How to Achieve Transfer

Transfer, the ability to apply learning to a novel situation, is a tricky thing. It’s not so hard for students to memorize, say, the definition of “subject” and “predicate” when working in a grammar textbook. Grammar exercises, however, are futile if students continue to produce incorrect sentences in their own writing because don’t know how to apply what they learned on a worksheet. Without being shown how to use new knowledge in other contexts, many students struggle to recognize how a concept can be applied to a slightly different problem or task. This is a very inefficient way to learn. Students who don’t understand how to transfer what they learn must memorize hundreds and hundreds of discrete skills rather than just a few core ones.

In response to the excellent publication from Deans for Impact, The Science of Learning, we have a few ideas about how to ensure that students can transfer what they’re learning to solve problems in all kinds of contexts.

Find Similarities Between Tasks that Appear Different on the Surface

In our own practice, we’ve observed that findings by Richland, Zur, and Holyoak (2007) hold true: students can transfer their knowledge and skills to new situations if they’re able to figure out what the new situation has in common with tasks they’ve already navigated successfully. For example, some kids would have a tough time determining what the following word problems have in common:

• Ryan sells 14 candy bars that cost $1.25 each. How much money does he collect?

• If each of the 5 players on a basketball team scores 8 points in the game, how many points does the team score altogether?

• A baker can decorate 40 cookies each hour. How many will he frost during a four-hour shift?

Many students struggle to know which operation to use in word problems because each scenario seems completely different. The problems above may seem disparate because one is about money, one about baskets, and one about cookies. In fact, though, each can be solved through multiplication. Comparing the problems can help students find similarities. Observant students, for example, might notice that “each” is repeated in all of the problems above. This word often signals multiplication. Students may find it useful to keep a sheet of notes on key words and the operations they often indicate.

Identify the Steps in a Multi-Step Procedure

Catrambone (1996 and 1998) suggests that students label the steps in multi-step procedures, such as conducing experiments, solving complex math problems, or writing. Such labels make it easier to compare the approach needed to work through similar tasks that appear different on the surface.

Producing a piece of academic writing is one of the most complex multi-step tasks asked of students. Generally the student must establish an argument, provide and explain evidence, rebut conflicting arguments, and all the while relate each point to the main point. Comparing one essay with another, though, might confuse some students. If one piece is about the use of allusion in a poem and another about the merits of recycling, what could the essays possibly have in common? Challenging students to label the steps each author conducted (stating thesis, referring back to thesis with each topic sentence, providing evidence, explaining evidence, etc.) makes it apparent that good essays, no matter the topic, have a lot more in common than might appear. Identifying the steps used to complete a task make the similarities between different tasks stand out, helping students to recognize how they can use the steps they already know to accomplish novel outcomes.

Provide Both Concrete Examples and Abstract Representations

Students often understand concrete examples best, but concrete examples can be hard to generalize so that the principles in them can be applied to other scenarios. Goldstone and Son (2005) found that learning is maximized when students are given a combination of concrete examples and abstract representations. The best learning outcomes occurred when students were exposed to concrete examples first, then gradually introduced to the abstract principle. Deans for Impact states that a combination of concrete examples and abstract representations “help students recognize the underlying structure of problems.”

To use this principle in a physics class, an instructor might begin teaching the concept of momentum by assigning the following word problems (i.e. concrete examples) and asking students to find the answers:

• Find the momentum of a 7-kilogram bowling ball traveling 17 MPH.

• Find the momentum of an 85-gram marble traveling 17 MPH.

• Find the momentum of a 450-kilogram car traveling 20 MPH.

• Find the momentum of a 283-ounce matchbox car traveling 20 MPH.

Next, the teacher should provide the following definition and formula:

“Momentum is determined by the mass of an object and its speed. The formula for calculating momentum at a given moment is p=(m)(v), where m=mass and v=velocity.”

Most textbooks provide concrete examples and formulas, but few ask students to do the work of defining the relationship between the two. This is where lasting, transferable understanding is formed.

In our example of the physics class, the instructor must help the students to see the relationship between their answers to the concrete scenarios and the abstract principle that defines momentum. Ideally, students will come up with a statement defining the relationship between the two, such as, “More mass and/or higher velocity results in more momentum. When mass, velocity, or both are decreased, momentum decreases.” That is transfer at work.

Math Apps for Building Spatial and Reasoning Skills

We recently took a look at spelling apps, recommending those that we think do the best job of providing critical orthographic skills while being fun and engaging. Today, we turn our attention to math apps we think are worth investigating.

Carstens Studios designs, creates and illustrates dynamic learning applications for young children. Daren Carstens, the owner of Carstens Studios, shares that his goal for each app is to inspire the love of math in children by giving them a new way to conceptualize mathematical concepts. Math requires spatial and sequential understanding, as well as a great deal of reasoning and logical thinking skill. 

There are an abundance of tools for building foundational numeracy and sequential skills. However, it can often be a challenge to find dynamic ways to build a child’s spatial understanding or reasoning skill. Carstens Studios offers a variety of fun, engaging tools that help students build their spatial understanding of math while encouraging reasoning and problem solving. We have detailed some favorite apps below that would add value to any math curriculum.

Math Doodles

In the introduction to this app, Mr. Carsten shares that when he was in school, math just looked like squiggles to him, leaving his mind to wander and him to doodle on the edge of his notebook. He soon realized that there were places for doodles in math, and that doodling helped him better understand the mathematical concepts. Building on this notion, each game in the Math Doodles app gives players choices for displaying the numbers in different ways (as words, Roman numerals, currency, etc.). Furthermore, each math puzzle encourages students to experiment while developing and strengthening their strategy and problem solving skills. Math Doodles builds a wide range of critical math skills, including building basic mathematical and number sense, as well as teaching basic operations, time, money and geometric concepts.

Attributes

It is true that math is more than numbers. It takes more to master math than merely mastering numeracy skills. Mr. Carstens states that, “The key to understanding math is understanding patterns.” Therefore, the Attributes app is comprised of a series of seven games, designed to challenge kids' mathematical thought processes through tables, patterns, and the concepts of attributes. The games help late-elementary children develop the abstract reasoning skills required in higher-level math. By playing the Attributes games, students will build their understanding of pattern sequences, Venn diagrams, logic, and classification.

Symmetry Shuffle 

As we have mentioned, spatial reasoning is an important element of mathematics, especially in geometry. The mathematical puzzle Symmetry Shuffle,allows students to explore line and rotational symmetry while developing their spatial sense. Through the game student will build her understanding of slides, flips and rotations of two-dimensional shapes, as well as the concepts of congruence, similarity and symmetry.

We can’t go into detail on every resource, but Carstens Studios has other apps that are also worth noting. So, if you are looking to provide students with extra addition and subtraction practice, check out Unknown Square, Sum Stacker or Connect Sums. Each of the apps has received numerous accolades and awards. But their biggest achievement is that they truly make math learning fun.