Posted in Mathematics education

Five task ‘types’ that create opportunities for conceptual understanding

Posted on by

In his paper The Design of Multiple Representation Tasks To Foster Conceptual Development, Professor Malcolm Swan of University of Nottingham presented five types of tasks  that foster conceptual understanding of mathematical concepts. This was developed through their work with teachers. This classification of tasks is a very good framework to use in designing instruction. I have used this framework in one of our lesson study projects.

Types of tasks for teaching with conceptual understanding.
  • Classifying mathematical objects

Students devise their own classifications for mathematical objects, and/or apply classification devised by others. In doing this, they learn to discriminate carefully and recognize the properties of objects. They also develop mathematical language and definitions. The objects might be anything from shapes to quadratic equations.

  • Interpreting multiple representations

Students work together matching cards that show different representations of the same mathematical idea. They draw links between representations and develop new mental images for concepts.

  • Evaluating mathematical statements

Students decide whether given statements are always, sometimes or never true. They are encouraged to develop mathematical arguments and justifications, and devise examples and counterexamples to defend their reasoning.

  • Creating problems

Students are asked to create problems for other students to solve. When the solver become stuck, the problem ‘creators’ take on the role of teacher and explainer. In these activities, the ‘doing’ and ‘undoing’ processes of mathematics are exemplified.

  • Analyzing reasoning solutions

Students compare different methods for doing a problem, organize solutions and/or diagnose the causes of errors in solutions. They begin to recognize that there are alternative pathways through a problem, and develop their own chains of reasoning.

Professor Malcolm Swan is also the author of the books Collaborative Learning in Mathematics: A Challenge to Our Beliefs and Practices and The Language of Functions and Graphs An Examination Module for Secondary Schools.

Posted in Curriculum Reform, Mathematics education

Why math education is failing

Posted on by

A backlink to my post What kind of mathematical knowledge should teachers have?   brought me to the essay by Matthew Brenner titled The Four Pillars Upon Which the Failure of Math Education Rests (and what to do about them). Here’s the quote from the essay posted in Wild about Math.

Kids are taught math as pets are taught tricks. A dog has no idea why its master wants it to perform. With careful training many dogs can be taught to perform complex sequences of actions in response to various commands and cues. When a dog is taught to perform a trick it has no need or use for any “understanding” beyond which sequence of movements its trainer desires. The dog is taught a sequence of simple physical movements in a specific order to create an overall effect. In the same way, we teach children to perform a sequence of simple computations in a specific order to achieve an overall effect. The dog uses its feet to move about a space and manipulate objects; the student uses a pencil to move about a page and manipulate numbers. In most cases, the student doesn’t know any more than the dog about the effect he creates. Neither has any intrinsic motivation to perform nor any idea why the performance is demanded. Practice, practice, practice, and eventually the dog can perform reliably on command. This is exactly how kids are trained to perform math: do a hundred meaningless practice problems, and then try to do the same trick on the test.

Mr.Brenner’s observation is as true in America as it is here in the Philippines. This is a painful truth but something that we all must take seriously. I strongly encourage our teachers, those writing our new curriculum framework (I think this is our third within the decade), textbook publishers and our DepEd officials to read the entire essay. The author outlined the reasons why math education is failing but he also offers solutions which I believe are doable even if our average class size here is 60! Let me list the 10 point solutions:

  1. Understanding Must be Central in Math Education
  2. Textbooks Must Not be Allowed to Undermine Math Education
  3. Teachers Must Stop Teaching Math as They Learned It
  4. Curricula Must be Coherent and Cumulative
  5. Worked Examples Must be Emphasized for New Material
  6. Curricula Must Include Examples of Excellent Performance
  7. Assignments Must Draw on the Old and the New
  8. Content Must be Meaningful and Contexts Must be Rich
  9. Metacognitive Activity Must Pervade Mathematical Activity
  10. Language Must be Taught, Used and Evaluated Fairly
I do not agree with #5 proposal because I believe that mathematics should be taught in the context of solving problems but I think this is a very good list. Find time to read it. Mr. Brenner also offers very good sample lessons. You may also want to read 10 signs there’s something not right in school maths and let us know your thoughts.
Posted in Mathematics education

What kind of mathematical knowledge should teachers have?

Posted on by

As a result of her research, Liping Ma developed the notion of profound understanding of fundamental mathematics (PUFM) as the kind of mathematical knowledge teachers should possess. She discusses this kind of knowledge in her book Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series). This book is now considered a classic by many mathematics educators. The ‘elementary’ in the title does not mean the book will be valuable to elementary teachers only or those engage in the training of prospective elementary teachers. The book is for all mathematics teachers, trainers, and educators. This book is a must-read to all that has to do with the teaching of mathematics.

