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Posts Tagged as "fifth grade"

Case Study: A 5th grader goes from believing “math doesn’t like me” to singing and dancing about math while wearing her purple tutu

Thursday, September 1st, 2016

When this fifth grade student first came to me, her mom told me, “My daughter is joyful about everything in her life – except for math.” This student was so anxious and uncertain about math that she refused to do her homework unless she was literally sitting next to her mom. She would tell her mom, “math doesn’t like me.”

This put a lot of pressure and stress on her mom, who was doing everything she could to try to help her daughter succeed at math, but she felt like she she was failing her daughter and being a “bad mother” because she couldn’t find a solution. The mom felt anxious picking her daughter up from school because she wasn’t sure whether or not her daughter would have a math temper tantrum. And even though when her daughter would express her feelings of math inadequacy, she was really just asking for help, it was so stressful for the mom that the mom sometimes would react with frustration just because she was so worn down from the seemingly endless math stress.


I started working with this student towards the end of her fifth grade school year. Because this student loves to dance and sing and has a great passion for musical theater, I started teaching her math songs to help her remember different concepts and formulas. We also really focused on filling in the gaps and building a strong foundation.

Midway through the summer, this student started spontaneously singing her math problems! She would make up these little operas about all the different math operations she was doing – as well as songs just about math concepts in general, with sophisticated lyrics that showed she really got the concepts. She would even come to some of her sessions wearing her purple tutu. I was overjoyed to see her expressing herself so confidently and creatively with math, even with her outfits. At the same time, her mom and I also weren’t yet sure how this would transfer to the classroom.

Her first day back at school, her first middle school math class of 6th grade, the teacher asked a question, and my student just couldn’t help herself – she shouted out, “It’s because of the commutative property!” It turned out that no one else in her class – even the students she thought of as being very strong mathematicians – had even heard of the commutative property before! This was a huge boost to my student’s confidence and enjoyment!


Since her first day back at school as a sixth grader, she has consistently made 90s or 100s on every single math test and quiz she’s taken – except for one! On this test, she got an 88%, and what is so interesting is that this absolutely didn’t defeat her.

When she talked about it with her mom, the focus was just about making sure to get the test back from the teacher, so we could go over what she didn’t understand in our tutoring sessions and learn from it. In some ways this was an even bigger victory than the tests where she scored higher, because it showed how much her mindset had shifted. We could see her resilience in how she dealt with a lower grade, and how her attitude had shifted to “I’ll get it, because I know I can get it.”

Just as important, the mom’s experience has shifted dramatically now that she isn’t the one who is helping her daughter with math. She shared with me that when she comes home from work, it’s easy for her energy to be fully engaged with her daughter because it isn’t sapped by worrying about helping her with her math homework right away. She can just decompress and regroup and be energized and be a good parent. And her daughter has become so much more independent that the mom can be reading a book in another room while her daughter is doing her homework on her own!

How did we create this totally awesome math transformation? Let me tell you all about it!


1. Positive, relaxed environment. We fostered an environment of trust and camaraderie. Our work together is committed and also relaxed; this student is totally free to make mistakes, ask questions, or go over whatever it is she needs to go over, no matter what.

2. Dealing with math feelings.
When this student is overjoyed, anxious, or heartbroken, we deal with it together head-on. There was one session very early on where she (quite understandably) cried because she was so disappointed and frustrated with a recent grade. Instead of squelching this or ending the session, we just talked it out, making a safe space for her to feel, express, and release her frustration and disappointment. Other times she was so happy with what she was learning and accomplishing that she would dance and sing with glee and pride!

3. Consciously fostering a “growth mindset” with math. This student has an awesome “growth mindset” when it comes to her work in musical theater. She will audition over and over again for the same Broadway show, and instead of getting discouraged if she hasn’t gotten a part yet, she is just really excited about the process and the experience.

At the same time, there have been periods where she has really expressed more of a “fixed mindset” about math – “you have it or you don’t,” and being worried that she wasn’t one of the ones who “had it.” We deliberately take time to talk about this together and draw parallels with her work in the theater so that she can pull that already-existing growth mindset into her math.


