Greg Ashman, an advocate of “explicit instruction”:

In an influential book for the National Academies Press in the US [

How People Learn], the constructivist position is explained in terms of the childrenâ€™s bookFish is Fish. In the story, a frog visits the land, and then returns to the water to explain to his fish friend what the land is like. You can see the thought bubbles emanating from the fish as the frog talks. When the frog describes birds, the fish imagines fish with wings, and so on.The implication is that we cannot understand anything that we have not seen for ourselves; each individual has to discover the world anew.

Meanwhile, here is how the cognitive scientists who wrote *How People Learn* *actually* interpret *Fish is Fish* (pp. 10-11):

Fish Is Fishis relevant not only for young children, but for learners of all ages. For example, college students often have developed beliefs about physical and biological phenomena that fit their experiences but do not fit scientific accounts of these phenomena.These preconceptions must be addressed in order for them to change their beliefs(e.g., Confrey, 1990; Mestre, 1994; Minstrell, 1989; Redish, 1996).

To illustrate this phenomenon we need only look at Ashman’s essay itself. Ashman came to his essay with the common misconception that constructivists believe that “each individual has to discover the world anew.” Even though the *How People Learn* authors interpret *Fish is Fish* explicitly, that explicit interpretation wasn’t enough to dismantle Ashman’s misconception.

Perhaps the *HPL* authors should have taken their own advice, anticipated Ashman’s misconception, and addressed it explicitly. It turns out they did exactly that in the next paragraph:

A common misconception regarding “constructivist” theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves.

A book is nothing if not a medium for explicit instruction and Ashman illustrates the limits of that medium here. Explicit instruction is powerful, certainly, and I can’t think of any influential scholars, least of all the authors of *How People Learn*, who would deny it. But it often isn’t powerful enough on its own to remedy a student’s existing misconceptions. Luckily, *How People Learn* offers many more powerful prescriptions for teaching and it’s free.

**BTW**. Always relevant: The Two Lies of Teaching According to Tom Sallee.

## 28 Comments

## Jim cibulka

September 8, 2015 - 4:48 pm -So true.

And How People Learn is the best education book ever.

Btw, I had a new teacher tell me all you can do in math class is lecture and practice.

The struggle is real.

## Ted Medina

September 8, 2015 - 5:21 pm -You wrote “Explicit instruction… often isnâ€™t powerful enough on its own to remedy a studentâ€™s existing misconceptions.”

Can you link any studies in which explicit instruction was compared with constructivist learning (of any kind) and found to be weaker when both methods attempted to directly change misconceptions?

## Brian Carpenter

September 8, 2015 - 6:05 pm -Ted,

I use modeling instruction, a particular form of guided inquiry for physics and chemistry instruction, in my own classes. There is an extensive body of research regarding it, but the following link will give you a sense of its effectiveness when compared with traditional science instruction. The results are measured using the Force Concept Inventory, a research implement specifically designed to reveal misconceptions.

http://modeling.asu.edu/modeling/Mod_Instr-effective.htm

## blaw0013

September 8, 2015 - 7:48 pm -Great post. I comment to distinguish two constructivist viewpoints, the second referred to (by Confrey, Steffe, Pat Thompson, Larochelle, von Glasersfeld, et al.) as Trivial Constructivism. A constructivist recognizes the unknowability of truth, “reality,” fact. Knowledge/knowing is considered to be true for the subject when that knowing is viable in the knower’s experiential reality – like fish is fish story. But taking constructivism seriously causes the observer to apply this same standard to her own observations. So an observers claim to knowing the knowing of another is restricted in the same sense, the observation is knowable only in the extent it is viable in the observer’s experiential reality. Trivial constructivists ignore the implications of the observer, and accept a Platonic notion of a knowable reality.

Do these distinctions have much to say about math teacher actions or behaviors? Maybe not. But it is my argument that they have significant ethical impacts for how a teacher interacts with students.

## blaw0013

September 8, 2015 - 7:52 pm -Ted, your question confuses constructivism as a theory for teaching. Explicit Instruction may be that, but constructivism is most certainly a theory for knowing/learning and suggests NO prescription for teaching.

## Greg Ashman

September 8, 2015 - 8:10 pm -I’d already blogged about this.

https://gregashman.wordpress.com/2015/09/05/how-people-really-learn/

## Marilyn Burns

September 9, 2015 - 12:32 pm -Thanks for this post. It brought up two issues for me. One issue is how to decide when explicit instruction is beneficial, even necessary, and when it isnâ€™t. The other issue is how to decide, in situations when explicit instruction isnâ€™t beneficial or necessary, what to do in the classroom.

