This is essential. On commission from the UK in 2005, Malcolm Swan wrote a guide to great math teaching [pdf] that’s as good as anything I’ve read at this length. It is, explicitly, a collection of activities â€” full explanations of resources a teacher can use for floatation in her first year â€” but the document also goes into exquisite detail about what *theory* motivates those activities. It outlines excellent pedagogy while at the same time keeping your head above the waterline.

This is a fast read â€” big type, thin margins, lots of color, etc. â€” and from now on, it’ll be the first thing I recommend to new math teachers. I hope you’ll do the same. But help them make sense of it also. Some of this only may look so brilliant in light of my abundant early-career failure.

## 23 Comments

## Phil

February 3, 2011 - 2:36 pm -Also check out the latest project he (Malcolm Swan) has been involved with: http://www.bowlandmaths.org.uk/index.htm

A package of ICT enhanced enquiry learning packages. Some brilliant stuff here.

## serafina

February 3, 2011 - 6:49 pm -Thanks for another great resource, Dan!

I look forward to reading this over the next couple of days.

## Nordin

February 3, 2011 - 10:53 pm -Thanks for the link – a brilliant resource for a new maths teacher (me)!

## Marilyn Knight Just

February 4, 2011 - 5:38 am -Dan & Phil — Thanks for the lead! I’ve included links to these resources on my home school web site & will explore methodologies & materials in search of ways to better reach my students :)

## simon walsh

February 4, 2011 - 5:59 am -The bowland approach sounds good. I think that giving kids and alternative way to learn is a big part of the battle won. I am a huge believer in activity based maths. You can see me teaching a class sequences with this method on our website. The lesson idea was not mine, it is a classic, but it shows the level of energy and enthusiasm that you can get in a maths class if you abandon text books.

video on this page..

http://mathsdoctor.tv/Maths-tutor/about-us/our-story/

## ncetm_administrator

February 4, 2011 - 7:42 am -The whole Improving Learning in Mathematics is available online. More details at https://www.ncetm.org.uk/resources/1442. We’re also on Facebook and Twitter (search on NCETM).

## blaw0013

February 4, 2011 - 9:56 am -Hi Dan. Thank you for the resource. I agree that this can be a very powerful tool for a new teacher as they work to develop concrete examples of pedagogical and curricular materials for the classroom toward a vision to transform mathematics education.

I hope all of my students who find value in this resource will recognize the fatal flaw in its organizing premise, that there is one mathematics and one way to think mathematically that is worth knowing. This author decries the notion of pursuing a “discovery” approach to mathematics, yet the entire pamphlet has students working to know someone else’s mathematics, i.e. to discover something already known.

A humane way to teach mathematics reverses the role of “discovery,” charging the teacher with discovering how the child/student is thinking. What did the student make of the problem situation posed? i.e. What is the question in the student’s mind? Then, what is the conclusion/solution to that question and what reasoning rationalizes/justifies that conclusion?

The next step in the teacher’s role is to bridge the child’s thinking with that of their own–that which is representative of the discipline we call mathematics.

Try Dewey’s 1902 essay “The Child and the Curriculum.” http://www.archive.org/stream/childandcurricul00deweuoft

## Elizabeth S

February 4, 2011 - 2:29 pm -Thank you for being in grad school (oh, and also for sharing this stuff!). :-)

Elizabeth (aka @cheesemonkeysf on Twitter)

## Pwolf

February 5, 2011 - 12:03 pm -“A humane way to teach mathematics reverses the role of â€œdiscovery,â€ charging the teacher with discovering how the child/student is thinking. What did the student make of the problem situation posed? i.e. What is the question in the studentâ€™s mind? Then, what is the conclusion/solution to that question and what reasoning rationalizes/justifies that conclusion?

The next step in the teacherâ€™s role is to bridge the childâ€™s thinking with that of their ownâ€“that which is representative of the discipline we call mathematics.”

This article says just that, and posits that there are concrete ways to get meaningful responses that will help teachers get a good picture of that student thinking.

It goes on to say that an effective teacher will be able to anticipate that not all students’ lines of reasoning will be correct, and will help students meet these shortfalls of reasoning head-on.

