In my small Twitter sphere recently there has been a resurgence of conversation around a controversial question:

Which is more effective for students: discovery learning or explicit instruction?

I entered this conversation by reading a post by Michael Pershan, called “Discovery learning vs. not discovery learning.” Michael structures his post as a conversation between a proponent of discovery learning and a proponent of much more guided instruction. I found the post fascinating even though I completely disagreed with most of it. But it might not be for the reasons you think. Here are some issues I might advocate for us as a community to clear up.

1. There is no such thing as “discovery learning.”

I can’t say this loud enough or often enough. And I’m not the only one. Mathematics education is not a field that advocates for discovery learning. Further, high quality mathematics instruction is not synonymous with any of the terms used commonly in these conversations. The source of much of this equivocation seems to be a paper by Kirschner, Sweller, and Clark published in the

So here we are. High quality mathematics instruction—that’s what I’ll call it from now on—is not discovery learning. It is not, strictly speaking, constructivist. Nor is it in any way “minimally guided.” The remaining three names have connections to high quality mathematics instruction, to be sure. However, it is more than any one or two (or even all three) combined. High quality—can I just call it HQMI, for short?—HQMI uses problematic situations to create mathematical experiences for students that involve making sense of those situations and using that to develop or choose a solution pathway or to consider an existing method. These experiences are highly social in nature, involving both individual and collaborative work and sense-making.

The role of the teacher is extremely important in every phase: planning, launch, explore, summary discussion, assessment. The teacher is pivotal in launching the task so that students’ prior knowledge is activated and the class understands the context well enough to begin work. The teacher structures the middle portion of the lesson so that students have both individual and collaborative thinking time, supporting them with questions that uncover and focus students’ thinking. For the final phase, the teacher decides the order of presented and discussed solution strategies. Here the teacher’s role is to clarify students’ understandings, engage them in comparing methods, and drawing connections between those methods (e.g., which might be most efficient, which is most “elegant,” etc.). Assessment isn’t a phase as it happens throughout the lesson, with the goal being to learn about how students are thinking about the mathematics.

None of this.

2. What the mathematics education community, generally, values is sense-making.

I’ve already hinted at this one, but it deserves some attention on its own. Michael is right in his post when he says that “[t]here comes a point where people just disagree on what they value. It’s hard to know what to say by the time someone gets to this point of clarity about what they care about.” James Hiebert[ii] makes this point very well when he says that “[d]ebates about what the research says will not settle the issue; only debates about values and priorities will be decisive. Until the value issue is settled, it will be difficult to find common ground for examining the research.” (p. 5).

So let’s talk about values. Let’s talk about priorities. The mathematics education community has been steadfast in its valuing of sense-making for more than 30 years at this point. Sense-making and flexible thinking are given priority in every document published by NCTM[iii], NCSM[iv], AMTE and many of the other professional mathematics organizations. Sense-making is given priority in the design of the Common Core State Standards for Mathematics, the Standards for Mathematical Practice, and nearly any derivative document (including a majority of derivative state standards). In short, the math wars are over. Sense-making is valued in our community.

This informs our conversation moving forward. Hiebert again helps us out: “We now know that we can design curriculum and pedagogy to help students meet the ambitious learning goals outlined in the NCTM Standards. The question is whether we value these goals enough to invest in opportunities for teachers to learn to teach in the ways they require.” (p. 16).

3. There is a body of research in mathematics education that supports HQMI as more effective at producing student learning than direct instruction.

Given the answer to the last question, there is some research worth examining. First, most research that compares approaches characterizes direct and/or explicit instruction as “traditional.” This will be important when you read the pieces below. One of my favorite pieces of research is Hiebert and Wearne (1993)[v]. This study, in short, studied traditional classrooms (teacher-centered, direct instruction) as compared to alternative classrooms—students “received fewer problems and spent more time with each problem, were asked more questions requesting them to describe and explain alternative strategies, talked more using longer responses, and showed higher levels of performance or gained more by the end of the year on most types of items.” (p. 393). The quasi-experimental design of this study offers robust support for the results.

Another example would be the work of Jo Boaler and Megan Staples in the Railside School study[vi]. This study found that a poor, highly diverse school could outperform more affluent schools through the use of accessbile curriculum and ambitious mathematics teaching practices. In short, a fundamental commitment to the use of problematic situations and the engagement of students in ways consistent with HQMI can produce profound results. Not only did performance increase, but also students’ mathematical identities and relationships with mathematics were starkly different[vii].

Then there are the curriculum studies. NSF-funded curriculum materials at all levels have been researched extensively[viii]. Each of these curricula use instructional models that are consistent with what I’ve described as HQMI[ix]. The sum of the research in this area is as follows: students who learn from curriculum materials that are reform-oriented (i.e., like the NSF-funded curriculum projects) generally outperform students who learn from traditional models along two axes. First, reform approaches tend to produce superior (or at least comparable) performance on standardized assessments. Second, these kinds of learning experiences also tend to produce better results on measures of problem-solving ability and nearly all affective factors associated with mathematics learning.

