{"id":7218,"date":"2023-08-21T11:00:14","date_gmt":"2023-08-21T16:00:14","guid":{"rendered":"https:\/\/braindevs.net\/blog\/blog\/?p=7218"},"modified":"2023-08-20T09:30:19","modified_gmt":"2023-08-20T14:30:19","slug":"using-worked-examples-in-mathematics-instruction-a-new-meta-analysis","status":"publish","type":"post","link":"https:\/\/www.learningandthebrain.com\/blog\/using-worked-examples-in-mathematics-instruction-a-new-meta-analysis\/","title":{"rendered":"Using “Worked Examples” in Mathematics Instruction: a New Meta-Analysis"},"content":{"rendered":"

Should teachers lets students figure out<\/em> mathematical ideas and processes on their own<\/em>?<\/p>\n

Or, should we walk students through<\/em> those ideas\/processes step by step<\/em>?<\/p>\n

\"3<\/a>This debate rages hotly, from eX-Twitter to California teaching standards.<\/p>\n

As best I understand them, the arguments goes like this:<\/p>\n

If students figure out ideas and processes for themselves, they\u00a0think hard<\/em> about those mathematical ideas. (“Thinking hard” = more learning<\/strong>.)<\/p>\n

And, they feel emotionally invested in their discoveries. (“Emotional investment” = more learning<\/strong>.)<\/p>\n

Or,<\/p>\n

If students attempt to figure out math ideas for themselves, they first<\/strong> have to contemplate\u00a0what they already know<\/em>. Second,<\/strong> they contemplate\u00a0where they’re going<\/em>. And third,<\/strong> they have to (basically) guess until they figure out how to get from start to finish.<\/p>\n

Holding all those pieces — starting place, finish line, all the potential avenues in between — almost certainly overwhelms working memory. (“Overwhelmed working memeory” = less learning<\/strong>.)<\/p>\n

Therefore, teachers should walk students directly through the mathematical ideas\/process with step-by-step “worked” examples. This process reduces\u00a0cognitive load and builds schema. (“Reduced cognitive load” + “building schema” = more learning<\/strong>.)<\/p>\n

Depending on your philosophical starting place, both argument might sound plausible. Can we use research to answer the question?<\/p>\n

Enter the Meta<\/h2>\n

One problem with “using research to answer the question”: individual studies have yielded different answers.<\/p>\n

While it’s not true that “you can find research that says anything<\/a>,” it IS true — in this specific case — that some studies point one way and some point another.<\/p>\n

When research produces this kind of muddle, we can turn to a mathematical technique called “meta-analysis.” Folks wise in the ways of math take MANY different studies and analyze all their results together.<\/p>\n

If scholars do this process well, then we get an idea not what ONE study says, but what LOTS AND LOTS of well-designed studies say (on average).<\/p>\n

This process might also help us with some follow up questions: how much do specific circumstances matter?<\/i><\/p>\n

For instance: do worked examples help younger students more than older? Do they help with — say — math but not English? And so forth.<\/p>\n

Today’s news:<\/p>\n

This recent meta-analysis<\/a> looks at the benefits of “worked examples,” especially in math instruction.<\/p>\n

It also asks about specific circumstances:<\/p>\n

Do students benefit from generating “self-explanations” in addition to seeing worked examples?<\/p>\n

Do they learn more when the worked examples include BOTH correct AND incorrect examples?<\/p>\n

So: what did the meta-analysis find?<\/p>\n

Yes, No, No<\/h2>\n

The meta-analysis arrives at conclusions that — I suspect — suprise almost everyone. (If memory serves, I first read about it from a blogger who champions “worked examples,” and was baffled by some of this meta-analysis’s findings.)<\/p>\n

In the first<\/strong> place, the meta-analysis found that\u00a0students benefit from worked examples<\/em>.<\/p>\n

If you do speak stats, you’ll want to know that the g-value was 0.48: basically 1\/2 of a standard deviation.<\/p>\n

If you don’t speak stats, you’ll want to know that the findings were “moderate”: not a home run, but at least a solid single. (Perhaps another runner advanced to third as well.)<\/p>\n

While that statement requires LOTS of caveats (not all studies pointed the same direction), it’s a useful headline.<\/p>\n

In the dry language of research, the authers write:<\/p>\n

“The worked examples effect yields a medium effect on mathematics outcomes whether used for practice or initial skill acquisition. Correct examples are particularly beneficial for learning overall.”<\/p>\n

So, what’s the surprise? Where are those “no’s” that I promised?<\/p>\n

Well, in the\u00a0second\u00a0<\/strong>place, adding self-explanation to worked examples didn’t help<\/em> (on average). In fact, doing so reduced learning.<\/p>\n

For lots of reasons, you might have expected the opposite. (Certainly I did.)<\/p>\n

But, once researchers did all their averaging, they found that “pairing examples with self-explanation prompts may not be a fruitful design modification.”<\/p>\n

