Education ∪ Math ∪ Technology

Month: September 2018 (page 1 of 1)

Handing out workbooks is not teaching

A number of times over the past few years, I have been confronted by a phenomena I do not understand: children being handed workbooks and, with very little to no instruction, expected to complete the workbooks and develop an understanding of the material contained within.

A variety of math workbooks


For some children this works out fine. The workbooks, mostly being designed for independence anyway, are sufficient for children who have some background in the material and see the patterns developed within the workbooks. For many children however, the experience is potentially one of frustration and humiliation: they have been given a task that their teacher clearly expects them to be able to do, and they cannot.

I have seen workbooks handed out with minimal to no instruction in two very different kinds of circumstances. In one case, the teacher sees a class with a diverse needs and although all of the students had the same overall goals for the class, students clearly have different levels of preparation and interest in the course. The other case is when teachers have a multi-age classroom (which also contains students with different levels of preparation and interest) and no time to actually teach anything at all. Both of these are compounded by the preparation of the teacher leading the classroom to deal with varied needs.

I have been in the first case, leading a group of students with varied needs but with the same goal, many times during my career. The best solution I have learned is the use of instructional routines with designs for interaction to support a varied set of needs, while working together towards the same mathematical goal.

As for the second case, when teachers have a multi-age classroom, I don’t know for sure what I would do. If I found myself teaching in such a classroom,  I think I would do two things; teach mini-lessons (using a variety of formats depending on the mathematical goals, but again instructional routines seem like a good fit) and assign problems from a somewhat independent workbook, and then while some students are working in the workbook, engage the other students in a mini-lesson on a different topic. I would have to organize the space so that students who were working on similar material sat together and I would need to develop classroom culture so that students knew to talk to each other after first trying the math themselves and to wait until I was available to assist. I would also look for places in the mathematical goals for the year where potential alignment between different grade levels occurs and use these opportunities to engage the entire classroom. For some students the activity might be review and for other students the activity might be somewhat new material, but both of these seem like helpful ways to use classroom time.

Imagine you are teaching a classroom with 20 students roughly evenly distributed from grade 2 through 7 (age 7 through 13). What would you do to meet the needs of these learners?


What is Conceptual Understanding?

According to Adding It Up  (H/T Dan Meyer):

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)


Essentially, conceptual understanding is knowing more than isolated facts, it is also knowing connections between those facts and having those facts well organized.

This image represents someone who understands some isolated facts.

Isolated facts as separated points

And this image represents what conceptual understanding looks like:

Points connected with line segments

I propose that those line segments between the points, representing the connections between isolated knowledge, are themselves a type of knowledge, consequently our second image is really more like the first image.

Connected dots transforming into disconnected dots


What is clearly missing in this analogy is that conceptual understanding is knowledge that is well-organized, but I claim that organization of knowledge is itself another type of knowledge. Also missing is knowing why the knowledge is important and the contexts in which an idea are useful, but again these also seem just like other types of knowledge.


Let’s take a look at a problem I had to solve recently and see if we can use it to unpack what conceptual understanding is.

Algebra II August 2016 Regents Exam, Question #1

According to the solution key, the writers of this question are assessing students’ ability to solve quadratic equations.  However, that likely requires applying the quadratic equation to all four equations given, which would be time-consuming and potentially error-prone.

When I solved this problem, I saw four quadratic functions, each transformed from y = x2, and based on those transformations, I was able to eliminate (1) and (3) as options since they have real solutions. I also know that if 1 – i is a solution, then so is 1 + i, and that the sum of the roots of a quadratic function are related to the coefficient of x in the equation. I could not remember if the sum of the roots is equal to the coefficient of x or to -1 times the coefficient of x, so I multiplied out (xa)(xb) to double-check and determined that if a and b are the roots of a quadratic equation, then -(a + b) is the coefficient of x in that equation (assuming a lead coefficient in the equation of 1). 1 – i  + 1 + = 2 so (4) must be the answer.

My actual solution strategy doesn’t matter all that much, what matters is that in order to produce it, I needed to know some smaller things and that those smaller things could be seen as connected together to create a larger idea. Instead of seeing this as the difference between an expert and a novice, I prefer to think of knowing things as being a gradient between knowing a little bit and knowing more than a little bit.


Conceptual understanding is both knowing ideas in an interconnected and organized way and knowing more ideas along the novice-expert spectrum.

The reason why conceptual understanding is an important goal is because otherwise we might be tempted to rely on teaching kids tricks instead of mathematics.

What mathematics does a child learn from learning this procedure for adding fractions?

Butterfly Method for Adding/Subtracting Fractions

Source: Pinterest


Very little! It is extremely difficult for a child to connect this procedure to other mathematics that they know, consequently using a trick like the butterfly method is likely to lead to students knowing an isolated idea and not much else. Note that I am not opposed to memory aids, I just think they should be based on solid mathematical reasoning and they definitely should not be taught instead of the mathematics.