I just found out that she placed into honors, and she was so excited when she told me that she screamed on the phone! I am SO proud of her hard work and persistence!

Here’s how we made it happen:

Openness and Trust. Throughout our sessions together, this student was extremely transparent about what she did and didn’t understand. This was enormously helpful, especially because while we thought the summer math packet consisted of review only, it turned out that a ton of material was stuff that this student had never learned. Her willingness to tell me whether she was elated or frustrated–frequently with a self-deprecating sense of humor–helped us build camaraderie and also made our work together much more effective.

Which brings us to…Adjust as you go. When we realized that we had a lot of material to cover from scratch, instead of just reviewing, we adjusted the plan and decided to meet more frequently.

Break it down. The packet was extra-challenging because each page was like a tossed salad, blending problems from all different parts of the curriculum. While this is a great strategy to use when you’re reviewing material, it is not an effective way to learn something new.

So we backtracked, and my student learned one prerequisite skill at a time, practicing it thoroughly until it felt comfortable and automatic. Then, we combined these skills in more complex problems, gradually building up to problems as hard as the ones in the packet.

Practice outside of sessions. I also gave this student individualized worksheets that gave her a chance to practice and internalize the skills we were working on, with answer keys so she could check her work as she went (instead of waiting to talk to me and then finding out that she had practiced something the wrong way). This was especially important because it was the summer and she wasn’t getting a regular dose of math from a school math class.

Feedback on solo work. After building up her skills, my student worked independently on chunks of the packet at the time. This way she got comfortable with problem sets where different kinds of problems are juxtaposed on one page, just like they would presumably be on the placement test. Then, when we worked together, we would go over all of her work so she knew she was on the right track.

Which brings us to, “What did I do wrong?” At first, my student just seemed annoyed with herself when she made a mistake, but I really emphasized to her that it’s okay if you make a mistake as long as you take the time to ask yourself why and learn from it. Scrutinizing and learning from errors gradually went from being an irritating chore to just a routine and helpful part of the learning process.

Enthusiasm. More than any other student I’ve ever worked with, this one has a great appreciation for mathematics’ dramatic resonance and poetic potential. When she learned how to find the solution to a system, she said that that would be a great name for a band. Frequently she remarked that new concepts we were going over would make the premise for a great science fiction story.

Her gleeful excitement about the greater meaning of what she was learning seemed to help her take the tough stuff more in stride, because even the “annoying” math procedures were part of something that was exciting to her.

Parental backup. The best tutoring happens when everyone works together as a team, and this student’s mom was totally focused on the process of learning. She made sure that her daughter completed assignments in between sessions (especially important during the summer). She asked me thoughtful questions about the material and her daughter’s progress that showed me she herself was deeply engaged with her daughter’s math material.

Because she was so organized and also willing to re-learn math and ask questions about the parts she wasn’t sure about, she was also a great role model to her daughter. Her involvement and support was instrumental in her daughter’s success.

I was so thrilled to hear that this student had rocked her placement test and placed into honors! Hooray!

***Update: I just found out today (12/7/2011) that my student got an A for the trimester in her honors math course! I love it when students become completely self-sufficient and continue to succeed after they “graduate” from tutoring. Hooray!!

Related posts:
The Rhyme and Reason of Making Mistakes
Five fun ways to help your kid learn math this summer
Case Study: An ADHD Student Raises Her Grade from a D to an A
Case Study: Regaining Love of Math

After about eight weeks together, she earned a 100% on a test, and her teacher sent her parents a note that she was doing really well and really seemed to be understanding the concepts in class.  After about six months of tutoring together, she just finished up the school year making more 100% percents on her tests!

Here’s how we did it:

Fill in the gaps. Algebra builds on everything that comes before, and a lot of 7^th graders struggle with algebra because they still feel shaky about decimals, fractions, and other prerequisites. Whenever we found a gap – like when she told me she’d rather convert fractions to decimals whenever possible – we’d go back to where it started to get murky and then work step-by-step through many practice problems until she had mastered the material and filled in the gap.  She also learned fun songs for all of the times tables to feel more secure with those foundational math facts.

Customize: make it visual. This student seemed to get a lot out of seeing the math.  When we went over decimals, we used grids to show how multiple decimals can add up to wholes.  When reviewing fractions, we would divide a square into parts to make the concept visual and concrete.  When her class started working on adding and subtracting negative numbers, we spent a lot of time using a number line to practice this.  Making it visual made the material less abstract and more clear (and also more fun).

Practice. Everyone needs to practice challenging material until you internalize it.  When she had questions about the material from class, we’d do lots of extra practice problems I’d make up for her on the spot.

For example, when she started working on order of operations problems, I’d create progressively more elaborate order of operations problems for her to practice.  This way all the steps became automatic—no more second-guessing or feeling confused.

