I’m hoping that as math instruction improves and becomes more “brain-friendly,” we’ll see fewer kids struggling in math.

When I was in my doctoral program, I was amazed at some of the research coming out on kids’ understanding of math concepts. We assume that children all learn pretty much the same math at roughly the same ages, and that they learn these concepts in math class.

In fact, there’s a wide natural variation, and not necessarily a lot of correlation between the math kids are taught in school and the math they actually know.

[I'm devoting my Monday blog posts to the topic of Learners with Special Needs, which, I find, describes us all in some way or another.] 

I work part-time at a school for students with all kinds of special needs. In addition to the usual academic subjects, kids also take classes in such topics as executive function, sensory integration and behavior therapy.

So much of the instruction is simple and useful and applicable to all of us!

When kids at the school have some conflict, they are required to fill out a Conflict Resolution Sheet:

[I'm going to devote my Thursday blog posts to the topic of All Things Academic: reading, writing, 'rithmetic and the other school subjects.]

Last week I said that I see value in having kids (and all learners) memorize a certain amount of factual information.

I also said that I’m not a fan of rote memorization of multiplication “facts.” Kids should also be learning when and how to apply all of the four operations to various situations.

I’m going to try devoting my Thursday blog posts to the topic of All Things Academic: reading, writing, ‘rithmetic and the other school subjects.

A home school mom of four writes:

The learning material that I struggle with is just that: motivation verses memorization. I have a hard time wrapping my mind around alternative methods to memorizing times tables and science facts and history dates. It just seems like there should be better ways to learn/teach.

Kids (and all people) learn best when information is relevant and interesting. Random facts that don’t connect with anything the student finds familiar or meaningful are tedious to memorize and soon forgotten.

Why do they have these timed tests, like 25 problems in 3 minutes?

This is an excellent question.

I currently work and have worked with quite a few students who receive extra time on standardized tests, and I know for a fact that colleges do not factor this into their decision.

Meaning, if we have two absolutely identical students A and B, and A scores a 2100 out of 2400 with regular constraints while B scores a 2200 with double the time, B gets in and A doesn’t.

So, first, does time really matter? And second, if it does, why does time matter?

My sister has argued that students should not freely be given extra time. I think my hypothetical identical students identified this problem. Her basic point is that in the real world (or a college environment), speed and time are factors. Take two engineers applying for a job: it’s obvious that the guy who’s faster at math has a practical advantage.

Yet, I’ve worked with students who just need more time, and for each of them, I’m so glad that they get the opportunity to let their true intellectual power show.

In case you don’t know, interactive whiteboards and “smartboards” are the hot, new classroom technologies. Replacing the old blackboard and chalk, these boards can show PowerPoint lessons, copy what the teacher writes so that kids don’t have to take notes, and enable a variety of Wow! kinds of presentations. Think of those interactive maps now used by TV weather reporters and you’ve got the rough idea.

Writing for Education Week, “Educator Bill Ferriter makes no bones about his distaste for interactive whiteboards, calling them ‘sad examples of careless decision-making and waste that are crippling schools.’” (His article is called “Why I Hate Interactive Whiteboards”):

Seen as the first step towards “21st century teaching and learning,” schools and districts run out and spend thousands of dollars on these gizmos, hanging them on walls and showing them off like proud hens that just laid the golden instructional egg.

I gave mine away last summer. After about a year’s worth of experimenting, I determined that it was basically useless.

Sure, my students thought it was nifty, but it didn’t make teaching my required curriculum any easier. I probably crafted two or three neat lessons with it, but there was nothing unique about those activities. I could have easily put together similar lessons using the computer stations I already have in my room and any number of free online tools.

I share Bill’s feelings. Not all technology is great or helpful or worth the cost.

We need to think first about what we are trying to accomplish educationally and then create the technology to achieve those goals. Instead, we often spend tons of money on cool-looking gadgets with little educational payoff.

The main goal of education needs to be to stimulate students’ brains…to get them to read with understanding and write clearly and research purposefully and problem-solve and think autonomously.

I question whether interactive whiteboards, as well as much educational computer software, does this. I fear that such technology often relies too much on exciting kids eyes and ears and doesn’t do enough to provoke deep thinking. I also wonder about the effects of such technology on attention span and the development of self-control. …

In my recent post, Midterm Exams and 21st Century Knowledge, I said that we ought to be focusing more on developing reasoning, problem-solving and researching skills, and de-emphasizing rote memorization of names, dates and formulas. Our sophisticated technology allows us to look up data and procedures easily, freeing up our “head space” for higher level thinking. Yet, every year, students are still required to cram for exams by stuffing lots of facts into their brains.

Here’s one of the replies I received: Yes, what the formula is may be important but as long as we know where to find it,it is much more critical to know why, when and how to USE it. So yes, I agree with you. The next thought is how do we change the system and the teachers entrenched in it?

I think that technology is going to be part of the key to a dramatic change for the better in our educational system. Right now there are experimental programs being piloted, including School of One, a system which uses computers and interactive software programs to deliver individualized, self-paced instruction. Kids learn best when they learn at their own pace, and technology can make this possible in ways large-group classroom instruction simply cannot.

