Developing Students into Deep Mathematical Problem-Solvers

Inside Mathematics identifies itself as “a professional resource for educators passionate about improving students’ mathematics learning and performance”. The goal of the site is to help educators continue to grow and transform their teaching practice. To accomplish this there are a lot of resources, including  lessons, challenging math problems, videos of classroom instruction, videos of teachers planning lessons, videos of teachers reflecting on lessons they just taught, etc.

Because it offers so much, it can be hard to get started here. The more time you have to invest, the richer your experience – particularly in terms of watching some of the classroom videos and the teacher conversations. But for a good sense of what is most immediately useful here for adult education teachers and their students, I’d recommend getting started with either the Problems of the Month or the Performance Assessment Tasks.

The Problems of the Month (POM) are non-routine problems intended to develop the themes of problem-solving, sense-making and perseverance in math instruction. Like all the materials at Inside Mathematics, the POMs were created for K-12 teachers, but because of the way they are written, they lend themselves towards easy adaptation to the adult education classroom. The Problems of the Month are designed to be school-wide explorations, and each problem is divided into five levels of difficulty – from Level A (Primary) to Level E (High School). With some effort, they could certainly be used across an entire, multi-level pre-HSE/HSE program, but they also lend themselves to a single classroom really well. The way the problems are divided into five levels is interesting. It is not the exact same problem simplified and made more challenging; it is more of a exploration of a particular concept at different depths. That said, the problems are non-routine at all levels. They do not start procedural and get more conceptual. Again, they are designed to develop a culture of problem-solving in math students at all levels. In adult education classes, teachers must often teach students with a wide range of abilities, often in the same class. These problems can help in a few different ways, depending on the level you teach. For an HSE class, you can target the Level C or D problems, but first use the Level A and B (and some whole-class discussions) to scaffold students’ explorations. Then you can use the Level D and and maybe even the E problems (those these are often very challenging) to extend the problem for students who are prepared for a further challenge. One thing that I imagine some teachers will not like is that there are no answers. I would recommend working on the problems on your own and with other teachers. Then discuss your answers, your different approaches and what students might gain by working on them. You can also post your answer in the comment section below and ask for feedback from other adult ed teachers. The POMs are all aligned to the Common Core and can be searched by grade level (K-12) or by math content area (more on this below). In addition to the 5 problems themselves, you can examine the Common Core Content standards, as well as the Mathematical Practices, involved with each. To get a sense of the POMs, check out The Wheel Shop or Growing Staircases.

The Performance Assessment Tasks (PAT) are formative assessments that use non-routine problems to evaluate your class and/or individual students. They come as packets that contain the following:

  • Both the Common Core content standards and the mathematical practices involved in the problem
  • An interesting problem, usually with several parts and including some explanation of student thinking
  • A rubric for assessing student answers
  • Some questions regarding what you might have seen in your students work to help you recognize common misconceptions and novel approaches
  • Samples of student work (both correct and incorrect) with commentary
  • Common mathematical understandings and misunderstandings, in terms of each part of the rubric
  • Teacher reflection questions
  • Implications for instruction

The PATs are meant to be given to students and collected, so that teachers can analyze them and gain some insight into where their students are, to help them develop future lessons. Adult Ed. teachers can certainly do that, and they can certainly use the rich materials above in a variety of other ways. I find the samples of student work particularly helpful, even though they are the work of K-12 students. They often represent the range of possible approaches and mistakes that our students come up with. In the end though, a teacher can use the PATs as a source of quality, non-routine math problems. A few that went well in an HSE class: Photographs (7th Grade), Picking Apples (8th Grade), Conference Tables (9th Grade).

Probably the easiest way to search through both the Problems of the Month and the Performance Assessment Tasks is the “Mathematical Content Standards” tab under the “common core resources”. This allows you to search by either grade level and/or by math content and the results will show both the POMs and PATs for any given grade/content.

There are lessons at Inside Mathematics, but they are not exactly easily searchable. They can be found under the “classroom video” tab in “public lessons”. At the time of this review, there are only 13, ranging from 2nd grade to 9-12th grade, though many of them would take more than one adult ed. length math class to do. You might look at “Comparing Linear Equations” (7th/8th grade), which could take as many as 4-5 classes (or more), and which includes the Picking Apples task mentioned above (and many related problems). Another interesting one is the 7th grade “Area and Perimeter” lesson. Unfortunately, there is no written lesson plan for that one – only several short videos showing each phase of the lesson. I found it interesting because it is a lesson that was developed based on how students did on one of the Performance Assessment Tasks. Students who worked on the task ran into problems with the questions that required more than a procedural understanding of perimeter and area. The lesson you can watch was developed to build up the conceptual understanding students lacked.

As I said, the POMs and PATs are great, but if you have time, the guided tours of reflective mathematics practice are interesting as well.

I look forward to hearing all the exciting things teachers do with these materials in their classes!

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About Mark Trushkowsky

Mark enjoys doing math problems that take weeks, family sing-a-longs and reading late into the night. At 16, he believed the next revolution would be waged through poetry. Now he believes it is adult basic education. But he still likes poetry. Mark has worked in adult literacy and HSE since 2001. He is a founding member of the NYC Community of Adult Math Instructors (CAMI). He was born and raised in Brooklyn where he lives happily ever after with his partner Sarah and their daughter Liv.

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