# Visualizing progress

Visualizing progress can have a powerful motivating effect. Here are some suggestions about how it may be used.

**Main message: expect fluctuation and watch the trendline**

Progress hardly ever happens in a straight line. The picture on the right shows a real life example of an improvement process. The red line shows the actual values found (for instance the sales at a certain point in time). As you see, the levels constantly fluctuate. The blue line is the trend line which shows that over time there is a slow but steady improvement. The arrows show the following: **Arrow 1**: fast first results, quick progress. **Arrow 2**: rather heavy fall back. **Arrow 3**: quick improvement again. **Arrow 4**: serious fall back again after which improvement picks up again. It would be very easy to get discouraged when focusing too much on the fluctuations, at point 2 and 4 for instance. Two things are important to remember: 1) It is normal for progress to show this kind of fluctuation, and 2) The trend line is an important line to watch. This line shows you that there is actual growth overall. The trend line is a very motivating line to watch.

**Have a mathematical look at progress**

A brief intermezzo before we’ll go to the second message about visualization. A few years ago, mathematics professor Steven Strogatz wrote two columns in the New York Times called Change We Can Believe In and It Slices, It Dices. The columns are about **calculus**, a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. Calculus is the mathematics of change. Its two basic branches are differential calculus and integral calculus. Without giving an in-depth explanation of these two topics, here is a brief introduction.

**Differential calculus is the study of the derivative of functions. Calculating the derivative is called differentiation. The derivative tells you how fast something is changing, how far youâ€™re going up or down a slope for every step you take. The derivate is the approximation to the slope of a graph. The derivate can be calculated for each point of a function but also for every point which leads to the derivate function (see picture on the right [from www.derivate.it]). When a slope is going up its derivate is positive, when a slope is going down it’s negative. At the peak and the bottom of a curve it is zero. At those points, change momentarily stands still.**

**Integral calculus is the study of integrals which tells you how much something is accumulating. It is the calculus of summation. Calculating the integral is measuring the area under a curve (see picture on the left). Functions can have some very irregular shapes which can make it quite hard to easily measure this area under the curve. What integral calculus does to solve this problem, is slice that area up into thin slices, calculate the volume of the slices and then cleverly adding them up again. On a side note, integrating is the inverse operation of differentiation.**

**Second message: watch the slope and the accumulation**

Applying the explanations of derivates and integrals to the original graph you get the picture on the right. Let’s apply them to the potentially most depressing point of the graph which is point 4. The **red line** shows the derivate of point 4, the colored areas show the integral of the graph until point 4. Both give reason for a positive and hopeful view on the change results. The derivate (red line) at point 4 shows the negative slope is getting less steep which says things are still getting worse but this decline happens more slowly which gives hope that the bottom is in sight and a turn to a positive slope may be near. The colored areas show the integral which, the area under the curve. The **red area** visualizes the accumulation of everything that has been achieved from point 0 to point 3. As you see it is all below 0 so the result is negative. This **blue area** visualizes the accumulation of everything that has been achieved from point 3 to point 4 and the absolute value is reaching 0 again, the Nett result of the change process is positive because the blue area is much larger than the red area. A lot has been achieved. The thought, at point 4, that every effort has been in vain is therefore unjustified. There is progress.

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