Maths, Automaticity & iOS Devices...

Using iOS devices such as iPads for 'skill+drill' is something we generally discourage in school, where we would rather these technologies are used for creating and collaborating, along with the many other skills that are described in the UWCSEA profile.

You see, skill+drill Apps don't need a teacher, what they do need is a device, time, and perseverance; so what this is, is an excellent productive activity children can easily engage in at home. This kind of practise builds the kind of 'automaticity' (instant recall without hesitation) that is fundamental to confidence in numeracy. Knowing mathematics facts frees up the mind to solve more complex math problems.  If a child has to struggle to solve 8 + 3, they have no mental energy (or desire) left to grapple with the types of problems that will increase their capacity as a mathematician.

In my experience spanning over twenty years, I find that teachers commonly (and traditionally) facilitate this through a relentless torrent of photocopied worksheets, something I myself have relied on over the years. However since the advent of the integration of digital technologies, I really struggle to understand how having kids complete a photocopied Maths worksheet can ever be seen as better than the kinds of differentiated, adaptive, multimodal practise offered by Maths apps, and Maths sites like Khan Academy. If teachers ceased to set these tedious sheet as homework, they could free up the time to plan better lessons, no need to 'mark', instead use the time to analyse the data—where are they struggling? Where are the gaps? Where should they go next? What group of kids do you need to conference with tomorrow?

Automaticity is the ability to do things without occupying the mind with the low-level details required, allowing it to become an automatic response pattern or habit. It is usually the result of learning, repetition, and practise.
(Wikipedia)

So while I generally discourage skill+drill in school, I can see its value at home. Below I have included the collection of Maths Apps we use (sometimes) at school that I believe are particularly powerful for this kind of learning, learning through practise. 

This is a small selection, no doubt a Google search would turn up many more, although I doubt they will be very different to these.

Top Tip: Ask your child to take a screenshot of their score after first attempt, then compare their progress after a week or so.

Disclaimer

These kinds of Apps are basically teaching mathematics in old ways using new technology, albeit amplified. 

Chocolate covered broccoli...

These Apps are essentially worksheets on steroids, so while your kids may be more engaged in the short term, don't expect this to last. These tools are essentially 'chocolate-covered broccoli'. That’s what designers of educational games call digital products that drape dull academic instruction in the superficially appealing disguise of a game, using the trappings of games “as a sugar coating” for what would otherwise be unappetising content—in short don't treat these games as a replacement for 'proper' games like Minecraft, but by all means treat them as replacements for worksheets.

What these Apps do offer that worksheets don't are features like, interactivity, capacity, range, speed and automatic, accurate, reliable responses at the speed of light.  These unique features make a contribution to the teaching and learning process, in that they motivate and interest children by interaction, allowing them to change the work in progress and facilitate a variety of paces of working.

We sometimes think of being good at mathematics as an innate ability. You either have "it" or you don't. But what these Apps can encourage, is what we call a 'growth mindset' it's not about ability it's about attitude. You master mathematics if you are willing to try.  Success is a function of persistence and doggedness and the willingness to work hard for 30 minutes to make sense of something that some people would give up on after 30 seconds.

Drill & Practise

"Of particular interest is the effect of drill and practise – and despite the moans by many adults, students need much drill and practise. However, it does not need to be dull and boring, but can be, and indeed should be, engaging and informative. Drill is a euphemism for practise: repeated learning of the material until it is mastered – this is the key ingredient in mastery learning, [...] and of deliberative practice. It does not have to be deadly, and a key skill for many teachers is to make deliberative practise engaging and worthwhile. Luik (2007) classified 145 attributes of drills using computers into six categories: motivating the learner, learner control, presentation of information, characteristics of questions, characteristics of replying, and feedback. The key attributes that led to the highest effect included learner control, not losing sight of the learning goal, and the immediate announcement of correctness or otherwise of the answer to the drill." (Hattie, 2013)

Many computer games are basically invested with high levels of drill and practise and many students can be thrilled and motivated to engage in these often repetitive tasks to attain higher levels of skill and thus make more progress through the game. Computer games include much engaging drill and practice with increasing levels of challenge that usually is mastered by overlearning or undertaking high degrees of drill and practice. So often, the evidence has shown positive effects from using computers to engage in deliberative practice, particularly for those students struggling to first learn the concept." (p 224)

Hattie J (2013). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. Routledge.