For example, just this week, this student expressed both concern and hope about a state-wide test she was taking the next day. She wanted to score high enough to be selected for state and national math events, and she was also worried that there would be stuff on the test that she didn’t know because she wasn’t in the “honors level class.”

We discussed at length how it’s like if she went to an audition and they asked her to play the bagpipes and do a Scottish accent, she wouldn’t beat herself up for not already knowing how to do those things – after the audition, she would just ask her teachers and coaches to help her learn, if that’s something she was interested in being able to do. Then she shared her philosophy of auditioning, which is that “it’s not just about the part, it’s about the experience, and if you’re not focused on the part, it will just naturally happen.” We drew direct parallels with what she tells herself during her auditions and what she can tell herself during her math tests.

4. Self-expression.
In the context of a supportive environment of trust where all of our work is super individualized, this student started to express herself more and more, whether it was singing the math songs she’d learned, making up her own original math songs, singing herself through the math problem she was working on, wearing her purple tutu, or decorating her problems with hot pink drawings (some of which are included in this very blog post)! Seeing her experience math as a vehicle of self-expression is absolutely encouraged, because it’s a huge sign that the student is getting way more comfortable and also really internalizing the material at a deeper level.


5. Support is normalized. Just like this student didn’t stop taking voice lessons or going to dance class once she started getting parts in musicals, math support that fosters her autonomy is now just part of her normal routine. Instead of saying, “Well, now her grades are higher, she’s done with math mentoring,” this student and her parents have recommitted to receiving support so that she can just continue to grow her math abilities and confidence more and more, and that her family can experience an even deeper experience of harmony around math.

I am so, so proud of this student, and how her persistence, vulnerability, and commitment has created such true mastery, confidence, and JOY with her math!

Are you tired of feeling like a bad parent because even though you’re doing everything you can to help your kid with math, it isn’t working?

Does it break your heart to see your own purple-tutu-wearing kid have meltdowns about math?

Are you ready to invest in high-level support?

Just click here to get started with your special application for my one-on-one math tutoring programs.

Once your application is received, we’ll set up a special phone call to get clear if my approach would be a good fit for your child.

I can’t wait to hear from you!

Sending you love,

Related Posts:
Case Study: A Rising 8th grader masters her summer math packet
Case study: A seventh grader goes from “I don’t get it” to getting 100 percents
Case Study: an ADHD student goes from a D to an A
I just can’t keep this a secret any longer

Posts Tagged as "fifth grade"

How to navigate the space-time continuum (or, a visual way to solve elapsed time problems)

Wednesday, January 5th, 2011

When I was growing up, I learned to do elapsed time problems by subtracting the start time from the end time (and when necessary, borrowing while keeping in mind that there are 60 minutes in a hour).

This technique always seemed convoluted, so when a fifth grade tutoring student of mine was working on elapsed time problems, I tried this visual way of solving them, which seemed to be much more intuitive for my student.

Let’s say you have a problem like this: A train departs at 2:55 and arrives at 5:18. How long is the train ride?

First, draw a timeline:

Draw in the beginning and ending times, marking the hours as you go:

Draw loops to count the hours:


Add up the hours:

Draw loops to count the minutes left at either end:

Add the minutes together:

Combine the hours and the minutes, and you’re done!


That’s it!

Once you do a few of these, there’s lots of different ways to draw the loops. For example, you could start at 2:55 and loop to 3:55 (1 hour), 4:55 (2 hours), and then from 4:55 to 5:18 (4:55 to 5:00 is 5 minutes; 5:00 to 5:18 is 18 minutes; 5 + 18 = 23 minutes) to get 2 hours and 23 minutes.


And after drawing some of these out, you can use the same process to do elapsed time problems mentally, too. Just think about counting from one time marker to another and adding up the different loops.

*Visiting from this week’s Carnival of Homescooling? Welcome, I’m glad to see you here! Below are some more posts you may enjoy.

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Gallon Man to the Rescue!
An easy way to remember how logarithms work
Mind meld is real!
Confused about fractions? Visualize brownies, not pizzas

Posts Tagged as "fifth grade"

Greater than / Less than signs – taking the alligator thing to a whole new level

Wednesday, October 6th, 2010

Growing up, I remember learning to remember the difference between the greater than and less than signs by imagining a hungry alligator with an open mouth getting ready to “eat” the bigger number.

