6 Making Mathematics Learning Visible Through Transfer Learning

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Copyright © Erin Null

Most of us have rented a car at one time or another. Rarely is the one we rent the same make and model as the one we drive at home. Yet after a few minutes of orienting ourselves to operating the mirrors, adjusting the seat, and figuring out how to turn the headlights and windshield wipers on, we pull out of the parking lot and set forth. How are we able to operate a complicated piece of machinery in an unfamiliar location? Because we were able to transfer what we know about cars and driving to a new situation. Sousa (2011) uses this as an example of transfer of learning in action. The driving example works for you because you engaged in a bit of transfer as you read this.

There are several reasons why this example works, and they have everything to do with methods for promoting transfer of learning. The first reason this works for you is because it is a situation nearly every adult has experienced, and it is therefore immediately relatable. Even if you have not rented a car, you are probably able to relate this to borrowing a friend’s car. As we mentioned earlier, close association between a previously learned task and a new situation is necessary for promoting transfer of learning. A second reason this works is because there is an analogy at play here, in this case comparing driving a car to complex learning. But this falls apart if we use this example with someone who has never operated a car. A limiting factor in fostering transfer is in pairing the experience with the knowledge base of the learner. In the case of a nondriver, we would need to find a different example. That brings us to the third reason: knowing the learner developmentally and experientially is essential when promoting transfer of learning.

But there’s one more factor, and perhaps the most critical one: we have the skills to detect the similarities and differences from one situation to another. In our car analogy, we know the key differences up front are the mirrors, seats, and headlights, and we know the key similarities are the brakes and accelerator. Having the skills to detect similarities and differences is key to transfer—without this, we may transfer the wrong ideas from our old car. Whoops, an accident waiting to happen!

In mathematics, much of transfer learning is the movement from learning the mathematics itself to learning to use the mathematics to describe information and situations in other disciplines and solve problems within those contexts. Algebra students learn about a variety of functions, including linear or exponential functions. Students can then transfer this learning about functions to describe and model situations that occur in science. How fast is this population of bacteria growing? How does the velocity of the ball change as it falls from the roof of a building? In Chapter 5, we shared an example of using slope to recognize that a variety of cell phone plans all have the same fee rate for the data usage portion of the plan. In that task, the context the teacher created gave the students a framework for thinking about creating a model. In chemistry class, studying the relationship between the pressure and volume of a gas in a balloon, students may have to create the tables and make the connections more independently.

The Nature of Transfer Learning

All of the work we do as teachers is for naught if students fail to transfer their learning appropriately by applying what they have learned in new situations. Generally speaking, John (Hattie, 2009) explains that transfer learning is about the ways in which students construct knowledge and reality for themselves as a consequence of surface and deep knowing and understanding. The distinctions are not clear-cut, as at all three levels we often learn in a haphazard manner. He argues that so much teaching is aimed at surface learning in the broader world of ideas and knowledge, and there is also much discussion about the importance of deep knowledge and thinking skills. But the task of teaching and learning best comes together when we attend to all three levels: ideas, thinking, and constructing.

Video 6.1 Teaching for Transfer Learning

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One concern is that mathematics instruction too often stops at the surface level of learning, and students (particularly struggling students) either fail to go deep and transfer, or they transfer without detecting similarities and differences between phenomena. When this happens, the transfer does not make sense, and too often students see this as evidence that they can’t do mathematics.

As we mentioned in Chapter 1, in mathematics, transfer is both a goal of learning and also a mechanism for propelling learning to the next level. Transfer as a goal means that teachers want students to begin to take the reins of their own learning, think metacognitively, and apply what they know to a variety of real-world contexts. It also prepares them to move through the progression of mathematical understanding as ideas build on each other across grade levels. It’s when students reach into their toolbox and decide what tools to employ to solve new and complex problems on their own. When students reach this phase, learning has been accomplished.

Linda was a straight-A mathematics student in elementary and middle school. She understood the hows and the whys of mathematical computation. She could even (and still can) explain why the “invert and multiply” rule for division of fractions works. She understood the mechanics, surface learning (how) mixed with a little deep learning (why). Yet given a page of applications in which fractions were used in a variety of multiplication and division situations, she had no clue which operation to use. She had surface learning, and she had some deep learning to go along with it, but not being able to apply that knowledge in real-world contexts left her short of transfer learning. And quite frankly, knowing the how and the why without being able to apply that knowledge leads kids to ask a valid question: “When are we ever going to use this?”

