“Technology and Noticing” – Psychology of Mathematics Education – North American Chapter

PMENA 2015 - Noticing and Technology - Final

I had a great time at PMENA-37 this past weekend! What a supportive, engaging, and productive space for learning about mathematics education – rather, immersing oneself into the mathematics education community.

Friday afternoon, I presented my poster, “Digital resources in mathematics: Teachersconceptions and noticing” and I learned a lot through conversations I had with visitors to my poster. You can read the overview of this poster in PMENA-37 Proceedings (p. 1256).

This poster represents a moment in time in my dissertation study. I am beginning to look at what teachers notice (or paid attention to, or found important) in online tools and resources by first considering how they responded in our pre- and post-evaluations in the online master’s-level course “Learning Mathematics with Technology.” We (the instructor and I, as a teaching assistant) gave students a broad prompt:

Assume you are considering using these tech tools in your teaching. Evaluate the three tools to decide whether you would recommend using them. Write an evaluation of the three tech tools for an audience of other teachers who might be considering using them.  Make your recommendation clear: which (if any) would you recommend? Argue your position. You will need to decide which elements, features, or characteristics of each tech tool to use in supporting your argument.

I considered the teachers in my study who chose the addition set of tools, and as I read and re-read their evaluations I created a list of tool features that the teachers paid attention to: Purpose of tool; Classroom use; Understanding (e.g., instrumental, relational); Thinking (e.g., critical, level); Engaging / Aesthetically pleasing / Distractions; Directions and Instructions; Customizable; Progress / Tracking; Preparation; Age / Grade-level; Interaction / Manipulative; Feedback / Responsiveness; Ease of Use / Accessibility; Differentiation / Learning styles / Learning needs; CCSSM / Standards Alignment; Mathematical representation; Teacher Support / Resources; Breadth / Variation. These were too many for the poster, so I narrowed them down to 12, and sorted them according to whether they seemed to be general, pedagogical, or mathematical characteristics (but recognizing the overlap in the categories, similar to the TPACK framework):

  • General: Customizable; Engaging / Aesthetically pleasing; Ease of Use / Accessibility
  • Pedagogical: Differentiation / Learning styles/needs; Progress / Tracking; Feedback / Responsiveness; Teacher Support / Resources
  • Mathematical: Mathematical Purpose; Standards Alignment; Understanding / Thinking; Interaction / Manipulative; Mathematical representation

In the tables, I noted whether teachers mentioned a particular characteristic was included or favorable included in the tool (Y), was not included or included in an unfavorable way (N), and whether the teacher did not mention a characteristic (–). Because of the limited space of the poster, I chose three mathematical features to show teacher comments about: Mathematical purpose, Understanding / Thinking, and Interaction / Manipulative. I included comments for each tool from each teacher in the pre-evaluation (left) and the post-evaluation (right).

Discussion of Results (which are shown in the poster above)

It goes without saying that the online tool itself seemed to impact the features that teachers paid attention to and I had chosen the three tools to be different in (hopefully) interesting ways. One tool is built around a ten-frame representation of addition, and includes a symbolic representation (horizontal addition statements). One tool is a quizzing software that includes a symbolic representation (horizontal statement) of addition only. One tool is built around a base-10 blocks representation of addition, and includes a symbolic representation (vertical addition statement). The tools differ in a number of ways and the teachers noticed many of their differences. There is evidence that teachers moved, not just to noticing more characteristics, but also to more profound noticing by the end of the course.

One interesting result was the teachers’ attention to thinking and understanding in the Math Trainer applet. In the pre-evaluation, none of the four teachers commented on opportunities for students to think or develop understanding. In the post-evaluation, all four teachers commented on Math Trainer’s support of instrumental understanding (we read Skemp (1976) early in the course), and on the lack of opportunities to “explore or gain a deeper understanding,” or for “critical thinking.”

Another interesting result is the change in Nicole’s use of words from the pre-evaluation to post-evaluation in her description of the mathematical purpose of each tool. She did not change words at all in her description of Math Trainer’s purpose “to practice math facts,” but changed from practice to explore for NCTM Illuminations: Ten Frame Addition and NLVM: Base Blocks Addition. Her description of the purpose of NCTM Illuminations: Ten Frame Addition in the pre-evaluation was “to practice using a 10-frame to work with numbers” and in the post-evaluation was “to work with a ten-frame (or five-frame option) to explore addition and subtraction of whole numbers.” For NLVM: Base Blocks Addition, she wrote “to practice addition with base-ten blocks” in her pre-evaluation and “to explore addition with base-ten blocks” in her post-evaluation.

PMENA 2015 – Noticing and Technology – Final (pdf of poster)

PMENA 2015 Proceedings – Noticing and Technology (pdf of proceedings paper)

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PMENA – November, 2015

PMENA-banner-1140

I’m excited to participate in PMENA this weekend – very conveniently held at my school, Michigan State University. I have helped out the Local Organizing Committee, but mainly I have enjoyed watching the “behind the scenes” action. Led by Drs. Tonya Bartell and Kristen Bieda, we have worked hard for about a year and a half. I’m excited to see the end result!

