Prevention Equals Treatment, Dept.

NOTE: For those interested in math education issues, the Science of Math organization has been formed to do for math education what valid research has done for the science of reading. Consider becoming an affiliate of Science of Math to support the efforts of this organization.

A recent paper by Sweller discusses how and why inquiry based approaches harm student learning and is largely ineffective. This is not the first paper that Sweller has written about the subject (not to mention those written by Kirschner), but it may be the most definitive so far.  The paper has sparked interesting observations. In particular, I was intrigued by a friend’s remarks that the prevailing mindset in schools and districts is that only students with learning disabilities need direct instruction and that it functions primarily as a student services support.

In math education circles direct (or explicit) instruction has been painted as the prevalent currency of traditionally taught math. And those who seek to reform math education tend to deride and mischaracterize traditionally taught math as  1) consisting solely of direct/explicit instruction with no engaging questions or challenging problems, 2) focusing on rote memorization and no conceptual understanding, and 3) failing to teach math in any complexity. In fact, traditionally taught math employs some inquiry based approaches, while reform math teaching generally relies more on discovery/inquiry approaches than direct. (A paper by Anna Stokke (2015) addresses what an appropriate balance of direct and inquiry-based instruction should be and states: “One way to redress the balance between instructional techniques that are effective and those that are less so would be to follow an 80/20 rule whereby at least 80 percent of instructional time is devoted to direct instructional techniques and 20 percent of instructional time (at most) favours discovery-based techniques.” This was corroborated in a paper by Adam Jang-Jones (2019) which quantified the “sweet spot” between inquiry and discovery based approaches.)

Reform approaches in math in the lower grades (K to 6) have steadily grown over the past three decades. I had long wondered whether some diagnoses of math learning disabilities (MD) of students were in fact incidents of low achievement (LA) due to lack of access to effective instruction. In other words, did the inadequacies of reform math mimic cognitive deficits?

I was therefore surprised and delighted to learn, when I took an Introduction to Special Ed class in ed school, that there was no beating about the bush when talking about students with learning disabilities and other disorders. We learned that students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction along with other methods. This was mentioned in our textbook (Rosenberg, et al., 2008) and often repeated by our teacher—a tremendously kind teacher named Carmen. (Interestingly, this course was one that was not required as part of my certificate program.)

Of particular interest in this course was the topic of “Response to Intervention” (RtI). RtI is a procedure in which struggling students are pulled out of class and given alternative instruction which includes direct instruction and other evidence-based approaches supported by randomized control trial studies. If they improve under RtI, then the student is presumed to not have a learning disability and is returned to the normal class. If they do not improve, that is an indication that they have an underlying learning disability. (The procedure was established under the Individuals with Disabilities Education Improvement Act (IDEIA) passed in 2004).

I recall the discussion we had about RtI in which I posed a hypothetical to the teacher. “Suppose someone is doing poorly in a math class that relies on an inquiry-based math program,” I said. “And they pull the student out and give him RtI using direct instruction and other techniques, and this student does well.  What happens next?”

“Then the student is placed back to the class during math.”

“But then suppose the student does poorly again? Wouldn’t that indicate that he needs more direct instruction rather than inquiry based approaches?”

“It doesn’t work that way,” Carmen said.

“So he’s stuck with a program in which he doesn’t do well.”

“Right,” she said.  “But if he did poorly in RtI, then that would be evidence he has a learning disability.”

What Carmen was telling me was the Catch-22 aspect of special ed. That is, in schools that rely on programs that follow math reform principles, approaches used in traditional math teaching are generally not an option unless a student qualifies as being learning disabled. And if under RtI, a student does well with direct instruction this is taken as evidence that the student does not have a learning disability.

I suspect that the use of RtI is higher in schools that rely on reform-based programs. I would like to see research conducted to see if that is true. From where I and many teachers and parents sit, the effective treatment for many students who are LA, is also the effective preventative measure.