We believe the true purpose of getting an education is to apprentice students into becoming their own teachers. We want them to be self-directed, lifelong learners, and to have curiosity about the world. We want them to have the tools they need to formulate their own questions, pursue meaningful answers, and, through metacognition, be aware of their own learning in the process. In other words, as their own learning becomes visible to them, we want the entire process to become the catalyst for ongoing learning, whether the teacher is present or not. However, we cannot leave these things to chance. Therefore, we must teach with intention, making sure that students acquire and consolidate the needed skills, processes, and metacognitive awareness that make self-directed learning possible. Recall John’s definition of learning from Chapter 1:

The process of developing sufficient surface knowledge to then move to deeper understanding such that one can appropriately transfer this learning to new tasks and situations.

In mathematics, this phase of transfer learning happens when students can make connections among mathematical understandings and then use those understandings to solve problems in unfamiliar situations, at the same time being intentionally aware of what they are doing. This often comes in the form of students asking themselves clarifying questions, and they can do this because in the deep learning phase, they had teachers who modeled how to ask those questions.

Chemistry students studying pressure and volume must deliberately think about what information to use in a data chart and what mathematical tools to use (equations? tables? graphs?) to identify the type of relationship (linear? exponential?) between the variables of pressure and volume. Students planting the school garden have to apply their knowledge of volume and some standard measures to determine how much mulch to buy to cover the garden plot. Using examples that relate to real-life situations not only provides students with opportunities to develop transfer learning; it also helps them to see the usefulness of mathematics.

In the next sections of this chapter, we will turn our attention to transfer mechanisms that help students go through this process. When we have a clear understanding of how transfer occurs, we can better establish the conditions for ensuring that students meet transfer goals.

Types of Transfer: Near and Far

Because learning is cyclical, not linear, transfer is actually happening all the time, not just at the end of a lesson or unit. In fact, the goal of all learning is eventual transfer (Bransford et al., 2000). By this we mean that the goals of instruction are not to leave students at the surface or even deep levels, but rather to ensure that they can take what they have learned and use it in the next unit, not to mention the next year and beyond. To help accomplish these goals, it helps to understand that transfer actually happens across two dimensions: near transfer and far transfer (Perkins & Salomon, 1992).

Near transfer occurs when the new situation is paired closely with a context students have experienced. For example, helping students become fluent with basic facts is one of the biggest challenges elementary teachers face. We know from experience that simply asking students to memorize their “times 8s” isn’t very effective. If we begin by providing students with situational contexts for multiplication, it provides them with the opportunity to make sense out of what multiplication means and to use different representations to work toward understanding each situation.

Let’s go back and visit Kate Franklin’s class from Chapter 4. Students used the In the Doghouse activity to understand physical models of equal groups. They recognized how to write that using symbolic notation, and they talked about their understanding. Students needed ample time to become comfortable with the equal-groups model. But it didn’t stop there. A few weeks later, Ms. Franklin continued to develop the concept of multiplication using situations that call for array models. Students connected that physical representation to real-life situations such as egg cartons (a 2 × 6 array) or cupcake tins (a 3 × 4 array). Again, surface learning and deep learning were closely aligned throughout all of these lessons.

Later in the year, as students began to explore area measurement, students could transfer understanding by extending their previous experiences to a new context. That is not all that was happening. Class and small group activities that supported deep understanding of multiplication went on throughout the year. Discourse, both in small groups and in the whole class, focused on looking for patterns that led students to recognize that changing the order of the factors in a multiplication example did not change the product. But they also learned that a 2 × 3 array would look different from a 3 × 2 array. Moving back and forth among surface, deep, and transfer learning helped students to become more fluent with multiplication facts as opposed to students who were told to simply memorize their multiplication tables.

The size of the leap is larger in far transfer, as the learner is able to make connections between more seemingly remote situations. For example, as Ms. Franklin began the study of division with her students, planning lessons that focused on connections between multiplication models and division models helped students move toward a deep understanding of division and recognize when a problem situation called for either multiplication or division. Fast forward to Grade 5 as students are beginning to multiply and divide fractions. Simply telling students to multiply numerators and multiply denominators provides no understanding of what it means to multiply fractions, not to mention why the product of two fractions is smaller than at least one of them. Perhaps an even better example is division of fractions. How many of us learned to flip one of the numbers and multiply, and how many of us quickly forgot or confused that rule because it made no sense? Worse yet, problems that called for one of those operations provided little context for understanding because all we had was a rule to depend on. Let’s look at an example.

