As we worked together, I observed my student slowly replacing her overwhelmedness with genuine interest and enjoyment. She started tackling difficult proofs, and her eyes would light up with excitement and understanding when all the pieces fit together. We were a few months into the long-term project of slowly building up her understanding when her dad made a decision, without my input, to pull her out of her geometry class because she was “in danger of failing.”

Even though my student understood the material, she got so nervous on the tests that if you just looked at her test scores it looked like she couldn’t do geometry. But she could! She consistently did it perfectly, by herself, in our tutoring sessions! When we reviewed her tests, the material made sense to her once she was outside the testing environment. And I was confident that she could pull up her grades if we continued working together.

In the sessions before her dad switched her math classes, I asked my student what she wanted to do. She told me that her choice would be to switch to another geometry class at the same level, but just with a different teacher. But for whatever reason, she didn’t perceive this option as being available to her—I’m not sure if it was a scheduling issue, a political issue, convenience, parental pressure, or something else.

What her dad decided to do was switch her into a “problem solving” class. My student and I met one last time after she switched into this class. Her book made me want to cry—it was a bunch of reasoning problems about things like Corey the Camel carrying bananas across the desert. (I’m serious. It really had problems featuring Corey the Camel.) The material was basically elementary-school level—no algebra, no geometry. Just simple word problems. Maybe the geometry class was 15% too hard for her, but this “problem-solving” class was about 100% too easy for her.

After that session, I did something I’d never done before. I wrote an email to the dad, explaining as diplomatically as possible and at great length that I really didn’t think this new class was appropriate for his daughter. I explained how much his daughter loved working on Geometry and was learning a lot even if she wasn’t yet testing well. And I expressed my concern that this class would limit her in the future, since basic algebra and geometry were prerequisites for so many other disciplines.

I wrote, wouldn’t it be better for her to take geometry and learn some geometry, even if she got a “failing” grade, than for her to take a class where she would learn nothing at all?

Her father’s response was vituperative. How dare I suggest that he allow his child to “fail!” And I never saw either of them again. I honestly don’t know how I could have handled this differently, but my heart still breaks for that student.

In comparison, another student’s family handled the perceived threat of failure very differently. I was working with a ninth grader who was struggling with Algebra 2 because her elementary school had failed to teach her basics like long division (she was supposed to “figure it out for herself”.) I believe when we started working together she was failing the class.

I was extremely proud of how hard this student worked, and she finished the year with either a low B or a high C. At the end of the year, her algebra 2 teacher suggested that she consider voluntarily repeating the class, just to strengthen her skills before moving on to more advanced math.

My student chose to repeat the class, even though she felt at least a little bit embarrassed to be the only sophomore in that class full of freshmen (at least I figured this was the case since she joked about it). She chose to learn instead of to look good. And her parents supported her. I was so impressed with her integrity.

By the end of her second time through algebra 2, the material that had brought her to tears the previous year did not phase her at all. But I think about the other family,
and how they didn’t want to let their daughter fail. Did that student ever get another chance to love geometry? Was she stuck in remedial math classes for the rest of high school? What did she did she do for her math requirements in college? I wish I knew. I hope she got another chance, instead of internalizing a message that she “couldn’t do math.”

Why do we protect our kids from failure, even to the detriment of their own learning?

Related Posts:
I cried myself to sleep over my algebra homework
Algebra Tears
“I Think I See A Mathematician!”

I split the video into 8 brief parts. Each part features one practice problem, fully explained and demonstrated on the whiteboard.

If you, your family, or your friends would like to see me make an instructional video about a particular math topic or type of problem, leave a comment to nominate your math problem for its very own video!

And if you like the video, please feel free to click on the “heart” to show that you “heart” it. <3

#1 – Multiplying Binomials with the Box Method (alternative to FOILing) from Rebecca Zook on Vimeo.

Click on “more” for the other parts of the video — four more examples, extra practice problems for you to test your mastery of FOIL, and answers to the extra practice problems!

#2 – Multiplying Binomials with the Box Method (alternative to FOILing) from Rebecca Zook on Vimeo.

#3 – Multiplying Binomials with the Box Method (alternative to FOILing) from Rebecca Zook on Vimeo.

