Learning by understanding

I just read responses on a Linked-In forum about how to teach factoring. The answers were full of steps and technical details. Not one linked factoring to something the students knew, or gave them a reason to learn factoring.This was my response:

Before you even start factoring, make sure students have a reason to use it, that they understand WHY they're factoring. Have them graph a simple polynomial equation, like the square of (x+3), using a T-chart for values, and find the zeroes.

I think it is extremely important for the students to understand factoring in polynomials is the same in factoring, say, 96.

They need to know that polynomials are the result of multiplication, so a good way to start is to have them multiply simple things, like the results of the graph they did and other squares, and then, for example, the sum and difference of 2 terms, to see if they discover the pattern, then give them the same problem, plus some similar ones, to factor. Then move on to things like (x+1)(x+3), saving ones with a coefficient other than one for later.

Using Algebra tiles is another way to visualize what's happening, and using the "box" method, which I like for multiplication of polynomials, because it helps keep them straight, is also a good help to reverse the multiplication, which is similar to the Algebra tiles.

But if they have no clue why they are factoring, it just adds to "when will we ever use this in real life?" which is a very legitimate question. They need to know what those zeroes can be used for, too. I'm not sure all that many Algebra I teachers can carry the discussion that far.

When students understand why they're factoring, what it's used for, how the polynomials are graphed and how they came about, I think they will be much more open to the fun puzzle of untangling them as factors.

Rigor mortis or rigor percipiare

OK, the Latin in the title is my own. The second half is supposed to mean "tenacity to learn" in my version of Latin. But the title was inspired by a very thoughtful article today by Linda M. Gojak,  President of the National Council of Mathematics Teachers, called "What is all this talk about Rigor?".

Evidently people have been writing that the Common Core requirements for mathematics include the word "rigor," although she says it is not there. She and a group of math coaches investigated the meaning of the word (as in rigor mortis, but more appropriately “thoroughness”and “tenacity”) to see how it can be applied to the teaching of mathematics. They came up with the following table, which I have borrowed intact from her article.

Learning experiences that involve rigor …	Experiences that do not involve rigor …
challenge students	are more “difficult,” with no purpose (for example, adding 7ths and 15ths without a real context)
require effort and tenacity by students	require minimal effort
focus on quality (rich tasks)	focus on quantity (more pages to do)
include entry points and extensions for all students	are offered only to gifted students
are not always tidy, and can have multiple paths to possible solutions	are scripted, with a neat path to a solution
provide connections among mathematical ideas	do not connect to other mathematical ideas
contain rich mathematics that is relevant to students	contain routine procedures with little relevance
develop strategic and flexible thinking	follow a rote procedure
encourage reasoning and sense making	require memorization of rules and procedures without understanding
expect students to be actively involved in their own learning	often involve teachers doing the work while students watch

This is what teaching should be about, although I wish they'd come up with a better word, since rigor also means "rigidity" and "suffering," according to their research! That sounds more like the drill & kill methods I experienced as a student teacher, and which they define as not having rigor!

The left column should apply to all learning experiences, not just in mathematics. Children are born with curiosity, a need to be challenged and a lot of tenacity. This I experienced this past summer as my year old granddaughter tried again and again to crawl across a very difficult door opening (threshold!) until she figured it out. She was enormously proud of herself as well. I was amazed when my teacher sister-in-law got impatient with my granddaughter's efforts and just lifted her over the threshold. But the child went right back to working it out after that.

We must provide thresholds for students to cross, where they can see intriguing unknowns that awaken their curiosity. Children who are helped to everything must lose their love of a challenge and their curiosity early on. As a high school teacher I find that I have to help students regain their curiosity and encourage them through a challenge until they proudly can see they have overcome it. That is how we all learn!