I recently got to work on this concept in an online tutoring session with a student of mine who’s a fifth grader.

First we had a regular < sign and we talked about the "alligator" idea. 2010-10-05_2349

He drew in some pointy alligator teeth:

Then he spontaneously drew a whole alligator:

As we worked on different inequality problems, he took it further. He drew a picture of a bird and explained that the bird’s closed little beak is shaped like an inequality sign. The bird would go for the smaller meal, while the alligator would go for the bigger meal.


And he topped it off – with sound effects.

The smaller number, which the bird would eat, has a “peck peck” sound. The larger number, which the alligator would eat, has a “chomp chomp” sound. Oh my gosh, I love it!

I’d never seen the alligator metaphor pushed this far before, and I wanted to share my student’s creative ideas!

What’s your favorite way to remember (or teach) the difference between the two signs?

Related posts:
Confused about fractions? Visualize brownies, not pizzas
Gallon Man to the Rescue!
Five fun ways to help your kids learn math (this summer)

Posts Tagged as "fifth grade"

Confused about fractions? Visualize brownies, not pizzas

Friday, September 3rd, 2010

Pizzas. They’re the best way to represent fractions, right? Everyone has seen a pizza. Everyone knows you can slice a pizza different ways. When you need to visually represent fractions, pizzas are the go-to metaphor, right?

Well, that’s what I thought until I tried using pizzas to teach equivalent fractions. Sure, it’s easy to use a pizza to represent quarters, sixths, eighths. Thirds aren’t even too bad.

But just try dividing a circle into five equal pieces. Or seven. Or ten. Even though I’m a grown-up and a professional math educator, it’s really hard for me to consistently do this.

And as for showing visually how two fifths equals four tenths? Unless I do an impossibly immaculate job of dividing those pizzas into pieces, forget about it.

Are they really the same?

Fortunately, I’ve found a much better way to represent fractions visually: the Math-U-See fraction overlays. If we’re going to use a food metaphor, they’re flat and square, like brownies from a square pan.

And like brownies, they are much easier to slice evenly than pizzas. In fact, because they’re reusable transparent overlays, you don’t even have to draw (or slice) anything. You just arrange them on top of each other.

Check it:


Awwww, yeah! Those fractions are DEFINITELY equivalent!

You can use the overlays to represent basic fractions with divisors from two through seven, or combine multiple overlays to build other denominators.

Not only do they make it really easy to see how equivalent fractions work, but using them is way easier than drawing pizzas and hoping the slices come out even.

The fraction overlays make the concepts extremely clear, and because they’re tactile, visual, and kinesthetic, they make fractions feel like a game instead of work. Whenever I’ve used them with my students, either in person or during online tutoring sessions with a webcam, they’ve react the same way: “This is fun!”

Now that’s how I want my students to feel about fractions!

Related Posts:
Doing fractions “In Chinese”?!
The best algebra book in the world?
Gallon man to the rescue!
Five fun ways to help your kids learn math this summer

Posts Tagged as "fifth grade"

“This is really neat”

Thursday, August 19th, 2010

Given the counterintuitive new research that has found that certain kinds of praise can undermine student motivation and achievement, I’ve been working over the past year to refine how I praise my students.

Here’s some very specific advice from NurtureShock co-author Ashley Merryman’s blog archive (to read the original, keep scrolling, scrolling, scrolling until you get to the post titled “How not to talk to your kids – Part 4”):

A common praise technique that people use (I know I did it with my tutoring kids… up til a few weeks ago, that is….) is to use a present success to control future performance. For example, if a typically-sloppy child writes an essay that’s atypically legible, a parent or teacher may say, “That’s very neat: you should write all of your papers like this.”

Even if it’s meant as sincere praise and encouragement, the research shows that’s not only an ineffective way to praise. In fact, like praising for intelligence – it can actually damage a child’s performance.

Here’s what is going on. While the first part of the sentence was positive, rather than focusing on that success, the latter part of the sentence (“You should write all like this”) was negative, doubly-so.