Extend. If we had extra time, we’d do more problems based on what she was doing in class, but take it to the next level.  I frequently asked her to create her own problems and was delighted to see that a lot of the time, the problems she made up were harder than the ones I made up for her – because she wanted to make it even more interesting!

I believe creating her own problems helped her feel like math was something that belonged to her, something that she could create, instead of a bunch of impersonal, arbitrary problems from a textbook.

Preview. This same principle of taking it to the next level meant that sometimes, instead of encountering a challenging new concept for the first time in class, we got to introduce it and explore it one-on-one.  Then, once it came up in class, this same student who used to feel like she “didn’t get it” knew exactly what was going on.

Immediate feedback. Throughout our work together, she got immediate feedback on whether or not she was doing the problems correctly.  This nipped potentially bad habits in the bud and also meant that she could learn the material right the first time without feeling disoriented.

Immediate feedback also meant that when she started to feel frustrated, we would talk about it, take a big deep yoga breath, and clear the air, which made her effort much more productive.

Working with this student was a great pleasure because she did such a good job of communicating what she wanted to work on and what she did and didn’t understand.  Because of her hard work, persistence, and open mind, she finished her year earning 100 percents!

Related posts:

Case Study: An ADHD student raises her grade from a D to an A
Case Study: Confused by math instruction in a foreign language
Case Study: Regaining love of math

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!)

Brainology
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!

Related Posts:
Case Study: Confused by Math Instruction in a Foreign Language
Case Study: An ADHD student raises her grade from a D to an A
Case Study: Regaining Love of Math
Case Study: A Homeschooler Prepares for the SAT
How to Find a Good Math Tutor

Here’s what worked for this student:

Address the fundamentals. Before we approached the SAT math test as a whole, we had to master basic algebra and geometry topics one at a time. We started at the beginning of an Algebra 1 textbook and moved at our own pace. We focused on what was important and what would be on the test.

Solo work and feedback. Most students that I work with are sitting in math class and doing math homework at least three times a week. But this student wasn’t in a math class. Tutoring was his math class. And he wasn’t getting homework assignments unless I gave them to him. So it was essential for him to have carefully planned homework assignments and get detailed feedback from me on each one.

Adjust the textbook when necessary. We started off using the Glencoe Algebra 1 textbook, but after several months of working together, I realized my student needed more drill and better sequencing. He needed to be able to do as many problems as necessary to master the material. And he needed to be able to check his answers without having to wait to see me. So, as a supplemental text, we added another algebra textbook that had better sequencing and more practice problems. In the end, we relied on it more than the Glencoe.

Adjust the pace when necessary. When we started working together, I’d demonstrate a technique and then give him a chance to do it himself, correcting him immediately if he made any mistakes. I wouldn’t move on to the next concept until he’d mastered the material. But at this pace, he wouldn’t learn enough of what was on the SAT. So I started assigning him sections of the book to read and teach himself. This worked for a while, but then we reached a point where he’d get stuck midway through the material and have to wait for our next meeting before getting a clear explanation.

So we changed our approach and aimed for a middle ground. I would demonstrate one or two problems from each section before asking him to do the work himself outside of tutoring. This gave him a preview of what to expect and let him learn more material. I just wish that I had known about Math U See back then. It would have been great if he could have used Steve Demme’s instructional videos as his “math class,” and then used our time together as a resource to discuss whatever he had questions about.

I was so proud that he was so willing to work hard to learn something that didn’t always come easily. And I was thrilled to hear that his work allowed him to meet his goal: he got into the art college of his dreams!

Related posts:
Case Study: An ADHD student raises her math grade from a D to an A
Case Study: Regaining Love of Math
Case Study: Learning Geometry with a Spatial Disability
The best Algebra book in the world?

This student first came to me the summer before ninth grade. The previous year she had struggled with focus, especially in math, and at the end of eighth grade, her math teacher had encouraged her to use the summer to review. So we started tutoring over the summer, which was perfect: tons of time, without the pressure of classroom tests or other school-year commitments.

Find the missing gaps and fill them in. Math is so cumulative that missing a single class or even spacing out for a few minutes can make a student feel totally lost! So a big part of our initial work together was retracing my student’s steps and seeing what skills were missing. Once those prerequisite skills were identified, she could master them and move forward.

Focus on conceptual understanding. A lot of students prefer to learn how to do something before learning why it works that way. However, this student craved conceptual understanding. Frequently, once the big picture became clear to her, her face would light up, and she’d exclaim excitedly. Off and running, she’d dive right into the problem, knowing exactly what to do even if I hadn’t told her first. Because this student thrived on big-picture teaching, we focused on that first in each session.

Adjust the curriculum. A easy but helpful psychological “trick”: when we started working together during the summer, we used the textbook for the upcoming year instead of using her old textbook. The material at the end of 8th grade and the beginning of 9th grade is usually the same. But she could start the year confidently, knowing that she’d already mastered the exact material that would be covered in the first few weeks of school. Also, after the school year began, when appropriate, we’d consult an alternative textbook for explanations better suited to her learning style.

