Tag: classroom advice

This meta-analysis, which looks at studies including almost 12,000 students, concludes that creating concept maps does indeed promote learning.

Specifically, it’s better than simply looking at concept maps, or listening to lectures, or participating in discussions, or even writing summaries.

The article summarizes several hypotheses to explain the benefits of concept mapping: it reduces working memory load by using both visual and verbal channels, it requires greater cognitive elaboration, and so forth.

So, let’s hear it: how do you get your students to map concepts? What successes have you had? Let me know in the comments…

(h/t IQ’s Corner)

Teachers hate (and love) multiple-choice tests. On the one hand, they seem dreadfully reductive. On the other, they’re blissfully easy to grade — and easy grading is never to be belittled.

In our recent conversation, Pooja Agarwal recommended multiple-choice tests as one kind of retrieval practice. Inspired by her guidance, you might be asking yourself: “what can researchers tell me about the best kind of multiple-choice test to give?”

If you’re asking yourself that question, look no further: the estimable Andrew Butler is on the case.

(For example: if you want to know how many distractors to include on your test, you should see what Butler has to say…)

 

 

When we walk into a classroom, especially an early learning or elementary school one, manipulatives are almost always within reach. Look to your left, and notice the group of children spinning the hands on a pretend clock, trying to figure out what 6:30 should look like. Glance to you right, and watch the students sort pretend money into the dollar slots of a dinging cash register. And peer over your shoulder, as students use square, circle, and triangle magnets to create geometric worlds on a magnetic easel.

In a previous article, I discussed some of the cognitive research on problem-solving and decision-making. And while that piece focuses primarily on how conscious and unconscious thoughts make sense of questions and choices, this article turns to another important aspect of problem-solving: classroom manipulatives.

How do physical objects help us make sense of questions and concepts?

Manipulatives in Mathematics

Manipulatives are a type of symbol that can take nearly any form. One of the most common types of manipulatives that we may come across are base-10 blocks; small foam squares that can be combined and separated to help students understand basic math concepts (e.g., addition). Other common manipulatives in the classroom include pretend money, model buildings, and modeling clay.

Now, fiddling with manipulatives can be pretty enjoyable; but, as a learning tool, they come with a fair amount of controversy. This is especially so with mathematics manipulatives.

The more traditional school of thought tends to suggest that manipulatives help children learn math by reducing the abstractness of math problems. [1] They do this by substituting mathematical symbols with concrete objects. For example, the symbolic character “3” can be represented with three blocks. And if you toss in another three blocks, you’ve represented both the concept of addition and “6”.

But, more recent arguments have asserted that manipulatives can only really promote mathematics learning when teachers assist children in understanding the symbolic relation between physical objects and the math concepts they represent. The dual-representation hypothesis posits that when children perceive manipulatives as only being objects (e.g., a single base-10 block as just a squishy square), it is challenging to understand their relation to the mathematical expression they represent (e.g., the number one).

Style vs. Substance

One study that demonstrated just how tricky manipulatives can be investigated the ways in which elementary school students used pretend money when solving math word problems. [2]

First, fourth, and sixth grade students were asked to complete ten world problems that involved money. Half of the participating students received manipulatives: realistic bills and coins along with the suggestion that these materials could be used to help solve the problems. The other half of the students did not receive any manipulatives.

At all grades, the students who did not have access to the manipulatives performed better on the word problems than the students who did. Access to the pretend money actually appeared to interfere with students’ accuracy.

But why?

In a second experiment, fifth grade students were asked to complete ten more word problems. This time, the students were assigned to one of three manipulatives conditions:

1. realistic, perceptually rich bills and coins
2. bland bills and coins
3. no physical manipulatives

The students were also asked to show their work on their answer sheets. This allowed the researchers to analyze students’ incorrect answers to determine whether they made conceptual or computational errors.

