Archive for Mathematics

Peer review by internet

There’s plenty of support and plenty of criticism for the Khan Academy videos.

Math Hombre have a sarcastic review  of a KA lesson about multiplying negative numbers that does make some valid points about the video’s shortcomings.

So a big thumbs up to the Khan Academy for their quick response to the negative comments. They pulled the video and replaced it with a much better version of the topic. It’s interesting to speculate what the end product might be like after a few cycles of this critique and response cycle.

There’s a marked contrast between the type of comments about the video on the math hombre blog site and the direct youtube site – people either love the KA videos (video users who’ve found the instruction helpful) or love to hate them (more likely to be an educationalist and the videos fail to develop conceptual understanding) – an internet version of Math Wars. Still there’s no escaping the fact that when it comes to an organised collection of video presentations of math topics, the Khan Academy video collection seems to meet a need, and has achieved it’s current status, for better or worse, because at present it doesn’t have any significant competition.

Dan Meyer suggested writing a critique on any KA video would be a good way to assess pedagogical knowledge and it seems that that idea has grown into a competition for the best Entertaining and Enlightening video posted to youtube featuring a Khan Academy video. It’s a great idea and I’m looking forward to seeing which videos come under more scrutiny, and what modifications are made to the Khan videos as a result.

The competition offically closes August 15, but who knows where it will go from there. Maybe peer reviews will be available for any type of educational material on the web and the standard of educational web resources will grow exponentially as a result. Fingers crossed!

Grappling with a concept

Concepts are slippery creatures.  Students seem to understand a particular topic one day, and then a week later the same student has apparently forgotten all the deep understanding recently acquired. Especially, it seems, when it comes to mathematics.  The videos of Cena and Jonathon are dramatic example of students who seems to understand place value, but on closer examination at a later date it is evident that this concept is still very vague for both students.

The initial knee jerk reaction here is to assume that this is evidence that the initial teaching was in some way flawed and the students probably never really had adequate conceptual understanding.

For me the real question is why children have so much difficulty developing their own understanding a concept that has become effortless for us.  If as teachers we have a better appreciation of why some concepts can be so elusive then we are more likely to provide appropriate learning experiences.  Current thinking in both neuroscience and artificial intelligence strongly suggests that concepts are the ability to identify patterns and make inferences from those patterns.

Here’s my attempt at explaining why conceptual understanding is fragile, and what can be done to help students acquire conceptual knowledge.

1) Provide many opportunities to manage the concept in various scenarios. The more often a pattern is identified, the less mental effort is required to recall the memory, and the greater the likelihood that the knowledge can be transferred to novel scenarios.

Each student seemed able to explain place value with numbers on the board, but subsequent questioning reveals significant conceptual gaps. This might indicate that either the previous learning was incomplete or partly forgotten, or it could be that the knowledge didn’t transfer to a different environment. Counting activities at the table are probably very familiar to the students, so it’s not so surprising that their previously well-established one-to-one correspondence type counting concept overpowers the newer concept of grouping things into tens and then counting those.

Brains learn by recognising patterns, and patterns don’t exist unless there is sufficient repetition for the pattern to become apparent. The flip side of this is that established learning – the already recognised patterns – is hard to change, which is handy for retaining existing learning but becomes a problem if the initial concept is incomplete or incorrect . When you think about it, it’s only logical that old learning should be hard to change. If it was easy to change, then most learning would only be transitory. Place value requires students to modify their previous learning of one to one correspondence and regard groups of tens and hundreds as special cases of one.

Our own familiarity with the decimal system of counting may make it difficult to appreciate the complexity of the place value concept. I thought I’d try myself out with a few simple addition, subtraction and basic multiplication operations in hexadecimal, and found myself struggling very quickly. In theory counting in hexadecimal or octal is conceptually no more difficult than counting using decimals, but in practice – well it’s something we need to practice before it can be done without considerable effort.

2) Conceptual knowledge usually take time to acquire and consolidate over a period of days or even weeks. Even those lightbulb learning moments can easily fade away from memory if they are not regularly reviewed. Learning seems to require sleep followed by reactivation of the memory of the pattern before it becomes readily available for use with minimal effort. Our experience of remembering jokes is an all too familiar example for most of us of the elusive quality of conceptual knowledge. A good joke requires conceptual understanding of the unexpected inference, innuendo or play on words in the punchline, yet even with this conceptual knowledge most of us struggle to successfully retell the well understood joke a few months later without an occasional review beforehand.

