The more traditional approach, has the teacher, in the classroom, explaining some basic turns and landmarks and modeling on the board how to get to a specific location. Then the car keys are handed out to find someplace specific for homework.

In this approach, separated from the actual experience of the city, the teacher is pulling on past experience, and the explanations and modeling are a short-hand for that experience. The student needing extra help (intervention) has no or at best a mixed experience to make accurate sense of the short-hand.

In a second approach, the teacher and student get in the car together with the teacher driving, pointing out the turns and landmarks he or she already knows, again asking student to find a specific location for homework. As in the first approach, we are learning more about the teacher’s competence than observing and developing the student’s ability.

The approach Mastery Learning Systems takes is to have the student drive with the teacher in the passenger seat not telling where or when to turn. But the student (or students, this works equally well one-on-one and in the classroom) is not left to wonder aimlessly around the city, possibly but not probably, finding a desired location!

In their approach, the teacher first orients the student to a map of the city to be experienced. With good questions, the student notices landmarks on the map and is able then to begin making connections between the map and what is experienced while driving, “Oh this turn on the map is coming up right here.” “Good,” says the teacher – - and that’s it – - no lecture on the history and use of gazetteers. The teacher keeps the student in the driver’s seat.

If the student makes a wrong turn, the teacher simply asks the student to show on the map what they did and where they wanted to go. The student retraces the steps and notices that a left turn was needed not a right turn. “Good,” says the teacher and the student turns left. The city is internalized as it is experienced, guided by the teacher and the map.

Mastery Learning Systems develops easily understood and remembered graphic representations of mathematical concepts and procedures. They are very effective for all students, but especially the “please don’t explain it to me again” learner. And this approach works equally well at home, in one-on-one tutoring, and in the classroom.

They integrated their approach throughout the math needs of K-8 students from adding and subtracting, multiplying and dividing, fractions and decimals, geometry and negative numbers to developing understanding and fluency with equations. The teacher or parent orients and guides the student through the graphic, the map, of the math and the student internalizes the experience of that math moving naturally from intervention to Math Mastery.

1. by good materials we mean those that, by their design, allow for ongoing observation of student performance. The materials need to be flexible enough to help discover uncertainties in the current lesson while uncovering prior skill gaps. And they need to allow the teacher to address what is observed

2. by good teaching, we mean keeping students in the active role so they can be observed. Good instruction is not missed by replacing some “teaching” time with active student engagement. And flexibility, as describe in point #1, also on the part of the teacher is essential to tap the potential in the materials and the students

We look for resources for our catalog that keep students in the active role as much as possible. This is easier for some subjects than others. Still, a teacher or parent can adapt materials by asking good questions along with “instructing.” Questions enable students to notice elements of a concept or procedure that later practice clarifies, reinforces, and extends. Depending on whether questions are narrow and focused or open-ended, a child’s attention is directed to a key element or it is invited to consider the context of a problem or how it relates to something already familiar.

Check our featured products of the week on the jd-commons.com homepage for examples of these resources from our catalog.

The function of a math book is to develop mathematical reasoning, not simply present problems that require its use. By forgetting what it is like not to already know the concepts and procedures, some textbook authors unwittingly exclude students from success. They actually stop the development of reasoning and force students to rely on mere memorization to cope with the work.

Yes we need to keep earlier concepts and procedures alive by integrating them into problems in subsequent chapters, and yes students need to explore multiple applications, and yes they need to use all of this to solve problems and not merely perform calculations. I am not arguing against any of this. But enrichment is enriching and higher level thinking is only thinking when students have access to it.

Why not take as much pride in opening up and developing that next level of higher thinking, whatever it may be, as in the creative, clever, complicated, and fun problems that already knowledgeable minds can conceive? Again, we should remember what it’s like for those who are struggling, hanging-on, or doing okay but presented with yet another new concept with which to grapple. What is higher level thinking for them?

We offer the math intervention materials from Mastery Learning Systems at www.jd-commons.com for this very reason. Whether the book is covering addition and subtraction, negative numbers, systems of equations, or graphing, the steps are incremental, developing a firm foundation each step of the way.

