Mathematics Teacher Education OER

In-depth blogs that tackle a lot of topics in meaningful ways.

National Council for Teachers of Mathematics– this blog is hosted by a premier mathematics education organization

Dan Meyer– one of my favorite blogs. You’ll find very practical resources as well as posts that handle bigger issues.

Marilyn Burns– amazing researcher and children’s book author. She transitioned to blogging extremely well.

Youcubed– though the entire site is amazing, this link focuses on writings that support teaching and learning

Illustrative Mathematics– great organization that has posts from a variety of writers.

Investigations– this the site from a solid textbook company that supports its work through blogging.

What do we know about how children think and make meaning in mathematics?

Embrace the Challenge– here’s a blog that does an amazing job at closely observing children’s thinking

Math is Figureoutable– solid site overall. The author does great work challenging the notion that we need to teach children algorithms.

Thinking Mathematically– Mark Chubb is an instructional coach who posts up-to-date research ideas in a very practical and student-focused way.

Graham Fletcher– in addition to providing some nice curriculum, Graham does great work in breaking down content and how children make sense of it.

What types of curriculum support in-depth student learning?

Math Coach’s Corner– This site focuses on providing practical resources for teachers

Talking Maths with Kids– Nice mix of practical resources with emphasis on student thinking.

Games– the whole site is great, but I’ve linked to the blog section to learn more about the games and their intent.

Joe Schwartz– practical ideas, often accompanied with student work samples.

Brian Bushart– nice vision of contemporary research with skill in creating new curriculum.

How do we teach mathematics in a way that supports equity?

Being Black at School– though not math specific, this resource is geared toward identifying and supporting the unique strengths of black children in schools.

Marian Dingle– she is wonderful at providing some very human reflection on teaching. She provides great insights how existing systems can be harmful to many students

Neplanta Teachers Community– This group approaches education with a social justice lens.

Hema Khodai– Hema is a practicing teacher who provides rich reflections related to issues of mathematics and race.

Lauren Baucom– practicing teacher and DESMOS fellow with great general reflections as well as reflections specific to mathematics and race.

Other

Tracy Zager– Tracy’s writings vary, though all her posts demonstrate a wonderful understanding of the research in mathematics education.

Visuals and Representations– Berkeley Everett is a K-5 math coach who has created an amazing bank of visuals to support learning.

Robert Kaplinsky– prolific writer with some great posts about mathematics education.

Shoot for Boring- What do we learn?

I opened this semester using a task for my elementary math methods course that I’ve used many times before. Each time I present it I’m struck by how engaged the PSTs are. They persevere for an extended period of time and I frequently hear that they continued to think about the task and work on it after the class has been dismissed. From an engagement standpoint, the task isn’t all that impressive. In fact, the side of me that really loves socio-cultural perspectives on learning gets mad every time I use this task. It’s not “real-life”, I mean, not really. It’s not a situation they’re going to run into again, thus giving students prior knowledge that would transfer. So why was it engaging? Why did they persevere? As I share the task I encourage the reader to avoid making a “Here’s why it’s engaging” list or “Here’s why they persevered” list. I feel like we all know what those lists would look like and that they may ultimately be hollow. Instead, I’m interested in how you might alter the task in a way that would kill engagement or cause persistence to falter. It’s a weird goal, I know. Attending to what hurts persistence can tell us just as much as attending to what helps it. Shoot for tiny alterations. The most minute alterations would tell us a lot about lesson design. In my experience, it’s the small differences that make it so one lesson thrives while another bombs. I mean, from one day to the next I don’t say, “You know, I think this technology thing is a fad and I don’t really want to build it into my lessons.” Telling me that technology use increases engagement and can support perseverance is the sort of broad-strokes advice that isn’t super helpful. I get it. That’s not a tool I’m likely to mistakenly toss aside. I’m interested in those small tweaks that I may not even be aware of. Bringing those to the surface is what improving teaching is all about.