A gasoline tank holds 16 1 2 Equation 33 gallons. If the pump delivers 3 1 4 Equation 34 gallons a minute, how long will it take to fill the tank, assuming it is empty at the start?

The Paths for Transfer: Low-Road Hugging and High-Road Bridging

So far we have discussed transfer across a continuum from near to far. Now let’s introduce mechanisms, or paths, for transfer: low-road and high-road transfer (Perkins & Salomon, 1992). In this case, the contrastive element is the extent to which the thinking involved is under the learner’s conscious direction. In the left column, the teacher provides structure to support students in transfer. In the right column, the students are using their own strategies and learning to lead the work of transfer. Figure 6.1 summarizes each of these mechanisms.

Let’s go back to the multiplication fact example. Developing fluency with facts helps to build automaticity—that is, as students continue to use those facts in a variety of situations, they will reach the point where they no longer have to think about a strategy or what the fact means. They just know it. And because they have learned it by building on conceptual understanding, they not only know those facts; they know when to apply them.

Applications in the real world such as the gas tank problem require higher level thinking. Without deeper understanding of the structure of mathematics—in this case, division—students would not be able to transfer learning to this new situation. Many students struggle with algebraic ideas because they have not developed the conceptual understanding of ideas such as the distributive property, and they do not have the deep learning to apply these ideas to new and more abstract situations.

It is important to recognize that transfer as a mechanism (1) occurs even among the youngest learners and (2) changes in appearance as the learner progresses developmentally. While our example is from the upper elementary grades, learners of all ages transfer their learning as they become more independent learners.

Selecting Mathematical Tasks That Promote Transfer Learning

When providing learners with opportunity and support for transfer, teachers should select tasks that encourage connections. As with deep learning tasks, these will be higher complexity tasks with higher difficulty. They may not have a clear entry point—in fact, they could have multiple entry points and multiple steps. They may have no one correct solution, but rather call for students to make some judgment about the best solution and be able to use evidence to justify their thinking. They could be tasks in which mathematical ideas are to be applied in another discipline; for example, the mathematics of algebra will be used to think about a geometry problem, or the mathematics of calculus will be applied to physics. Tasks that are meant for transfer learning may take more than one lesson to solve. Teachers must select tasks that provide opportunities for near transfer and far transfer. For some learners, near transfer might be a familiar problem in a new context.

Video 6.2 Transferring Learning to Real-World Situations

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Let’s look at an example from high school. Mr. Assof gives his eleventh graders a three-part task on exponential and logarithmic functions to work on in small groups. In this task—designed to take several class periods—student groups are asked to be a team of financial advisers who must help clients decide how to spend, save, and invest their money. All three clients are seeking advice on their upcoming vehicle purchases and want to know if they should pay cash for a new car, or finance the car and invest their cash. Part 1 of the task focuses on discussions about how auto loans and investment accounts operate, and students work to determine how their clients qualify for different interest rates and what down payment requirements they will have based on their credit scores. Mr. Assof has assigned each of the three fictitious clients a different profile that includes the price of the car they want to buy, their credit score, the length of the loan they want, and whether they are members of a credit union. Students must calculate each client’s car payment based on these factors, using a table they’ve been given that compares the interest rates banks and credit unions offer for different time spans at different tiers of credit scores.

Once students have determined how much each client will pay in a monthly car payment if he or she chooses to finance the purchase, they can move into Part 2. This is where they will determine if each client should pay cash for the car, or if the client should finance the car and invest the cash in a mutual fund called the Bond, James Total Return Bond. This fund offers between a 5.8 percent and 6.2 percent per-year return (compounded monthly) to those investing their money between three and five years, respectively. Here students need to use what they have previously learned about exponential and logarithmic functions to determine how much money each client would make if he or she invested in the mutual fund, and if this amount is more or less than the total amount paid for the financed vehicle purchase, and then make a recommendation about what the client should do. In all cases, they needed to use specific evidence from their calculations to justify their response.