The researchers found that the students who used the perceptually rich pretend money made more errors than both the children who used the bland money and the children who did not use manipulatives.

The students who used the bland money performed at the same level as the students who had no access to the manipulatives.

Further, when analyzing the pattern of errors made by students in each condition, it appeared that strategy selection was influenced by the students’ access to the perceptually rich money. Compared to the students in the other two conditions, students in the perceptually rich condition were more likely to select a particular strategy (such as multiplication or division) that often resulted in an incorrect answer.

However, even though these students made more errors overall, their written work indicated that their conceptual understanding of the word problems was the strongest of the three groups.

Thus, there appears to be somewhat of a trade-off when using manipulatives. While these materials can help students relate their learning to real-world experiences, as well as promote conceptual understandings, perceptually rich manipulatives may distract children–and that distraction ultimately results in computational errors.

Two Sides to Every Coin

Interestingly, although research suggests that physical manipulatives can be distracting in a not-so-good way, it also seems that symbols can sometimes distract in a not-so-bad way.

This finding has been shown in preschoolers who participate in the Less is More task. In this tricky game, children must point to a small tray holding two candies in order to receive a larger tray with five candies. To succeed, children must inhibit their urge to point to the tray with more candies on it when asked which one they would like.

Given that young children generally have difficulty inhibiting themselves under such conditions, one study asked whether variations of the Less is More task might reduce the affective component of the game through symbolic distancing. [3] That is, would three year olds’ performance on the task improve if the large and small quantities of candy were represented by something else?

Children were randomly assigned to one of four conditions:

1. the traditional representation of smaller and larger quantities of candies (real treats)
2. rocks representing the candies, with children shown one-to-one correspondence between the rocks and candies (i.e., if children chose the tray with two rocks, they got five candies)
3. arrays of dots to represent the candies without one-to-one correspondence (i.e., one set of dots was larger than the other, but the number of dots was not the same as the number of candies)
4. one picture of mice and one picture of an elephant to represent small and large rewards, respectively

It turned out that the preschoolers’ performance on the mouse/elephant condition was significantly better than on the real treat condition. In other words, children more often pointed to the mice (small symbol) in order to get the elephant (large reward) than they did the two candies (small quantity) in order to get the five candies (large quantity).

Performance on both the mouse/elephant condition and the dots condition were significantly better than the real-treat and rock conditions. It appears, then, that the use of symbols can also distract in a helpful way. In particular, symbols with greater psychological distance from their referent (i.e., the mouse and elephant seem less related to the candies than the one-to-one corresponding rocks do) can reduce the emotional component of the Less is More task.

With this buffer from the emotional temptation of the larger tray of candies, children seem better able to inhibit their instinct to point to it.

Use ‘Em or Lose ‘Em

Despite the controversy that surrounds manipulatives and symbolic reasoning, most researchers seem to agree that there is a time and a place for each. And, most certainly, each has its own learning curve.

In order for manipulatives to be beneficial, researchers generally suggest that teachers:

a) should strive to explicitly connect the manipulatives to the concepts they represent; and

b) should select objects that easily allow children to understand their relation to concepts.

For example, the best math manipulatives tend to be objects that are only used for math learning (e.g., base-10 blocks); are not particularly interesting or familiar; and possess an internal structure that explicitly represents the relevant math concept.

But, when aiming to distract from emotionally-charged situations, symbols that seem unrelated to the emotionally charged object or event generally set students (especially young children) up for success.

References:

[1] Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37-54.

[2] McNeil, N. M., Uttal, D. H., Jarvin, L., & Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on math problems. Learning and Instruction, 19, 171-184.

[3] Carlson, S. M., Davis, A. C., Leach, J. G. (2005). Less is more: Executive function and symbolic representation in preschool children. Psychological Science, 16, 609-616.

Most of us have heard the adage about the two ways that someone can get into a swimming pool: jump right in, or enter slowly to acclimate to the temperature a few inches at a time.

