Some Problems with "New" Math

We're constantly helping students with math. One trend we're seeing is that some of the "new" math curricula isn't giving students a strong foundation of basic math facts. Math builds on itself over time. As a student advances in math, tackling increasingly complicated problems and skills, undetected holes in their foundational knowledge can cause huge problems later.

What is "new" Math?

Everyday Math is one type of "new math" widely used throughout Chicagoland elementary schools. It has grown in popularity because it's thought to teach "higher level" problem solving skills. Problem-solving skills are important, and the truth is, many students do quite well with Everyday Math. The consistent problem that we see is that foundational, basic math skills are not taught, which really leaves some kids behind.

What is it about this "new" math that leaves some students with gaping holes in their math knowledge? This youtube video, Math Education: An Inconvenient Truth, sheds some light on the problem. It compares the traditional methods for solving a multiplication problem with the new methods. It's not difficult to see why the new methods are confusing for kids learning math for the first time. Watch this if your child's school is using the Everyday Math curriculum.

What do we do about it?

If your child is using a "new" math program, the important thing is to supplement!

• Make sure your children learn their basic math facts to mastery. Flash cards, dice games, leap frog toys, and online tools like aleks quick tables are great for learning math facts.
• Check out ixl.com or aleks.com if you want to do an online math program at home.
• Tutoring is always an option! Feel free to give us a call. We've encountered many of these situations in tutoring, and we're happy to help!

Jumping into the Deep End: moving from surface structure to deep structure

In previous posts, Sara introduced the concept of surface structure & deep structure when it comes to problem solving. She then went on to describe the ways the surface structure of problems can help students understand certain aspects of those problems or concepts. Now Sara goes into more detail about how, through knowledge transfer, we can help students move beyond the surface structure of problems to understand their underlying deep structure.

Transfer (applying past knowledge to new situations or problems) is an essential part of how we define learning. We can’t say that we’ve really mastered a concept or skill until we can apply our knowledge in different situations. The question is: how can we promote transfer?

As I explained in my last post, students have an easier time recognizing deep structure if the surface structure is familiar. The key point here is that familiar surface structure is not necessary to solve a problem - but it is helpful. I’ve noticed that when working with ACT students over the years, most of them have an easier time tackling percentage word problems when they have a familiar context. Take these examples:

1) Sheila had lunch at a restaurant. Her bill was $20, and she wants to leave a 20% tip. How much is her total?

2) A business purchases its inventory at lower wholesale prices and then sells items to the public at higher retail prices. Sheila, the owner of a shoe store, purchases shoes at $20 per pair, and she sells them for 20% more than she paid. What is the retail price of the shoes?

Even though the calculations are identical, most high school students have an easier time with the first problem because they’ve had experience with leaving a tip at a restaurant. It’s rare that you’ll find a high school student who has experience determining retail prices based on wholesale cost.

In the same vein, if my son had had more experience with track and field, he probably wouldn’t have balked at the ‘laps around the track’ addition problem. He’d probably have more confidence tackling this problem: “If an artist uses 2 cans of paint to make one mural, how many cans of paint would he need to make 2 murals?” than he would this problem: “If there are 2 hydrogen atoms in one molecule of H2O, how many hydrogen atoms are in 2 molecules of H2O?”

What we can take away from this is that the more exposure a student has to different subjects, the easier it will be to transfer knowledge (and the easier it will be to learn new information, period!) Encourage your children to expand their base of knowledge by going to museums, reading books, participating in extracurricular activities, and engaging them in conversation on different topics. This is a case where more is definitely better!

As helpful as a wide base of knowledge is, a student doesn’t need to be familiar with the surface structure of a problem in order to recognize its deep structure. How can we promote knowledge transfer to unfamiliar contexts?

The answer: lots of practice!

This may seem oversimplified, but the truth is that it’s easier said than done. I think most of us know that practice is a necessary part of learning, and that the more practice a student has with different types of problems, the more apparent the deep structure of those problems becomes. After all, when I think of the variety of addition problems that I’ve seen in my lifetime, the amount of exposure my 7-year-old has had seems miniscule. With more practice, the ‘laps around the track’ problem will become as simple to him as it was to me, and the deep structure of addition problems will eventually be unmistakable.

Practice has other benefits in addition to promoting transfer. It helps students retain information for longer periods of time. It also helps basic skills become automatic, which frees up “thinking space” in the brain for performing higher-level problem solving.