Based on conversations I’ve had with education professors, I believe the educational establishment will likely continue to resist recognizing the merits of traditional math teaching and direct instruction. The following statement from James McLesky (2015), one of the authors of the textbook we used in the special ed class and a professor at University of Florida’s College of Education, is typical of what I’ve been told:

If we provide only (or mostly) skills and drills for students with disabilities, or those who are at risk for having disabilities, this is certainly not sufficient. Students need to also have access to a rich curriculum which motivates them to learn reading, math, or whatever the content may be, in all of its complexity. Thus, a blend of systematic, direct instruction and high quality core instruction in the general education classroom seems to be what most students need and benefit from. 

Statements such as these imply that students who respond to a diet of more direct instruction constitute a group who may simply learn better on a superficial level. I fear that RtI will evolve to incorporate some of the pedagogical features of reform math that has resulted in the use of RtI in the first place.

I am hoping that the publication of Sweller’s latest paper and the reaction to it that I’ve seen so far, will result in an increasing recognition of the benefits of direct instruction specifically and traditional instruction generally, as well as the harm that can result from inquiry-based approaches. Unless and until this happens, the group-think of the well-intentioned educational establishment will prevail. Parents and professionals who benefitted from traditional teaching techniques and environments will remain on the outside — and the methods that can do the most good will continue to hide in plain sight.


McLesky, James. (2015) Private email to Barry Garelick. November.

Rosenberg, Michael; D. L. Westling, J. McLesky (2008). Special Education for Today’s Teachers: An Introduction. Pearson. New York.


Doing it wrong, Dept.

Talking about teaching math opens one up to choruses of “You’re doing it all wrong” among those who have been indoctrinated into the various catechisms of math education. One of those is “Never tell a student they made a mistake”. I guess this is because it affects their confidence and self-esteem and therefore is anti-growth-mindset. (On the other hand, we have Jo Boaler telling us that students should be encouraged to make mistakes because it makes their brains grow.)

I have no problem telling a student they made a mistake, though I do it by saying “That’s not what I got. Anyone else get an answer?” When many students make a mistake I capitalize on this and say “OK, so far we have …. ” (I rattle off the various answers), and then many hands go up among those who want to be part of what is now perceived as a fun game. But if only one person makes the mistake, I’ll try to see if they know what they did wrong. Sometimes the student knows; other times, I might know and I’ll give my opinion. And still other times we don’t know, but I’ll give another similar problem and the student who made the mistake usually will try again. At least that’s been my experience. But to make the math ed progressives reading this entry feel better, I’m no doubt doing something wrong.

For eighth graders, it’s a little trickier, because they are very self-aware at this stage of their lives and can be very guarded. Some teachers use mini-whiteboards on which students write the answer and hold up the boards for the teacher to then say “Right, right, nope–try again, …” etc. I do a variation of this. I don’t use mini-whiteboards. Instead, I’ll tell them to do the problem in their notebooks, and then I go around. If someone has the wrong answer and they write out their steps, I can point out the mistake, and they can then re-do it. For those who get it right, I’ll tell them so. If the person who got it wrong initially then gets the right answer, I call on that person to tell the answer to the class. In this way, the person is not singled out for making a mistake, and they feel confident in giving the answer to the class, knowing it’s correct and not fearing the teacher saying that it’s wrong in front of their peers.

But when time pressure is an issue, you sometimes have no choice but to tell someone they are wrong. I make note of those who are not getting it, and during the time that I allot for students to start on their homework (a term which has now morphed into “practice problems”–I guess “homework” is too risky a word in view of self-esteem and growth mindset fantasies) I spend the most time working with them.

For those students in eighth grade who really should not be in such class but who are placed there because of parents’ insistence, there are a number of options I exercise. I may recommend to the parents that they hire a tutor. Another alternative (which may occur even if the student has a tutor) is to recommend that the student repeat algebra in 9th grade. Some go along with this, but others do not.

If these ideas are offensive to some of you, please realize that I wear my shirts tucked in, avoid Apple products, and use a point-and-shoot digital camera rather than take pictures with my cell phone. I am semi-anachronistic and am determined to stay that way. It’s only a matter of time before my out-of-date habits become the latest fad. By that time, I’ll likely be dead, in which case they’ll probably name a brick-and-mortar bookstore after me.