First, rather than simply focusing on the present achievement, the second half of the sentence reminds the child about all the past mistakes. Second, it’s an expression of pressure to continue at this level in the future. But the kid may think that the work he just completed was very difficult, and he doubts he can live up to these new expectations.

Even worse, a child who suddenly wrote more legibly did it on his own volition. But if the praiser qualifies the praise with the expectation of future performance, now if the child continues to perform, he’s not doing it because he wanted to: he’s doing it to fulfill the praiser’s expectation.

Basically, the whole exchange kills the kid’s intrinsic motivation to improve. Furthermore, studies have shown that children’s performance actually may go down: they will even intentionally underperform, just to show that they refuse to follow the attempted control. In other words, yes, they do badly just to spite you.

The better thing to have said was, “This is really neat,” and left it at that.

I have been waiting for a year for a chance to try this out with one of my own students. I finally had a chance to implement this a few days ago while tutoring a rising fifth grader online.

He did a particularly neat job of writing out a problem on the online whiteboard, so I told him, “You did a good job of writing that out neatly and lining up the decimal points and the columns.” That’s it. I didn’t say anything about how he should write future math problems.

When he wrote out the next problem much less neatly than the last, I didn’t say anything.

Without me saying anything at all, he scratched out the messy version. And then he started over and wrote out a new, neat version, all my himself.

As a tutor, I am so excited that this style of feedback encouraged him to manage this on his own, without any cajoling or controlling from me — just an objective assessment of what he did well.

And I love having this clear guidance from Ashley Merryman’s archive on how to praise my students without worrying that I’m doing it the wrong way.

Related Posts:
Tips on Effective Praise from Ashley Merryman

What a Balinese dancing queen taught me about praise and encouragement

Praise and Intrinsic Motivation: An Answer?

Posts Tagged as "fifth grade"

Case Study: A 5th grader emerges as a successful student and enthusiastic mathematician

Sunday, August 8th, 2010

When this student first came to me, her dad was concerned that she had lost interest in learning math. During the school year, it also emerged that the student was in danger of not passing fifth grade.

Here’s what worked for this student:

Supporting the student’s own efforts to be proactive
During one of our first math tutoring sessions, I pointed out to this student that numbers that end in zero are even. Somehow she hadn’t learned that before. To help herself remember this new fact, she spontaneously made up new lyrics to the tune of Michael Jackson’s “Beat It.” They went like this: “Even! Even! Numbers with a zero are even!”

The next time we met, I added to her original lyrics: “Even! Even! Numbers that end in zero are even! So are the numbers 2, 4, 6, 8. They are all even, and they’re all great! They’re even!”

She would sing the zero song whenever this topic came up. Not only did my student create a great way to remember this fact (and inspire me too), but singing also allowed her express her enthusiasm for math and let off a little steam.

Another time, she suggested we create a “Mistakes Log Blog” to help her analyze what mistakes she had made on a test that we were reviewing. I ran with this idea. When she wrote down where she’d made mistakes, the patterns became much clearer to her. In later sessions, she’d refer back to the “Mistakes Log Blog” when analyzing errors.

“Field trips”
In order to make concepts more concrete, we’d take field trips—to my living room, where we’d practice perimeter and area by measuring my rug, or to the kitchen, where we’d measure a round plate to show where the number pi comes from.

At my kitchen sink, we poured water between different containers to show the relationships between units of measurement. And we acted out word problems using food from my refrigerator. Field trips were way more engaging to her than sitting with a worksheet, so I tried to maximize this.

Multi-sensory learning
From taking all those field trips during math tutoring, I noticed my student benefited from hands-on learning. So we also used fraction overlays and math blocks from Math U See to build fractions and do “fraction of a number” problems. Using the manipulatives made abstract concepts concrete for my student, and really helped her “get” the material. Plus it was fun!

When I realized my student didn’t know her 9s times table yet, I taught her the Rockin’ the Standards song for the 9s, to the tune of the hokey pokey, so she would remember them forever. I also taught her the Place Value Rap to remember key facts about place value. Not only were these songs a great chance to stand up and play air guitar, but they were also an excellent way to internalize crucial material and build on the success of the Zero Song.