I think that the issues rely on the source of the knowledge for the student; that is, whether the knowledge is based in logic, where a student can come to understanding through reasoning, or whether the knowledge is based in a social convention, where a student needs information thatâ€™s not available through reasoning. Explicit instruction (teaching by telling?) is appropriate, even necessary, when the knowledge is based in a social convention. Then I feel that I need to â€œcoverâ€ the curriculum. We celebrate Thanksgiving on a Thursday, and that knowledge isnâ€™t something a person would have access to through reasoning without external inputâ€•from another person or a media source. Thereâ€™s no logic in the knowledge. But when we want students to develop understanding of mathematical relationships, then I feel I need to â€œuncoverâ€ the curriculum. I love the Fish Is Fish childrenâ€™s book example.

Yes, students need to memorize multiplication tables, but Iâ€™ll never forget the fourth grader, Paul, who proudly told me one day that he knew that â€œsix times eight is forty-eight.â€ I asked him how he knew that, and he recited a rhyme, â€œGoing fishing, got no bait, six times eight is forty-eight.â€ I asked him what six times nine was. He didnâ€™t know and had no idea how to figure it out. Of course, the two sources of knowledge are often intertwined in math instruction, which makes it complex and is what keeps me searching for better ways to teach.

## Greg Ashman

September 9, 2015 - 2:42 pm -Hi Marilyn, I think your view is entirely consistent with my reading of How People Learn and its use of the Fish is Fish story. Thanks for sharing.

## Chester Draws

September 12, 2015 - 3:18 am -But when we want students to develop understanding of mathematical relationships, then I feel I need to â€œuncoverâ€ the curriculum.You expect student to “uncover” caculus?

This is what confuses me about constructivist approaches. How is someone remotely going to get to calculus without being a Newton or Liebnitz? Do constuctivist teachers quietly just directly instruct for such topics? Do they teach it, but pretend the students found it out for themselves? I can’t even begin to imagine how I could teach differentiation by constructivist techniques.

I’ve seen teachers try to get students to “understand” trigonometry. I remain baffled by the approach. I don’t understand the Law of Sines. I can “prove” it, naturally, but application of it is just an issue of manipulating numbers to the formula. And how would understanding it help me solve a problem anyway?

## blaw0013

September 12, 2015 - 1:41 pm -Chester, I appreciate your candor in noting you do not possess the curricular expertise to design a sequence of questions/puzzles/prompts that may support student invention of concepts like instantaneous rate of change (derivative) and accumulation (integral) (or limits or the Fundamental Theorem of Calculus). It is a very specific knowledge base and talent that can write this curriculum. I am familiar with the Interactive Math Program (IMP) in which students are given contexts to develop meaning for derivative and integral — in the year 3 unit “Small World” and the year 4 unit “How Much, How Fast.”

Similarly, the IMP curriculum provides opportunity for children to develop meaningful understanding of right triangle trig then circular trig, and even generalizations like the Law of Sines.

Of course a curriculum is not constructivist teaching (whatever that is), and neither have much to do with the initial meaning of constructivism–a theory for learning.

But you end with a provocative question – How would understanding [some math topic] help a person solve a problem?

Trying to guess what you might mean by that, it seems you suggest that it takes something different than (more than?) understanding to solve a problem. It is reasonable to me to make a statement such as this, when a definition for understanding is structured in such a way that strongly distinguishes understanding for skill or procedural knowledge. That seems to me like a false dichotomy. But if I took that definition, I could not imagine how someone would be able to solve a problem with this procedural knowing only, no understanding. It seems they would have absolutely no idea if they solved the problem or not.

## Elizabeth (@cheesemonkeysf)

September 13, 2015 - 4:44 pm -This whole discussion got me reflecting on how I use the “How People Learn” framework in my teaching and learning processes. So I wrote a blog post about how I think about it and use it â€” and how I integrate different components into the four stages of the learning model.

My blog post is at http://cheesemonkeysf.blogspot.com/2015/09/how-people-learn-and-how-people-learn.html .

– Elizabeth (@cheesemonkeysf)

## Chester Draws

September 13, 2015 - 9:34 pm -I could not imagine how someone would be able to solve a problem with this procedural knowing only, no understanding.My 14-year-old kids use a known side and a known angle of a right angle triangle and find another side. They do so entirely by a process of selecting which trig function to use, and whether to divide or multiply.

I ask “why does this process work?” They have no idea — not even an inkling. They couldn’t even tell you how sine relates to cosine (they find that cos 30 = sin 60 to be a miracle).

