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,

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

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.

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 isn’t. The other issue is how to decide, in situations when explicit instruction isn’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 that’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 “cover” the curriculum. We celebrate Thanksgiving on a Thursday, and that knowledge isn’t something a person would have access to through reasoning without external input―from another person or a media source. There’s no logic in the knowledge. But when we want students to develop understanding of mathematical relationships, then I feel I need to “uncover” the curriculum. I love the Fish Is Fish children’s book example.

Yes, students need to memorize multiplication tables, but I’ll never forget the fourth grader, Paul, who proudly told me one day that he knew that “six times eight is forty-eight.” I asked him how he knew that, and he recited a rhyme, “Going fishing, got no bait, six times eight is forty-eight.” I asked him what six times nine was. He didn’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.

But when we want students to develop understanding of mathematical relationships, then I feel I need to “uncover” 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?

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 – 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)

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.

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, 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.

@Chester:

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?

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.

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.

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.

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.

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.