Managing focus
During the year, we met twice a week for either 60 or 90 minutes. If I noticed my student was losing focus, we’d take a break to jump up and down to rejuvenate ourselves. After a while, my student would ask to jump when she was having trouble concentrating. It might sound silly, but I was proud that my student was starting to pay attention to whether or not she was paying attention and that she knew how to refocus herself. (Thanks to Gretchen Rubin for inspiring me to try this!)

When I realized my student was in danger of not passing fifth grade, I decided to use Carol Dweck’s Brainology curriculum, one of the most powerful motivational tools I know of to address one of the underlying cause of low achievement: low motivation. For several weeks, we would spend part of each tutoring session doing Brainology, which uses basic neuroscience to teach students that their brains are plastic and they can grow their intelligence.

My student enthusiastically embraced the Brainology program. She talked about the characters like they were her personal friends, and she responded to questions like “what is happening in your brain when you think?” with answers like, “Neurons are sending messages within a trillion connections.” She also used Brainology concepts like getting enough sleep and eating “brain food” while she was taking her end-of-year standardized tests (the CRCT).

The results

Three or four weeks after we began working together, her teachers reported a positive change in this student’s attitude. She started sitting in front, participating, and speaking up when she didn’t understand.

After about sixth or seven months of meeting twice a week, this student mastered the material, pulled up her grades, and successfully passed fifth grade. Her final math test score was so high that she was only either 10 points or 10 questions away from placing into the advanced math class in sixth grade. I am so proud of her!

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Posts Tagged as "fifth grade"

Gallon man to the rescue!

Thursday, June 3rd, 2010

Do you need a way to remember unit conversion effortlessly and forever? Or just a way to calculate how many cups there are in a gallon?

Here’s how to figure it out. Draw a gallon man!

First, draw a really big capital G. (This is the gallon.)

Inside the G, draw four big Qs. (These are the quarts.)

Inside each Q, draw two Ps. (These are the pints.)

Inside each P, draw two cs. (These are the cups.)

For the final flourish, draw an arrow to one of the cs and write “8 ounces.” (There are eight ounces in every cup.)

When one of my students, a fifth grader, taught me about Gallon Man, I thought, I wish I had learned about this in fifth grade! My entire life, I’ve had to look up each of the conversions and never really internalized how they all fit together.

Since I’ve been introduced to Gallon Man, I’ve gleefully shared him with a fourth grade tutoring student (online), a friend who is a professional organic farmer (in person), innocent bystanders (at a restaurant), and most recently, my Mom (over the phone…”first, draw a really big G…”)!

They’ve all found Gallon Man helpful. Responses have included: “Can I take that drawing home with me?”, “Oh…I get it!”, and “I’m going to hold onto this.”

Gallon Man is totally visual and works for many learning styles. You can SEE how many quarts are INSIDE a gallon. Gallon Man is intuitive for all grade levels (unlike dimensional analysis, you don’t have to worry about the numerators or denominators). Gallon Man is practical. You can use it in your kitchen or in the grocery store. Gallon Man is easy to remember. And Gallon Man is fun to draw!

Gallon Man has recently gotten some airtime from other math bloggers, including Sam J Shah, who pointed out that it really helped him to see someone drawing Gallon Man. Here’s Sam’s post and video.

Yay for mnemonic devices!

*Are you looking for an online math tutor who uses multisensory methods? I’d love to help! Give me a call at 617-888-0160 to discuss your situation.

*Visiting from the Math Teachers at Play Carnival (Adventure Edition)? Welcome, I’m glad to see you here! Below are a few other posts you might enjoy!

Related posts:

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Case study: a homeschooler prepares for the SAT

Posts Tagged as "fifth grade"

“I think I see a mathematician!”

Wednesday, April 28th, 2010

A little while ago, I was working with a fifth grader on a problem that challenged her. In the middle of the problem, she looked up at me, smiled, and said, “I’m struggling, but that’s OK. My teacher told me that mathematicians struggle.”

What an exciting moment! I am so glad that my student has a classroom teacher who is teaching her that struggling doesn’t mean that you’re “stupid,” and to persist in the face of a challenge!

