geoff in math

A classroom with quality, complex problems as its cornerstones can support English Language Learners. First let’s check out a few “ground rules” about supporting English Language Learners. The following ground rules are not exhaustive, but are pulled strategically from English Language Learners and the New Standards by Margaret Heritage, Aída Walqui, and Robert Linquanti.

Use authentic and meaningful tasks to build student agency and sense of purpose, as well as background knowledge and schema
Learning needs to be social. Scaffolding should be interactive and occur through discourse.
Learners need to use language themselves in meaningful and purposeful ways.

Problem-based learning can go hand in hand with these principles, provided you establish strategies to promote them. The use of authentic, quality tasks paired with discourse can yield significant gains in language acquisition. However, you must be conscious and strategic. Here are five strategies to help that process.

Use word walls. Post challenging vocabulary on your classroom walls to offer visual reminders of content terms. Be sure to provide not only the word itself, but pictures and examples when possible.

Use Sentence stems. Provide students sentence frames and sentence stems. For example,

Be judicious about sentence stems as time goes along. You don’t want to stifle student creativity and their ability to flex their English going forward. Still, sentence stems can be an effective way to bridge that gap.

Double Entry Journals. Double entry journals are a strategy of writing to learn. Student divide their notes with a vertical line. On the left is the source material. It may be a problem, a vocabulary work, or a strategy. On the right they can write their interpretation or process or even additional questions.

Use Graphic Organizers. Thankfully in math we have a wealth of graphic organizers at our disposal. Graphic organizers can include concept maps, Venn diagrams, bar charts, etc. Be sure to take your time with them. If you’re placing items in a particular part of a Venn diagram, encourage a discussion and come to a consensus as a class.

Amplify, don’t simplify. Don’t shy away from using sophisticated vocabulary. English Language Learners need multiple and varied exposures to rich, and even technical vocabulary. Offer the challenging vocabulary word and support it by providing synonyms, definitions, and examples.

This is only a small bank of strategies to support pupils. What are some of your most effective strategies to support students for whom English is not their primary language?

(Note: While I use the term English Language Learners (ELLs) in this blog post, I recognize that there are other – probably better – terms for students for whom English is not their first language. By putting “English” first in the term and implying English acquisition, the term promotes anglo-supremacy. It is the accepted term from the U.S. Department of Education. Other similar but distinct terms include Emergent Bilinguals (EBs) and First Language Not English (FLNE) students.)

What Khan Academy Gets Really, Really Right

Posted by Geoff in math

There is no shortage of criticism of Khan Academy around these parts. In fact, Khan Academy criticism was among the first unifying themes of the math blogosphere. Since then, however, KA has made their platform more robust and useful. And those of us who swore it off might want to take another look.

I’m taking another look because my son is using it – not because his school is assigning it, but because he wants to learn the material. This in and of itself was perplexing to me, so I decided to figure out why he was so invested that he’d want to use KA as his limited screen time for the weekend.

Part of the hook is that it comes through a screen. For whatever reason, he’s more apt to do work when it’s on a computer. He’ll struggle to write an essay on physical paper, but struggle to keep his word count low in Google Docs. He doesn’t love to draw with a pencil, but will spend hours creating an interactive image in Scratch. Or even in the KA computer science section.

So that’s one hook. There are also probably lessons to take away around customization and visualization. For example, here is my son’s “bio.”

But there are two crucial things that make KA engaging and even pedagogically sound (or at least, can push us to be more pedagogically sound).

It lowers the pain threshold for incorrect answers. By allowing instant retakes on problem sets, there’s less of an admonishment for incorrect answers. Students who are troubled by a “7/10” on the top of a paper may be more prone to give the assignment another shot if they know that score will essentially disappear.
It offers immediate and actionable feedback. There’s no feedback more timely than immediate. Even the hardest working teachers return tests and quizzes the following day, if not several days later. And with the immediate feedback comes suggested instructional how-to videos.

Zaretta Hammond in Culturally Responsive Teaching and the Brain offers a few characteristics of quality feedback:

[Quality feedback] is specific and in the right dose… It is timely… It is delivered in a low stress, supportive environment.
Hammond (2015) p. 103

Hammond also cites research suggesting that students of color in particular often do not receive timely, quality feedback from teachers. I’d recommend checking out the book and especially this section about the types of feedback students receive and what type of feedback they should receive. While KA is not the solution to problems around timely feedback, it is instructive.