Transforming Maths Practice & Practise

Why Use Digital Tools in Mathematics?

• Immediate feedback
• Infinite patience
• Personal (individual//automated) differentiation
• Less marking, more monitoring (dashboards)
• Dynamic interactive models (what if)

SAMMS
Situated: work anywhere, any place any time. No carting around text or exercise books, all you need is scrap paper and a pen or pencil. Students can work out the own pace in their own space without having to do work pitched at a group of students in order to make the management of the task practically feasible for the teacher. No more having to set 'homework', now the homework is the classwork continued, and vice versa. As the results feed into one dashboard, you can see the results of an entire class/classes in one view, updated in real time.

Access: videos and tutorials from some of the greatest Maths teachers on the planet is only a click away. Not to mention access to a wider range of strategies, and ways of explaining. Leverage the computer processing power of automated marking; faster, and more efficient than a human, freeing teachers to focus on marking the stuff computers cannot, and freeing time for teaching/planning. No longer do students have to wait several days to find out whether the work they did is correct or incorrect, they know as soon as they submit an answer and are able to work on each problem until they get it right without the need for teacher intervention.

Multimodality and Mutability: beyond text and static images to illustrate, they can use video to explain, and animations (animated gifs) to demonstrate visually/aurally, in ways that allow rewind, repeat, retry, as often as is needed. Interactive dynamic models allow students to really explore mathematical models, with 'what if' experimentation. Got it wrong? Try again. No limits, no stress, no strife. Undo, try again, repeat.

Socially Networked: via an online space, students can share their questions, clarification, celebration. Teachers and students alike can can help one, help many. The fact that students can receive so much of the mathematical support via digital resources and via each other means the teachers actual face-to-face time can be used far more efficiently to work with smaller groups that would benefit more from the personal touch that computers cannot replicate.

Who Says?

Well, there’s lots of research, but let's just focus on a few for the sake of brevity. I reckon the points they made (some time ago, I might add) will convince anyone who has any passion for the teaching of Mathematics that their argument make sense.

In Principles and Standards for School Mathematics (NCTM 2000), the Technology Principle asserts: “Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning” (p 24). More specifically, a technology-rich environment for mathematical learning influences five critical features of the classroom (Hiebert et al 1997): the nature of classroom tasks, the mathematical tool as learning support, the role of the teacher, the social culture of the classroom, and equity and accessibility. An essential question when working in a technology-rich mathematics environment is how technology can be used (appropriately) to enhance the teaching and learning of mathematics.

An effective way to optimize the mathematical thinking opportunities presented by technology is to plan the mathematics task focused on the five Process Standards (NCTM 2000): Problem Solving, Reasoning and Proof, Communication, Connections, and Representation.

...

Learning environments that take advantage of virtual manipulatives offer a number of ways for students to develop their mathematical understanding. The authors identify the following as five primary benefits:

1. Linked representations provide connections and visualization between numeric and visual representations. 
2. Immediate feedback allows students to check their understanding throughout the learning process, which prevents misconceptions. 
3. Interactive and dynamic objects move a noun (mathematics) to a verb (mathematize). 
4. Virtual manipulatives and applets offer opportunities to teach and represent mathematical ideas in nontraditional ways. 
5. Meeting diverse learners' needs is easier than with traditional methods. 

Enhancing Mathematical Learning in a Technology-Rich Environment
Teaching Children Mathematics / November 2008


Then there’s this from the Centre for Research in IT in Education (CRITE) Bray & Tangney (2013):

An examination of the extent to a which recent technological interventions in mathematics education make use of the educational opportunities offered by the technology and the appropriate pedagogical approaches to facilitate learning, focused on digital tools classified as follows:

• Outsourcing of Processing power 
• Dynamic Graphical Environments (DGE) 
• Purposefully Collaborative 
• Simulations/Programming 

These are the guiding principles that have the potential to form the basis of a 21st Century model for the integration of technology into mathematics education. An appropriate and innovative technology intervention in mathematics education should:

1. Be collaborative and team-based in accordance with a socially constructivist approach to learning. 
2. Exploit the transformative as well as the computational capabilities of the technology. 
3. Involve problem solving, investigation and sense-making, moving from concrete to abstract concepts. 
4. Make the learning experience interesting and immersive/real wherever possible, adapting the environment and class routine as appropriate. 
5. Use a variety of technologies (digital and traditional) suited to the task, in particular, non-specialist technology that students have to hand such as mobile phones and digital cameras. 
6. Utilise the formative and/or summative assessment potential of the technology intervention. 