I also believe that once teachers are freed from their roles as classroom lecturers and disciplinarians their energy will flow in productive directions. Once technology takes over the bulk of the lesson-delivery process, teachers will have time to help individual students or pursue special-interest topics with small groups. Teaching will become a very different profession from what it is now.

A few teachers, the ones “entrenched” in their old-fashioned roles, will not welcome the change. But I can’t help believing that most real educators…people who love knowledge and are excited about learning…will be overjoyed to spend their careers actively empowering kids to develop into strong thinkers and communicators.

Many schools use a spiral review approach in their curricula. In the British system, for example, kids get one trimester each of biology, chemistry and physics every year, instead of taking these courses separately over full years.

Math texts always include review of the previous year’s skills before launching into the new work. Homework, summer review packets, mid-terms and finals, are all examples of spiral review.

There are unique challenges when applying a spiral approach to math learning. Math, unlike other subjects, is hierarchical. Concepts build on top of earlier concepts, and if any layer is weak the next layer will be even shakier.

Spiral review in math, therefore, MUST be individualized in order to be effective, and it must dig back to foundational concepts and reinforce these core understandings.

Unfortunately, math curricula which use a spiral approach often befuddle students by touching too lightly on new topics and then flitting away before students can get a handle on them. Students are often left with only vague notions of the new concepts, plus feelings of confusion and distress. No one likes to be taught something new and then left with the feeling that they “didn’t get it.” These kinds of experiences can contribute to math anxiety, disliking of math, and negative self-image.

Here are some ways you can use the spiral review technique to help your student or your own child:

• Revisit / review material periodically to refine and strengthen cognitive connections

• Practice regularly, not just right after the lesson or right before the test (think of it as “mental weight-lifting,” building mental muscles through consistent, moderate exercise)

• Do a few review problems that now seem easy, but used to be soooo hard! Make sure you point out how tough these problems used to be, and how much progress has been made.

My favorite part in Catcher in the Rye is where Holden Caulfield visits the Museum of Natural History. He finds it reassuring because the exhibits stay the same while he is different every year he returns.

The technique of spiral review is like this. The material is the same and the student revisits it periodically. And with every visit, the student learns more deeply and thoroughly. This is due to two, interrelated mechanisms called “assimilation” and “accommodation.”

Every time a learner revisits material, she reprocesses it; she “rechews” it; she “thinks it through again.” In other words, she assimilates, or “digests,” more than she did on her last pass.

Then, she accommodates, or incorporates, the material into her brain structure, which has already been altered from the last experiences with the material. New neural connections are made and existing connections are strengthened.

So, with every loop of a spiral review, the student is processing the material more thoroughly and then incorporating it into a brain more ready to receive it. The student’s grasp of the material becomes more accurate and more permanent with every revisit.

Sometimes we use the phrases “surface knowledge” and “deep knowledge” to distinguish between things we know in a perfunctory way and things we know inside-and-out. To be an expert on something implies that one has deep knowledge, from long experience with the subject matter in his field. This deep knowledge, this expertise, came from repeated exposure to and use of, the body of knowledge. It came from spiral review.

I see the difference in surface knowledge and deep knowledge all the time in my students. How could I not? They are taught new concepts or procedures and then asked to apply them. This goes fairly well as long as demands remain modest. Most kids can, for example, apply new math procedures to the evening’s homework, so long as the homework sticks closely to the exact lesson they were taught. Where kids run into trouble is with the application of new procedures to novel situations; word problems, SAT problems, etc. What makes these sorts of problems “hard” is that they require kids to take knowledge that they …

Here in Connecticut it’s midterm (first-semester final) exam season and I’ve been working extra hard with students who are frantically trying to prepare.

I like the underlying philosophy behind midterms and finals, which is that learners should expect to retain what they were taught. Otherwise the focus is only on remembering information just long enough to regurgitate it for one test and then forget all about it. Cumulative exams force students to revisit material and, hopefully, entrench it more permanently in their heads.

But there’s also a lot of unfairness and counter-productivity in this system. I have one student, for example, who works very, very hard but has trouble remembering details. She does well on individual tests and quizzes but cumulative exams overwhelm her with the sheer load of material to be memorized.

We began studying together for her Algebra II midterm well in advance, and we’ve been working steadily ever since, plus she’s been doing tons of practice on her own. We’ve been using every study skills and memory enhancement technique available, and she’s come so far! I’m so proud of her!

Yet…”I wish I could carry a note card into the exam!” she sighed yesterday. I wish she could, too.  I truly can’t see why students shouldn’t be able to use their notes (isn’t this what note-taking is for?) to help them remember formulas and procedural details. Making students memorize such things can put tremendous stress on mental recall capacity and clutter the brain’s ability to process.

I wish I could spend my instructional time working on concepts, abstract reasoning and problem-solving. I would much rather explore why a formula works, or how the formula was derived or when to use it in real-life applications.  Instead, I spend way too much valuable tutoring time training students on mnemonic techniques.

There is a lot of talk lately on training kids to have “21st-century skills.” Usually this means learning to use and understand computers and other technology.

To my mind, 21st-century skills must include the age-old, basic skills of note-taking, note-using, and fact-seeking. With laptops available to everyone, who needs to memorize facts and figures anymore? Yes, students should be conversant with …