Comments on: Creating a Need for Coordinate Parentheses & Combining Like Terms

By: Kristen Peterson

Kristen Peterson — Thu, 19 May 2016 21:18:25 +0000

“Why did we invent this skill?” is such a powerful question to begin lesson planning with. Whether the answer leads to a practical ‘real-life’ skill or a necessary mathematical skill, offering our students a purpose for why we are learning promotes more engagement and better retention. I love the idea of presenting the headache to the students first, before revealing the aspirin. I also applied this idea to like terms, but took it a little further with polynomials. I started teaching polynomials as representations of numbers. I started by expanding a three digit number into scientific notation (965 = 9 x 10^2 + 6 x 10^1 + 5 x 10^0) and then replacing the 10 with a variable. I told my students that polynomials to numbers in different bases, we just don’t know what the base is — that really baffled them! Their connection between mathematical representations and numbers has been growing stronger since I implemented this strategy. Now my lessons on polynomials are able to constantly be connected back to what we would do with numbers. Like when we add two numbers we add the place values, so when we add two polynomials we can only combine terms with the same exponent since that represents the place value. Thanks for the inspiration!

By: education realist

education realist — Sat, 14 May 2016 15:42:10 +0000

While I often use activities to reveal the need for a particular process, I had an idea last month on showing how two dimensions expand to three, and why we need it. It was an ad hoc lesson that turned into a pretty epic discussion for one of my algebra 2 classes.

https://educationrealist.wordpress.com/2016/05/06/great-moments-in-teaching-the-third-dimension-part-i/

So epic that I’m working on part 2. It’s long, but an easy read.

By: Bill S

Bill S — Sat, 14 May 2016 02:04:58 +0000

Or could it be that

-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3

is meant to be

(-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3),

a single point in 17-dimensional space?

By: Dan Meyer

Dan Meyer — Fri, 13 May 2016 21:38:26 +0000

Miles Calabresi:

You correctly pointed out that the need already exists, but you continue using the term “create” – why not say something like “highlighting” the need?

I suppose my meaning is "creating the need in the student's head." The need may already exist in the world, or the development of math, but not in the student's head. I'll have to think harder about my choice of words. Thanks for the comment.

By: Miles Calabresi

Miles Calabresi — Fri, 13 May 2016 16:14:06 +0000

I remember on an old post that a previous comment suggested avoiding the expression “creating need” so we don’t look like we’re rationalizing the existence of an otherwise useless set of skills or facts (and indeed, we’re not). You correctly pointed out that the need already exists, but you continue using the term “create” — why not say something like “highlighting” the need? We’re not so much creating the need for them as pointing it out in a plausible context. It already exists; they might just not know about it yet because they haven’t faced a problem where the need is apparent.

The headache needs a cure, but students might not know the headache exists until they try a worthwhile activity (for instance, running track, if I may stretch the metaphor) that brings it into focus. Is that not more of the business we want to be in? Guiding the student through worthwhile activities, letting them discover already existing headaches and then presenting the aspirins that have been developed so that they can see the usefulness of these cures themselves? Sorry to bring up an old topic, but “creating headaches” sounds a little more dishonest than what you’re actually doing.

By: danburf

danburf — Fri, 13 May 2016 15:56:43 +0000

I blogged on this about a month or so ago also! I hooked students a little more by telling them I would race them to see who could finish the fasted, I gave ’em a 30 second head start just to even out the playing field.

https://danburf.wordpress.com/2016/04/11/adding-polynomials/

By: Lane Walker

Lane Walker — Fri, 13 May 2016 13:13:48 +0000

Anne’s Youtube example is priceless in my experience. The emotional hook provides a faster refresh of the “need” than the symbolic alone. For the value of an emotional connection, research is on Anne’s side. A 60-second portion of the video Anne provided is likely to save time in recall later.

Emotion enhanced retention of cognitive skill learning
by Steidl, Stephan; Razik, Fathima; Anderson, Adam K
Emotion (Washington, D.C.) (1528-3542), 02/2011, Volume 11, Issue 1, pp. 12 – 19

Brain mechanisms of emotion and emotional learning
by LeDoux, J E
Current opinion in neurobiology (0959-4388), 04/1992, Volume 2, Issue 2, p. 191

By: Mia L.

Mia L. — Fri, 13 May 2016 01:55:17 +0000

This is a real interesting idea. As a college student persuing a career as a mathematics teacher, I have been struggling to find ways to look at some concepts in the real world so my future students will find it important. This is an interesting viewpoint. Thanks for sharing!

By: l hodge

l hodge — Fri, 13 May 2016 01:04:32 +0000

It seems to me that the first question is more of an extension of 10 + 200 + 30 + 500 than oranges and apples. Oranges & Apples would be more like 3x + 2y + 5x. Just as we wouldn’t say 10 + 300 is 400, we wouldn’t say x + 3x^2 is 4x^2.

By: Fred Thomas

Fred Thomas — Thu, 12 May 2016 22:00:39 +0000

Dan, I understand and agree with your comment that reality (and familiarity) are relative. One of our biggest concerns with using color was the fact that an average class has at least 1 or 2 students (usually male) who are color blind. That’s part of the reason we always display graphs as well. The only part of color theory that comes into this at all is the principle most students learned with crayons that red + green appears yellow. The typical confusion is in the understanding of mathematical functions–that 2 expressions which appear very different can actually be equivalent for all values of x. Nothing motivates every student, but we have found that a large number enjoy producing dynamic color displays–even people like me with very little artistic sense.

Sarah: I put a screen-capture jpg at http://www.mathmachines.net/temp/ColorFun.jpg

Kevin: That precisely what we want. “Simplifying expressions” and such is really all about selecting a suitable form for communicating. That includes communicating to other math students/teachers (which is what many seem to mean by “standard form”), but we think it should also include communicating mathematically with computers. It also includes getting things into a summary form for one’s own mental processes.