math

My Philosophy On Math Pedagogy, And Other Tidbits [Edutopia]

by Jose Vilson on March 21, 2013

Here’s an excerpt from my latest at Edutopia (including a diss on Robert Marzano and the like). It’s about engaging math teachers:

Keep This Rule of Thumb: Complete, Consistent, Correct

By “complete, consistent, correct,” I mean we should allow multiple pathways to a correct answer that a) allow for full understanding of a given procedure, b) can be used time and again without fail, and c) actually have a sound basis in math. While it sounds constricting, it removes some of the limitations we’ve set for ourselves when looking at student work.

For instance, when finding 25% of 80, the most basic thing we can do is turn the percent into a decimal (0.25) and multiply that decimal by 80. The result is 20. Yet when I presented this problem to a seventh grade class just learning this, one of the students astutely observed that 10% of 80 is 8, and 25% is just 10% + 10% + 5%. They doubled 8 (16), then took half of 8 (4), and added the results (16 + 4 = 20).

Some teachers might mark that incorrect because it doesn’t follow the exact procedure they asked for, but we really should accept such a response fully, not just because of the answer, but because the procedure the student used works time and again.

Read more here. Share with your friends. Comment. Thanks!

Mr. Vilson, who wants Friday to be over already, and it’s only Thursday …

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Why Learning Math Is Political

by Jose Vilson on October 23, 2012

Paul Ryan Cartoon

For my own professional development, I picked up the book Radical Equations: Math Literacy and Civil Rights by Robert Moses. The book equates the struggles Moses had with developing voter representations amongst the most underrepresented in the South with developing math knowledge / pedagogy into the curriculum in America’s classrooms. Observe:

So algebra, once solely in place as the gatekeeper for higher math and the priesthood who gained access to it, now is the gatekeeper for citizenship; and people who don’t have it are like the people who couldn’t read and write int he industrial age. But because of how access to – the learning of – algebra was organized in the industrial era, its place in society under the old jurisdiction, it has become not a barrier to college entrance, but a barrier to citizenship.

When people tell me that they weren’t born to do math, a small part of me wonders about the ramifications of any student who consistently tells themselves that they don’t have either the capacity or the potential to do any of the maths we learn in schools. Because of the changing economy, the entire way our communities view math needs to change.

Equally as important, we have to tell our communities that we can and will learn math.

See, the most dangerous thing about education is that it has the potential to dispense knowledge to others. When people actually learn about their histories, their legacies, and their worth on the planet, they become critical thinkers and agents for change.

It’s a small part of the reason why those of us who think critically seriously wonder if the confusion, bureaucracy and diminishing budgets in education serve to assure inequity rather than relieve it.

This is also why math is the answer. Governments, media, and corporations cloak their most important operations in advanced mathematics. We can no longer settle for our communities only getting the four operations. Unlike literacy, people generally consider math a subject that no one needs to master unless they’re a specialist of some nature. Yet, without a solid foundation of math, our most impoverished students have less options for their futures economically and politically.

We will do better.

Jose, who thanks each and every one of you for voting this as the best Latin@ Education Blog in all the land …

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Teaching Exponents In Eighth Grade, Part 1: [Old To The Now]

by Jose Vilson on September 11, 2012

India Power Outage

Every so often, I’m inserting some posts on pedagogy, especially for those of us who aren’t as math-inclined. If you find this helpful, just let me know in the comments.

Today, I’m differentiating between the old way I used to prepare the students for a lesson on exponents and the way I do it now.

Old Way

1. Ask “What’s an exponent?”
2. Here are the five mail rules you need to know.
3. Apply them.

New Way

1. Ask “What’s an exponent?”
2. Use their definitions with a bit of refinement.
3. Get their thinking about a few cases involving exponential, expanded, and standard forms.
4. Have them deduce what 3¹ and 3° power would look like.
5. If they can see the operation of division, then prompt them to ask if this works for every case in general.
6. Use the same thinking from 3-4 to get the values for 3¹ and 3°.
7. Let them establish the (real) first two laws of exponents.
8. Use similar tabular thinking to deduce product, quotient, and power of a power laws.

Usually, when students are just given the laws, there’s rarely reason why we give it to them. It gets shoved on their laps, especially x to the zero power. Some of it might be because of a lack of understanding on the part of the teacher, but I also suspect it’s because we don’t see the importance of taking them through similar steps that an actual mathematician would go through.

But I changed that last year, and into this year.

My main objective was to make sure to establish the foundation for every argument we will make about exponents from here until October.

Here’s a sample of what I mean:

Exponential Expanded Standard
x3 x · x · x n/a
x2 x · x n/a
x1 x n/a
x0 1 1

 

I’m hoping for a couple of things here. First, I hope they see that, as the exponent decreases, the amount of x’s in the expanded column decreases, just to solidify our definition of “exponent.” Also, I hope they see some sort of division (making the connection between multiplication and division matters here). I use a specific case first, and it’s usually a positive base, just so they’re familiar with the numbers, then work my way towards this.

By the way, once they see the 1 at the end, they should also be able to see that Icould have had a multiplication of 1 all along since 1 times any number equals that number.