A recent review of one of their books by The Old Schoolhouse Magazine read in part,” If I had to describe this program in two words, they would be “comprehensive” and “thorough.” The simple, straightforward approach is sure to help many struggling students build a strong foundation in math.”

Mathematics explores basic principles or structure, what’s “under the hood” of our day-to-day experience. It requires exacting levels of observation, attention to detail, analysis, synthesis, relevant question asking, and problem solving.

We invite children deeper into or higher up these modes of thinking year-by-year, subject-by-subject. So how and when do we develop higher level thinking? I’ll use mathematics as a model?

A look at some of the approved and adopted texts suggests that one answer is, “Algebra,” that is, introducing higher level activity further down the grade levels. Constants, variables, coefficients, expressions, equations, real, rational, and irrational numbers, and combining like terms…  If we can just get upper and even lower elementary students to start thinking about some of these abstract ideas, wouldn’t higher level thinking and reasoning take place? Or are “higher level activities” and “higher level thinking” not necessarily the same thing?

I tutored a high school geometry student recently who had memorized a formula stating that amount “A” was half of amount “B” and applied it correctly, each time. But given the reverse to solve, where “B” was twice “A,” demonstrating an understanding and not merely a memorization of this formula and prior math lessons on fractions, he was stumped. For this student, working with basic fractional relationships was higher level mathematical thinking—higher than the current level of understanding.

Higher level thinking is a fluent or functioning understanding of whatever is the next step. The relationship between 1/2 and twice, or that a group can be both one and many, or that a “1″ sitting in the tens column has a different meaning or value than a “1″ in the ones column are all examples of higher level thinking for children who do not yet have a functioning understanding of them.

What is important is not when to introduce Algebra, but whether or not whatever work the children are doing is developing a fluency and memory based on understanding. Those who can factor quadratic equations because they have memorized the appropriate rules cannot be said to be applying higher level thinking necessarily.

What does this have to do with the lessons we present to our children and the methods we use in teaching?

Children can give the appearance of higher level thinking when performing activities through mere memorization or repetitive activities devoid of real understanding. “Tests” are given to “prove” that this “works.” But what do they really test? Present children with an application of that knowledge in an unexpected or even expected but new setting, and the appearance is shown for what it is.

When “lower level” activities are taught in ways that build fluent understanding, then children are invited into higher levels of thinking and reasoning. It happens right where they are.

Again, Algebra (and fill in here “higher level” activities for other school subjects) is not the problem. The problem is assuming that higher level activities assure higher level thinking. This can force us into accepting thinner and thinner teaching methods, devoid of the richness of developing understanding and fluency (real higher level thinking) just to keep up.

The next post will look at levels of complexity as a stimulus for higher levels of thinking.

The education catalogs we research are filled with colorful activities for children. These activities remind them of what they already know in a fun and engaging way. For instance, children do practice their math facts or grammar rules through whimsical illustrations. But is this good education?

There is no real mathematical or linguistic thinking being developed. These materials also take up time that could engage children in both memory building and conceptual development. And, color and whimsy can weaken the motivation to take on lessons that really engage a child in a subject.

Right beneath the surface color, not hidden at all, is usually nothing more than simple computations or drills. Does this spoonful of sugar help? It can give the illusion that learning is taking place. It can give the child, parent, or teacher the false satisfaction that the child knows more than the activity demands.

Learning is developmental, one skill underpinning another. A weakness along the way compromises later understanding, performance, and language, math, or science-esteem, in other words, the desire to keep going. Color and whimsy are not designed to uncover and address prior skill or knowledge gaps. Equally important, they are not designed to fix them.

The maze of hype we navigate to find real lessons vs. surface color and diversion is daunting. Good lessons, well delivered are engaging and can be colorful and whimsical. But the opposite is not true, that engaging color and whimsy assure good lessons.

To support intervention programs, parents, and children who need extra help, we are careful to avoid mere entertainment and colorful drills in our catalog. We appreciate your feedback on these ideas and your experience with the resources we offer. Our catalog is strengthened by it and everyone benefits.