In addition to our summer meetings, we continued to meet periodically during the year. After barely four months working together, I was thrilled to learn that my student earned a grade of 108 on her algebra test: 100 plus the 8 point extra credit problem. The highest grade in the class!

Related Posts:
Case Study: Confused by Math Instruction in a Foreign Language
Case Study: Regaining Love of Math
Case Study: Learning Geometry with a Spatial Disability

We started with a lot of math drill, reviewing arithmetic concepts that were unclear from years of math instruction in French. Then we moved on to mixing that with a review of what she was working on in class. We worked very slowly, and at the end of every page or every problem I would give her a high five and a special sticker. (Now, after all I’ve learned about rewards and motivation, I might not give her a sticker every single time.)

Another helpful strategy was paying attention to her emotions of frustration and anxiety, and modeling how to handle them. When she got frustrated or anxious, I would stay calm, just like I hoped she’d learn to stay calm in the face of a challenge.

One day she got really upset about some things in her life that were stressing her out, and I could tell she needed a break. (I was trying to build on what I’d learned from working with another student who broke down during tutoring once.) So we packed up our work and spent the rest of the session leisurely exploring the beautiful library where we met for tutoring.

Very gradually, things improved to the point where she even told me that she “loved” certain kinds of problems. This made me so proud of her! It was amazing to see her going from feeling scared and confused about math to actually being comfortable and delighted with it. Overall, I think what worked for her was just personalized and caring attention with a stress-free vibe.

Related Posts:
Case Study: Regaining Love of Math
Case Study: Learning Geometry with a Spatial Disability

Since every student is different, I wasn’t sure until we started working together what would help her regain her love of math. She was already very organized and would come to each session with a plan for what she wanted to discuss.

It quickly became apparent that she really just needed some time one-on-one to go over the things she had questions about. The way that her classroom teacher explained things wasn’t always the way that made the most intuitive sense to her. (This isn’t unusual, considering that every single human has a unique way of approaching their own learning).

Another thing that worked was introducing alternative ways of thinking about particular math concepts. This student was great at evaluating what options worked best for her. She’d explain which approaches made total sense and which ones really didn’t help her. She’d also use her synaesthesia to create her own mnemonic devices.

This student would tackle tough problems with gusto. Once, after she cracked a particularly challenging problem, I drew a star with shining rays next to her final answer to show how proud I was. We jokingly named it “The Star of Vanquishment”—vanquishing seemingly impossible problems! This became a running joke. We’d draw it when we felt like we needed inspiration to get through something unfamiliar, or to celebrate when we solved a tough problem.

My student’s school year ended later than any other schools in the area. I was concerned because before I’d committed to working with her, I’d made plans to be out of town for a music festival during her final exams. So she was one of the first students to test-drive my online tutoring technology with me.

During our final session online, she told me that her past three quiz grades had been an 100, an 103, and a 93—“but the 93 was the highest grade in the class on that quiz.” I was so proud of her!

Most importantly, it seemed from her confident and enthusiastic attitude that she had regained her love of math, or at least was well on her way. Overall, I think the “secret ingredient” here was just supporting her and personalizing her instruction in a relaxed and encouraging environment.

Related Posts: Case Study: Learning Geometry with a Spatial Disability
Case Study: Confused by Math Instruction In a Foreign Language

One of the first things I tried with this student was using erasable colored pencils to label different parts of the diagram in different colors. I hoped the different colors would help him distinguish different parts of the diagram, un-jumble them, and process the information better. But he didn’t seem to be into the colored pencils, so we stopped using them after a while.

However, I knew he must have excellent kinesthetic-spatial intelligence in order to be such an awesome hockey player. I mean, he specialized in creating and responding to vectors on ice, right? So I tried to talk with him about visualizing things in motion. I would tear up pieces of notebook paper and create animated versions of the diagrams by moving the pieces of paper around.

In the end, I think the teaching strategy that helped him the most was just really breaking down the geometry diagrams. I realized that he was missing a lot of crucial information about how to interpret diagrams that most teachers probably never explain, most likely because it seems so “obvious.”

For example, someone without a spatial disability would look at a diagram of a triangle and just infer that if a number is tucked inside an angle, then it is the measure of that angle. Similarly, it would be easy for them to intuit that if a number is next to a line, then it’s the measure of that line.

But because of his disability, these things weren’t obvious to my student, and no one had ever explained them to him before. So we filled in the missing pieces. We broke down the different parts of those diagrams so he’d know exactly what to look for and which numbers affected which part.

The awesome part is that after a couple months, he went from failing to getting As and Bs!

Related Posts: Case Study: Regaining Love of Math
Case Study: Confused by Math Instruction in a Foreign Language