*This article was originally published on The Conversation. Read the original article.*

Despite decades of reform efforts, mathematics teaching in the U.S. has changed little in the last century. As a result, it seems, American students have been left behind, now ranking 40th in the world in math literacy.

Several state and national reform efforts have tried to improve things. The most recent Common Core standards had a great deal of promise with their focus on how to teach mathematics, but after several years, changes in teaching practices have been minimal.

As an education researcher, I’ve observed teachers trying to implement reforms – often with limited success. They sometimes make changes that are more cosmetic than substantive (e.g., more student discussion and group activity), while failing to get at the heart of the matter: What does it truly mean to teach and learn mathematics?

Traditional middle or high school mathematics teaching in the U.S. typically follows this pattern: The teacher demonstrates a set of procedures that can be used to solve a particular kind of problem. A similar problem is then introduced for the class to solve together. Then, the students get a number of exercises to practice on their own.

Students in these kinds of lessons are learning to follow a rote process to arrive at a solution. This kind of instruction is so common that it’s seldom even questioned. After all, within a particular lesson, it makes the math seem easier, and students who are successful at getting the right answers find this kind of teaching to be very satisfying.For example, when students learn about the area of shapes, they’re given a set of formulas. They put numbers into the correct formula and compute a solution. More complex questions might give the students the area and have them work backwards to find a missing dimension. Students will often learn a different set of formulas each day: perhaps squares and rectangles one day, triangles the next.

But it turns out that teaching mathematics this way can actually hinder learning. Children can become dependent on tricks and rules that don’t hold true in all situations, making it harder to adapt their knowledge to new situations.

For example, in traditional teaching, children learn that they should distribute a number by multiplying across parentheses and will practice doing so with numerous examples. When they begin learning how to solve equations, they often have trouble realizing that it’s not always needed. To illustrate, take the equation 3(x + 5) = 30. Children are likely to multiply the 3 across the parentheses to make 3x + 15 = 30. They might just as easily have divided both sides by 3 to make x + 5 = 10, but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.

A key missing ingredient in these traditional lessons is conceptual understanding.

Concepts are ideas, meaning and relationships. It’s not just about knowing the procedure (like how to compute the area of a triangle) but also the significance behind the procedure (like what area means). How concepts and procedures are related is important as well, such as how the area of a triangle can be considered half the area of a rectangle and how that relationship can be seen in their area formulas.

Teaching for conceptual understanding has several benefits. Less information has to be memorized, and students can translate their knowledge to new situations more easily. For example, understanding what area means and how areas of different shapes are related can help students understand the concept of volume better. And learning the relationship between area and volume can help students understand how to interpret what the volume means once it’s been calculated.

In short, building relationships between how to solve a problem and why it’s solved that way helps students use what they already know to solve new problems that they face. Students with a truly conceptual understanding can see how methods emerged from multiple interconnected ideas; their relationship to the solution goes deeper than rote drilling.

Teaching this way is a critical first step if students are to begin recognizing mathematics as meaningful. Conceptual understanding is a key ingredient to helping people think mathematically and use mathematics outside of a classroom.

Conceptual understanding in mathematics has been recognized as important for over a century and widely discussed for decades. So why has it not been incorporated into the curriculum, and why does traditional teaching abound?

Learning conceptually can take longer and be more difficult than just presenting formulas. Teaching this way may require additional time commitments both in and outside the classroom. Students may have never been asked to think this way before.

There are systemic obstacles to face as well. A new teacher may face pressure from fellow teachers who teach in traditional ways. The culture of overtesting in the last two decades means that students face more pressure than ever to get right answers on tests.

The results of these tests are also being tied to teacher evaluation systems. Many teachers feel pressure to teach to the test, drilling students so that they can regurgitate information accurately.

If we really want to improve America’s mathematics education, we need to rethink both our education system and our teaching methods, and perhaps to consider how other countries approach mathematics instruction. Research has provided evidence that teaching conceptually has benefits not offered by traditional teaching. And students who learn conceptually typically do as well or better on achievement tests.

Renowned education expert Pasi Sahlberg is a former mathematics and physics teacher from Finland, which is renowned for its world-class education. He sums it up well:

We prepare children to learn how to learn, not how to take a test.

*This article was originally published on The Conversation. Read the original article.*

The current system of mathematics teaching, described in the paper as procedural teaching with overtesting, has the great advantage of being a cheapest socially acceptable option. Anything cheaper will be of even lower quality. Anything bettter is likely to be much more expensive. “Teaching for conceptual understanding” is a very expensive option: it requires much better (and much more expensive) teacher education, retraining of the army of existing teacher, higher salaries for retention of teachers, smaller classses. As simple as that.Teaching for conceptual understanding

I totally agree which the concept of teaching mathematical concepts, rather than simply “This is how you do this” or worse, “This is which keys on your calculator that will give you the answer when you are taking the standardized test”. Learning to use mathematics concepts is useful even if the student is not currently on a path to a career in a technical field, simply because it exercises the brain, and I have yet to see a student harm because he was forced to actually think about something. Many seemly menial tasks have an element called “how much material do I need?”, and often the answer an be reformed as a mathematical exercise. So yes, once you know which mathematical formula you need your calculator can give you the answer, but does your calculator know how to reform the question as a mathematical formula, probably not. A former student though who has learned mathematical concepts will be able to bridge that gap.

Excellent article equally relevant to the UK. However the pattern of exposition, example and exercise is not intrinsically at odds with teaching for conceptual understanding but is most effective when the goal is conceptual understanding (why it is solved this way) rather than simply this is how we do it.

Majority of the reason we cannot include more conceptual methods is due to the high demand of curriculum. In the USA, we cover an insane amount of material while other countries probably cover half, if not less. That is because they teach the WHY, not the WHAT (ie: formulas, testing kids the same questions just different numbers, etc.).

We must distinguish between several tasks: calculate the result given procedure and numbers to substitute; to turn known procedure to equation and solve it for missing parameter; to build procedure by yourself ( and here you cannot avoid revealing interconnection between all the parameters of the problem in mathematical form, choosing the proper means and use them in the order you can prove it`s convergence to solution ). The question is – to what extent we need the pupils` society to master the skill of how to build procedures? Life shows that very often there will appear student who will ask loudly ” Mr. Teacher, who built the procedures you teach us? I READ in TEXTBOOK WHAT the AREA is, and I BUILT at home another procedure to count area – check me, please”. Such a student is ready for conceptual learning, but should the society seek to grow up as many such students as possible? Maybe some countries will answer “YES”?

To make teacher change the way the teach just change the questions in “external” examinations. Questions have to contain two sections 50% on basics and another 50% contain questions that are 3 times longer than the basics. Longer questions allow conceptual understanding, deeper understanding and more advanced skills in problem solving to be included. Teachers will naturally improve themselves to prepare their students to stay competent – there is no need for expensive retraining. In my teaching experience there have been occasions questions with new ideas and majority of us the teachers’ response were always how to prepare their students for such surprises.