What are High-Level Tasks?

Preservice elementary teachers, both when they are seniors and interns, have opportunities to read about, discuss, and create high-level tasks. Even though we have several readings, and discuss ways to create high-level tasks or to adapt curriculum in high-level tasks, our preservice teachers are not always able to answer the question “What is a high-level task?”

This Fall, my co-instructor and I decided to have our interns come up with a list of what they see as the most important features of high-level tasks. I think it went well! What are other features that should be included? (acknowledging, of course, that it isn’t only the task itself that makes the activity high-level but also the discussion after and the teacher interaction with students as they explore)

When you create high-level tasks, consider whether they have the following features:

  • Leads to a deeper mathematical understanding
  • Authentic – should connect to students’ real-world (which may be very different than your real world
  • Open-ended
  • Collaborative
    • Open to / requires multiple smartnesses (so that students are all valuable members of the team)
  • Multiple entry points
  • Multiple strategies and valid solutions
    • Not just knowing strategies, but applying strategies as needed
    • Allows for students to see why some strategies don’t work (sometimes or all the time), others work but might be easier / faster / make more sense
  • Multiple possible representations (manipulatives, diagrams, numbers, words, pictures, etc.)
  • Integrates multiple topics, strategies, and mathematical understandings
    • Connecting to grade-level standards that are above your own grade level
    • Ties to future knowledge and tasks
    • Builds on the knowledge students bring from outside school
  • Avenue for productive struggle
    • Knowing your students, helping your students know themselves, and giving them number choices can help you and them choose just the right amount of struggle – so they aren’t bored becaues it isn’t challenging enough but not frustrated because it’s too challenging
    • Task can be easily modified (by you or students) to be more or less challenging

Fraction Multiplication and Division Using Virtual Measurement Models

Session Presented at the National Council of Supervisors of Mathematics (NCSM) annual conference in April, 2015, in Boston, MA. Because it was NCSM, our participants were professional development facilitators and other teacher educators. We presented a similar session in February, 2015 at regional practitioner conference – we deliberately changed very little between the two sessions, so that we could talk to the teacher educators about what we had seen in the teachers’ interaction with manipulatives in the session

In this session we explored various web-based applets created to support conceptual understanding of fraction, multiplication, and division using spatial measurement models (length, area, volume). We also discussed how these applets can be used in professional development or classroom settings with students.  (View Session Site; Download Session Hand-out and Session Slides)

Overview of session:

We hoped to stimulate discussion focused on the complex interactions of:

  • Students’ understanding of mathematical ideas, such as: measurement models, number lines, and fraction operations
  • Relationships between: quantities and numbers / actions and operations
  • Benefits and limitations of instructional models: manipulatives / applets

Exploration of physical manipulatives:

We provided participants with several physical tools, including rubber bands, linking cubes, WikiStix, and graph paper. We asked them to use the tools to make sense of representing fraction multiplication with discrete quantities and with continuous quantities.

Exploration of virtual manipulatives:

We asked the participants to interact with virtual manipulatives (http://tinyurl.com/STEM-NCSM2015) that we had created in an attempt to explore discrete versus continuous quantities in multiplication and division of fractions.

We created the applets specifically to support thinking about measurement dynamically and continuously because research has shown students see measurement as static and discrete most often, and yet dynamic and continuous experiences can support students develop flexible understandings of multiplication and division that may support them in understanding later covariational relationships.

Final discussion:

We asked participants to talk about: How can focusing on these ideas support students’ and teachers’ thinking about measurement and fractions? When are virtual manipulatives appropriate? When are physical manipulatives appropriate?

Stehr, E.M., Satyam, V.R., Smith, J.P., & Gilbertson, N. (2015, April). Fraction multiplication and division using virtual measurement models. Presentation to the 2015 Annual Meeting of the National Council of Supervisors of Mathematics, Boston, MA.

Modeling 5 Practices – Monitoring

Attachment: Measurement Applets – Monitoring Sheet

In Spring 2015, I taught one section of a mathematics methods course for senior preservice K-8 teachers (PSTs). The seniors spent about 5 hours each week in a classroom – most were in the same classroom all year. In two large assignments, they created high-level tasks, wrote lesson plans, and then implemented their lesson plans. Throughout the process they used, as resources: Van de Walle (2013), Smith, Bill, and Hughes (2008), and Stein, Engle, Smith, and Hughes (2008). One requirement of the lesson plan was that it explained how they would incorporate the “5 practices” to lead a productive mathematics discussion.

