### How Should We Make Lessons More Challenging

Learning materials are tools that assist a student. Lessons become more concrete, for instance, with manipulatives. Providing too much, however, is not ideal since lessons can become too specific that critical characteristics of a material to be learned can be easily misinterpreted. Oftentimes, superficial traits become incorrectly equated with distinguishing features. As a result, going to the next lesson or transferring what is learned to a different situation becomes much more difficult. Effective lessons therefore require a balance between accessibility and difficulty. There is indeed a continuum between direct instruction and discovery-based learning. Finding just the right amount of scaffolding allows for students to find an activity doable and at the same time, challenging. The key is introducing "desirable difficulties". These are difficulties that should extend the lesson and avoid cosmetic scaffolds that may introduce irrelevant or incorrect generalizations.

One time, I saw my son and his friend doing this math activity on the computer. It was the Battleship Numberline game from BrainPop. The game basically requires a child to estimate on a number line. The following is a screenshot:

The game first tells a child where the ship is. In the above example, the ship is supposedly spotted at "10". At the beginning of this game, there is no submarine drawn on the picture. It only shows up after a guess is made along the number line. In this particular case, the guess I made is indicated by a missile falling. It is a fairly good estimate of where "10" is along the number line. Below is an example of a bad guess I made:

I can make this game easier by simply adding markers at the quartiles of the number line:

This is actually realistic since, with measuring tools like a ruler, it is common to find finer divisions. Unfortunately, by making it more concrete, the exercise has become too specific. This can be alleviated by making one more revision:

The number line now goes from "0" to "180". The quartiles are present, but in this case, the situation is not limited to estimating numbers between a scale that goes from "0" to a "100", which is a bit more general. Still, this may not be as widely applicable in real life.

It is amazing how small changes in an activity or lesson can affect the difficulty. Whether these changes actually matter in learning outcomes is the subject of a research paper published in the Journal of Educational Psychology:

### Citation

Applying grounded coordination challenges to concrete learning materials: A study of number line estimation. Vitale, Jonathan M.; Black, John B.; Swart, Michael I. Journal of Educational Psychology, Vol 106(2), May 2014, 403-418. http://dx.doi.org/10.1037/a0034098

### Abstract

Do concrete learning materials promote strong learning outcomes, or do they simply make learning tasks more initially accessible? Although concrete materials may offer an intuitive foothold on a topic, research on desirable difficulties suggests that more challenging tasks facilitate greater retention and transfer. In the approach introduced here, grounded coordination challenges (GCCs) are embedded into the design of concrete learning materials to deliberately increase the difficulty of the learning task. More specifically, these challenges are intended to promote a deliberative process of mapping between perceptual elements of the materials. In 2 experiments the GCC approach was tested in a number line estimation task by comparing training with an “incongruent ruler”—which was designed to mismatch the length of an on-screen number line—to a “congruent ruler” (both experiments), or no ruler (the 1st experiment only). In both cases participants with the incongruent ruler were more likely to transfer knowledge to spatially transformed number lines. These results indicate that desirable difficulties facilitate learning in mathematical activities. Furthermore, the difficulties should emphasize a deliberate coordination process between critical features of the learning tool and the task. Implications for the design of learning activities that balance instructional support with conceptual challenge are discussed. (PsycINFO Database Record (c) 2014 APA, all rights reserved)

In this work, Vitale and coworkers introduce an incongruent ruler to make estimation on a number line more challenging. An incongruent ruler is one that does not match the total length or dimension of the image a student is studying, as illustrated in the following figure:

 Above copied from Vitale et al.
Having an incongruent ruler obviously raises the difficulty level of this exercise. On top of the scale not being nicely rounded like 0 to 100, an incongruent ruler is not of the same size. The difficulty of the exercise shown above can be measured by how many tries a student needs to accomplish a given number of correct estimates. Having an incongruent ruler is still better than not having one, but having a congruent ruler is clearly much easier:

 Above copied from Vitale et al.

The activity is composed of "blocks". Each block is a set of eight pictures in which a randomly generated position needs to be estimated. In the above figure, the number of blocks a student has attempted before achieving a perfect score (getting all eight positions in a block correctly) is shown for each condition. Clearly, the congruent ruler makes the activity easy. Easy, unfortunately, comes with a price. Using a posttest that requires students to locate target values on a number line with varying lengths and orientation (Examples of questions are shown below),

 Above copied from Vitale et al.

it becomes apparent why "desired difficulties" are necessary. The results of the posttest demonstrate the usefulness of the incongruent ruler:

What is shown above is a measure of the number of mistakes (not the number of correct responses) made by the students in each subtest of the posttest. Those who were trained with a congruent ruler clearly made a larger number of errors, even more than those who did not have any ruler.

Students who participated in this research are from second, third and fourth grade classrooms in a public school that serves a predominantly low income Hispanic community. The experiment is well designed and the conclusions drawn from the study are strongly supported by data.