At this point, I’ve talked enough for one blog post. These arguments are based on my own experiences and my own understanding of the research I’ve read. I do not claim that this is a comprehensive case, but I believe that I’ve addressed many of the concerns brought up in Michael’s blog post. I have some other thoughts that I’ll get down in a sequel to this post:

[i] Kirschner, P., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of construtivist, discovery, problem-based, experiential, and inquiry-based teaching.

[ii] Hiebert, J. (1999). Relationships between research and the NCTM standards.

[iii] See, e.g., NCTM. (2000).

[iv] See, e.g., NCSM. (2020).

[v] Hiebert, J. & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic.

[vi] Boaler, J. & Staples, M. (2008). Creating equitable futures through an equitable teaching approach: The case of railside school.

[vii] Boaler, J. & Selling, S. K. (2017). Psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adult’s lives. Journal for Research in Mathematics Education, 48(1), 78-105; Boaler, J. & Greeno, J. G. (2000). Identity, Agency, and Knowing in Mathematics Worlds. In J. Boaler (Ed.)

[viii] See, for example: Huntley, M. A., Rasmussen, C. L., Villarubi, R. S., Sangtong, J., & Fey, J. T. (2000). Effects of Standards-based mathematics education: A study of the Core-Plus Mathematics Project algebra and functions strand.

[ix] Examples include Everyday Mathematics (K-5), Connected Mathematics Project (6-8), Core-Plus Mathemetics Project (9-12). For a complete list, see https://www.nsf.gov/pubs/2002/nsf02084/chap1_4.htm.

Which is more effective for students: discovery learning or explicit instruction?

I entered this conversation by reading a post by Michael Pershan, called “Discovery learning vs. not discovery learning.” Michael structures his post as a conversation between a proponent of discovery learning and a proponent of much more guided instruction. I found the post fascinating even though I completely disagreed with most of it. But it might not be for the reasons you think. Here are some issues I might advocate for us as a community to clear up.

1. There is no such thing as “discovery learning.”

I can’t say this loud enough or often enough. And I’m not the only one. Mathematics education is not a field that advocates for discovery learning. Further, high quality mathematics instruction is not synonymous with any of the terms used commonly in these conversations. The source of much of this equivocation seems to be a paper by Kirschner, Sweller, and Clark published in the

*Educational Psychologist*entitled “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching.[i]” I find the arguments in this paper to be unconvincing at best. But the authors are correct about some things. Discovery learning did fail. It took the human race thousands of years to develop even the basic ideas of school mathematics, to say nothing of the full scope of modern mathematics. The idea that we can sit a student down and have them, individually, discover all of that in 13 years is ludicrous. Which is why we don’t do that.So here we are. High quality mathematics instruction—that’s what I’ll call it from now on—is not discovery learning. It is not, strictly speaking, constructivist. Nor is it in any way “minimally guided.” The remaining three names have connections to high quality mathematics instruction, to be sure. However, it is more than any one or two (or even all three) combined. High quality—can I just call it HQMI, for short?—HQMI uses problematic situations to create mathematical experiences for students that involve making sense of those situations and using that to develop or choose a solution pathway or to consider an existing method. These experiences are highly social in nature, involving both individual and collaborative work and sense-making.

The role of the teacher is extremely important in every phase: planning, launch, explore, summary discussion, assessment. The teacher is pivotal in launching the task so that students’ prior knowledge is activated and the class understands the context well enough to begin work. The teacher structures the middle portion of the lesson so that students have both individual and collaborative thinking time, supporting them with questions that uncover and focus students’ thinking. For the final phase, the teacher decides the order of presented and discussed solution strategies. Here the teacher’s role is to clarify students’ understandings, engage them in comparing methods, and drawing connections between those methods (e.g., which might be most efficient, which is most “elegant,” etc.). Assessment isn’t a phase as it happens throughout the lesson, with the goal being to learn about how students are thinking about the mathematics.

None of this.

**, would happen without the presence and active engagement of the teacher. The goal is not for each student to “discover” everything. The goal is for the class to engage, as a group, in sense-making and to construct a group understanding of important mathematics.***Absolutely none of it*2. What the mathematics education community, generally, values is sense-making.

I’ve already hinted at this one, but it deserves some attention on its own. Michael is right in his post when he says that “[t]here comes a point where people just disagree on what they value. It’s hard to know what to say by the time someone gets to this point of clarity about what they care about.” James Hiebert[ii] makes this point very well when he says that “[d]ebates about what the research says will not settle the issue; only debates about values and priorities will be decisive. Until the value issue is settled, it will be difficult to find common ground for examining the research.” (p. 5).

So let’s talk about values. Let’s talk about priorities. The mathematics education community has been steadfast in its valuing of sense-making for more than 30 years at this point. Sense-making and flexible thinking are given priority in every document published by NCTM[iii], NCSM[iv], AMTE and many of the other professional mathematics organizations. Sense-making is given priority in the design of the Common Core State Standards for Mathematics, the Standards for Mathematical Practice, and nearly any derivative document (including a majority of derivative state standards). In short, the math wars are over. Sense-making is valued in our community.

