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

Paying a tithe (10% of earnings)

In Sunday School, I teach kids who are turning 7 this year. There is a lesson manual and a schedule to follow, but I like bringing in authentic tasks (especially when it is math-related!) whenever possible.

Today the lesson was “I can pay tithing.” We talked about the meaning of tithe (Merriam-Webster says: “to pay or give a tenth part of especially for the support of the church”).

I brought a sack of pennies (about one-year’s accumulation!) and gave each of the kids a pile. I asked them, “Find out how many pennies you have and then tell me how many pennies you’ll give for tithing.”

Multiple Answers

One child said, “I’ll give them all.” I responded, “Sure. You don’t have to give all of your pennies, but you can. Do you still want to?”  “Uh-huh.” “Why do you want to give all of them instead of just 10%?” “Well. I don’t need them. I could keep some.” After church was finished, he came running back to tell me “I kept some of my pennies but I put some on the ground so that someone can find it for a lucky penny!”

Another child said, “I’ll give 4 because I’ve got 43 pennies.” I responded, “Okay. How did you figure that out?” “It’s easy! Every time I count 10 pennies, I take one.”

Another child said, “Can I keep the extra pennies?” I responded, “Sure!” “Okay, I’ll give 17 because I have 39 pennies.” “Okay, how did you decide to give 17?” “Because it looks like half.” “Ok. You can decide how much you want to give. The church asks for 10% but it’s okay to give more. Do you still want to give 17?” “Uh-huh!”

I gave them each a tithing envelope, they filled out their own slip (mostly), put the pennies in the envelope, and brought them to the bishop.

Assumptions

Sometimes in textbooks a word problem is given and the question asks: “Which should you choose?” or “What’s the best choice for Rohit?” Those problems make me cringe because it is making an assumption that “optimization” means the same thing for everyone, especially when some context is given. In this example, the kids all have different reasons for wanting to keep or give away the pennies – some told me they have $20 or $30 at home to spend, so maybe they don’t value pennies much. Others were already making plans for how they would spend their pennies. Some kids are just too darn nice.

But, if I pushed them to only pay 10%, then it feels like I’m making an assumption that you should always give the minimum. At the same time, by not pushing them to pay only 10%, I feel like I’m telling them they should always pay more than the minimum. So, even for me, I’m not sure what the “right” response is in this real-world math problem.

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.

Creating Technologies to Support Measurement

I worked with a student teacher who had been an undergraduate research assistant for the Strengthening Tomorrow’s Education in Measurement project, and was a recent graduate of Michigan State’s Teacher Education program. She is currently interning in a nearby classroom. She agreed, along with her mentor teacher, to design and implement a lesson with my support. Two redesigned applets resulted, and a third is possible. The link below connects to her lesson.

The student teacher had taught a lesson that incorporated van de Walle’s Crooked Path and Broken Ruler activities two weeks before the technology lesson (Van de Walle, 2013). She created crooked paths in her classroom using masking tape on the floor and on some tables. Students measured them, moving from path to path as they completed each measurement.

The student teacher felt that the STEM Jagged Path applets (see link above) would support the learning goals of her previous lesson by encouraging students to practice skills they had developed as well as deepening their earlier thinking. She asked all students to explore Activity 1, using Activity 3 as an extension for students who finished quickly.

The student teacher and I talked above another applet that could support students’ thinking. She said that her mentor teacher worked hard to encourage her students to confront misconceptions and change their thinking. We designed and created a third applet (shown in Activity 2, using the site link above) that would show the student two “wrong” ways to measure. We asked the students to explain what they thought was the thinking behind each method, and to explain which method they thought was correct – or if neither was correct, to explain what a correct method would be.

Fractions as Lengths

Session presented at Math in Action annual conference February 22, 2014 at Grand Valley State University in Allendale, MI.

Download session slides here.

The main take-away from the session was that: If students are going to understand the number line in rich and meaningful ways, they should understand how numbers represent accumulated quantities of lengths. To support our participants in their understanding and future teaching, we created opportunities for them to explore connections between fraction operations and fraction representations using a length model.

Connections between measurement models and fraction operations

FractionsAsLengths-01

We reminded teachers that often an area model is used to represent fractions and fraction operations. We also brought an example of a word problem that connected fraction operations to a length model (as represented by lengths of string).

We asked participants to represent fractions as numbers, and then operations on fractions, using sentence strips (as a length model).

Stehr, E.M., Gilbertson, N., & Clark, D. (2014, February) Fractions as Lengths. Paper presented at the Math in Action Conference, Grand Valley State University, Allendale, MI.