#4 – Multiplying Binomials Using the Box Method – extra problems for you to practice with on your own from Rebecca Zook on Vimeo.

Multiplying Binomials with the Box Method – #5 – check your answer to the first practice problem from Rebecca Zook on Vimeo.

Multiplying Binomials with the Box Method – #6 – check your answer to the second practice problem from Rebecca Zook on Vimeo.

Multiplying Binomials with the Box Method – #7 – check your answer to the third practice problem from Rebecca Zook on Vimeo.

Multiplying Binomials with the Box Method – #8 – check your answer to the fourth practice problem from Rebecca Zook on Vimeo.

Two words: Nancy Oliver.

My amazing ninth grade geometry teacher.

Nancy taught in a classroom where a former student had painted a colorful mural of the trig mnemonic “SOH CAH TOA” as a tribute to her on the back wall. In her room, I felt relaxed, focused, and safe. I had just spent three years of middle school algebra feeling panicked, utterly frustrated and incompetent in the math department. But with her instruction, I finally felt like math was something I was completely capable of doing.

How did she do it? Like any good teacher, she showed us what to do, and then gave us a chance to do it. At the beginning of each class, she’d demonstrate a new type of problem. Then, after answering our questions, she’d assign practice problems so we could practice what she’d just shown us. With her, even challenging proofs seemed like enjoyable puzzles to figure out. My brother and I still talk about what an amazing math teacher she was, over ten years after we took her class.

But when I reflected on my friend’s question, I realized something I’d never thought of before. Nancy Oliver, the only math teacher I had from 6th to 12th grade who was a woman, was also the only math teacher I had from 6th to 12th grade who really made sense to me. Coincidence?

Obviously there are some great male math teachers out there. I’ve worked with some of their students (Byron Parrish’s, at the Winsor School), I read their books and watched documentaries about them (Rafe Esquith), and I follow their blogs (Sam J. Shah). I was just never lucky enough to actually have one of them as a teacher myself! (Disclaimer: I also know from experience there are bad female math teachers out there.)

Maybe my personality and teaching/learning style was just more compatible with Nancy than with any of my other teachers. But it’s also possible that the fact that Nancy was a woman was a big part of why math finally started to make sense to me, a girl, when she was my teacher.

Maybe the secret ingredients were:

I felt completely comfortable asking her for help—more comfortable than I did with any other math teacher. I never, ever felt stupid or ashamed, no matter how confused I was. (In comparison, I often felt embarrassed asking my male teachers for help, even though I knew most of them wanted to be patient and kind with me.)

I understood her explanations. Nancy consistently explained things to me in a way that made sense to me. (I often felt discouraged even approaching my male math teachers for help. Not only did that mean I couldn’t figure it out by myself, but also, their explanations didn’t clear up my confusion as consistently as hers did.) It’s possible that Nancy approached math in a particular way as a woman that made it easier for me as a girl to understand her. Or, maybe she just had a larger repertoire of explanations than my male math teachers did.

She was a role model to me. Maybe I thought—even subconsciously—“if this awesome lady can do geometry, maybe I can too.”

Now that I’m a math tutor, I feel a special bond with many of my students who are girls. (I bond with my male students too, just over different things, like biking through Boston in the snow.) At first I thought that girly bonding—over the release of Mean Girls, or Betsey Johnson handbags shaped like strawberries, or mutual admiration for each other’s style—was just part of establishing rapport and helping my students feel comfortable. But now I wonder if maybe some girls just feel more comfortable with me as a role model because I’m female.

So, thank you, Nancy Oliver, for being my female math role model, and helping me turn everything around. I hope I can carry your torch!

Related Posts:
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Case study: regaining love of math
Case study: confused by math instruction in a foreign language

I loved math as much as any other subject until I hit 6th grade and was introduced to pre-algebra for the first time. Isolating for a variable, balancing an equation, the order of operations—none of this made any sense to me. I would go to my teacher for help, and he would patiently try to explain it to me, but it still didn’t make any sense. I made the same mistakes over and over and over without gaining any understanding or insight.

I have absolutely no memories of seventh grade math, but eighth grade math burns in my memory: sitting in class, trying to do the problems, approaching my teacher’s desk, asking him to explain it to me, dutifully nodding even though I still really didn’t understand, returning to my desk, and feeling overtaken by numb despair.

