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.

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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.

Meeting Paul Cobb

What advice do you have for new researchers?

Young researchers that go to research universities fixate on tenure and the checklist they need to complete to earn tenure. The first few years can easily become an identity-shredding experience. The wrong game is: “How can I get ahead?” Also, don’t write a proposal just to get a large grant. Think about your pre-tenure years, instead, as six years to work on something that you actually care about. Ask yourself: What you want to understand? What aspect of your practice do you want to improve? Because, if you care about it, you will work hard because it’s something you want to do. Don’t entirely forget about tenure – but don’t fixate on hitting a number of papers. Write quality papers that result from interest rather than weak papers resulting from “gotta”. As you plan, think about: Why did you come into this program to start with? What do you care about? Be aware of the danger of focusing on yourself in an ego way. Instead of thinking about how you can be a star, do work that you are interested in, task-involved, focused on doing good work. Also, in your first years don’t try to get a large grant but instead start with a small enough project so you can be intimately involved in all aspects of it.

What do you know about writing now that you wished you had known?

I wish I had know to outline, in detail, my ideas and argument. You should do this detailed outlining, pretending you are giving a presentation. You can use PowerPoint, for example, to create the outline. You should share your outline with others – meet with someone and talk with them through the outline. As you talk through the outline, pay attention to the structure of the argument: Does the structure make sense? Struggle with sentences: What is the next step in the outline? How do I say it appropriately? Instead of focusing on the question: What am I going to do next? focus on What do I want to figure out? Why am I doing this? What do I want to get a handle on conceptually? As you think about these questions, write a purpose statement for yourself: By the end of this, this is what I want to learn. This is what I’m doing to support that. This is why what I’m doing will help me reach my goal. 

As a grad student, how do I know when I should publish something?

As you publish papers, you will develop a better sense of when something is ready for publication, but there are no guarantees that you will be accepted. The main thing is, as you develop a reputation, you don’t want to submit a paper that is truly awful and then the editors know who you are and will remember how bad the paper is. But as a graduate student and new researcher, you have no reputation and you have no sense of when a paper is ready. So just try to publish it. If it gets rejected, or if it’s terrible, no one knows you and they won’t remember. Then you can develop your sense of when the paper is “good enough” that it won’t embarrass you, and when it is a “good fit” for the journal.