3) Manipulatives and concrete aids provide opportunities to learn in a specific context, while conceptual knowledge requires the ability to generalize from specific contexts. Hands on manipulatives are a great aid to understanding, but not a panacea. The activities and materials need to be carefully managed to provide opportunities to be sure that the focus is on the desired learning experience. Both Cena and Jonathon count accurately and confidently, but their previous experience of counting without grouping into ten seems to overwhelm their developing concept of place value. They both quickly forget about using groups of ten and revert to their already well established of counting sequentially one at a time starting with one.  While we easily generalize the concept of place value to any scenario requiring numbers more than ten, to these children it is quite likely counting stars on the board and counters at the table are almost two different concepts that have yet to consolidate into a larger general principle. It would be interesting to know how the students would have answered the questions with the counters if they had had more practice in both counting the counters by groups of ten and creating larger numbers from multiples of ten at the table immediately prior to the session.

4) Once short term memory is overloaded processing information becomes overwhelmingly difficult. It’s virtually impossible to form a concept from connected dots if you can’t keep track of all the required dots. At the board students were accurately answering fairly specific questions about the tens numbers, but the next day when the presenter’s references to the multiples of ten were less direct both students seemed to struggle to keep track of the information. This was particularly evident when students were asked to take away a number that required the subtask of breaking a ten group into ten ones.  Both students found it difficult to keep track of the tens and the units at the same time, focusing on one to the exclusion of the other.  By switching to using the less mentally taxing counting procedures that they have already mastered they were able to reduce the mental effort.

The take away message from these two videos is that if we to want to develop students’ conceptual understanding, then we need to understand what processes need to occur for that conceptual understanding to develop. The videos also highlight that conceptual understanding is more likely to be gradually developed than acquired through a sudden insight. As Marilyn Burns, the interviewer in these videos, observes,  “confusion and partial understanding are natural to the learning process” . (Marilyn Burns is passionate about improving how mathematics is taught in school. She discusses her vision in the second half of this video clip).

A different approach to teaching fractions

Fractions have an unenviable reputation of being difficult to teach.  Although most students have extensive practice shading in parts of a rectangle and counting pieces of pie, many students still lack a conceptual knowledge of fractions. In a multiple choice question asking Year 9 students to estimate the nearest correct answer to  11/12 + 7/8, only a minority of students correctly answered 2. The most popular answers were 19 and 20.  It seems that for many students fractions are a confusing nightmare, but why?

Mastering fractions requires a change in thinking about numbers. Moving from the whole number counting system to fractions (also known as rational numbers, i.e. numbers which are not whole numbers) takes students beyond their existing concept of whole numbers which are used to represent the number of things, to a more powerful number concept that can be used to describe not just whole things, but parts of things.  Read more

Hooked on maths

Challenge: Is it possible to find a student who would not be engaged by  Vi Hart’s brilliant  math doodling videos.?  These are about having fun while  discovering mathematical relationships.  Here’s a sample – there’s more on her site.

Doodling in Math: Sick examples


There’s plenty other mathematical stuff  on Vi Hart’s  blog.   Stuff with balloons,and how to make platonic solids out of fruit.  This is mathematics that meets Paul Lockhart’s1 description of the way mathematics needs to be taught – using playful “serendipitous exploration”  to discover that mathematics is about weaving ideas into patterns.

1. Paul Lockhart is author of  the book “A Mathematician’s Lament”   -  introduction is online

Learning to think

In an era that emphasizes the need for “deep understanding” in learning, it’s alarming that most of the students I meet requiring tutoring are totally reliant two ineffective strategies to find an answer – 1) guess and hope, or 2) don’t even try – that way you can’t get it wrong. Until recently my typical strategy would be to identify areas of  misconceptions,  and make students aware of what they needed to know to correct their knowledge of basic mathematical concepts . I stressed the importance of reasoning over guessing, and provided students with opportunities to practice and apply their new-found knowledge.  I figured I was helping them acquire the “deep understanding”  they had previously missed out on, and that this newly acquired knowledge would improve both their self-confidence and their motivation.

It didn’t work  very well. Read more

Over 20 of the best online fraction teaching resources

Attack fraction frustration with this hand-picked selection of the best  fraction resources on the web.   They have all passed the selection criteria of providing easily and clearly understandable accurate explanations of fraction operations, and/or providing effective practice opportunities.   The list includes interactive sites for visualizing fractions, and  games and worksheets.


Most of the games in this list do not involve timed activities, as time induced stress often often reduces performance  resulting in a frustrating and discouraging experience.  Game activities which do rely on timed activities should only be attempted by students who are already confident of their number skills. Read more