Inquiry and Maths

[Elizabeth Hutchinson Participant]

I am running an inset day at a local school in January and am looking for some ideas about linking FOSIL to maths. If anyone is a maths teacher and can give me suggestions of examples or ideas I can share that would be great.

If not, could we discuss how we can talk to maths teachers and help them see that FOSIL is also for them and their students? Thanks

[Elizabeth Hutchinson Participant]

I just wanted to share something that was posted on twitter by Dr. John Taylor.  I was wondering how I could use it as a simple example that I could talk about. It really shows how maths is more than just the numbers. Here is the link

[Jenny Toerien Keymaster]

It’s a really important question Elizabeth, and one we also struggle with. Responses from my maths department colleagues generally fall into two categories – “inquiry doesn’t work in maths” and “we already do a lot of inquiry – we call it ‘investigation'” (and it is true that, when done well, maths ‘investigations’ can also be inquiries and can be enhanced by viewing them through the lens of the FOSIL cycle). As a maths graduate myself, who also taught Year 9 maths briefly, I have been trying to work out how to address this. I don’t believe that inquiry is inherently unsuitable for any subject, but I do appreciate that it seems harder to apply in some subjects than others and have been trying to work out why.

Ever since we started the IB MYP here last year, my colleague Lucy has been working hard on how to ‘do’ inquiry in MFL with Year 7 and 8 (11 and 12 year olds). Lucy has an MFL background and is an outstanding librarian with considerable background in inquiry. The stumbling block here was that the students did not yet know enough French (in this case) to conduct a meaningful inquiry using resources entirely written in French, but it was not productive for them to spend too much time using resources in English when the goal was for them to learn French. With Year 6, the French department was happy for them to spend some time learning about French culture using English resources, but by Year 7 they really wanted them to be focussing on the language. Lucy and the French department have done an excellent job of designing an inquiry which meets all these needs for Year 8 (which we will write about separately), but it got me thinking that the problem with maths is actually a language problem too.

As a maths graduate, I often describe my experience to non-mathematicians by telling them that school maths is like learning a language. When you get to university you then use that ‘language’ to essentially study the mathematical equivalent of literature and poetry. If you are that way inclined, school maths is fun but university maths is beautiful and exciting. The problem is that we cannot ask students (and particularly younger students) to “do research on” a particular topic (e.g. inequalities) because the likelihood of them finding something useful that they actually understand is slim. The advantage of this is that maths hasn’t traditionally suffered from the rash of lazy, poorly designed, copy-and-paste-masquerading-as-research tasks that plague other subjects (“go away and find five facts about…”). The disadvantage is that many maths teachers think inquiry doesn’t work in maths.

So why bother? I think this image of maths as a language is important in thinking about students’ perceptions of maths. I have a six-year-old son who loves stories and books, but has only recently started to enjoy reading for himself. Why? Because until his reading reached a certain level, he found the kinds of stories he could read for himself really tedious. Maths can be like that for some students, who might otherwise become excellent mathematicians. At the necessary basic ‘language learning’ level, it can seem tedious and pointless – but what if we could give them a glimpse of where it was going and why it mattered? Effectively ‘read aloud’ some of those novels that they might be able to access later. Also, one of the biggest stumbling blocks I found for a certain group of students in maths was when they could follow a pattern and get a right answer, but didn’t understand why it worked. This created two problems – they couldn’t deal with questions that were essentially along the same lines but were very slightly different so their ‘pattern recognition’ approach didn’t work, and they had to learn so many patterns and rules that they inevitably made mistakes and ran out of ‘brain space’. The students who succeed in maths are the ones who want to understand why a rule works – and inquiry is ideal for this.

I realise that this doesn’t really answer your question, Elizabeth, and I don’t entirely have an answer (- yet, as Carol Dweck would say)  but I think at this stage the why is more important than the how. Changing practice is hard work and in order to get colleagues to ‘buy in’ enough to want to collaborate on this, you have to first convince them why it is going to be worth the effort. I have, however, found a few interesting resources that might be worth a look in terms of demonstrating what inquiry looks like in maths and giving some concrete examples, and I will try to post these tomorrow.

[Elizabeth Hutchinson Participant]

Thanks for your interesting reply Jenny! I had been thinking along the same lines although I would not have been able to put it nearly so well as you…Learning why you are are doing something is really helpful for those students who really don’t get it. I think I am probably one of those students myself as maths really is not a subject that was good for me. If I had a fuller rounded picture it may have helped.

I like the language analogy it is a great way to explain it and I am sure that a maths teacher would appreciate the link so that is a great start. I also agree that MFL has a problem too so linking them together could help us solve this problem for both of them. It is good for them to see that they are not the only subject where this does not easily fit.

I look forward to seeing your examples…

[Jenny Toerien Keymaster]

As promised…I am fairly sure that I could find more, and if I do I will post them, but these are a good place to start:

•  How Can We Teach Math To Encourage ‘Patient Problem Solving’?  (by journalist Katrina Schwartz on the KQED news website) led me to Dan Meyer’s excellent TEDx Talk, Math class needs a makeover. Dan explains with great passion and skill both why inquiry is important in Maths, but also how to go about reshaping your class around inquiry starting from traditional problem sets and working backwards to something more interesting. This is definitely one to watch, and one to recommend to your Maths Department colleagues.
• Dan Meyer’s excellent blog contains a wealth of ideas and suggestions. I was particularly taken with the Three-Act Math section. It is also worth noting that he freely shares his resources under Creative Commons Licences.
• How inquiry-based learning can work in a math classroom from The University of Texas at Arlington is a much more general article (but is actually what led me to the KQED article and thence to Dan Meyer).
• The Inquiry Maths website contains large number of examples of inquiry prompts, suggestions on how to run inquiry in Maths and articles about inquiry in Maths. I was particularly interested in the blog posts The differences between investigations and inquiries and Levels of inquiry (which is strikingly similar to Callison’s (2015) description of inquiry as “Controlled, Guided, Modelled and Free”) – in fact I would recommend looking at the blog as a whole.