The tricky part is keeping students engaged during all that practice! Here are some tips from Willingham in Why Don’t Students Like School?:

• Break up practice into small chunks of time over a long period. There’s no need to do two hours of practice in one sitting. Instead, try fifteen or twenty minutes a day for a week or two.
• Vary the problems. Different surface structures not only help with transfer, they can help a student stay interested, too.
• Mix up the skills in the practice. If a student seems to have a pretty good handle on multiplication word problems but needs more practice, try throwing an occasional addition or subtraction problem into the mix. It’ll keep her actively thinking about the problems.
• Practice “old” skills in the context of new ones. If students need continued practice with reading aloud, it isn’t necessary to just have them read story after story. Reading aloud could be involved in a lesson about plot or character, or it could be a natural part of acting out a dramatic work.

At Nurturing Wisdom, our guideline is to provide an immense amount of practice, distributed over time, and across varied contexts. Looping, a technique we use when tutoring math, is one way we’ve managed to apply these guidelines with excellent results. I’ve begun using looping with my son for addition, subtraction, and now multiplication problems, whenever teachable moments arise. As a rising third grader, he’s hardly seen the end of stumping math problems – and I’m sure I’ve hardly scratched the surface of moments that I will be surprised and confused as a parent! – but we’re at least one step further along the path of his growth as a student, and I have a much better understanding of how to guide him along in the future.

Achieving the Impossible!

Walking in to your first tutoring session, you have so much emotion running through you: nerves, excitement, curiosity, anticipation. You are so excited to meet your new student and you can’t wait to see what the future holds! Walking out of the first session, you know that you can help this student achieve their goals. Not once do you think about how this student is going to affect your life and change you. Well, here’s the truth that I’ve learned: every student will change your life and make you a better person.

Enter Ben. Ben was a junior who needed tutoring for the ACT. Ben was one of my first students and after hearing all about him, I was so excited to work with him! Ben’s mom, however, was a little skeptical. She knew how new I was, and she also knew that Ben needed to do really well on his ACT to get into his top choice schools. My emotions went from excited to extremely nervous. (I mean, no pressure or anything to make sure Ben does really well on his ACT.) Ben and I started working together and it was a perfect match! We worked really well together; he responded well to my teaching techniques and feedback and always did EVERYTHING I asked of him without one ounce of complaining! I was always so excited for Sunday tutoring because I got to work with Ben! To make a long story short: Ben went up 6 points on his first real test, but then went down on his second.

After his second test, Ben, his parents, and I all sat down and had an amazing talk. We knew Ben was capable of surpassing his highest score, and he really needed to get one point higher to be a “shoo-in” to his top choice school. Ben was on board! He said he would do whatever it took to get that extra point. And if he didn’t get it in June, he would try again in September. Ben worked his tail off prepping for the June test, even with the load of finals weighing on him. But again, not one complaint came out of his mouth. A few weeks ago I got a phone call from Ben’s mom. I had been having dreams about Ben’s ACT scores - clearly I couldn’t wait to hear what she said. She told me that Ben had gone up two additional points from his first real test, raising his total score increase to 8 points - above and beyond what he needed to get into his top choice school. Then she started talking about how I have changed Ben’s life. She said that a year ago, she never thought that any of this was possible for Ben, but that we proved her wrong! Hearing her say all these wonderful things brought tears to my eyes. It was nice to hear how I had affected Ben’s life, but what about how he changed mine?

Ben made me realize why I love my job so much! He made me realize that life is all about the hard work you put into it. There’s no such thing as an easy path, and if there were, I wouldn’t want to take it. Ben and I worked so hard to achieve the impossible. The road was tough at times, but man does it feel good to look back and say, “We did that!”

Lauren Feldmann, Staff Development Assistant

The Lightning Round: Tips from our Productivity Maven!

Jeanne Rerucha has a nickname on our staff: we call her "Lightning" because she gets things done so quickly, and so well! Here, she shares some of her productivity secrets.

Have you ever wondered, "how will I get all this done?" or better yet, "how will I get all this done well"? I have found that it is all about technique, strategy, and a little multi-tasking to get through many of my days. As a mom of three girls and a tutoring director who works from home, I used to feel stress just getting through each day. I want to be there for my kids, I want to get my work done well, the laundry is everywhere; you get the picture. It's all about organization and a little flexibility.

