In my part of our presentation I plan to talk about Teacher Training and Curriculum development. Here are some links I plan to use / reference:

• Overview of the eNLVM
• eNLVM eModules
• Curriculum development / teacher training case study
• How to rite good applets
• eModule research materials
• Transformation of Functions (example observations)
• Probability and Relative Frequency (example of simulations)
• What’s My Line (example of reinforcing group discussion with hands on experience)
• Patterns, Relations, and Functions (example of adapting by adding questions)
• Digging Dirt (example of open-ended, manipulative supported, grounded instruction)
• THEORY FOR AUTHORING TOOLS THAT SUPPORT TEACHER ADAPTATION OF MATHLETS
• Presentation on the eNLVM
• When Teachers Reuse and Remix Interactive Online Resources
• Interesting Interactive Math Websites

Teacher Training. A process I have used with teachers:

1. Demo a lesson
2. Discuss / critique the lesson together based on a rubric
3. Have teachers observe students while the lesson is taught
4. Debrief the experience
5. Revise the lesson
6. Reteach

Training Goals. My goals are for teachers to learn:

• Content knowledge
• Instructional strategies
• Familiarity with using high quality materials
• Where to find high quality materials
• Material evaluation / selection
• Help them enter a community
• Implementation issues
• Barriers / workarounds
• How to adapt existing resources
• How to evaluate student learning
• Processes for continuous improvement

Developing Interactive Math Curriculum

When developing curriculum, one of the first thing designed are the elements and structures of design and the processes of design, whether they be ad-hoc or structured. Curriculum design structures include time, learning outcomes, content models, instructional strategies, messages, representations, and media elements such as pages, displays and controls. Important design activities include:

• analyzing content,
• identifying desired learning outcomes,
• recognizing common student errors and difficulties,
• developing an overarching design,
• using design tools that allow for quick inexpensive prototyping and iteration,
• putting off expensive development as long as possible,
• using modular approaches,
• expert review,
• testing with real users,
• iteration

Considerations for Designing Interactive Math Curriculum

The design of interactive math curriculum should take into account the affordances that the medium can offer. These include efficient rich, dynamic, and linked representations, exploration, simulation of physically in-accessible situations and events, guided practice, immediate feedback, easy revision, recording sharing and replay, collaboration at a distance, linkages to real-time data, data sampling, and complex computation.

Contexts of Use

• Group presentation – projection, activity, discussion, worksheet
• Classroom station
• Computer lab (self-paced, pairs)
• Home work (self-paced)

Three Levels of Adaptation

• Activity instructions (via web or worksheet)
• Problem sequence (via web or instructions)
• Virtual manipulative (via configuration or code)

Multi-Disciplinary Teams

• Mathematician – Make sure the concepts / content / terminology / representations are correct
• Instructional Designer – Translate ideas into concrete technology designs
• Programmer – Develop virtual manipulatives
• Educator – Instructional sequences, strategies and types of activities
• Classroom Teacher – Access to students, anticipate student responses, guide implementation

Some Observations

• When an interactive model is on the screen, students ignore the text
• Text must be short and to the point
• Questions that require responses can help focus attention
• Transitions (going to lab, setting up equipment, getting people started, etc) waste time and need to be taken into account when designing experiences
• Always test out exact usage before going live and check again in the morning
• Developing virtual manipulatives is expensive
• Leverage existing manipulatives
• Choose areas where the impact will be greatest
• Teachers are much more likely to adapt than create activities

Problem solving is what you do when you don’t know what to do. Problem solving requires recognizing and defining the problem, selecting an approach, breaking the problem down into sub-problems, selecting procedures for solving those sub-problems, executing those procedures, evaluating and diagnosing progress, recognizing when a solution is satisfactory, and interpreting results. Note, if practice makes perfect, we better give students opportunities practice in all of these aspects of problem solving, not just simple recall and algorithmic procedures.

Common wisdom says that we should wait until people have developed the basics before we ask them to solve problems and do higher level thinking. I reject that notion. Higher level thinking may not so much be “higher level” as it is “different level”. Kids at the youngest ages can and need to be given opportunities to engage in real problem solving. Maybe, part of why kids learn to hate math is because we spend so much time focusing on “repeat what I just said” and “do what I just did”, to the exclusion of authentic problem solving.

“Theories are like toothbrushes, everyone has a theory, but no one wants to use someone else’s theory”

This is true within a given field such as artificial intelligence / cognitive science in which the PSLC work is based. It is even more true if you look across multiple fields such as education (teacher preparation), cognitive science, instructional design, and math education. Each of these fields have something to say about learning and teaching math, but the languages of their literatures are as different as English, Urdu, Chinese, and Russian.

I recently presented on the PSLC and this framework at a recent USU Math & Stat Journal Club meeting (see my PSLC Theoretical Framework Trailfire Trail). In our discussion it was brought up that sometimes teachers object to the type of instruction that is typical of the PSLC model tracing intelligent tutors because it is not open-ended or exploratory. My reactions to this criticism is that the tutors are not meant to replace all instruction. This type of instruction is very effective at teaching procedural (algorithmic) skills consisting of a sequence of steps. It turns out that a lot of the math what we expect middle school and high school students to learn (and demonstrate on standardized tests) is of this nature.