Most of us have probably also witnessed (or experienced) the varied ways that someone might approach an assignment: one could start and finish it right away; work on it in small chunks over an extended period of time; or wait until the last moment to start, likely rushing to finish.

And for those that are keeping an eye on back-to-school sales events, there are of course different ways to shop: one could impulse purchase an item, or do some research beforehand to get the best possible deal.

The common thread in all of those scenarios is that different methods, strategies, and thought processes can be employed to solve problems or complete tasks. And each has its own time and place. So how do we decide exactly which ones to use in a given situation?

Algorithms and heuristics

The science behind problem solving and decision-making comprises a robust portion of cognitive research and involves the study of both conscious and unconscious thought.

Overall, there are two primary ways that a problem can be tackled: with algorithms or with heuristics. [1] An algorithmic approach refers to a series of steps that are more or less guaranteed to yield the solution. While this approach is most easily thought of in the context of mathematics (e.g., following a mathematical formula), an algorithmic approach also refers to such procedures as following a recipe or backtracking your steps to find a lost object.

Heuristics, on the other hand, are associative strategies that don’t necessarily lead to a solution, but are generally pretty successful in getting you there. These include conscious strategies (such as solving a maze by making sure your path stays in the general direction of the end point) and unconscious strategies (such as emotional instincts). Because heuristics are more subjective and less systematic than an algorithmic approach, they tend to be more prone to error.

In the classroom, solving problems with an algorithmic approach is fairly straight-forward: students can learn the needed procedural steps for a task and identify any places where they might have gone wrong, such as a miscalculation or a typo.

Heuristics are more complicated, however, and much of the research on problem solving aims to understand how children and adults solve problems in complex, confusing, or murky situations. One question of particular interest involves transfer: how do children apply, or transfer, their knowledge and skills from one problem-solving scenario to another?

Six of one, half-dozen of the other

Research suggests that students tend to have trouble transferring knowledge between problems that share only the same deep structure. For example, two puzzles that can be solved with the same logic, but that have different numbers, settings, or characters, are tricky.

In contrast, problems that share both their deep structure and shallow structure can be solved with relative ease.

A seminal study that illustrates the challenges of transfer asked students to solve the Radiation Dilemma: a medical puzzle of how to destroy a tumor with laser beams. [2] Some of the students were first told to read The General: a puzzle (and its solution) based on the common military strategy of surrounding an enemy and attacking from all sides. The solution to the Radiation Dilemma was analogous to the solution for The General: radiation beams should target the tumor from all sides until destroyed.

The researchers found that the students who first read the solution to The General successfully solved the Radiation Dilemma more often than those who did not.

However, students who received a hint that the solution to The General problem would help them solve the Radiation Problem were actually more successful in solving it than those who read both problems but received no hint.

This finding suggests that analogies can certainly be a helpful guide when children (or adults) are trying to make sense of a problem or find similarities between different contexts. But, they can also be confusing. Presumably,  people become distracted by or hyper-focused on shallow structural features (e.g., reading the Radiation Dilemma and trying to remember what medical strategy was used on a TV drama) and thus overlook the deep structure similarities that are present.

So, when we ask students to make connections between two problems, scenarios, or stories that have surface-level differences, a little hint may just go a long way.

The less the merrier?

In addition to better understanding how to make decisions or think about problems, researchers also aim to understand how much we should think about them. And, contrary to popular thought, it appears that reasoned and evaluative thinking may not always be best.