Here are a a few tricks that have really helped me:

1. A calendar: this is so important!! I use my calendar religiously. Now, honestly, I haven't gotten into the online google calendars yet because I need to physically be able to see my calendar at any time, and my computer is not in my car. I keep track of all appointments, meetings for work, and playdates in my calendar. Because I am ultimately keeping track of 5 schedules, this is needed.

2. Maximize your help: I have a baby-sitter who comes for a few hours during the week. I know that's the time I need to use for phone calls and items requiring full concentration. Some things can be done at the same time as others, while some things need full attention.

3. Know your multi-tasking limit: Sure, I can be on the phone, wash dishes, and help with homework; but add in one more activity and it is too much. It's important to know how many things you can do while making sure they are still getting done well.

4. Reflect at night: I find it is easiest to do reflection in the evening. For me, the kids are in bed, it is much quieter in the house and I'm not rushing around getting things done. I like to save the items that need some thought or even some idea brainstorming for the end of my day. I also feel this is a good time to chat with co-workers, as they too are not trying to do a bunch of things.

5. Charts: I love my "at a glance" sheets. At work, I do this for my list of students, so that I can keep track of what tests I am expecting, who is tutoring them, etc. I know I can go deeper in the file system to get more information, but this sheet saves me time. I do this for some personal items as well. I have a sheet of phone numbers/addresses that I use most. Of course I have an address book to refer to when needed, but the doctor's number is easily accessible on my fridge for quick reference.

Do you have any productivity tips to share? Every little bit helps!

Tutor Spotlight: Kristi Harreld, "Practicing to Be"

Kristi Harreld has been tutoring for nearly eight years, four of them with Nurturing Wisdom. She holds an M.A. in English from the University of Illinois (Chicago). When she isn’t teaching or tutoring, she enjoys kayaking, riding waterslides, and of course, taking photographs.

Teacher. Tutor. Writer. There are many words I would use to describe myself, but I recently encountered one I wasn’t so comfortable with. After opening the Easter card I’d made for him, my father-in-law exclaimed, “I didn’t know you’re a photographer!” I managed a smile but mentally protested. A photographer? I just took a photo of a lily and pasted it to a white card! I then toyed with the new label. What does or does not make me a photographer?

I often experience such confusion, except I’m usually the “labeler” and my students are the ones who are confused or even sometimes insistent. “I’m not good at writing” turns up frequently in sample paragraphs. “I’ve never liked to write” typically masks a lack of confidence. Many times, though, the self-proclamations run deeper; the worst of them are simple yet profound: “I’m not a writer.” Students usually laugh if I point out that they just turned in a typed something, a thing called a piece of writing, which makes the individual who produced it a writer. Still, the declaration lingers: I’m not a writer. Why not?

I’ve been writing since I could hold a pen. In the third grade, I was “published,” if you count the two slim short stories I bound together using scraps of hideous, discarded wallpaper samples (still good reads, if you ask me). Early on, I identified writing as fun, a task that gave me a sense of myself. And then I practiced. A lot. Hidden inside the polished writings I produced were hundreds of errors, revisions, and newer, better things that grew from those earlier attempts. In short, it’s the act of writing that has made me a writer. The quality of my writing has been described in various ways; nevertheless, I continue to identify myself as a writer because I keep writing. Many theorists believe that if we want students’ writing to improve, we must teach them to value the act of writing by having them do a lot of it. I’ve seen this theory proven many times, but I’d add a little twist: it helps to accept any labels that go with that practice, even if you can only do so temporarily. It may further help to tell students to try on such labels like those red-and-white “Hello my name is” stickers and see what comes of it.

Since the Easter card incident, I’ve applied this theory to myself a little more. A few weeks ago, I cleaned out the flash drive full of photos I’ve taken in the last six months. Several hundred shots were deleted because they didn’t hold value for me anymore. Actually, I realized their value wasn’t in their clarity, focus, or beauty, and it wasn’t meant to be; it was in helping me practice to be something new. Teacher. Tutor. Writer. Photographer. I kind of like the sound of that.

Photos copyright 2011 and 2010 by Kristi Harreld. Copied by permission only.

A Closer Look at Surface Structure

Applying past knowledge to new situations or problems is called transfer.