1. In primary grades the major emphasis is on recall of basic math facts (e.g. 3 + 5).
2. Many kids aren’t wired for simple recall (they aren’t good at memorizing). I’m not.
3. Commonly used instructional approaches don’t give the immediate feedback essential to developing simple recall (e.g. do this worksheet which will be graded later today or tomorrow).
4. Because of the heavy emphasis on simple recall kids perceive that it represents all of math.
5. This Simon-Says approach to math instruction often continues throughout a student’s public education.
6. By the time students are given the chance to engage in other types of mathematical thinking, they have developed a negative self-perception that pre-disposes them against trying or learning math.

I attended a discussion group facilitated by Damon Bahr of BYU that focused on finding the balance between teaching for procedural and conceptual knowledge. I also attended a presentation by Blake Peterson of BYU that reported research comparing the presence in US and Japanese preservice training of discussion related to classroom management.

Summary

• Searching and Publishing – People come to the MTDL to find and share resources
• Overcoming Isolation – People come to the MTDL to help overcome isolation
• Discuss Development – Talk with others about the development of math software

Personas

• Teacher Developer
• Professional Developer
• Educational Researcher
• Inexperienced Developer
• Hobbyist Developer

Metaphor and Principles

• Workshop metaphor
• Design for multiple roles
• Design for multiple levels of expertise
• Provide activity indicators

I like the workshop metaphor, though I think that perhaps there are better. I can’t really discern the implication of designing for different roles and different levels of expertise. I lilke the idea of activity indicators. I realize that this is an area of recent interest throughout the field.

What are your thoughts as a teacher? Which would rather have, a prepared lesson (not a description of a lesson, but the actual materials to use such as black line masters, manipulatives etc), or a catalogue of pieces (e.g. CDs full of clip art, handouts, and worksheets) from which you can assemble lessons?

Does your answer change if we are talking about web based lessons that can be easily adapted?

I think a careful look at how technology could change the role of the teacher would alleviate these fears and in fact excite teachers. I have teachers responsd to Saxon and similar heavily scripted curriculums by saying that it turns them into a robot. Some teachers react to to scripted online instruction in similar ways.

Technology contrasts heavily with this. I believe that a major opportunity that technology offers is to allow us to deliver mastery learning (teach students where they are at and don’t move on until they do) in a way that also encourages understanding as opposed to rote learning of facts and procedures.

My perception is that direct instruction approaches to mastery learning try to narrow the gaps between learners, but often results in rote learning (and higher test scores by the way). Side note: I take it for granted that understanding is much harder to measure than recall and for that reason we assess recall and results based learning (NCLB) values instructional methods that produce it.

I believe technology offers an alternative to trying to homogenizing all learners, it can help teachers manage the logistical chaos of trying to let large numbers of learners go at their own pace. The role change I see of teachers is from dispenser of information to diagnoser of student understanding and customizer of instruction. To me this does not denigrate or demean the teacher, rather it seems to be immensely freeing. It allows you to be a fellow sojourner for understanding. I’d be interested to hear the opinions of lots of teachers on this? I’m sure one reaction, is yeah, that should great but it is pie in the sky.

• Repetition
• Visualization
• Procedures for generating the answers
• Counting by __ (for multiplication)
• Using reference values

If anyone has other strategies or pointers to online resources that are good examples of these strategies to teach basic math facts, please share.

The most common strategy it seems is recall by drilling and killing like in my boy’s school. One effective way to do this is by using flash cards. The Surfnetkids homework help newsletter lead me to their math facts flashcards page. The problem with most flash cards is that they don’t teach any alternative besides recall. It seems like flash cards could be created that taught alternate strategies.

If you were given a small amount of resources, what would you do?

Who should be targeted? Those that are flunking out, the average student, the high achiever? What age level?

A number of opinions have been expressed:

• Technology can help, but in itself it is not the solution.
• Integrated curriculum can help motivate students, but in itself it is not the answer.
• Highly structured, programmed instruction is needed to help students gain basic skills, but if it is used in isolation it results in students loosing interest and not learning higher order skills. It should be used in conjunction with other exploratory and discovery methods.
• We need to start early with students. If we wait until middle school or high school it is probably too late. A contrasting opinion is that we can in fact help change the culture of middle school and high school classrooms, which results in improved student perspectives of mathematics (though possibly not improved math scores).
• A professor of secondary ed stated that he thought what made the biggest difference (in student learning and attitude?) is the interaction pattern used by teachers. He deplores the IRE (teacher initiates a question, student responds, and teacher evaluates the students response). He believes that this results in students feeling like everything they say will be evaluated, and so they stop saying anything.
• We need to support individualized instruction (a la Mastery Learning). Teach each student where they are at, and don’t move on until they learn it.
• In order to support individualized instruction, a detailed model of what each student knows should be developed and follow students through their educational career. This model should be used by teachers to help individualize instruction.

To me, the challenge seems to be to develop a model, materials, and support system that teachers will buy into, that allows them to individualize instruction. Does anyone out there have examples of how this has successfully been accomplished? I am also interested to hear opinions on the points I have listed.