Teaching Practice: Mathematics
Craig Barton recently published How I Wish I’d Taught Maths: Lessons Learned from Research, Conversations with Experts, and 12 Years of Mistakes (2018 John Catt Educational Ltd.). Craig has two websites that are worth visiting – Mr. Barton Maths and Diagnostic Question. He is a leading mathematics educator in the United Kingdom. His books is part confessional, part analysis, and part exhortation with lots of technique thrown in. There are many directions we could go with this book but in this article we just consider his list of “10 things I used to believe” (p. 21). Maybe one or two might surprise you too!
- The best lessons have little teacher-talk and lots of student-talk
Barton’s point is not that students should not be engaged in their own learning; it’s that explicit or direct instruction (there seems little difference between the two) is key in the initial stages of learning when the students know nothing. Discovery learning seems to be a dead end when there is no “automacity of foundational knowledge” (p. 33). In The Effectiveness of Direct Instruction Curricula: A Meta-Analysis of a Half Century of Research (2018), the researchers describe direct instruction as “in opposition to developmental approaches, constructivism, and theories of learning styles, which assume that students’ ability to learn depends on their developmental stage, their ability to construct or derive understandings, or their own unique approach to learning”. Their analysis finds that “It is clear that students make sense of and interpret the information that they are given—but their learning is enhanced only when the information presented is explicit, logically organized, and clearly sequenced. To do anything less shirks the responsibility of effective instruction (p.24)”.
- Where possible, students should ‘discover’ things for themselves
Barton writes an entire chapter of the book on ‘Making the Most of Worked Examples’. This is just one of several places where he questions the whole process of discovery learning. He says that he used to believe that “it was important to get students practicing on their own as quickly as possible” (p. 191). However he notes, referencing back to the first chapter of the book about how students think and learn, that Cognitive Load Theory shows that overloading working memory is counter-productive. Students have to develop schemas in long-term memory to avoid cognitive overload – worked examples that allow them to study the solutions and work backwards help them to do that. He cites research showing that worked examples only result in declarative knowledge and that it has to be immediately followed by problem-solving. Thus students discovering for themselves (and this is only one dimension of his discussion) is replaced by worked examples interleaved with problem solving. Barton actually splits his board into two with the worked example on one side and the problem to be solved on the other (p. 196). This is also a practice that utilizes retrieval, something we will look at later in the article.
- We can teach problem-solving skill
An immediate issue is to define what a problem is. (Schoenfeld (Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics.) D. Grouws (Ed.) Handbook for Research on Mathematics Teaching and Learning (pp. 334-370) offers one definition as “In mathematics, anything required to be done or requiring the doing of something” (p. 10). Given the levels of math anxiety in our classrooms, that’s a pretty good definition! What is not a problem? However, this just makes problem-solving a means to good technique. But solving a problem requires you to be an expert (another issue Barton addresses in the book) and since students are not experts, even the term problem-solving fills them with anxiety. It requires “domain-specific knowledge” (p. 292). He summarizes his view: “The reason many of my students were not good problem solvers was not because they lacked some magic problem-solving, it was because they lacked strong domain-specific knowledge, storied, organized and automated in their long-term memories” (p.300) i.e. solving problems is a function not of repeating the exercise of problem-solving but of providing deep domain-specific knowledge.
- Effective differentiation means giving students different work to do
Differentiation are a set of practices that teachers use to be effective in heterogeneous classrooms. A variety of techniques have been suggested from meeting learning styles to engaging students with content that they find interesting. There has been a drumbeat over the past little while pointing to the lack of any reputable research showing that attending to students’ learning styles makes any difference. And the idea that different content makes a difference also has little evidence to support it. Barton believes that differentiation means neither of these things. To him, “the most effective form of differentiation is by time not task” (p. 59).
- The maths we teach should be relevant to our students’ lives
The issue of motivation is a complex one that Barton explores in chapter two. He quotes Nuthall (The Hidden Lives of Learners 2007) who says that “students are constantly on their guard against being conned into being interested”. To some extent, Barton is just skeptical that something he teaches can be meaningfully be made relevant, whatever that means, and relevant to every student in the classroom. That seems like a lot of trouble to go to for a questionable end. He finally puts it this way: “I have a theory about the question, When will we ever use this in real life? What I really think students are saying is I don’t understand this….If I teach my students well so they can achieve success, that rather annoying question seems to disappear” (p. 64). This fits well into the idea that success builds motivation – Dan Pink is not wrong when he identified mastery as a key to motivation (Drive 2011).
- Students should always know why they are doing something before they learn how to do it
This is another motivation issue that Barton spends time being skeptical of. Marzano in The Art and Science of Teaching (2017) lists 8 motivators on page 75 including personal projects, altruism projects etc. all leading to self-actualization. It is unlikely that Barton would be excited about their use in mathematics. As he writes: “when I am standing there claiming that David Beckham considers the properties of the resultant quadratic curve when lining up his free-kick, I am fooling no-one apart from myself” (p. 61). Another problem he identifies is that so much of what they learn they have met before in some form – “a large proportion of our teaching time is spent revisiting topics” (p. 324). Knowing the why is less important than having what he describes as ‘purposeful practice’.
- The more feedback we give the students the better
Barton cites a 2015 study (Learning versus Performance by Soderstrom and Bjork) that disputes the notion that constant feedback during the acquisition stage of teaching/learning is effective: “Empirical evidence suggests that delaying, reducing, and summarizing feedback can be better for long-term learning than providing immediate, trial-by-trial feedback” (p. 23). This seems to be truer for the learning of motor than the learning of verbal skills. It is clear that a great temptation for the teacher is to engage in processes that lead to good immediate performance because that is so gratifying. However, there is no proof that performance is the same as learning. In fact, if we are interested in long-term learning, we have to think differently about performance. Note: Barton actually has a section on the Silent Teacher (pp. 176-78)!
- Tests are predominantly tools of assessment
On page 422, Barton writes: “viewing tests not just as tools of assessment but as tools of learning has been the single biggest change to my teaching over the last two years”. This notion that tests contribute to the student’s learning (without denying that formative assessment is powerful in helping the teacher attune practice to the need of the student) is profound. It is somewhat amazing that the well-known effects of this kind of testing know as retrieval practice are still so little known and practiced in classrooms. There is a website – retrievalpractice.org – and a newsletter associated with it. Tests or retrieval practice make use of the finding that when we ‘forget’ something and then have to remember it, the subject becomes more strongly embedded in long-term memory. Add spacing and interleaving to retrieval and the outcome is even more powerful. Testing is more powerful than memorizing!
- Doing lots of past papers is the best way to prepare for an exam
Very simply, just doing past papers is not targeted enough.
- If students are struggling then they are learning
Barton continuously returns to the distinction between novice and expert. “Before I set students off to work independently, I ensure they have enough domain-specific knowledge to solve problems on their own. Counter-intuitively, I help them become more independent by using the techniques of teacher-led explicit instruction discussed throughout this book” (p. 319). It is clear that students are not universally expert or novice. The student who is brilliant at algebra may struggle with the spatial elements of geometry. The problem for the novice is that ‘struggling’ too often means not having enough working memory to deal with all the elements of a problem, failing to understand the underlying mathematical system, suffering cognitive overload and/or being able to solve the individual problem but unable to transfer that knowledge to new problems. For the expert with significant domain-specific expertise, ‘struggling’ may have benefits since working memory is not consumed by managing ‘fundamental skills and procedures’ and thus problem solving is possible.
Enjoy his book! Agree or disagree, you will certainly be illuminated.