My student also told me that when she was eager to stand up and give the answer to a math problem, her teacher said, “I think I see a mathematician!” I’m thrilled that this teacher is encouraging her students to identify themselves as mathematicians—people capable of deeply engaging with mathematics!


Thank you to Choco Leibniz for creating and sharing this beautiful, inspiring image of mathmaticians! The Famous Mathematicians are from left to right, top to bottom: Euclid, Boole, Al-Kwarizmi, Newton, Leibniz, Turing.

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Posts Tagged as "fifth grade"

Doing Fractions “In Chinese” ?!

Friday, March 5th, 2010


I was so excited to discover that in Malcolm Gladwell, his recent book, Outliers, presents a bunch of new research on learning math!

There’s so much good stuff in there that I can’t even begin to tell you all about it. But one thing that struck me in particular was Gladwell’s discussion of the cultural differences between Asian and Western attitudes towards learning math. (You can read an excerpt from the chapter here.)

To start, language differences give Asians a linguistic advantage. In Asian languages, numbers are more transparent. For example, when an English speaker has to do mental math, they need to translate words into numbers first. Before we add “forty-three” to something else, we have to break it down into “four tens and a three.” By comparison, in Chinese the word for “forty-three” is already broken down: “four-tens-three.”

Similarly, we say “three-fifths” to describe a fraction in English. But the Chinese for the same number literally translates as, “out of five parts, take three”: the definition of how a fraction works is built in. These linguistic differences make calculation easier in Asian languages. And because it’s easier to figure out what things mean just from the words, there’s an attitude that it’s normal to be able to figure math out.

This creates what Gladwell calls a “virtuous circle”: because the names for numbers are a little bit easier to understand, arithmetic is a little bit easier to do, which means that maybe students like math a little bit more, which means that maybe they take more math classes and ultimately achieve more in math. In contrast, Western children, by third and fourth grade, start to feel that “math doesn’t seem to make sense; it’s linguistic structure is clumsy; and its basic rules seem arbitrary and complicated.” And the trouble begins…

When I mentioned this to my friend, the Future Doctor Jones, she said, “We’re stuck with this language! What are we supposed to do with it?” Her question is valid—if I tell my tutoring students to say “two-tens-seven” for 27, will they just get beat up on the playground for talking crazy numbers?

So recently I was working with a fifth grader on fractions, and I casually mentioned that in Chinese, they say fractions like, “out of four parts, take one,” instead of “one-fourth.”

I was totally surprised when later in the lesson, this same student spontaneously started saying fractions “the Chinese way.” “Out of seven parts, take four!” “Out of two parts, take one!” When I slipped up and said, “Out of two parts take five,” she corrected me immediately, which meant she completely understood the concept.

Most importantly, she didn’t want to stop doing fractions. She was begging for more!

I’m grateful to my student for spontaneously showing me how we, as English-speakers, can adopt a “Chinese” way of thinking about numbers.

Related Articles:
I cried myself to sleep over my math homework
The Downside of Always Telling Students to Try Harder (1)
The Downside of Always Telling Students to Try Harder (2)
Malcolm Gladwell on Math and Persistence (1)

Posts Tagged as "fifth grade"

Tiny Garlic Melons

Friday, January 29th, 2010

This summer, one of my students got to go to Video Game Making Camp. My student explained to me that he wanted to make a video game where you killed vampires by throwing garlic at them. But there was no “garlic” graphic available to build into the game. So he took a graphic of a giant melon and made it so tiny that it looked like a head of garlic!

Part of what I want to teach all my students is how to customize their education when I’m not around. So later in the session, I seized the teachable moment. This particular student has dysgraphia, ADHD, and a really unique brain. I told him that everyone, whether or not they have dysgraphia or ADHD or whatever, has learning situations where they’re not getting what they need. And we all have to learn how to invent our own ways to work around it.

“It’s just like the tiny garlic melons,” I concluded. “Sometimes you don’t get what you want and you have to turn it into what you need.”

So when life gives you melons, make….tiny garlic melons!!!!

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A Cosmic Imperative to Customize!
The Downside of Always Telling Students to Try Harder (2)
Ana Reynales earns her BA at age 82