Following some conversations with George Haines on Twitter, I attempted to embark on a very complicated topic: transfer of learning. The literature is full of unanswered questions and the research is equally equivocal or sparse.

**What does “transfer of learning” mean? **

The definitions seem to branch out with every paper that I read but, despite this variety, the basic meaning can be resumed to *the ability to extend what is learned in one situation to new contexts*. The major classification is between:

*near*transfer – when knowledge is applied in a__similar situation__(e.g. adding in a class math –calculating change in a store)*far*transfer – application of knowledge or general principles to__a more complex or novel situation__(e.g. learning about the scientific method –applying its principles in designing and conducting an experiment, testing hypotheses, critiquing other experiments etc.)

Transfer is implied, to some extent, in any new learning otherwise we wouldn’t be able to learn anything new (you can’t really learn, say, how to conjugate verbs unless you have some previous knowledge about verbs). Yet the ability to transfer information or ideas is not a given. Quite often, information learned in a **specific way**, or in a **particular context**, does not transfer to another. For instance, students may very well ace your vocabulary quiz yet fail to use the very same words in their writing. Or they may have very well learned a mathematical fact but do not know how to apply it in a new problem.

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I will focus on mathematics because that is where I see most issues arising. I also read more papers on transfer in mathematics because I teach it so it was essential for me to understand why this problem is so complicated. However, I am sure that some of the solutions that I give at the end can be used in any subject or at least some principles can be applied when designing learning tasks and progressions in other subjects.

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**Why is transfer in mathematics so difficult?**

The most obvious difficulty students encounter is in secondary school and it is no secret that there is an underground war between primary and secondary teachers on the knowledge and skills students need to demonstrate. Most of this is not the primary school teachers’ lack of knowledge although this can be a major drawback in any math teaching *(“Teachers who make mathematical errors not only fail to deliver accurate academic content but also likely lower students’ self-efficacy in math.”* – Teacher and Teaching Effects on Students’ Academic Behavior and Mindsets, David Blazar, Mathematics Policy Research, December 2015).

The main reason is actually the unique structure of mathematics, its hierarchical nature and the close interdependence between concepts. Richard R. Skemp noted that back in 1962, in his paper The Psychology of Mathematics Teaching and Learning:

*“This hierarchical arrangement appears in other subjects to varying degrees, but to a less extent. When learning a language, ignorance of (say) one page of irregular verbs does not preclude learning the next page. Ignorance of the history of the 10 ^{th} century matters little to a pupil learning that of the 15^{th}. In the sciences the interdependence is greater but still less than for mathematics. Take physics as an example: most of the theory of electricity and magnetism can be acquired without knowing anything about mechanics or hydrostatics. Mathematics is probably the most interdependent and hierarchical of any structure knowledge currently taught.”*

Moreover, a leap in mathematical thinking needs to occur for students who move from arithmetic to algebra; the complexity and the level of abstractness in mathematical thinking required in secondary are high:

*“The importance of this distinction lies in the fact that certain mental processes are required for algebra which are not needed for arithmetic. The distinction is in the level of abstraction involved. For example, the number 7 can be represented by seven physical objects – which cannot be done the variable x. One can draw a triangle having sides 3,4,5 cm but one cannot draw ‘AC, any right-angled triangle’. Similarly, the operation ‘adding 3’ can be represented concretely by a physical movement of 3 objects – but not ‘differentiate with respect to x’, nor ‘project AB to infinity’. Anticipating a little, this means that the subject matter of mathematics consists of purely mental objects; and it is from this that many of its particular difficulties arise.” *

I know it is easier to blame primary school teachers for either using ineffective methods or for lack of knowledge in mathematics. Sometimes it can be true but that is also applicable to secondary math teachers (I can tell you about two of my math teachers who consistently failed at that!). I consider, therefore, that it is more productive to see how to provide a better transition so that our students experience success and continue to enjoy mathematics even after the “didactic cut”.

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**What can impede transfer or higher levels of abstractness?**

Three of the most consistent finds are:

- the overuse of
**concrete**materials - highly
**contextualized**tasks **authentic**, real-life problems

Previous research had established the importance of using concrete materials in teaching – it helps students gain immediate comprehension, it helps them form connections with previous or at least familiar knowledge, and it keeps them more engaged in learning new material. Concrete materials are also easier to process and thus they help reduce cognitive load to some extent. *“It is therefore no surprise that meaningful, concrete examples are encouraged in pedagogy—they are typically associated with substantial improvements in comprehension, memory, and reasoning.”* (1) Similar reasons can be given for contextualized tasks and authentic problems.

The issue is that while they provide short-term benefits they can be disadvantageous in the long run.

Filloy and Rojano (1989) found that *“various concrete models did not significantly increase students’ abilities to operate formally with equations”.* De Bock et al (2000), who have explored the influence of authentic and realistic contexts as well as self-made drawings in solving math problems found *“no beneficial effect of the authenticity factor, nor of the drawing activity, on pupils’ performance.” *Furthermore, the researchers suggest that *“realistic problems may, in fact, steer pupils away from the underlying mathematical structure of a problem.”* (Handbook of Research on the Psychology of Mathematics Education, 2006)

Interestingly, in another paper, Boulton-Lewis et al (1997) noted that, *“Most students did not use the concrete manipulatives (cups, counters, sticks) that were made available for solving linear equations. That is most likely due to the fact that concrete representations increased processing load.” *

As mentioned in another post, this finding was also signaled by OECD with data from PISA 2012 where the more real-life mathematics students were exposed to the lower their mathematics performance.

The Educational Evaluation and Policy Analysis (June 2014) found that “manipulatives, calculators, and physical activities” in math actually had a *negative* statistically significant coefficient (-0.3).

Two other reasons that can limit transfer are:

- teaching facts and procedures in
**isolation** **rote**learning- overly
**simplified**tasks **block**practice

I mentioned those, however, in the previous post so I won’t insist on them.

In part 2 (tomorrow) I will give concrete solutions for breaking these barriers in our daily teaching. Some strategies can be applied to subjects other than mathematics while others are too domain-specific to be extended to other areas of learning.