That’s not a problem. How would having any more understanding of trig functions aid them in any way, given the sort of problems they have to solve?

Later, if they stay on the calculus path, they might gain a bit more knowledge of trigonometry, and perhaps the clever ones some understanding. But that’s much later.

## Grabe

September 14, 2015 - 5:30 am -Note that in considering options to facilitate conceptual change one option includes the instructor/information source listing common misunderstandings. Some might consider this a form of direct instruction. The personal effort to reconcile a long held model and new information can be prompted in several ways.

## Grabe

September 14, 2015 - 5:39 am -The key in conceptual change is to activate and then confront the existing flawed model. Without activation, it is possible for individuals to maintain incompatible models with stimuli from the environment activating different models (the danger would be if one model is only activated in “educational” settings).

## blaw0013

September 14, 2015 - 6:08 am -Chester – to solve a problem to me means that I have resolved a puzzle/problem/question for myself. It sounds as though your 14 year olds can compute a solution but have no idea if they’ve solved the problem or not.

My sense is that you are not arguing for such shallow mathematics instruction; rather you remain curious how children can develop such an understanding for many of the topics of the HS math curriculum–in a timeframe faster than the human species did themselves.

I will still say that it is accomplished in many classrooms, in part with well designed curriculum and teaching strategies that press students to expect mathematics to make sense.

## Dan Meyer

September 14, 2015 - 7:53 am -@

Brian, given your sniffy reply toChesterearlier, I find it disappointing that youaren’tproviding examples of direct instruction-less calculus and that youareshifting the goal posts. His question was about learning calculus without direct instruction. Your response here is about “pressing students to expect mathematics to make sense,” which is a much lower bar to clear.## blaw0013

September 14, 2015 - 9:37 am -Dan, I did provide curricular examples that do well to give students an opportunity to understand key concepts of calculus in a classroom environment that values student-generated mathematical understanding using rich problems, as opposed to treating fundamental ideas of calculus as social knowledge, which Marilyn described well. Look at my response in item 10.

It might be a lower bar to clear, from certain vantage points, that HS students (and teachers) expect mathematics to make sense. The challenge is to teach calculus in a manner that students experience it as making sense, having meaning. Chester himself suggested that direct instruction is not likely to achieve this–even if it can achieve shallow, procedural understanding.

Dan, as your entire post sets out to describe–it s not likely through words alone (written or spoken) that understanding of another’s ideas can be achieved. Words do not carry meaning; the listener or reader interprets meaning through their own lens. Fish is fish.

Do the mathematics of the IMP units I suggested, better do them with a small group of people; better yet do them with an instructor who is knowledgeable of pedagogical strategies paired with knowledge of the curriculum.

I am thinking of the Stanford effort to revise the IMP Year 1 Shadows unit, and the ways in which I think it likely misses the mark in terms of providing students an opportunity to make sense of mathematics. I do not think the Stanford authors shared the curriculum vision as the IMP authors–hence a fairly different curriculum. One that may not yet fully achieve the goals for students to know mathematics with meaning, in an educational environment that has as its goal (moral & intellectual) autonomy.

## Dan Meyer

September 14, 2015 - 9:56 am -I should clarify that I took exception to Brian’s comment at length in a new post.

## Jason Dyer

September 14, 2015 - 11:35 am -@Chester:

Thatâ€™s not a problem. How would having any more understanding of trig functions aid them in any way, given the sort of problems they have to solve?I find the students will tend to mix things up terribly with the kind of understanding you are throwing out there, and multiply or divide more or less at random as memory fades.

There’s also later stuff that’s easier or harder to learn based on basic understanding. The fact sine starts at 0 and cosine starts at 1 is tied in with the unit circle meaning in a way that makes it instantly comprehensible to a student with understanding and just another thing to memorize and mix up to a student who is strictly procedural.

Same goes for later identities like sin^2(theta) + cos^2(theta) = 1.

As a pre-calculus teacher I see all the time students clinging to some procedure from geometry (which only has time to do a surface understanding of trig; I know because I also teach geometry) and making my life harder, not easier. I have to actually “de-program” their prior algorithms so they’ll get it right second time around.

Also regarding “what kind of problems they’re going to be responsible for solving”: this one on the PARCC has a fair chance of bowling over any student with only a surface knowledge of sine and cosine.

## educationrealist

September 14, 2015 - 12:22 pm -“My 14-year-old kids use a known side and a known angle of a right angle triangle and find another side. They do so entirely by a process of selecting which trig function to use, and whether to divide or multiply.”

But do they know what a trig function is?