There remain countless valid criticisms of Khan Academy. The problems are rote and allow for little in the way of problem solving. It’s anti-social. Left in the hands of a less discerning teacher, it can reinforce negative attitudes about math and about one’s self as a mathematician. Even the timely feedback mechanism has pitfalls: a student can retake tests easily and quickly, which allows them to bypass reflection about the content and why the answers are incorrect. If KA is a cornerstone of a classroom, I’d have concerns.

But there are elements that we can learn from to make our own instruction better, particularly in the realm of supportive, timely, actionable and low-stress feedback. And for pet drawings

What teacher training and PD can learn from Trader Joe’s

Posted by Geoff in math

When we got a Trader Joe’s in our humble little burg of Fort Collins there was much rejoicing. Now we have a place to get all sorts of goodies, which I’ll describe in more detail in a moment. Sadly, due to some byzantine Colorado laws they cannot carry Two-Buck Chuck. Nevertheless, I hit up TJ’s once or twice a week.

Trader Joes are pretty funky. There are a few distinguishing characteristics. They are small grocery stores with seemingly bizarre, diagonal aisles. It’s actually quite an inefficient layout when compared to a big box grocery (which, as we’ll learn, is part of the success). Most of the products in a Trader Joes are branded as a Trader Joes product.

I came across this article about what makes Trader Joe’s so successful. And boy is it successful (it sells twice as much as Whole Foods on a per square-foot basis). Naturally, my mind turns to professional development and teacher training. Naturally.

What are some lessons we can take from the success of Trader Joes to make our professional development more effective

Simplify the supply chain, get rid of decision paralysis.

Whereas in a typical grocery store you’ll find over a hundred pasta sauces, Trader Joes has less than ten. In the past when I’ve wanted to show how much awesome math material is out there, I just blasted out dozens of dozens of websites. Attendees are left not knowing exactly where to go because there are too many places to go. It would have been more effective if I’d just given participants three to explore.

2. Open it up, explore leisurely without that blast of cold air.

The frozen section in most grocery stores consist of a few aisles of glass doors, keeping the food frigid. At TJs, the frozen food is right there out in the open in super cold bins. A shopper will walk by and pick up and hold the food and wind up buying it. By letting customers interact with the food without barriers, they’re more likely to buy it. Professional development is so much more effective when we can get our hands on it immediately. Let participants play around with the concepts quickly (for instance, via an engrossing math task).

3. Create an element of discovery.

The reason most TJ branded products are hand-drawn is because it gives customers a folksy sense of discovery. Like, they’re the one who discovered this bag of russet potatoes.

How can we incorporate more discovery into professional development teacher training? Spend some time ideating with one another. Ensure participants are quick to share tasks and strategies that have worked well for them in the past. And for goodness sake, let attendees explore those (three) websites. Don’t just throw them a link and expect them to check it later.

And my personal favorite reason to shop at Trader Joes:

4. Offer a mix of pre-prepared items and fresh produce

If I want to bring the groceries home and have something ready in five minutes, I can grab a bag of fried rice and go with that. Or I could snag some produce and meat if I wanted to cook something. Most of the time, I do a bit of both: something quick for tonight and something that requires a bit more effort for tomorrow. While most grocery stores have both, TJs’ pre-prepared foods are cheap and good.

When I conclude professional development, I want teachers to have something quick-n-easy to implement tomorrow while also knowing where to turn for more intentional concepts. In a math PD, I may introduce number talks (interacting with them) so they’ll be able to turn around and facilitate one the next day. I’ll also often showcase a Problem-Based task or Project Based Learning unit. These require more planning than our PD often allows, so it’s something I can follow up with afterwards.

Similarly, I like to offer a task to facilitate as well as a pathway to build additional tasks. It’s a nod to a framework I wrote about earlier: Find, Adapt Create.

To conclude this blog post, as a Trader Joes sycophant, I’d like to pivot entirely away from professional development and share Geoff’s Trader Joe’s Essentials shopping list. These are the things I purchase every time I go there, because it’s always good to have these in stock:

Potatoes: russet, yukon gold, and sweet
Coffee (the $4.99 whole bean tins)
Fried rice (one chicken, one veggie)
Orange chicken
Chicken Schwarma Thighs ($4.99/lb)
Dried sweet mangoes
Yogurt
Bananas
Either the pollo asada or beef bulgogi, whichever I’m feeling for later in the week
Brie cheese