My seniors struggled with anticipating and monitoring, so I developed an in-class task where they could experience the monitoring, selecting, sequencing, and connecting practices. I did the “anticipating” and created monitoring sheets. The seniors engaged in five tasks in pairs, using applets from https://www.msu.edu/~stemproj/ (links are found in the attached monitoring sheets) on the classroom computers. They were in five groups, so for each task one group monitored their classmates and four groups engaged in the task.

After the five tasks, the seniors had ten minutes to talk in groups about how they would select and sequence the strategies they had seen to support a particular measurement learning goal. Then each group led a (unfortunately too-brief) five-minute discussion to support their classmates in making connections.

The applets were designed by the Strengthening Tomorrow’s Education in Measurement research project to support K-8 students in confronting misconceptions about particular aspects of measurement. I helped design the applets, and (for most of them) wrote the code to create the applets. My seniors thus were able to talk deeply about measurement strategies as well as to experience the decision-making process around the 5 practices.

References

Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a Lesson: Successfully Implementing High-Level Tasks. Mathematics Teaching in the Middle School, 14(3), 132–138.
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340.
Van de Walle, J. A., Karp, K. S., Lovin, L. H., & Bay-Williams, J. M. (2013). Teaching student-centered mathematics: Developmentally appropriate instruction for grades Prek–2 (Vol. 1). Boston, MA: Pearson.
Van de Walle, J. A., Karp, K. S., Lovin, L. H., & Bay-Williams, J. M. (2013). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Boston, MA: Pearson.

Fraction Multiplication and Division using Virtual Measurement Models

Session Presented at the Math in Action annual conference in February, 2015, at Grand Valley State University in Allendale, Michigan. We presented a similar session in April, 2015 at a national conference (National Council of Supervisors of Mathematics) – we deliberately changed very little between the two sessions, so that we could talk to the teacher educators about what we had seen in the teachers’ interactions with manipulatives in the session

In this session we explored various web-based applets created to support conceptual understanding of fraction, multiplication, and division using spatial measurement models (length, area, volume). We also discussed how these applets can be used in professional development or classroom settings with students.  (Visit Session Site; Download Session Slides and Session Hand-Out)

Overview of session:

We hoped to stimulate discussion focused on the complex interactions of:

  • Students’ understanding of mathematical ideas, such as: measurement models, number lines, and fraction operations
  • Relationships between: quantities and numbers / actions and operations
  • Benefits and limitations of instructional models: manipulatives / applets

Exploration of physical manipulatives:

We provided participants with several physical tools, including rubber bands, linking cubes, WikiStix, and graph paper. We asked them to use the tools to make sense of representing fraction multiplication with discrete quantities and with continuous quantities.

Exploration of virtual manipulatives:

We asked the participants to interact with virtual manipulatives (http://tinyurl.com/STEM-MIA2015) that we had created in an attempt to explore discrete versus continuous quantities in multiplication and division of fractions.

We created the applets specifically to support thinking about measurement dynamically and continuously because research has shown students see measurement as static and discrete most often, and yet dynamic and continuous experiences can support students develop flexible understandings of multiplication and division that may support them in understanding later covariational relationships.

Final discussion:

We asked participants to talk about: How can focusing on these ideas support students’ and teachers’ thinking about measurement and fractions? When are virtual manipulatives appropriate? When are physical manipulatives appropriate?

Stehr, E.M. & Gilbertson, N. (2015, February). Fraction multiplication and division using virtual measurement models. Paper presented at the Math in Action Conference, Grand Valley State University, Allendale, MI.

Spinning your wheels and productive struggle

A couple of years ago, my parents bought a cabin a few miles up Blacksmith Fork Canyon in Cache Valley, Utah. It’s about a 40-minute drive. My dad asked to drive me up there today to see the progress he’s made on the cabin – it’s supposed to snow tomorrow so this would be our last chance for the winter.
2014-12-19 16.30.32

We drove on clear roads for some time, but eventually got high enough that the roads were covered in snow and, eventually, ice. 

At the first transition from snow to ice, my dad lost control of his truck and we slid around and landed in some trees. The front tires were off the road, but the rest of the truck was at a 90-degree angle to the road.

My dad switched gears to reverse out and get back on our way. The wheels just spun. My dad is very calm in emergencies and, even though my first thought was, “We’re in the middle of nowhere. We are going to die.”, he calmly began telling me what he was doing as he spun his wheels.

He told me that the friction caused by spinning the wheels would melt the snow or ice down until the tires would hit the ground and we would be on our way. So, he would spin the wheels for a minute or two, and then let the truck cool down a bit and let the ice settle a bit, and then spin again.