Part 3 of this task asks students to generalize their thinking by answering two questions: (1) Given the statement Car salespeople use the rule “Your payment will increase by $200 for every $10,000 you finance,” determine if this is accurate and explain why or why not using specific examples, and (2) Do you agree or disagree with the statement “It is always better to pay cash for a car instead of finance”? Justify your position by giving specific mathematical examples.

This task is a great illustration of transfer in that it requires leverage of prior surface and deep learning about exponential and logarithmic functions; it is a high-complexity, high-difficulty task with real-world application that will be a scenario many students will face in the near future. It requires students to think about the strategies in their tool belt that they need to utilize to solve a problem that may have no right or wrong answer, but requires some judgment on their part. It also requires them to engage deeply with many of the mathematical practices.

Conditions Necessary for Transfer Learning

It should come as no surprise that relevancy is a major condition for transfer learning. Learning becomes more meaningful when learners see what they’re learning as being meaningful in their own lives. Relevancy doesn’t have to be at the world peace level, but it does need to have implications that are developmentally appropriate and are seen as being useful in their learning lives. Also, the learning intentions and success criteria that we discussed at length in Chapter 2 are just as important for promoting transfer as they are for fostering initial learning. Although students are engaged in more self-directed learning during this transfer period, they need goals and ways to measure their own progress.

One of the problems for many students is that they rush to apply their learning to a new situation. It is time well spent to require that they pause and consider the similarities and differences between the new and a recently completed problem. Considering the specifics of the problem in front of them, the nature of the question, and any additional details are all worth attention. Too many students, especially struggling students, jump into a new problem without thinking about how it is different from or similar to a previous problem.

A team of middle school teachers used an engineering design challenge to provide an opportunity for students to transfer their learning from math and science class to solving a real-world problem. These teachers had heard conversations among their students about the amount of food thrown away in the school cafeteria, so they opened the unit with a discussion of the problem and how students might help to solve it. By beginning with a challenge they knew was relevant to the students, the teachers were creating good conditions for transfer learning. The teachers’ instructional intentions were to teach their students about using design thinking (Crismond & Adams, 2012) to solve a real-world problem that required application of their math and science learning. They established small groups of students to work as design teams across all classes and taught the students about working as a team to communicate effectively, establish goals, develop a timeline, and set deadlines. These team collaboration skills are essential for transfer learning and for much of real-world problem solving.

While the teachers did not teach the students how to solve this problem, they intentionally selected a problem that highlighted concepts students had been studying in math and science and building on their prior knowledge in developmentally appropriate ways. The teachers knew that the students had learned to collect, organize, and analyze data. They understood about food webs and decomposition and how living things depend on one another in an ecosystem. The students were able to measure and calculate using those measurements. Students were ready to transfer this learning to a problem in their own school, managing cafeteria waste.

As student teams began to work, they approached the problem in different ways. One team wanted to collect data from students in the school about how serious they thought the problem was and what ideas they had to solve it. Another team decided that building a compost bin was the way to go and set about collecting information about the volume of food waste collected each day and using that to figure out the dimensions of the compost bin. A third team decided to focus on encouraging the cafeteria to serve meals that more students were likely to finish so there would be less waste in the first place. While each team had a different approach, all of them had to use their collaboration and problem-solving knowledge, along with their math and science learning, to design a solution to a real problem in their school.

The students used problem solving and reasoning to propose and test possible solutions. The practical criteria and constraints on a workable solution (for example, limits of space and budget and the fast turnover of students moving through the cafeteria at lunchtime) provided immediate and relevant feedback on potential solutions. Teachers circulated among the groups, asking questions about their strategies and helping them find effective ways to manage their sometimes conflicting data. Seeing the mathematics and science in this context provided opportunities for re-teaching in a new way if they found students were missing requisite surface or deep knowledge.

The conditions created by these middle school teachers allowed for teams to think conceptually, especially to identify problems and propose solutions, testing their proposals, making adjustments, and thinking of alternatives, all dispositions identified by Bereiter (2002) as evidence of transfer of learning. The teachers found a relevant and appropriate problem that built on students’ prior learning. They deliberately taught teamwork skills so students could use their reasoning skills in collaborative ways. The teachers and the problem provided feedback, and teachers were close at hand to reteach when necessary.

