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** By: Guest Author Pamela Weber Harris**

Part 1: Important Questions

“This is good. You are right. We’re convinced. Why isn’t this the way math is taught in more places? Why don’t more people get this?”

I get this question often at workshops, presentations, and keynotes from teachers, administrators, and my university students, but recently I have gotten this question poignantly from parents.

I work with 2 independent schools, one in Austin, Texas, and one in San Francisco, California. Both schools know that it’s wise to keep their patron parents well informed, especially when something is potentially controversial. Both schools have opted to hold parent meetings to explain their approach to teaching mathematics. One school is holding parent classes. These are highly successful people in very tech savvy, entrepreneurial places. They begin the conversation this way:

“Why does my kid’s math instruction look different than mine did? If it was good enough for me, why isn’t it good enough for my son/daughter? Why mess with a good thing? I’m not always sure anymore how to help my kid because it’s so different—that can’t be good.”

I get it. Good questions. And to be very clear: I would never want to drive a wedge between parents and their children. That’s not me. In fact, I want to empower parents to help their children be young mathematicians. To that end, we do a little real math together. I walk them through some of the advantages of really owning mathematical relationships (real math versus fake math), especially the way their student could understand higher math better, and something amazing happens. The attitude in the room shifts dramatically. No longer is the question ‘why would anyone teach math this way’. Instead, emotional parents demand to know ’why doesn’t everyone teach math this way? Why would you teach it any other way? What went wrong in our schools?”

Remember, these are highly successful people – the traditional way of math instruction “worked” well for them, right? It is true that many of these folks were in my group X, they constructed real math relationships on their own despite traditional teaching. They are the ones who don’t see the reason to change anything. They think:

•Doesn’t everyone look at math the same way I do?

•The teacher shows you a rule but you don’t have to use it, not when you can just so obviously figure it without the rule.

•Of course you would never add 99 + 47 by lining them up – who would line the numbers up when you can so obviously just add 100 + 46 to get the same answer? No one would mindlessly apply a rule like stack the numbers and add right to left if you can so obviously just see an easier way. And everyone can see that, right?

•Of course you don’t need to rote memorize multiplication facts (like you would a password or in the old days, a phone number) — it’s perfectly respectable to refigure those you don’t have at your fingertips as long as you can do it fast enough to not get bogged down. Everyone knows that.

•When solving x + 4 = 9, why would you subtract 4 from both sides? Silly, just reason that you know that 5 and 4 is 9. Show your work? Nah, I don’t need to in this case.

•And so on….

HOWEVER, even in this group, most of them reach a point when they say, “Well yeah, that stuff, that stuff there you do need to rote memorize” But they say that in places where they somehow were not able to reason about the relationships anymore (even though the relationships are still figure-out-able), they just suddenly weren’t easily enough figured out without expert help, so these folks reason that since it doesn’t come to them naturally, it must needs be a thing to not understand but to just take on faith (a place where group Z folks are very familiar)

There are some group X folks who don’t reach that point – I think they are the folks who go on to be mathematicians, physicists, computer scientists – for them it’s all still so reasonable and related and figure-out-able. They just keep on reasoning about it all. AND they don’t understand that for many, many of us it doesn’t happen so naturally.

The good news is that it CAN come! The rest of us can also do real math! We just need someone who knows some real math to actively help us construct those relationships. At a minimum, it would be helpful if our teachers knew the difference between real and fake math and at least acknowledged that in their teaching.

It’s always a little (a lot) fun to have the less “mathy” spouse pull me aside, and with this almost secretive, embarrassed look, tell me that they have always felt slightly less than because they were only good at fake math (group Z) or not good at fake math at all (group Y). They’ve felt marginalized either because all they had were the memorized rules that didn’t seem relative to life at all, or they never even had that—just mixing and matching rules because none of it ever made sense. This is fun because then the spouse smiles, saying, “I can do what we did today. I can think that way!”

So, the group X parents say, “You mean I could’ve gone even farther if someone would’ve actively helped me make more connections? You’re planning to help my student really own these relationships—they won’t be on their own to figure them out, so they can go farther too? Yeah, I want some of that. I absolutely want my child to be able to think and reason the way I did and if they can even go farther, faster than I did, bring that on!”

And the group Z parents say, “You mean I could have understood all of that stuff that I rote memorized? Math could have been something relative and interesting to me? It could’ve changed the way that I view the world?” and sometimes, “You mean, I could’ve chosen the life path that I actually wanted but it was cut off from me because I couldn’t do the math? Well, I want that for my child (and me!)! Let’s do it! Do you teach parent classes? I wonder if our school will offer parent classes?”

And the group Y parents say, “You mean the tricks that I did to get by because I never understood what was going on – some of that was real math? I was actually doing real math while I was being told that I was bad at math? You can help my student use the same relationships and more and better ones and not make them feel like an idiot in the process? PLEASE do that for my kid! I’ll sign up for your parent classes!”

Now, undertaking how to teach this real math is not a trivial pursuit. That’s fodder for future blogs, but for now, I just wanted to clarify the conversation between (at least) three radically different perceptions of math. I hope this has helped.

Stay tuned for Part 2 next week – Rote Memorize versus Knowing!

*Pamela Weber Harris is co-author of the third edition Discovering Advanced Algebra and Discovering Algebra Problem Strings. She is also the author of Discovering Advanced Algebra Problem Strings and various other math books. A former secondary math teacher, she currently teaches at Texas State University, is a K-12 mathematics education consultant, a T ^{3} (Teachers Teaching with Technology) Instructor, and an author and coauthor of several professional development works. This blog was previously published on her website Math Is Figure-out-able.*