Hope that helps. Best,

Mr. Vilson, who will have a post like this about once a week …

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Great. Another Non-Math Person Complaining About Algebra.

by Jose Vilson on July 30, 2012

Holy cow, Andrew Hacker. Shut up!

OK, that was a bit harsh. Warranted, but harsh. Say what you want to, but lower your voice a few decibels. Frankly, I didn’t care much for your rhetorical question, but you had to write it in the New York Times, adding a semblance of legitimacy (if not outrage) to your argument against teaching abstract math to kids. The crux of your argument, that we shouldn’t teach algebra except to those of us who want to get deeper into the math and that we should instead focus on useful life math, whatever that means. I won’t spend time debunking any of your claims because Dan Willingham did so rather convincingly today.

Rather, I’ll ask: if we taught the humanities (presumably English, arts, and yes, political science) in the same way you’re suggesting for math, how would that look like?

If you’re OK with English and language arts taught this way, then let’s focus 100% fully on non-fiction texts … like “How To Operate Your VCR” and “The Intricacies of Setting Up Your Passport.” Useful, and often complicated, these texts would surely be of worth post-college, especially for kids who never get out there. They’d have no need for Jon Steinbeck, William Shakespeare, or Julia Alvarez; their texts aren’t very relevant to what students actually encounter on a daily basis, so we’d leave it alone in the hopes that they don’t have to think abstractly.

If you’re OK with political science taught “with relevance,” you’d teach them about a few politicians here and there, their initiatives, and ow the media views the two prominent candidates. You’d never have to discuss political ideology, how bills are created, and about the Electoral College. About 58% of citizens of voting age actually vote anyways, and they don’t use with strategy in mind, but their message, and how their favorite news channel views the candidate. They’d never see The Federalist Papers, or the Emancipation Proclamation; all people need to know is that they’re free … so long as they’re not imprisoned. It doesn’t matter anyways, because that’s not very useful to the everyday citizen. Just the ones it affects.

If you’re OK with the arts taught this way, then … we wouldn’t have use for the arts, really. Unless you show a gravitation towards creativity early on in age, then everyone should get to paint the fruit basket with the shadow to show a level of mastery in the arts until they get to college.

Most math pedagogues would agree that we need to reexamine the way we approach math for our students, especially those who get taught math as a way of passing the deluge of tests at the end of the year. But you didn’t see that, Hacker. You chose instead to spew what I’m dubbing a humanities elitism that perpetuates attitudes this culture espouses about math. If you’re not interested in math, that’s really up to you, but don’t call for a disbanding of math if you don’t want anything to do with math. Instead, join us in saying that we need to infuse more mathematics into the socio-cultural discussion of civilization.

Because that’s a much more compelling argument. Every civilization’s greatest contributions involved math, from the architecture to medicine, fostering a love of math meant having an understanding of how society’s form, whether abstractly or otherwise. Don’t be the loser who asks us to dummy the math down to “usefulness.” Be a voice that asks us to transform the teaching of math.

Jose, who strives for relevance as a teacher …

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Stay In Your Lane [A Math Teacher's Lament]

March 27, 2012 Mr. Vilson

Why do people stigmatize math teachers? It’s bad enough we teach people that they’re either math people or they’re not (patent lie, I promise you). Now, we’re even limiting math teachers to the fields in which they can excel. They stereotype (!) math teachers as having hobbies like playing piano (fractions!) and read xkcd (might [...]

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Beware The Calculator

October 23, 2011 Mr. Vilson

As a math coach, people always want to catch my ear about the use of a calculator. I ought to put my voice into the argument since … well, that’s what I do here. The two sides to the argument go as follows: 1) Students should use calculators because the machine can already do it [...]

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Love Is The Base, Not The Exponent (A Theorem Semi-Proven)

May 2, 2011 Jose

Recently, I’ve been going over exponents with my students for the big math state test next week. Not sure if it was the chalk hitting my nose or the positivity I’ve surrounded myself with, but I’ve been thinking lots about this idea of love. Thus, I was prompted to put out this thought a couple [...]

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Finding A Needle in A Stack of Needles: A Solution to the Racial Achievement Gap

April 25, 2011 Jose

In the category of “Yes, That Makes Complete Sense,” David Kirp of The LA Times reported that scientists have figured out a 1-hour “fix” for our most disadvantaged students: encourage them while they do it. Combined with the latest article from The New York Times regarding Jump Math, this piece almost made me smack the [...]

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Creativity Is Almost Dead … and Non-Educators Want To Kill It

January 18, 2011 Jose

After posting a few crosswalk documents between the New York State math standards and The Common Core Standards on this site and at my place of employ, I’ve been very involved in understanding how these mandated national standards will transform our way of teaching students, and how we need to get parents and students involved [...]

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The Idea of Race As A Number Line

November 14, 2010 Jose

I recently had the delightful experience of hanging out with the alumni of Nativity Mission Center, the Lower East Side based Catholic middle school, most of whom I only get to see once a year at these events. This time felt different in a good way; most of their discussion was around my blog and [...]

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