In fact, there is evidence for the deliberation-without-attention effect: some problem-solving situations seem to benefit more from unconscious cognitive processing. To investigate this, scholars at the University of Amsterdam set out to determine whether better decisions result from unconscious or conscious thought. [3]

In their experiment:

• participants (college students) read information about four hypothetical cars
• the descriptions of the cars were either simple (four features of the car were listed) or complex (12 features were listed)
• some of the features were positive and some were negative; the “best” car had the highest ratio of positive-to-negative features
• four minutes passed between participants reading about the cars and being asked to choose the best one
• some participants spent those four minutes thinking about the cars, while the others were given a puzzle to solve in order to distract them from such thinking

When asked to choose the “best” car, two groups stood out:

• Group A: participants that (1) read the simple car description and (2) consciously thought about the cars were more likely to identify the best car than those who read the simple description and then worked on the puzzle
• Group B: participants who: (1) read the most complex car descriptions and (2) were then distracted by the puzzle were more likely to identify the best car than those who read the complex description and consciously thought about the car options

The participants in Group B actually had a higher overall success rate than those in Group A.

Thus, it appeared that conscious thinkers made the best choices with simple conditions, but did not perform as well with complex circumstances. In contrast, the unconscious thinkers performed best with complex circumstances, but performed more poorly with simple ones.

Buyer’s Remorse

Of course, the cars that the participants evaluated were fictional. The researchers therefore wanted to see if their results would hold up in similar real-word circumstances. They traveled to two stores: IKEA (a complex store, because it sells furniture) and a department store (a simple store, because it sells a wide range of smaller items, such as kitchen accessories).

As shoppers were leaving the store with their purchases, the researchers asked them:

• What did you buy?
• How expensive was it?
• Did you know about the product before you purchased it?
• How much did you think about the product between seeing it and buying it?

The researchers then divided the shoppers into two groups: (1) conscious and (2) unconscious thinkers, based on amount of time they reportedly spent thinking about their purchased items.

After a few weeks, the researchers called the shoppers at home and asked them about their satisfaction with their purchases. In a similar vein to the first experiment, here the conscious thinkers reported more satisfaction for simple products (department store) and the unconscious thinkers reported more satisfaction for complex products (IKEA).

Thus, these experiments indicate that conscious thinking is linked to higher satisfaction with decisions when conditions are simple (less to evaluate), whereas unconscious thinking leads to higher satisfaction when conditions are complex (many factors to evaluate).

Why don’t you sleep on it

While these studies are only a snapshot of the problem-solving and decision-making research field, they offer some valuable thoughts for how we can support students in the classroom.

First, we know that students need to understand problems in order to solve them. It is likely a good habit to continually remind ourselves that our students do not all make sense of the same problems in the same way or at the same rate. Thus, as we saw in The General, when we offer students problem guides, strategies, or templates, a little nudge as to how to use them can be enormously beneficial.

Second, we often push our students to think deeply and critically about problems and context. And that is probably true now that, more than ever, thoughtful, evidence-based, and logical reasoning is critical for tackling both local and global issues.

But there is also much to be said about instinct, conscience, and whatever it is that goes on in our subconscious. So if we see our students dwelling on a problem, or sweating a decision, the best way that we can help them delve into a solution may just be to first have them step away for a little while.

References:

[1] Novick, L., & Bassok, M. (2006). Problem solving. In K. Holyoak & R. Morrison (Eds.), The Cambridge Handbook of Thinking and Reasoning (pp. 321-349). London: Cambridge University Press.

[2] Gick, M. & Holyoak, K. (1980). Analogical problem solving. Cognitive Psychology 12(3), 306-355.

[3] Dijksterhuis, A., Bos, M., Nordgren, L., & van Baaren, R. (2006). On making the right choice: The deliberation-without-attention effect. Science, 311, 1005-1007.

Should young children count on their fingers when learning math?

You can find strong opinions on both sides of this question. (This blog post uses 4 “No’s” and 5 exclamation points to discourage parents from allowing finger counting.)

Recent research from the University of Bristol, however, suggests that finger counting–when combined with other math exercises–improves quantitative skills more than either intervention by itself.

The study design is quite complex; check the link above if you’d like the details. But, the headline is clear: for 6- and 7-year-olds, a taboo against finger counting may well hinder the development of math skills.