Here’s what Liping Ma says in the introduction:

Chinese students typically outperform U.S. students on international comparisons of mathematics competency. Paradoxically, Chinese teachers seem far less mathematically educated than U.S. teachers. Most Chinese teachers have had 11 to 12 years of schooling – they complete ninth grade and attend normal school for two or three years. In contrast, most U.S. teachers have received between 16 and 18 years of formal schooling-a bachelor’s degree in college and often one or two years of further study.

In this book I suggest an explanation for the paradox, at least at the elementary school level. My data suggest that Chinese teachers begin their teaching careers with a better understanding of elementary mathematics than that of most U.S. elementary teachers. Their understanding of the mathematics they teach and -equally important – of the ways elementary mathematics can be presented to students continues to grow throughout their professional lives. Indeed about 10% of those Chinese teachers, despite their lack of forma education, display a depth of understanding which is extraordinarily rare in the United States….

Why the word ‘profound’? Profound has three related meanings – deep, vast and thorough – and profound understanding reflects all three. From the paper delivered by Liping Ma and Cathy Kessel in the Proceedings of the Workshop on Knowing and Learning Mathematics for Teaching conference, Liping and Cathy offered the following explanation:

  • A deep understanding of fundamental mathematics is defined to be one that connects topics with ideas of greater conceptual power.
  • A vast or broad understanding connects topic of similar conceptual power.
  • Thoroughness is the capacity to weave all parts of the subject into a coherent whole.

A teacher should see a ‘knowledge package’ when they are teaching a piece of knowledge. They should know the role of the current concept they are teaching in that package and how that concept is supported by which ideas or procedures.

To further explain the kind of mathematics knowledge a teachers should possess, Liping and Cathy used the analogy of a taxi driver  who knows the road system well. The teachers should know many connections so that they are able to guide students from their current understandings to further learning.

I think this is how designers of curriculum, writers of curriculum materials, and teachers should interpret the standard “Making connections”.  It is not simply linking.

 

Posted in Algebra, Assessment

What are the big ideas in function ?

Posted on by

Function is defined in many textbooks as a correspondence relationship from set X to Y such that for every x (element of X), there is one and only one y value in Y. Definitions are important to know but in the case of function, the only time students will ever use the definition of function as correspondence is when the question is “Which of the following represents a function?”. I think it would be more useful for students to understand function as a dependence/covariational relationship  first than for them to understand function as a correspondence relationship. The latter can come much later. In dependence/co-variational relationship “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (Sfard, 1991, p. 15)

The concept of change and describing change is a fundamental idea students should learn about functions. Change, properties, and representations. These are the big ‘ideas’ or components we should emphasize when we teach functions of any kind – polynomial, exponential, logarithmic, etc. Answer the following questions to get a sense of what I mean.

1. Which equation shows the fastest change in y when x takes values from 1 to 5?

A.     y = 4x2               B.     y = -2x2                C.     y = x2 + 10              D.     y = 6x2 – 5

2. Point P moves along the graph of y = 5x2, at which point will it cross the line y = 5?

A. (5, 0) and (0,5)      B. (-5, 0) and (0,-5)     C.  (1, 5) and (-1, 5)    D. (5, 1) and (5,-1)

3. Which of the following can be the equation corresponding to the graph of h(x)? 

A.  h(x) = x3 + 1           B.   h(x) = x3 – 1

C. h(x) = 2x3 + 1          D. h(x) = 2x3 + 4

4. The zeros of the cubic function P are 0, 1, 2. Which of the following may be the equation of the function P(x)?

    A.  P(x) = x(x+1)(x+2)       B. P(x) = x(x-1)(x-2)        C. P(x) = x3 – x2         D. P(x) = 2x3 – x2 – 1

5.  Cubes are made from unit cubes. The outer faces of the bigger cube are then painted. The cube grows to up to side 10 units.

The length of the side of the cube vs the number of unit cubes painted on one face only can be described by which polynomial function?

A. Constant    B.  Linear       C. Quadratic      D.  Cubic function

Item #1 requires understanding of change and item #5 requires understanding of the varying quantities and of course the family of polynomial functions.

Of course we cannot learn a math idea unless we can represent them. Functions can be represented by a graph, an equation, a table of values or ordered pair, mapping diagram, etc. An understanding of function requires an understanding of this concept in these different representations and how a change in one representation is reflected in other representations. Items #2 and #3 are examples of questions assessing understanding of the link between graphs and equations.

Another fundamental idea about function or any mathematical concept for that matter are the properties of the concept. In teaching the zeroes of a function for example, students are taught to find the zeroes given the equation or graph. One way to assess that they really understand it is to do it the other way around. Given the zeroes, find the equation. An example of an assessment item is item #4.

You may also want to read  How to assess understanding of function in equation form and Teaching the concept of function.