For example, I build from special right triangles, where they derive the ratio of the sides, and know that the ratio of short leg to hypotenuse is 1 to 2, etc. Then when I get to right triangle trig, I show them that the ratios are unique to the angle.

## Chester Draws

September 20, 2015 - 2:40 pm -The fact sine starts at 0 and cosine starts at 1 is tied in with the unit circle meaning in a way that makes it instantly comprehensible to a student with understanding and just another thing to memorize and mix up to a student who is strictly procedural.They still have to memorise that sine starts at 1, don’t they? How do they deal with Tan?

On the whole, the unit circle is the hardest way to teach “understanding” in my experience. For higher level students a trig graph from 0 to 2pi is more effective.

The SOH CAH TOA method of trig triangles pretty much removes any thinking about whether to divide or multiply. It really is foolproof.

## Chester Draws

September 20, 2015 - 2:43 pm -But do they know what a trig function is?No, of course not.

We teach trigonometry well before we teach functions. It’s not a problem. You can do trig perfectly well without any understanding of functions at all.

I deliberately avoid any mention of ratios when teaching trig. They don’t understand them properly, and it only confuses them.

## educationrealist

September 20, 2015 - 4:26 pm -I wonder what Greg Ashman would make of Chester.

Chester, let me rephrase. Do your kids know what a trigonometric RATIO is. You used the word function, not me. But the function, in right triangle trig, is just returning the ratio. I was asking if your students understood that they were using a ratio of the legs or leg/hypotenuse.

## Chester Draws

September 21, 2015 - 9:52 pm -No, they don’t know they are using a ratio.

They know that sin is used when they have a hypotenuse and need an “opposite”, or vice versa. The choice of division or multiplication is determined by the SOH triangle.

https://ma610.files.wordpress.com/2013/10/trig-equations.gif

People in general have extremely poor understanding of ratios. They struggle to recognise that 1.2 : 1 is 6 : 5, for example. That sin(57) means the “opposite” to hypotenuse is in the ratio of 0.83867 : 1 is just some random word salad that Maths teachers like to use.

## educationrealist

September 22, 2015 - 5:47 am -No, it means that the opposite leg is 84% of the hypotenuse. And while they might struggle to realize that 1.2:1 is 6:5, it’s not outside the realm of possibility that they’d recognize the first number is 20% greater than the second.

And it is possible to get kids to think about it. I do it all the time. Here’s an early version: https://educationrealist.files.wordpress.com/2012/03/trigquestion.jpg

I often give a triangle with clearly distinguishable leg lengths (but no measurements) and ask them to order the six ratios from largest to smallest. This forces the kids to think about the ratios. The kids should always understand why sec >= cos, why csc >= sin, and why cot and tan are not as predictable.

So no, it’s not just some random word salad.

## Jason Dyer

September 22, 2015 - 7:10 am -They still have to memorise that sine starts at 1, donâ€™t they? How do they deal with Tan?Not really; if they know the sin is the y coordinate of the unit circle, just visualizing or drawing the unit circle will give the information sine starts at 1. If they can’t do this, I would say they don’t understand trig well enough to handle identities.

So yes, they have to memorize at some level, but it is the arbitrary convention of sin = opp / hyp, cos = adj / hyp, tan = opp / adj. Everything else can be pulled straight off the unit circle.

Tangent is sine divided by cosine because it is opposite over adjacent, that is, the y value divided by the x value on the unit circle. And yes, I would expect them to be able to visualize that, because otherwise they get lost in the different varieties of Pythagorean identities.

There is an alternate way to draw the unit circle which directly has tangent as one of the sides. I only use it if students seem to be really interested in the geometric aspect.

https://numberwarrior.wordpress.com/2008/03/27/trig-identities-and-the-unit-circle/

## Mr_Young

September 22, 2015 - 6:01 pm -When Mr. Ashman provided a defense of his outlook in the link in post number 6, he uses an excerpt from the book that Mr. Meyer is recommending. The excerpt describes the efforts by one classroom teacher to produce an entirely student driven method of solving problems. I had to re-read his post to see that this description is given in the book “How People Learn” as an exemplar of ideal practices. He provides a link to an entry to the “Encyclopedia of the Sciences of Learning” that seems to refute exemplar from “How People Learn.” The encyclopedia entry says that “Extensive research has shown that for novices in particular, this pairing methodology of study-solve, leads to superior performance …” It does not site the source of this conclusion, but I found it to be compelling.

If you carefully read Mr. Ashman’s defense of his interpretation of the book, it is tough to argue that he is totally wrong. It did detract from my previous opinion that “How People Learn” might be a valuable resource.