2014-12-19 16.30.53After he repeated this process over and over for about 15 – 20 minutes, the wheels finally caught and we reversed back on to the road, made up to the cabin. We got stuck one more time, but luckily it was closer to the cabin and some of his neighbors were there with a truck and chains that they could use to pull him out. He calmly explained that if they weren’t there, he would go get his tractor from the cabin and use that to pull the truck out.

The reason I include this story on a math ed blog is because, even though the content is not strictly mathematical, we often talk about the importance of “productive struggle” for our students. The trigonometry book I taught from mentioned that students should be encouraged to regulate their work time and try to keep from “spinning their wheels uselessly.”

This experience taught me a couple of things: First, that spinning your wheels can actually be a productive thing to do: Even when working on mathematics, it can be helpful to dive in and struggle for a while and then take a break to go on a walk, take a shower, or do the dishes. Spinning your wheels and taking a break and then jumping back in can actually help you reach solid ground. Second, I have not had as much experience getting stranded as my dad. So, I can imagine I would spin my wheels for a few minutes and then just give up. But because my dad has had experiences where he was forced to find a way out, he has learned that he just needs to persevere and eventually it will work. So helping our students have experiences where they work hard for a while and reach a solution is important. Third, my dad asked for help and had back-up plans in his mind in case asking for help didn’t work. He suggested a solution to his neighbors, showing them how they could be useful. He also considered other resources available to him, and what he could use if his neighbors were not able to help. So giving our students opportunities to plan for multiple eventualities and to flexibly consider potential strategies is also important.

Preparation of Algebra Teachers

Session presented at the Michigan Council of Teachers of Mathematics annual conference in Holland, MI.

We presented preliminary findings from the Preparing to Teach Algebra (PTA) project in the following areas:

  • Overview of Findings from PTA national survey
  • Opportunities to learn (OTL) about Equity and Algebra to support teaching and learning
  • CCSSM and Algebra to support teaching and learning

Download slides here

Overview of Findings

  • Significant variation in preparation programs
  • Extensive coursework for OTL algebra in advanced mathematics courses
  • OTL equity issues in algebra are generally not provided
  • OTL about (or to teach) the algebra, functions & modeling strands in CCSSM are less likely
  • Many programs have made changes to address some aspects of CCSSM

Equity and Algebra

  • Readings and discussions: Some instructors used articles to generate awareness and promote discussions about equity.
  • Secondary Mathematics Methods: Each one of the 4 universities incorporated at some extent equity aspects in (at least one of)  their Methods courses.
  • Conceptual and cognitive aspects: Instructors addressed issues such as conceptual understanding and how to make it more accessible to students and high cognitive demand tasks (not necessarily high difficulty) as a means of providing diverse options to solve problems.

CCSSM and Algebra

  • Content and Practice standards: Understanding how these two pieces of CCSSM interact
  • Using/Dissecting CCSSM: Working to unpack the standards in order to understand and implement
  • Connecting CCSSM and Teaching:  Personalizing the standards document as applicable to practice

Discussion

Throughout the session, we asked teachers to reflect on their own experiences and experiences in their teaching programs.

Acknowledgements:

This study comes from the Preparing to Teach Algebra project, a collaborative project between groups at Michigan State (PI: Sharon Senk) and Purdue (co-PIs: Yukiko Maeda and Jill Newton) Universities. This research is supported by the National Science Foundation grant DRL-1109256.

Stehr, E.M., Craig, J., & Medel, L. (2014, August) Preparation of Algebra Teachers. Paper presented at the Michigan Council of Teachers of Mathematics Conference, Holland, MI.

Supporting spatial measurement tasks with technology

Session presented at the Michigan Council of Teachers of Mathematics annual conference in Traverse City, MI.

Download slides here.

In this hands-on session, we introduced several sample tasks that included use of computer applets to strengthen understanding of length and area concepts. Many students struggle with length and area, using only pen and paper or physical manipulatives. In this session, we provided materials to support their learning (that will be available for use during and after the session).

We asked participants to use their devices to work through one sample lesson (Area of Rectangular Regions) including Launch / Explore / Summarize components. We gave participants two sample lessons that they could in their classrooms, along with applets designed and created by Strengthening Tomorrow’s Education in Measurement project.

Stehr, E.M., Gönülateş, F., & Siebers, K. (2013, August) Supporting spatial measurement tasks with technology. Paper presented at the Michigan Council of Teachers of Mathematics Conference, Traverse City, MI.