Tag: mathematical knowledge

Posted in Number Sense

Math Knowledge for Teaching Addition

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This post is the second in the series of post about the Math Knowledge for Teaching (MKT) where I present task/lesson that teachers and interested readers of this blog can discuss. The first is about Tangents to Curves, a Year 12 lesson. This second post is for young learners.

The task

How many small cubes make up this shape?

cubes

This is a pretty simple task.  Any Grade 1 pupil will have no difficulty giving the correct answer. All they need to do is to count the cubes. Yesterday, in one my workshop with teachers about lesson study, we viewed a Japanese lesson using the same task but was used in such away that children will learn not just counting.

The lesson

Before this lesson the class already learned that putting together concept and the symbol + and =.

The pupils were given small cubes to play with on their tables. After a minute, the 2x2x2 cube was shown on the TV screen and the teacher asked the class to predict how many small cubes make-up the shape. Some used their cubes to make a similar shape without the teacher encouragement to do so. The cubes were only there to help those who might have trouble imagining the bigger cube were some parts are not shown. The pupils counted the visible cubes one-by-one and then those not seen in the drawing (a drawing of the cube is posted on the board). But, the teacher was not just after the answer 8, he was after the learners’ counting strategy. So he asked: Can you use the + sign to show us your counting strategy? Some of the students answers were: 4+4 = 8, 2+2+2+2 = 8, 6+1+1=8. But, the teacher was not only after this, he wanted the class to realize that this number expressions may have come from a different way of looking at the cube. He started with those who wrote 4+4 to show the class how this counting was done. There were two different strategies: halving the cube vertically and the other horizontally which the students demonstrated using the cubes. All throughout the teacher was asking the class, “Can you follow the thinking? “Do you have a different idea?” “Who has another idea?”

After the summarizing the different ideas of the pupils in the first task, the teacher gave the second task:

What is your idea for counting the small cubes in this shape? Show your idea in numbers and symbols.

cubes

The shape was projected on the TV screen as the teacher rotated the shapes. The pupils came-up with different combinations of visible and not visible cubes like 7+3 = 10, 4+6 = 10, etc. They were invited to explain these expressions and their thinking using the drawing on the board. The teacher did not have any difficulty getting the answer he wanted from the pupils: “We already know that this shape (the big cube) is 8 so we just add 2  (8+2 = 10).

Questions for Teachers Discussion/Reflection:
  1. What about numbers will the pupils learn in the lesson?
  2. What is the role of technology and visuals in this lesson?
  3. What about mathematics is given emphasis in the lesson?
  4. What mathematics teaching and learning principles underpin the design of the lesson?

Remember this quote from George Polya: What the teacher says in the classroom is not unimportant, but what the students think is a thousand times more important.

math knowledge

For further reading:

Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (Studies in Mathematical Thinking and Learning Series)

Posted in Curriculum Reform, Mathematics education

Why math education is failing

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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?

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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 Mathematics education

Forms of mathematical knowledge

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Anyone interested to understand how mathematics is learned should at first understand what mathematical knowledge consist of. The book Forms of Mathematical Knowledge: Learning and Teaching with Understanding describes various types of knowledge that are significant for learning and teaching mathematics. It defines, discusses and contrasts various types of knowledge involved inthe learning of mathematics. It also describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas. The book is a collection of articles/papers from well known mathematics educators and researchers.


Top in the list of forms of knowledge presented in the book is a discussion about intuition and schemata.
While there is no commonly accepted definition, the implicitly accepted property of intuition is that of self-evidence as opposed to logical-analytical endeavor. Now, what is the role of intuition in the learning of mathematics?

While in the early grades teachers are awed by intuitive solutions by our students, those handling higher-level mathematics would find intuitive knowledge to constrain understanding of mathematics. In the book, the author of offered examples of these. He also defined the concept of intuitions and described the contribution, sometimes positive and sometimes negative, of intuitions in the history of science and mathematics and in the teaching process. The author argues that knowledge about intuitive interpretations is crucial to teachers, authors of textbooks and mathematics education researchers alike. The author further argued that intuitions are generally based on structural schemas.

My favorite article in the book is about the description of mathematical knowledge as knowing that, knowing how, knowing why and knowing-to

Knowing why, meant having “various stories in one’s head” about why a mathematical result is so. For example, when partitioning an interval into n subintervals, one might recall that n+1 fenceposts are required to hold up a straight fence of n sections. Knowing why and proof are different — in many cases, the proof doesn’t reveal why. As an example, the author suggested that when primary teachers ask why (-1)(-1)=1, they want images of temperature or depth, not a proof, or even a consistency argument that negative numbers work like positive numbers.

Knowing to means having access to one’s knowledge in the moment — knowing to do something when it’s needed. For example, in evaluating a limit, a student might just know to multiply by a certain quantity divided by itself. This kind of enacted behavior is not the same as writing an essay explaining what one is doing — it often occurs spontaneously in the form of schemas unsupported by reasons, whereas explanations require supported knowledge.

Forms of Mathematical Knowledge: Learning and Teaching with Understanding is a must-read for teachers, educators, and those doing research in mathematics teaching and learning.