# Are there science-backed effective teaching strategies?

As a math teacher, I am always trying to self-assess my teaching methods. I am trying a lot of different methods but I would like to organize my study on the subject without weighing too much on the kids with a trial and error approach, so I'm looking for some scientific research on effective math teaching strategies and practices for high school students. There are a lot of ideas out there online but based on what I found either a) these ideas are really narrow-focused only on small children or b) the ideas are "common sense" ideas that might work but are not really backed by any scientific study.

Any help?

In terms of controlled experiments, then, yes. Note that most are opposite or orthogonal to virtually all pedagogical norms in math education.

1. Active recall. "Put away all your notes and textbooks and computers. Here is a blank piece of paper. Show me everything you've learned in the past couple of weeks."

3. Elaboration. "Don't just copy written steps. Tell me: How do you know this is true? How do you know it's useful? How can you check your work? What does it really mean?"

4. Interleaving. "Here are 12 word problems about a bakery. All the word problems involved the numbers 32 and 8, but you'll have to figure out whether it's division, multiplication, addition, subtraction, exponents, brackets, negatives, fractions, whatever. You need to learn to make those distinctions." More resources: SSDD Problems.

5. Multiple Representations. I assume you're already doing this, but in lower grades, it might be: "How can you represent the number of forks needed using those manipulatives? Can you draw it? Can you write the number in words? Can you write it using math symbols?" In upper grades, it's "What does this relationship look like as a graph? An equation? A story problem? Words? A table? A diagram? Can you tell me how they all represent the same information?"

6. Concrete examples. This has two implications. First, you should START with the concrete example. NOT "Here's two weeks of studying transformations and propreties of $$f(x)=b^x$$... and now here's a (magic) formula for compound interest. Now plug in these numbers." But rather, "How do credit cards work? What happens if you don't pay over time? Is it good to just add 2%/month to calculate what our debt will be in 10 years?" See Problem-Free Activity and Intellectual Need for more info. To this I'd add that students must be able to distinguish examples from superficially similar non-examples, plus they should be able to generate examples and non-examples on their own. Note that virtually NO textbooks or math courses emphasize any of this.

7. Alternating worked examples and exercises. Note that virtually ALL math textbooks have an information & example section first, then exercises after. At most they sprinkle some "Can you regurgitate/mimick?" tasks in the information sections. Most math lessons follow the same structure. [Edit: I am unaware of scientific evidence on this, but it seems obvious that frequently walking around the class to see if your students are actually following the lesson should be helpful. Frequent alternating is one structure that allows this.]

8. Ask probing questions. My take on this: Fight confirmation bias. Do NOT ask "What can my students do?" [fertile ground for confirmation bias]. Instead, imagine a brilliant math teacher saying "I can prove right now that the apparent learning you perceive is all an illusion. The students didn't learn anything." How would that teacher prove their case? [And how often can most of us answer that question confidently?]

Problems with these recommendations include [edited list]

• They ignore the problem of many students arriving in class with grossly inadequate prior knowledge.
• They surface lots of bad news and discomfort. Students and teachers are often shocked at the low performance. They infer struggles that these teaching methods are ineffective - if not outright harmful - despite overwhelming evidence from controlled experiments. If you implement these things, prepare for tremendous pushback.
• They don't help much with tight deadlines or most standardized tests.

So, I consider it a design problem to best implement those strategies into my pedagogy. And it's one helluva tough problem.

Looking elsewhere, JUMP Math does an extraordinary job of breaking math into small steps. This approach has been tested in an RCT against my country's former mainstream math textbook series and JUMP came out on top, by far.

IMHO, many of the problems generated by: (a) that mainstream math textbook series and (b) progressive approaches towards math education generally - are solved by the work of Peter Liljedahl. I am not personally familiar with the science or controlled experiments on this, but he has pretty vast experience and data.