Metacognition Promotes Transfer Learning

Having students practice self-reflection on their level of understanding, as it relates to a target, has been shown to be a powerful strategy to increase both understanding and motivation (National Research Council, 2005). This is a form of self-talk that promotes transfer learning. Metacognition is the ability to think about our thinking, and it is vital to the learning process. It first appeared in the education literature in the 1970s (e.g., Flavell, 1979), and it has become more of a buzzword ever since. The hype is well deserved: metacognitive processes have been shown quite clearly to boost student achievement. Marzano (1998, p. 106) called metacognition the “engine” that drives thinking. Garofalo and Lester (1985) went so far as to suggest that mathematics instruction without metacognitive instruction is inadequate. Palincsar (2013) describes metacognitive awareness as consisting of three parts:

  1. Knowledge about our learning selves
  2. An understanding of what the task demands and necessary strategies to complete it
  3. The means to monitor learning and self-regulate

In other words, it describes our ability to observe and monitor our own thinking. But students need guidance in how to become more metacognitively aware. To develop their metacognitive skills, students need to learn the art of self-questioning.

Self-Questioning

As learners complete an assignment or work through a rich mathematical task, they will be most successful if they think about their thinking as they work. They may realize that they’ve reached a dead end with their strategy or that they are missing an important piece of information. These realizations come from a background series of questions we continually ask ourselves: “Does this make sense?” “Am I making progress finding a solution?” This self-questioning is a critical element of our metacognition as it allows us to track our understanding and catch ourselves when we realize we are off target.

There is also something about writing that clarifies students’ understanding, though. It also makes them better writers. At the beginning of a lesson, middle school teacher Nate Hernandez uses prompts to have students write in their math journal.

  • How much do I already know about today’s learning intention? What makes me say that?
  • How difficult do I think this lesson will be? Why do I think that?
  • How do I rate my desire to be successful in this lesson? Why do I want to be successful?
  • How much effort will I put into today’s lesson?
  • What strategies do I think will be helpful for me in meeting my learning objective today?

Mr. Hernandez decides which questions will be the most useful based on his ongoing evaluation of student cognition, metacognition, and attitudes toward learning. He also thinks about how much class time he wants to spend on this part of the lesson. At several points during Mr. Hernandez’s lessons, he pauses from the mathematics instruction to have students rate their understanding of the learning intention again, and have them explain why they gave themselves their ratings. This leads to some interesting student writing, as they reflect, analyze, and clarify for themselves their levels of understanding. It also helps to keep Mr. Hernandez’s kids focused on the learning objective as they note their progress.

Mathematical Talk That Promotes Transfer Learning

The tasks discussed in this chapter that can be used for this ultimate level of transfer are fueled by the discourse that occurs because of rich class discussions. We’ve shared throughout this book the importance of intentional math talk, and nowhere is the language, thinking, and reasoning that happen when extended discussion happens more powerful than in the transfer of learning. Research and experience both point to the benefits of being able to speak coherently and thoughtfully during meetings and discussions, and in settings that are larger than four or five people. Furthermore, class discussions have been shown to help students learn content. These discussions provide students a chance to shine, to help each other, to hear ideas and make connections, and to make sense of mathematics. But not all class discussions are helpful. Some of them devolve into unrelated storytelling; random opinion sharing; or too commonly, as mentioned in Chapter 5, the teacher doing the majority of the talking.

Classroom discussion, especially for the purposes of transfer, should be so much more, and it is definitely not the time to play the “guess what’s in the teacher’s brain” game that researchers call I-R-E. Discourse is an essential part of the mathematics classroom, so much so that NCTM has featured it in its standards since 1991. Mathematics students are expected to be able to represent their thinking to others, pose questions, and engage in disagreements without being disagreeable. You’re looking for mathematical discourse, not chatter about mathematics. But these practices can’t be realized without significant changes in how discourse is fostered, taught, and supported. This doesn’t mean that the teacher stands back and lets her students do all the talking. The teacher needs to be an active facilitator who uses questioning moves that elevate students’ thinking, presses them for evidence, and requires them to link to concepts. These talk moves are an extension of the accountable talk techniques we discussed previously in this book.