So, here’s the task.

Launch: I start by telling a (mostly) fabricated story about watching my little niece play with blocks. I notice that she was building a pattern, as kids are wont to do, so I watched more closely. I present the pattern as my adult brain made sense of it. As a pattern that started with a single blue block and then grew outward in concentric rings. I label each new ring a “step”. I use a simple presentation to show the first 3 steps as they develop.

I ask students to list any questions they might have if they were watching this pattern unfold, giving them a voice in the problem formulation. If they didn’t ask it, I’ll usually ask a question about whether or not my niece would run out of blocks. I then ask PSTs to find how many total red blocks would there be in the entire figure once step 5 has been completed. I generally have multi-link cubes, tiles, and graph paper available. I let students work alone or in groups.

 

Side note: My elementary math methods students generally don’t have go-to strategies regarding linear or quadratic functions, so this task tends to put them at the edge of their comfort level without pushing them overboard.

 

Explore: As students problem solve, I walk around asking questions about how they found their answer, making note of the different strategies they tried. I only redirect their thinking if I think it will lead to unproductive failure. I’m more thank okay with incorrect answers, but some incorrect approaches do not support productive struggle.

Discuss: I choose students to share with the attempt to build toward greater efficiency. So, if I had a student who built the whole figure and then counted each red block, I have them go first. I ask if that strategy will always work. We talk about strengths of the approach as well as any potential obstacles. A student who built it (or drew it) and only counted up a quarter of the figure and then mulitplied by 4 might go next. We identify similarities and differences, strengths and challenges. Some students start noticing patterns, we get those out there. Some start tracking information through tables, we talk about those. Pretty foundational stuff.

 

We then repeat the launch, explore, discuss cycle 2 more times. During the second cycle they answer the same question for step 10 (does it work just to double step 5?) and on the third, they tackle step 100. Each Discuss section involves making connections, justifying work, asking questions to presenters (student to student interactions), and challenging one another’s thinking.

 

There are quite a few fits and starts. Strategies are tried and then abandoned. No one builds step 10 in its entirety. Many find that each new row of red grows by 8 blocks. That makes it easy to track the new red and then add each layer together. It’s fairly efficient for 10 steps, but not 100. By step 100 they either need an algebraic expression (formal or informal), or enough patience to number crunch.

 

So… Nothing special. Students really persist, though. Why? Is it just because they’re wonderful students? I mean, probably, but that can’t be the only reason.

 

So, let’s hear those alterations. Shoot for boring. What would shift this from a dynamic experience to a mundane or tedious one? I’ll start the ball rolling with some obvious, broad-strokes alterations.

  1. Jump right to step 100. Skip the steps 5 and 10 cycles. By requiring abstraction right off the bat I would lose all of my less confident students. Frustration would replace persistence.
  2. Remove the tools and manipulatives. This would also implicitly communicate that the only type of thinking that is valued is abstract representation.
  3. Expect all students to work independently, because, you know, who cares about Vygotsky? No need for all this ZPD stuff.

 

These would surely be terrible alterations, but they’re also not ones I’m likely to make. Can you think of subtle changes that could really impact engagement and persistence?

 

Other thoughts about the task generally:

Things I don’t like about the task:

  • The problem is not compelling. If she runs out of blocks, who cares? Is this something that really needs to be solved?
  • As far as PSTs’ mathematics is concerned, it doesn’t function well as a stand-alone experience. Students do not gain deep insights into different functions. They would need follow up experiences to deepen and connect their knowledge.

 

Reasons I’ll likely use it again next semester:

  • Did I mention students persisted?
  • It gives me great insights into their problem-solving approaches. Some students needed very concrete approaches. Some loved to delve into the abstract but got lost in their own thinking. Some were brilliant, but lacked confidence. As a formative assessment piece, it was very rich.
  • It engages students in a number of important mathematical practices.