This informs our conversation moving forward. Hiebert again helps us out: “We now know that we can design curriculum and pedagogy to help students meet the ambitious learning goals outlined in the NCTM Standards. The question is whether we value these goals enough to invest in opportunities for teachers to learn to teach in the ways they require.” (p. 16).

3. There is a body of research in mathematics education that supports HQMI as more effective at producing student learning than direct instruction.

Given the answer to the last question, there is some research worth examining. First, most research that compares approaches characterizes direct and/or explicit instruction as “traditional.” This will be important when you read the pieces below. One of my favorite pieces of research is Hiebert and Wearne (1993)[v]. This study, in short, studied traditional classrooms (teacher-centered, direct instruction) as compared to alternative classrooms—students “received fewer problems and spent more time with each problem, were asked more questions requesting them to describe and explain alternative strategies, talked more using longer responses, and showed higher levels of performance or gained more by the end of the year on most types of items.” (p. 393). The quasi-experimental design of this study offers robust support for the results.

Another example would be the work of Jo Boaler and Megan Staples in the Railside School study[vi]. This study found that a poor, highly diverse school could outperform more affluent schools through the use of accessbile curriculum and ambitious mathematics teaching practices. In short, a fundamental commitment to the use of problematic situations and the engagement of students in ways consistent with HQMI can produce profound results. Not only did performance increase, but also students’ mathematical identities and relationships with mathematics were starkly different[vii].

Then there are the curriculum studies. NSF-funded curriculum materials at all levels have been researched extensively[viii]. Each of these curricula use instructional models that are consistent with what I’ve described as HQMI[ix]. The sum of the research in this area is as follows: students who learn from curriculum materials that are reform-oriented (i.e., like the NSF-funded curriculum projects) generally outperform students who learn from traditional models along two axes. First, reform approaches tend to produce superior (or at least comparable) performance on standardized assessments. Second, these kinds of learning experiences also tend to produce better results on measures of problem-solving ability and nearly all affective factors associated with mathematics learning.

At this point, I’ve talked enough for one blog post. These arguments are based on my own experiences and my own understanding of the research I’ve read. I do not claim that this is a comprehensive case, but I believe that I’ve addressed many of the concerns brought up in Michael’s blog post. I have some other thoughts that I’ll get down in a sequel to this post:

- Direct instruction prevents access to mathematical content for some students
- Consider an example.

[i] Kirschner, P., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of construtivist, discovery, problem-based, experiential, and inquiry-based teaching.

*Educational Psychologist, 41*(2), 75-86.[ii] Hiebert, J. (1999). Relationships between research and the NCTM standards.

*Journal for research in mathematics education, 30(*1), 3-19.[iii] See, e.g., NCTM. (2000).

*Principles and Standards for School Mathematics*. Reston, VA: NCTM; NCTM. (2014).*Principles to Actions: Ensuring Mathematical Success for All*. Reston, VA: NCTM and any associated books.[iv] See, e.g., NCSM. (2020).

*NCSM Essential Actions: Framework for Mathematics Education Leadership*. NCSM; NCSM. (2019).*NCSM Essential Actions: Coaching in Mathematics Education*, NCSM; NCSM. (2019).*NCSM Essential Actions: Instructional Leadership in Mathematics Education*. NCSM;[v] Hiebert, J. & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic.

*American Educational Research Journal, 30*(2), 393-425.[vi] Boaler, J. & Staples, M. (2008). Creating equitable futures through an equitable teaching approach: The case of railside school.

*Teachers College Record, 110*(3), 608-645. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings.*Journal for Research in Mathematics Education, 29*(1), 41-62.[vii] Boaler, J. & Selling, S. K. (2017). Psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adult’s lives. Journal for Research in Mathematics Education, 48(1), 78-105; Boaler, J. & Greeno, J. G. (2000). Identity, Agency, and Knowing in Mathematics Worlds. In J. Boaler (Ed.)

*Multiple Perspectives on Mathematics Teaching and Learning*(pp. 171-200). Westport, CT: Ablex Publishing;[viii] See, for example: Huntley, M. A., Rasmussen, C. L., Villarubi, R. S., Sangtong, J., & Fey, J. T. (2000). Effects of Standards-based mathematics education: A study of the Core-Plus Mathematics Project algebra and functions strand.

*Journal for Research in Mathematics Education, 31*(3), 328-361; Grouws, D. A., Tarr, J. E., Chavez, O., Sears, R., Soria, V. M., & Taylan, R. D. (2013). Curriculum and implementation effects on high school students’ mathematics learning from curricula representing subject-specific and integrated content organizations.*Journal for Research in Mathematics Education, 44*(2), 416-463; Tarr, J. E., Grouws, D. A., Chavez, O., & Soria, V. M. (2013). The effects of content organization and curriculum implementation on students’ mathematics learning in second-year high school courses.*Journal for Research in Mathematics Education, 44*(4), 683-729;[ix] Examples include Everyday Mathematics (K-5), Connected Mathematics Project (6-8), Core-Plus Mathemetics Project (9-12). For a complete list, see https://www.nsf.gov/pubs/2002/nsf02084/chap1_4.htm.