The subjects taught in school tap different thinking skills: observing, analyzing, synthesizing, interpreting, generating, improvising, honing, taking the long view, attending to the detail, defining the problem, and solving the problem. This is most effective when students are knee-deep in the subject, exploring, discovering, and experiencing it. The skills are then rooted in meaning, context, and application. This is not a revolutionary idea, but it is easy to forget about in the midst of day-to-day subject teaching and learning.

The danger in forgetting about it is that it can lead us into thinking that activities disconnected from the experience of the subject and usually devoid of meaning, like drills and tricks, are useful. I’ve read promotional material for educational products that tout all of the hours of “proof” that drills are a good educational tool. Depends on what you are trying to achieve I think.

Facts or skills drilled into students do not constitute knowledge or thinking ability. While true, at the same time thinking needs to think about and with something. If the students’ minds are burdened with having to reestablish basic facts and concepts, they’ll be like music students not fluent with their scales, burdened and frustrated later when they come across those scales in a piece of music. The essential experimentation and discovery in student focused learning, and memory and skill building are not mutually exclusive learning objectives. Nor do they need to be or are they best taught in isolation.

Given our time constraints, why try to teach them in isolation, why talk about concepts and then drill facts or skills later? For our online store, jd-commons.com, we look for educational materials that weave concept with practice, so practice is rooted in meaning. Presenting concept with practice is not only good time management, it is also good education, it sets students free without setting them adrift.

Last winter I came across a blog post stating something that seemed logical and helpful. Before we start working with students to help improve math performance, it noted that we needed to test them to know the areas to focus on. A quick search on the internet reinforced this. A number of sites discussed different types of assessment: diagnostic or prior assessment, formative or ongoing assessment, and summative or post assessment. But when looking for examples of each, ongoing formative assessment received less attention compared with more examples each of diagnostic and summative testing.

This reminded me of when I made a presentation before a state curriculum adoption committee. I explained that since my business partner and I publish intervention materials designed to support students struggling in math, formative assessment is built into the fabric of each lesson in the program. While there are pre, post, and mid-chapter tests, the emphasis is on the ongoing support for these students of formative assessment. One member of the committee laughed and said, “Well that requires good teaching!”

It appeared that everyone in the room recognized that good assessment is systematic and ongoing.  But several members of the committee saw repeated testing as the means of insuring ongoing assessment.

I won’t go into possible reasons for the apparent preference for testing. Rather, in this post, I want to lay out some problems with relying on testing as a primary means of assessment, especially for students challenged by math or any subject for that matter.

Going back to the opening paragraph, diagnostic or prior assessment is essential. We’re flying blind without it. But favoring tests over integrated ongoing assessment may miss part of the potential power and value of assessment and even create some of the very problems our better teaching are trying to address.

When taking tests, students challenged by math will continue to write some incorrect answers. They will leave some answers blank. And a test is usually graded by someone else, not the students, and is usually handed back at a later time disconnected from the experience of that assessment. So what’s the problem with this?

Students who are not doing well may not give us the best indication of what they know and can do by taking another test. Continuing to write incorrect answers can reinforce memories of those wrong answers, reinforce faulty procedures leading to the incorrect answers, and reinforce partially understood or misunderstood concepts. Having to leave some answers blank can reinforce a negative self-image, and frustration with and resistance to the study of math. It also means that needed practice on those problems is missed. When the test is graded by the teacher and handed back at a later time, the students are no longer engaged in the problems and many will not retrace and fix the steps that led to the wrong answers.

Assessment has multiple uses for teacher and student. It gives the teacher some evidence of student preparation and performance, which informs what is taught and how it is taught. It also provides students with feedback. While testing is an essential component, should it be the primary form of feedback for the intervention teacher? The teacher may not know from a test, which problems the student labored over or solved quickly whether they got them right or not. The test may not reveal which prior conceptual, procedural, or simple nomenclature gaps led to an incorrect answer. Depending on when a test is given, it may be too late to go back and address certain problems. And because of all of this, it may also not be the best form of feedback for a student struggling in math.

What do you think?