[My student learning goal in this activity is to have them translate a word problem into visual and/or symbolic (i.e. algebraic)representations that give each one of them the opportunity to demonstrate how they would solve different types of linear equations/inequalities.]

Week Two Question:

From the other projects you have seen in this seminar, which projects have given you ideas about your own activity?

From Angie M's poster, her creative problem solving techniques leading to creative thinkers (I believe one of the articles described this as moving from novice to expert thinking) and the clever way she is getting to know her students better (both motivationally and mathematically) by using the discussion function in WebCT Vista.

From Carolyn S's poster, her neat ideato use electronic porfolios to save her language journals, research papers, and "thought papers." This seems to coincide with some of our readings to date on reflective learning and a compilation of documents into an electronic archive.

From Gordon W's poster, his special insight to integrate technology into the classroom, not make it some bullet points but rather to "embed" these skills into specific teaching techniques and methodologies. Also, his idea on going "public" matches some of the readings to date.

From Jean D's poster, I likeher idea that different "maps" (data) lead to different conclusions based upon the information presented and the choice of solving used (deductive vs. inductive reasoning).

From Alison + Jackie's poster, both their primary source focus combined with using WebCT Vista are just right for where I am striving to introduce students to the idea of both gathering information and ways to manage said information.

From Patricia H's poster, I am looking forward to seeing how she will use the constructivist approach as I am attempting that same technique to some degree in my mathematics courses.

From Sherri L's poster, her focus on communication (chat, oral and group presentation) as opposed to regurgitation of stale facts sounds both fun and interesting.

Week Three Question:

What will constitute as evidence that students have reached this goal?

Reference: My evidence template for thinking about student processes.

This question is the toughest to date as it challenges some of my traditional thinking about assessement and evaluation. Evidence to me in this context will mean the documented strategies that students use to solve linear equations/inequalities. (This would involve BOTH qualitative (accuracy) and quantitative (number of possible ways to derive a solution) demonstration (I am thinking about Gardner's work on multiple intelligences). Some will simply translate a word problem directly into an algebraic expression but I am thinking that I want to see more visual in the sense that they "draw" the word problem into a visual representation, THEN try to construct their own equation/expression that may or may NOT use traditional algebra variables. Finally, then to graph said solution and to explain (in words) what that graph describes relative to the question posed in the word problem.

Week Four: Cognitive Apprenticeship

Where and how will you model in this activity?

One of the modeling techniques I will use is to have students watch the "holistic thought processes" I employ to solve a complete problem. This begins by having students "drop their pencils/pens" and simply WATCH as the entire process reveals itself to them. This is a good modeling strategy (particularly in mathematics) for a number of reasons. Many students have copied board work as it is presented and never engage higher level (Bloom's Taxonomy) thinking skills because they are busy copying what I am writing...and then cannot understand why they STILL do not understand the process when they review their notes later. Also, during this "holistic thought process", I welcome ideas from the audience and intentionally begin using a process that does NOT work. This helps/models for my students want I want good logistics/mathematicians to note: namely that the "correct" solution process does NOT always make itself evident in the initial problem solving process.

Another area I pay attention to model well is the ability to "draw" the problem (I also somewhat sarcastically point out that although they cannont possibly draw as well as me that they should still make the effort) as this helps to visulize both the information, the problem, and the possible solution strategies. Translating this picture into symbols is the very gist of what makes an expert problem solver (as opposed to that student who simply copies an equation and cranks out an answer) versus a novice.

Finally, I model defending my answer (or rather the answer that a student gives) when they come up with a particularly creative or alternative solution to the problem at hand.

Week Four: Cognitive Apprenticeship

What kind of scaffolding will you provide to students?

Once a problem has been articulated and then "drawn out" visually, I model jotting down the pertinent data and considering what we may already know about how that data can be linked. For instance, in linear algebra, we discuss that while two points make a line, one needs at least THREE points to assure that arithmetic mistakes are not committed and that the resulting graph is not ONLY linear but visually answers the question and makes sense.

This idea of scaffolding is used in ways such as AREA computation. For instance, while all the students know that base times height equals area for a square, I extrapolate that idea for other shapes (rhombus, parallelogram, trapezoid, triangle) and then ask them to SUPPOSE / WONDER if it also works for circles...and then SHOW them that it does! This reinforces the concept that AREA is scaffolded for multiple shapes and is in fact the beginning idea for Newton's calculus.

Finally, I use the scaffolding principle to reinforce algebraic ideas involving squares, roots, and absolute values. We consider a difficult problem in one of those areas by reducing it to a simpler problem and then SCAFFOLD that concept to the problem being attempted.

Answer the following questions:

What big questions are you asking?

[How does one begin to think mathematically?]

What narrow questions are you asking?

[How does a portfolio help in demonstrating that mathematical prowess?]

How will you gather evidence?

[Reflections in WebCT's discussion board, PowerPoint Presentations of synthesis, and graphing of spreadsheets.]

How will you analyze the evidence?

[How succinct, clear, and competitive a response as compared to their peers.]

How will your evidence and your questions interact?

[Evidence is the factual data that answers my questions.]