If I think of anything else, I'll add it. :-)

Edit in response to @NotThatGuy 's comment. Here's more on confirmation bias. (Please read the hypothetical example linked before moving on.) Often, teachers think "Can my students do these calculations correctly? If yes, then they must have mastered the concepts!" That inference is, in a huge share of cases, an example of confirmation bias that pervades math education from kindergarten to middle and high school to community college up to physics at Harvard to Nobel Prize laureate-picked graduate students. Confirmation bias is practically everywhere in math education. Every math educator should be aware of this, they should stop inferring conceptual mastery from calculations, and prepare tasks and assessments that detect such illusions of learning.

• Your point 8 cannot be parsed. Feb 16 at 16:40
• @user21820: seems perfectly parsable to me. It says to imagine what someone would ask your class, if they were trying to demonstrate your class hadn't learned anything. Then ask them that. Feb 17 at 1:36
• What do you mean by "What can my students do"? What can they do for or about what? Why is that confirmation bias? You need to start asking how you can teach them (better) and come up with a teaching strategy before you're able to have any bias to confirm about that. "What would that educator do next" - is "that educator" the brilliant math teacher or the "you" they were speaking to? If it's the former, they'd probably refine the strategy based on the proof (which we wouldn't have). If it's the latter, they'd probably ask to see the proof. In any case, how does that help anyone reading this? Feb 17 at 12:39
• Concerning 7. It is neither reasonable nor wise to put homework into class. Unless you have so much time that they are going to do all their homework in class the expectation should be they do exercises after seeing the concepts and calculational techniques in lecture. The reason the exercises are at the end is an organizational convenience. Books are not learning life coaches. Students need to take responsibilities for how they study. Common sense indicates to do problems to gain understanding. But, common sense also dictates to not clutter your presentation with hwk in class. Do hwk at home Feb 17 at 16:20
• @JamesS.Cook - First, I made exactly zero claims about homework. Second, I understand your objections about my sequencing recommendations, but: (A) The OP asked for "science-backed teaching strategies" and the superiority of alternating examples/exercises is about as proven as you can get in the social sciences. (B) Many students lack the prior knowledge (or life situations) to take responsibility for their study methods. Alternating examples/exercises, IMHO, helps them the most. (C) See JUMP Math and its teacher resources for an example of how to satisfy both of our views. :-) Feb 17 at 21:37

Maybe not so concrete on practical strategies, but here goes: One key concept that has always resonated strongly with me is in David Tall's work where he talks about compression being the prime mover in understanding mathematics, e.g. this one: https://files.eric.ed.gov/fulltext/ED489653.pdf

Mathematics is extremely hierarchical. To understand multiplication, you need to understand addition first, that's just common sense. But really you need more than that, you need to compress the concept of addition such that you can do addition using only a small part of your intellectual capacity. Only then can you learn multiplication.

The same concept goes all the way up. You have no hope of learning Itô calculus if you have not compressed the ordinary theory of Riemann integration.

You can use this concept as a lens with which you view your teaching. Is the student struggling because it is difficult to understand the new thing, or because they have not yet achieved compression of the old thing?

• I haven't read that paper (I've added it to the bottom of my reading stack). Does the author claim that humans have a linear "intellectual capacity" and that each thinking process uses a discrete portion of that "intellectual capacity"? I can see how someone might come up with such a hypothesis, but it's not how the human brain works. That said, it is perhaps a model that can illustrate how to more effectively teach certain topics. But as an accurate model of real cognitive processes... um, it's about as accurate as saying a vehicle's speed is limited by how hard one can press an accelerator. Feb 16 at 21:13
• @RockPaperLz-MaskitorCasket I would say this is similar to many theories of expertise: at the beginner stage, one learns the "rules of the game" and applies them consciously, and with great effort. At the intermediate stage, one learns patterns, or you might call it "the rules of the meta-game" and applies them consciously (e.g., thinking through the possible moves in chess). At the expert stage, the rules and tactics have become internalized, and conscious effort is expended searching through the highest-level meta-space. Feb 16 at 23:12
• Learn multiplication? I thought the only way was by rote. That worked for mr (like so much other stuff) and probably millions of others. What does 'learn multiplication' actually mean? Serious question!
– Tim
Feb 18 at 16:56