They hypothesize that — more often than not — students’ self explanations just weren’t very good, and might have included prior misconceptions.<\/p>\n

The Third Place?<\/h2>\n

In the\u00a0third<\/strong> place came — to me, at least — the biggest surprise: contrasting\u00a0correct<\/em> worked examples with\u00a0incorrect<\/em> worked examples didn’t benefit students<\/strong>.<\/p>\n

That is: they learned information better when they saw the right method, but didn’t explore wrong ones.<\/p>\n

I would have confidently predicted the opposite. (This finding, in fact, is the one that shocked the blogger who introduced me to the study.)<\/p>\n

Given these findings and calculations, I think we can come to three useful conclusions: in most cases, math students will learn new ideas…<\/p>\n

… when introduced via worked examples,<\/p>\n

… without being asked to generate their own explanations first,<\/p>\n

… without being shown incorrect examples alongside correct ones.<\/p>\n

Always with the Caveats<\/h2>\n

So far, this blog post has moved from plausible reasons why worked examples help students learn (theory) to a meta-analysis showing that they mostly do help (research).<\/p>\n

That journey always benefits from a recognition of the argument’s limitations.<\/p>\n

First<\/strong>, most of the 43 studies included in the meta-analysis focused on middle- and high-school math: algebra and geometry.<\/p>\n

For that reason, I don’t know that we can automatically extrapolate its findings to other — especially younger — grades; or to other, less abstract, topics.<\/p>\n

Second<\/strong>, the findings about self-explanations include an obvious potential solution.<\/p>\n

The researchers speculate that self-explanation doesn’t help because students’ prior knowledge is incorrect and misleading. So: students’ self-explantions activate schema that complicate — rather than simplify — their learning.<\/p>\n

For example: they write about one (non-math) study where students were prompted to generate explanations about the causes of earthquakes<\/em>.<\/p>\n

Because the students’ prior knowledge was relatively low, they generated low-quality self-explanations. And, they learned less.<\/p>\n

This logic suggests an obvious exception to the rule. If you believe your students have relatively high and accurate prior knowledge<\/em>, then letting them generate self-explanations might in fact\u00a0benefit students.<\/p>\n

In my own work as an English teacher, I think of participles and gerunds<\/strong>.<\/p>\n

As a grammar teacher, I devote LOTS of time to a discussion of participles; roughly speaking, a participle is “a verb used as an adjective.”<\/p>\n

During these weeks, students will occasionally point out a gerund (roughly speaking, a “verb used as a noun”) and ask if it’s a participle. I say: “No, it’s something else, and we’ll get there later.”<\/p>\n

When “later” finally comes, I put up sentences that include participles, and others that include similar gerunds.<\/p>\n

I ask them to consider the differences on their own and in small groups<\/em>; that is, I let them do some “self-explanation.”<\/p>\n

Then I explain the concept precisely, including an English-class version of “worked examples.”<\/p>\n

Because their prior knowledge is quite high — they already know participles well, and have already been wondering about those “something else” words that look like<\/em> participles — they tend to have high quality explanations.<\/p>\n

In my experience, students take gerunds on board relatively easily.<\/p>\n

That is: when prior knowledge is high, self-explanation might (!) benefit worked examples.<\/p>\n

TL;DR<\/h2>\n

A recent meta-analysis suggests that worked examples help students learn algebra and geometry (and perhaps other math topics as well).<\/p>\n

It also finds that self-explanations probably don’t<\/em> help, and that incorrect examples don’t<\/em> help either.<\/p>\n

More broadly, it suggests that meta-analysis can offer helpful and nuanced guidance when we face contradictory research about complex teaching questions.<\/p>\n

\n

Barbieri, C. A., Miller-Cotto, D., Clerjuste, S. N., & Chawla, K. (2023). A meta-analysis of the worked examples effect on mathematics performance.\u00a0Educational Psychology Review<\/i>,\u00a035<\/i>(1), 11.<\/p>\n","protected":false},"excerpt":{"rendered":"

Should teachers lets students figure out mathematical ideas and processes on their own? Or, should we walk students through those ideas\/processes step by step? This debate rages hotly, from eX-Twitter to California teaching standards. As best I understand them, the arguments goes like this: If students figure out ideas and processes for themselves, they\u00a0think hard […]<\/p>\n","protected":false},"author":18,"featured_media":7222,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[15,183,208],"class_list":["post-7218","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-lb-blog","tag-classroom-advice","tag-mathematics","tag-worked-examples"],"_links":{"self":[{"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/posts\/7218","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/users\/18"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/comments?post=7218"}],"version-history":[{"count":5,"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/posts\/7218\/revisions"}],"predecessor-version":[{"id":7224,"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/posts\/7218\/revisions\/7224"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/media\/7222"}],"wp:attachment":[{"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/media?parent=7218"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/categories?post=7218"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learningandthebrain.com\/blog\/wp-json\/wp\/v2\/tags?post=7218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}