## Matt W.

February 5, 2011 - 1:52 pm -I agree with both pwolf and blaw003. It seems like there’s a disconnect between the underlying principles (which are great), suggested teaching strategies (also helpful), and many of the tasks – especially the classifying and matching ones and the ones where kids have to put somebody else’s steps in order.

## louise

February 5, 2011 - 5:19 pm -I think the classifying items comes from some other parts of the UK curriculum for elementary students, which are really rather wonderful as they are not language dependent. I picked them up a couple of years ago on a trip “home”. Someone who is from the UK might tell us what they’re called (mine are at school). Non-verbal reasoning? I have used them in my classes to great effect with ELL students (high school geometry)

I think that if you see the classifying and matching in this context, it will be more cohesive with the whole premise.

## Pwolf

February 6, 2011 - 1:09 pm -I want to add in here that I’m using lots of these strategies on Monday: using the white boards, having students create problems, and making posters with linear equations.

I spent most of last few semesters trying to include techniques like this in my classes, to little success. I think I’m finally getting the hang of what direction my activities need to go. It’s kind of hard in the beginning to know what type of activity goes with what topic or current level of understanding, and what’s the right ratio of information given to information left out of an activity.

## blaw0013

February 9, 2011 - 6:30 pm -@Pwolf. First, let me say (restate) with great emphasis, the classroom strategies advocated in the pamphlet are GREAT, and the pamphlet itself I predict will be a great resource for the preservice teachers that I teach.

My disagreement with the pamplet’s author is not captured in the passage you quoted, rather in the paragraph before: “the fatal flaw in its organizing premise, that there is one mathematics and one way to think mathematically that is worth knowing. This author decries the notion of pursuing a â€œdiscoveryâ€ approach to mathematics, yet the entire pamphlet has students working to know someone elseâ€™s mathematics, i.e. to discover something already known.”

My naming “someone else’s mathematics” is a deeply troubling (form me?) aspect of what we are charged to do as public school teachers. Are we to create divergent or convergent thinkers? A convergent goal would agree with the “discover how you’re supposed to know/think mathematically.” This convergence sees one mathematics, probably those represented by “the standards.”

A divergent goal would engage kids in the mathematical activity of being human, thinking, reasoning, generalizing, deducing, etc. Maybe the Common Core Standards for Mathematical Practice point well to this defining quality of what might be “everyone’s mathematics.”

I don’t know… it is a very slippery idea for me. I know I don’t want to create a roomful of young adults who believe they don’t think right, that is like me.

## liz

February 11, 2011 - 4:11 pm -Hey Dan! I’m so happy to have found this blog. I’ve only poked around, but what a great resource! Thank you especially for this. I’m in my pre-service year, gearing up for next year, so I appreciate the wealth of knowledge here. I hope your studies are going well. I’ll be checking your site often!

## Sam Critchlow

February 13, 2011 - 3:52 pm -Thanks for the post – we have apprentice teachers at our school each semester, and I have been in the process of collecting readings, videos, etc. specific to math education to supplement their teaching seminar. Will add this to the collection.

## KYoung

February 14, 2011 - 12:36 pm -we refer to this pack as the Big Blue Box (as that’s what the folders come in)

It has been an invaluable resource to a department who want to encourage pupils to think about maths.

Also it isn’t just about the resource its about how to use it effectively.

## lfarrington

February 24, 2011 - 7:35 pm -Just finished reading this tonight. Lots of highlighter ink was used! Thanks for the pdf.

## Audrey Mc^2

March 12, 2011 - 5:52 pm -Hi Dan – thanks so much for this. I’ve now made, and used in my class, 2 activities based on this document. Huge improvement in engagement, even with lots of room for improvement on my design. The kids ended up discussing exactly those things that I wanted them to, only without any overt prompting from me. The activites are embedded in my Mar 11 and Mar 7 posts at:

http://audrey-mcsquared.blogspot.com/

(in case anyone wants to use/improve on them – hey, have at it!)

Audrey

## pepsmccrea

November 16, 2011 - 1:18 pm -Dan et al.

If you want a greater insight into Malcolm’s thinking underpinning the design of the standards box then this is a good start…

http://www.educationaldesigner.org/ed/volume1/issue1/article3/

## Kirk

November 17, 2011 - 2:54 pm -Wow. Just finished reading it. That is powerful stuff. I think I am going to create a working group at the high school level around this.