I will be participating as a presenter on a brief research report and three posters:

  • Friday, November 6, 9:20 AM – 10:00 AM (Kellogg: Conference 62)  Assessing Teacher Knowledge and Practice
    • Mathematical Knowledge for Teachers: Opportunities to Learn to Teach Algebra in Teacher Education Programs
      Presenters: Eryn M. Stehr, Jeffrey Craig, Hyunyi Jung, Leonardo Medel, Alexia Mintos, Jill Newton
  • Friday, November 6, 4:00 PM – 5:30 PM (Kellogg: Lincoln)  First Poster Session
    • Digital Resources in Mathematics: Teachers’ Conceptions and Noticing
      Presenter: Eryn M. Stehr
    • Building Algebra Connections in Teacher Education 
      Presenters: Hyunyi Jung, Jill Newton, Eryn M. Stehr, Sharon Senk
  • Saturday, November 7, 10:30 AM – 12:00 PM (Kellogg: Lincoln) Second Poster Session
    • Secondary Preservice Teachers’ Opportunities to Learn About Modeling in Algebra
      Presenters: Hyunyi Jung, Eryn M. Stehr, Sharon Senk, Jia He, Leonardo Medel

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.

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.

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.

Technology and Algebra in Secondary Mathematics Teacher Preparation Programs

Session Presented at the Research in Undergraduate Mathematics Education annual conference in February, 2014, in Denver, CO.

Abstract: Most recently, the Conference Board of the Mathematical Sciences has advocated for incorporating technology in secondary mathematics classrooms. Colleges and universities across the United States are incorporating technology to varying degrees into their mathematics teacher preparation programs. This study examines preservice secondary mathematics teachers’ opportunities to expand their knowledge of algebra through the use of technology and to learn how to incorporate technology when teaching algebra in mathematics classrooms. We explore the research question: What opportunities do secondary mathematics teacher preparation programs provide for PSTs to encounter technologies in learning algebra and learning to teach algebra? We examine data collected from a pilot study of three Midwestern teacher education programs conducted by the Preparing to Teach Algebra (PTA) project investigating algebra. Our data suggest that not all secondary mathematics teacher preparation programs integrate experiences with technology across mathematics courses, and that mathematics courses may provide few experiences with technology to PSTs beyond strictly computational.

In this session, we presented results from an analysis of opportunities to use technology to support algebra teaching and learning in secondary teacher preparation programs. The data was collected during the PIlot phase of the Preparing to Teach Algebra project.

Presentation of Findings

We found that instructors responded in interesting ways regarding their use or non-use of technology in support of algebra teaching and learning:

  • Practical Concerns:
    • Not useful in certain courses:
      • …It just doesn’t strike me as really helpful… [University A, Structure of Algebra]
      • It’s abstract for a reason. [University C, Abstract Algebra]
    • Issues of access:
      • …we don’t have money in our department to buy them. So, we don’t have those and our students… need to know those things. [University A, Secondary Math Methods]
    • Not enough time or support:
      • I do not have time to work all the [PowerPoint] slides… [University B, Linear Algebra]
      • …I think …we should have a nice computer-simulated programs that make you see the difference [between convergence and uniform convergence of functions]. … For me, I see it in my head….But I can’t see how. I really can’t see how. [University B, Analysis]
  • Impeding Learning:
    • Blocks Students from Developing Memory
      • … because of the calculator and all these technologies [people] don’t … develop their memory.  But then you are asking them to develop their memory on something that is harder than adding or subtracting, you know? [University B, Analysis]
    • Computational Use Blocks Concept Development
      • But I also want them to know the concepts involved so sometimes … I make a point to tell them that they shouldn’t use technology… [University A, Linear Algebra]
      • [A]t a college level we’re now quite concerned because … we have students who can’t multiply…because they have always had a calculator, you know.  There are students who can’t tell you what the graph of y = x looks like.  …[T]o be able to think about what y = x and y = x2 looks like — they can’t do without a machine.  …So, we are actually moving to not using technology. [University A, Secondary Math Methods]
  • Enhancing Learning:
    • Making the Abstract More Tangible
      • [Technological tools] can bring some of these more abstract things to make them more tangible for students. [University C, Middle School Math Methods]
    • Allowing Different Perspectives
      • I think it …gives them a way to see the problem from a different perspective…understand it from a learner’s perspective and …to think about how to instruct students in multiple ways… [University B, Secondary Math Methods 1 and 2]
    • Conceptualizing Mathematics
      • All of these tools represent ways to represent and conceptualize mathematical ideas that go beyond the symbolic. They’re important tools to really develop a conceptual understanding of mathematics. Moreover, it’s critical that our students are prepared to use these same tools … to foster the same sorts of understandings. [University B, Secondary Math Methods 3 and 4]
  • Whether or not to use technology is complicated:
    • Which courses could use technology?
      • In this course none. …In other courses that I teach I do use technology… I know that that is kind of counter-intuitive because textbooks always have technology stuff in there and some textbooks are even focused on technology. To me that is not what this [course] is about and the more technology you have in a course like this the less that there is for algebra. [University C, Differential Equations]
    • What are instructional consequences of technology use?
      • … there are times where instructionally it may be not the best thing to always use technology and so making that kind of judicious choice is something we talk about as well. [University A, Secondary Math Methods]

We also found some examples of activities to support preservice teachers in decided whether, when, and how to use technology:

  • Affordances of technology, e.g.:
    • engagement
    • enhances some concept development
  • Constraints of technology, e.g.:
    • instructor’s/instructional time
    • impedes some concept development
  • … you don’t just use a tool or technology just because it’s going to be fun; but you really have to think about – What does this particular tool or technology afford me in terms of students’ understanding the content? …sometimes when we’ve used technology it didn’t really offer us any more than if we had just drawn [on] a piece of paper…. [University B, Secondary Math Methods 1 and 2]

We shared two examples of use of technology to support algebra teaching and learning, one from a mathematics course and the other from a mathematics methods course:

RUME-01RUME-02

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. & Guzman, L. (2014, February). Technology and algebra in secondary mathematics teacher preparation programs. Paper presented at the Seventeenth Annual Conference on Research in Undergraduate Mathematics Education (RUME). Denver, CO.

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.