Helping Students Connect Mathematical Understandings

During the transfer phase of learning, students take ownership of their work. Learners are in the driver’s seat about what they want to learn and what tools they will use. This requires learners to have a sense of how the mathematics that they know is organized. The process of creating an organizational structure for their knowledge—in terms of both conceptual understandings and procedural skills—is a powerful tool.

In order to help others understand mathematics more effectively, a peer tutor (indeed, any teacher) must be able to connect what we want the learner to master with what the learner already knows and understands. When students serve as peer tutors, they must understand how their own knowledge is organized well enough to create a path between what is already understood, or misunderstood, and the mathematics the person being tutored is learning.

The local high school has a tutoring club, where students who volunteer to be tutors are matched with students at nearby elementary and middle schools for tutoring. Ian is an Algebra II student and a math tutor; he is working with Maria, a seventh grader. Maria is having trouble mastering operations with integers, and Ian is working to help her. Guided by the teacher who leads the tutoring program, Ian has a set of six problems about adding and subtracting integers for Maria to solve at their first meeting. After introducing themselves, Ian asks Maria to solve the problems he brought so he can see how she thinks about adding and subtracting integers. As Maria works, she talks out loud about what she is doing. “7 – -3 . . . I’m subtracting so I find seven minus three, that’s four, and the sign is from the big number so it’s positive. That one is positive four.”

Ian listens and realizes Maria is mixing together several shortcuts for adding and subtracting integers. This means she gets some problems right and some problems wrong. After talking with the teacher who leads the tutoring program, Ian decides to take algebra tiles to their next session and model the math using this manipulative and making connections to addition as joining groups of objects and subtraction as separating groups of objects. Maria and Ian work through a number of problems together, and Maria builds understanding of what is actually happening rather than trying to follow shortcuts blindly.

This tutoring experience has helped Ian solidify his own understanding of adding and subtracting integers. He has to recognize the shortcuts Maria is using and find a path to help her make sense of the disconnected “rules” she has tried to learn. He begins to select problems for Maria to solve without the assistance of his advising teacher; this requires him to predict how Maria might solve the problem and what errors he might see and help her with. Not only has Maria’s learning improved, but Ian’s own understanding is stronger because he had this transfer learning experience.

Connected Learning

As students claim ownership of their learning and build their network of connections between their school learning and life during transfer learning, many learners are inspired to wonder about new problems. One strategy for providing opportunities for transfer is to create time and space for learners to explore their own ways of using what they have learned. “Connected learning is realized when a young person is able to pursue a personal interest or passion with the support of friends and caring adults, and is in turn able to link this learning and interest to academic achievement, career success or civic engagement” (Ito et al., 2013, p. 4). It is the connection of personal passions and academic achievement that makes this idea powerful for transfer.

Students are not going to benefit from opportunities to research and investigate their own problems and questions if they don’t have a solid foundation of knowledge to use as a springboard for their investigations. Notably, Google’s lauded investment in creativity, the Genius Hour, is directed at its highly accomplished engineering staff who already possess deep wells of knowledge to draw from. It’s not likely that you’ll get the same effect from simply announcing that every Thursday afternoon is dedicated to an hour of being a genius. Structure, of course, is essential. You can’t just turn students loose without some procedures in place and expect that everything will be fine. On the other hand, we shudder at the bloom of “Genius Hour worksheets” that seem to be gaining popularity.

Opportunities for, and expectations of, transfer of knowledge should be woven into classroom life. We understand that setting aside sacred time to devote to this is both appealing and plausible. However, nothing magical happens at 1:30 p.m. if there hasn’t been preparation. Students should be continually challenged to develop projects and investigations across the learning day. Asking students, “How could you use this?” near the end of a lesson can spur the kind of metacognitive thinking essential to learning. “How will you know you are successful?” reminds students (and ourselves) that one’s internal measure is as important as an external evaluation.

Helping Students Transform Mathematical Understandings

Problem-Solving Teaching

Earlier in this book, we noted that our challenge as educators is not just to identify what works, as almost everything works for some students at some time, especially when zero growth is expected. Rather, we need to match what works to accelerate student learning, and then implement it at the right time (Hattie, 2009). Problem-based learning (PBL) is one such practice. The evidence is that when PBL is used too early in the learning cycle, before students have had sufficient experience with learning the declarative and procedural knowledge needed, the effect size is very low: 0.15. This is surface-level knowledge, and they just aren’t equipped with enough knowledge to pursue inquiry. But when problem-solving teaching is employed, the effect size skyrockets. Unlike conventional PBL, where the problem is presented to students in advance of knowledge acquisition, problem-solving teaching is deployed when students are already deepening their knowledge.