As many of us teachers and parents have experienced, getting a child to transfer knowledge isn’t always easy. The ‘laps around the track’ problem that I gave my 7-year-old son was a poignant example for me. Students will often focus so much on the unfamiliar surface structure of a problem that they miss the deep structure, and they don’t realize that they actually have the skills to solve the problem.

Why do students put so much emphasis on surface structure? Because it’s the clearest part of the problem. The surface structure is written out and obvious, whereas the deep structure is hidden and unclear. It seems more efficient to work with obvious and concrete information rather than take haphazard stabs at what the deep structure might be.

It is important to remember, however, that paying attention to surface structure – the content of the words, the meaning of the story – is actually a necessary step when we’re learning new information. We learn new information by connecting it to information we already know.

For example, the other day my son was reading a children’s non-fiction book about murals. I know that he already has some background knowledge on the concept of art: painting and sculpture are kinds of art, Picasso and Van Gogh are famous artists, paintings are done on paper or canvas, etc. After reading the book, he hopefully fit the new information about murals into his conceptual understanding of art.

Here’s a simplified idea of how some of that information might be organized in his mind. The left and middle boxes represent what he already knew about art, while the box with dotten lines on the right represents new information.

Since he already had some context of what art is, he was able to fit this new art form into his previous understanding of art. As time goes on and he gains more knowledge and exposure to this topic, this understanding will grow and deepen.

If there’s no background knowledge or context for understanding a new fact or idea, the chances of a student learning is pretty slim.

When a student focuses on the surface structure of a problem (like laps around a track), he’s recalling related background knowledge (laps, tracks, runners) to see if it will help to solve it, or to see if there’s anything new that can be attached to what he already knows. If this is how we learn and understand, we can hardly fault a student for placing so much emphasis on the surface structure. Though surface structure can be helpful for absorbing new conceptual information, it wasn’t enough to help my son solve the “laps around the track” addition problem that I presented to him.

How can we help students recognize deep structure, which is also a very important skill, and promote transfer?

The short answer: lots of practice.

In my next post, I’ll go into more depth about how to help transition from surface structure to deep structure.

-Sara McGuinn, Northshore Tutoring Director

Optimizing for Ownership

“Optimize for ownership” is a phrase we use often here at Nurturing Wisdom. With ownership comes a sense of control, of investment, and of pride. We believe that optimizing for ownership ultimately results in engaged, high-quality work amongst our team. But we don’t limit this approach to the adults at Nurturing Wisdom; we want all of our students to take ownership, too! How do we accomplish this?

ESTABLISH HIGH EXPECTATIONS AND CLEAR GOALS. When I meet with students, I tell them that the tutor will teach them content and strategies, but it is dedicated and consistent work on the student’s part that will lead to improved grades and scores. Meeting regularly, completing homework, and taking practice tests are some of the major steps students take to achieve their goals. But working with a can-do attitude, asking questions, and learning from mistakes are just as important. Students take an active, rather than passive, role in their learning.

ESTABLISH ROUTINES. We teach students how to effectively use their assignment notebooks, we help them create a system of folders and binders, and we show them how to study a little bit every day. They will learn interactive study strategies, and they will learn how to self-advocate. Working through a process of modeling, guided practice, and finally independent practice, our goal is for students to internalize and apply what they've learned from their tutor. Students get to know their own process of learning, which helps them take charge of their learning.

ESTABLISH A GROWTH MINDSET. We sometimes hear from parents and students that they're disappointed in a score or a grade. At Nurturing Wisdom, we avoid using the dreaded D-word! We encourage families to look beyond that brick wall to figure out what the student needs to work on. A shift in attitude will often reveal what a student has done well.

One example of this happens with ACT tutoring. On the ACT, students may have a lower reading score on their mid-test than their pre-test. But when we look closer at the student's mid-test, we may find that errors are clustered in the final pasage. When errors are evenly distributed throughout the reading section of the pre-test, students show a need for comprehension and test-strategy work. But when errors are clustered at the end of the section, we see that timing is the real issue. Rather than feeling disappointed in a lower score, we will celebrate this shift and focus on timing strategies. Students are armed with a sense of control and optimism when we take this tack.

I love hearing about improved grades and scores as much as anyone, but what is really music to my ears is hearing about a student's increased confidence and ability to transfer skills. By optimizing for ownership, we create lifelong independent learners.

-Erin Doyle, Tutoring Director in the Western Suburbs