Something important to point out here is that there is perhaps a subtle difference between “Inquiry-Based Learning” (IBL) (particularly in Science and Maths) and “inquiry” as pedagogical approaches. Kuhlthau, Maniotes and Caspari define of inquiry as:

“an approach to learning whereby students find and use a variety of sources of information and ideas to increase their understanding of a problem, topic or issue. It requires more from them than simply answering questions or getting a right answer. It espouses investigation, exploration, search, quest, research, pursuit and study.” (2007, p. 2)

But Buchanan, Harlan, Bruce and Edwards, in their review of the literature on “Inquiry Based Learning Models, Information Literacy, and Student Engagement” say:

“IBL is characterized by some or all of the following key components: 1) a driving question 2) authentic, situated inquiry 3) learner ownership of the problem 4) teacher-support, not teacher-direction and 5) artifact creation” (2016, p. 9)

Notice that “use a variety of sources of information” does not appear in this list, and I think that is another part of the problem. We sometimes think that consulting secondary sources is a vital component of inquiry and that Maths and Science investigations which rely on some form of ‘experimentation’ and don’t leave the lab don’t constitute true inquiry. However, we might be happy with a History inquiry that worked solely with primary sources. Do we need to start to view experimental data in Maths and Science as the equivalent of primary sources in History? It is striking that earlier in the same paragraph Buchanan et al say:

“Despite the fact that IBL has been in the school library world for more than forty years…, IBL is most commonly implemented in science and math classrooms” (2016, p.9)

I would say that in my experience we have almost the opposite problem here – most of my colleagues view inquiry as the natural territory of the humanities. Given the opposite is the case in the States, does it say more about what we consider to be inquiry than about a genuine problem in Science and Maths, if we are having a problem in the UK understanding how to ‘do inquiry’ in Maths?

As a final thought, returning (eventually) to your original question Elizabeth, I would regard FOSIL as an ideal support structure for inquiries of this type – we might just need to think more creatively about the activities that happen during the Investigate stage. While students may still need to consult secondary sources, they may not need to do this in every inquiry – and their ‘experimental data’ is a perfectly valid form of Investigation. As a Librarian I still feel I have a lot to offer in supporting this type of inquiry because my expertise is in the inquiry process itself, and not limited to helping students to locate, evaluate and use resources.

[Elizabeth Hutchinson Participant]

Thank you, Jenny. This is all really useful and I will definitely be listening to the TED talk. I would agree that I have seen the same problem with inquiry being linked to humanities and would have assumed that most librarians would too so it is really interesting to read that this is different in the States. I need to get my own head around this but I am sure that I will be able to include this in my inset presentation and be able to include both maths, science and language examples to engage those departments. I will share my presentation once it is done.

Thanks!

[Darryl Toerien Keymaster]

Thanks, Elizabeth – looking forward to the presentation.

[Darryl Toerien Keymaster]

More Maths resources may be found on the Galileo Educational Network website, which is the professional learning arm of the Werklund School of Education at the University of Calgary.

See Designing Learning, and in particular Designing For Deep Mathematical Understanding and Math Fair Problems.

There are also a number of relevant Classroom Examples and some Math Investigations.

I don’t know how useful these resources are for Maths, but I flag them because there are some very helpful articles on inquiry, especially What is Inquiry?, which addresses some of the misconceptions about inquiry. There is also the excellent Focus on Inquiry (see below).

[Darryl Toerien Keymaster]

I realise that much has happened since this Topic was started, but for the record …

I am watching Anthony Bourdain: Parts Unknown. While in Spain (Season 2, Episode 1), he and cinematographer Zach Zamboni visit the Alhambra (28:20 – 32:19), palace and fortress of the Moorish monarchs of Granada,* which, according to Bourdain, is “one of the most enchanted, inscrutable, maddeningly beautiful structures ever created by man.”

The Alhambra, as Zamboni explains, depicts and is geometric systems. Bourdain continues, “How did nature unfold, pattern itself? Could the basic patterns of nature, even if divine, be replicated in this magnificent structure?”

Visually, given that Zamboni is a cinematographer and “a bit mad about the place,” the section is arresting, and the mathematics of the building and its decorative features very cleverly revealed (I will see if I can find a screenshot to illustrate, but have included an image from Britannica below**).

The section is (SFW) safe for work.

I have not had a chance to look, but I imagine  that there must be other visual sources that illustrate the mathematics of the Alhambra in an equally spellbinding way.

—

*Alhambra. (2020). In Encyclopædia Britannica. Retrieved from https://school.eb.co.uk/levels/advanced/article/Alhambra/5709

**Alicatado. [Image]. In Encyclopædia Britannica. Retrieved from https://school.eb.co.uk/levels/advanced/assembly/view/4153

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