I’m not sure if his explanations didn’t make sense to me because he always explained everything the same way, or if he had a variety of explanations but none of them clicked with my learning style. He was a sweet, patient man, but his explanations did not help me to learn.

Now that I’m a math tutor, when I remember all those eighth grade nights, crying myself to sleep over my algebra book, I ask myself, why didn’t I think of getting a tutor? I never thought about asking anyone but my math teacher for help. I didn’t ask my friends, I didn’t ask my parents, I didn’t ask other teachers. It never even crossed my mind to try to switch to another teacher, or get another book. Why?

Maybe I wasn’t aware that these options were available. Or maybe I felt somewhere deep inside that, as a student who had a passion for learning and a capable reputation, asking for a tutor would be an admission of defeat. Or maybe it seemed “easier” to think of those nights of algebra tears as isolated incidents instead of taking on the “larger project” of trying to find a better solution for myself.

But paradoxically, I think this experience made me a better tutor. Many of the students who come to me might be completely frustrated and far behind. Maybe they don’t have anyone else they can turn to for help. Maybe they’ve never found a textbook that works with their brain. Maybe they are crying themselves to sleep over their algebra homework. Just like I did.

Related Posts:
When Persistence Isn’t Enough
The Downside of Always Telling Students To Try Harder
The Downside of Always Telling Students To Try Harder (2)
Algebra Tears

Words: “For any real numbers a and b, if a^2=b, then a is a square root of b.”

Huh? I can tease the definition apart if I slow my reading speed down to about one mile per hour. But usually things make sense to me a lot faster if I see an example.

Example: “Since 5^2 =25, 5 is a square root of 25.”

Phew… so much better!

What I like about Glencoe Mathematics Algebra 2 book is that it includes both kinds of explanations in the explanation box—Words, Example, and when appropriate, Symbols and/or a Model. I love how this maximizes the chances that students can see the kind of explanation that makes sense to their own brain!

For example, I was working with a student from a very progressive high school, but her Algebra 2 book only had verbal explanations, with no symbols or examples. We pulled out the Glencoe book and found the “explanation box” for the concept we were discussing, and it made SO much more sense to her than just the words did.

This book doesn’t go as far as to include examples for tactile or kinesthetic learners (like Math U See does) but it’s definitely a step in the right direction!

Disclaimer: The sequencing in this book has been confusing to many students, so it’s not perfect.

Related Posts: The Best Algebra Book In the World?
I am SO excited about Math U See!

[technorati claim code: ZFH7S9WFAK33]

This book doesn’t have a ton of frills—there are barely any pictures or “extras.” But
what makes this book exceptional is its GREAT sequencing. It does an excellent job of breaking the math down without dumbing it down. The problems get harder very incrementally. There are so many practice problems to choose from that you can really practice until each procedure becomes second nature. And the book only introduces new concepts once you’ve already mastered the prerequisite skills.

For example, when this book introduces factoring trinomials, it introduces each pattern that you might encounter one at a time. You practice that pattern extensively before facing a new pattern. Once you’ve practiced all the different patterns separately, THEN it mixes all the different patterns together in one problem set. But by now you know how to recognize the different patterns and what to do differently for each pattern. So when faced with a page full of different types of factoring patterns, you can just think, “OH—difference of squares!” or “OH—perfect squares!” instead of having to do trial and error until you erase a hole in your paper!!

The students I’ve used this book with acquire very, very strong algebra skills without getting bored or frustrated. And I think it’s because the sequencing forces students to learn how to “chunk,” a concept I learned from Daniel T. Willingham’s book, Why Don’t Students Like School?

For example, take two algebra students. One is still a little shaky on the distributive property, the other knows it cold. When the first student is trying to solve a problem and sees a(b + c), he’s unsure whether that’s the same as ab + c, or b + ac, or ab + ac. So he stops working on the problem and substitutes small numbers into a(b + c) to be sure he’s got it right. The second student recognizes a(b + c) as a chunk and doesn’t need to stop and occupy working memory with this subcomponent of the problem. Clearly the second student is more likely to complete the problem successfully. (p 31)

Thank you, thank you to those who wrote this book so chunk-fully: Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, and William L. Cole!