The engineering design process is an iterative process with several components. Notice the similarities to problem-solving teaching above.

  1. Define the problem by asking questions about the situation.
  2. Explore the elements of the problem, imagine potential solutions, and develop a plan.
  3. Make and test a model or prototype.
  4. Reflect on the testing and identify improvements to your solution. Revise/retest the plan.

This is an iterative process that can continue as long as time and resources permit. As the middle school students working on a solution to the problem of cafeteria waste learned, even in the real world, engineers do not typically find the perfect solution to a problem; they find the best solution given the available time and resources—the optimal solution.

Teachers can often identify design challenges for their students by listening thoughtfully to their wondering about the world around them. Preschool children might wonder about how they can go faster or slower down the slide. An elementary classroom might wonder about how to use their limited classroom storage space more effectively or the best way to design a school garden. By high school, student wonderings are often connected to the problems they will solve as adults in their careers.

Reciprocal Teaching

A common complaint among mathematics teachers is that the students cannot comprehend word problems well enough to solve them. Reciprocal teaching (Palincsar & Brown, 1984) is a literacy strategy used to build reading comprehension by having students work in structured collaborative groups to build their understanding of texts. Mathematics education researchers have adapted the strategy (e.g., Reilly, Parsons, & Bartolot, 2009; van Garderen, 2004) for use in the mathematics classroom to help students understand word problems.

As we have discussed, transfer learning is about working with difficult and complex tasks. If students do not comprehend the task, they cannot begin to solve it. In one study, Reilly et al. (2009) adapted the four phases of reciprocal teaching to mathematics as predicting, clarifying, solving, and summarizing. In the predicting phase, students use their knowledge of mathematics and the information provided in the problem to make predictions about what is happening in the problem and what mathematics they will need to solve the problem. In the clarifying stage, students make lists of information they need to solve the problem: unfamiliar vocabulary, known facts, and information they will need to determine to solve the problem. While the first two lists are fairly straightforward, the last requires some inference and more abstract mathematical thinking. Students are encouraged to work through this process in collaborative groups and to reread the problem once they have clarified the items on their lists.

During the solving phase, students use their problem-solving strategies to find one or more solutions to the problem. With the foundation laid by the predicting and clarifying stages, they are more successful in this process. Learners are encouraged to use multiple representations as they find solutions to the problem. Finally, the summarizing stage serves as individual self-reflection. For learners, this includes justifying the solution, reflecting on the effectiveness of the strategies selected, and evaluating their participation in the work of the group. Throughout the process, students keep a record of their thinking and work.

Too often, teachers jump straight to solving problems before ensuring students really understand the problem. When students are unsuccessful, it can be difficult to know if this is the result of not understanding the problem or not understanding/applying the mathematics correctly. With a strategy like reciprocal teaching, mathematics teachers can support students in understanding the problem well so they have the best opportunity to use their mathematics to solve the problem.

Conclusion

This chapter has focused on transfer, the application of what has been learned in mathematics to unfamiliar situations. It’s nearly impossible for students to transfer if they don’t also have surface and deep knowledge. In other words, it’s hard to apply knowledge to unknown situations if you don’t understand the ideas, procedures, or strategies. In many ways, transfer is all about answering the age-old question “When am I ever going to need this?” Transfer learning represents a leap from one context to another; these leaps can be smaller or larger. Tasks are generally both difficult and complex when they are selected to facilitate transfer learning. And, importantly, transfer allows students to develop and use their metacognitive skills.

Transfer also allows students to assume responsibility and ownership of their learning, essentially putting them in the driver’s seat. From this perspective, self-reflection and self-questioning are critical elements of transfer learning. We shared several strategies for building comprehension of tasks, making learning engaging and relevant, and providing structured opportunities for learners to make sense of what they themselves know and can do. Like surface and deep learning, transfer learning should not be left to chance. Rather, it is up to caring teachers to deliberately design these opportunities by using effective strategies and an understanding of their students. But when transfer learning becomes visible, teachers can have no doubt whether they are making an impact.