Students often wait days or weeks after handing in classroom work before receiving feedback. In contrast, research suggests that learning proceeds most rapidly when learners have frequent opportunities to apply the ideas they are learning and when feedback on the success or failure of an idea comes almost immediately (Anderson, 1996).

The sheer autonomy that online tools like Khan Academy provide for students, is radical in and of itself, a radical departure from the traditional textbook centred, non adaptive model that anyone older than 40 would have experienced at school. As Jo Boaler explains, these tools allow students to 'become self-regulatory learners, so that they are not dependent upon following somebody else's plans':

Assessment for Learning
A4L is based upon the principle that students should have a full and clear sense of what they are learning, of where they are in the path towards mastery, and what they have to do to become successful. Students are given the knowledge and tools to become self-regulatory learners, so that they are not dependent upon following somebody else's plans, with little awareness of where they are going, or what they might be doing wrong. (My emphasis)

Boaler J (2009)

References

Aibhin Bray, Brendan Tangney Centre for Research in IT in Education (CRITE), School of Education and School of Computer Science & Statistics, Trinity College Dublin, Ireland

Anderson JR, 1996. The architecture of cognition. Mahwah, NJ: Lawrence Earlbaum Associates, 1996.

Boaler J (2009). The elephant in the classroom: Helping children learn and love maths. Souvenir Press)

Bray, A., & Tangney, B. (2013, May). Mathematics, Technology Interventions and Pedagogy-Seeing the Wood from the Trees. In CSEDU (pp. 57-63).

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann, 1997.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000.

Calculators in the 21st Century

What place do calculators have in the 21st Century?

Some would argue, none, and that they never should have had a place even in the 20th Century. Certainly one of my earliest memories of TEL (Technology Enhanced Learning) was the replacement of the humble 'log book' by calculators in my Maths classes on the west coast of Ireland, back in 1986 or so.

This post is not about to debate an argument that is at least 20 years old, but I am happy to point you to this post which discusses the nuances of matter very well indeed.

I would like to highlight some comments made about calculator use by the renowned John Hattie, in his magnum opus, Visible Learning, where he says:

Calculators + Good Pedagogy = d = 0.72

"Ellington found that the effects were much higher when calculators were involved in the teaching process; for example, when used for composition problem solving, the effects were d = 0.72: "When compared with students who did not use calculators, students in treatment groups were able to solve more problems and make better decisions with regard to selecting methods for generating solutions"(Beddington, 2003, p 169) 

For more on Hattie's meta - analyses, have a look at my other post here, but for now, take it from me that a rating of d = 0.72 is very good, no, not very good, it's astounding.

Now, I've been thinking for some time, what if you took the pedagogically sound use of calculators and applied them to a technological tool that is favoured by adults who work with numbers the whole world over? What do financial professionals use to manipulate numbers? Calculators?—you must be joking, no they use spreadsheets, why? Because,

Spreadsheets are the calculators of the 21st Century

Maths Problem Solving with Numbers (click to enlarge)

I can tell you that my work with students confirms this, in fact, if I take the liberty of tweaking Hattie's profound eulogising about the wonders of spreadsheets calculators I think the observations hold true, no, they are truer, more emphatic in my experience than ever:

"Hembree and Dessart (1986) found that the pedagogical use of calculators spreadsheets improved students basic skills both in completing exercises and problem solving. Across all grades (and particularly above grade 5, when calculators spreadsheets become more prevalent) and across all ability levels, students using calculators spreadsheets lead to greater effects in students' basic skills in operations and particularly in problem-solving."  

Visualize Data with a couple of clicks (click to enlarge)

"The effect on problem-solving seems to relate to improved computation and lower cognitive workload demands. They also found that there was a better attitude towards mathematics and an especially higher self-concept of mathematics for those using calculators spreadsheets compared to those not using calculators spreadsheets.

Transform number manipulation by moving it online, collaborative, social. 

"... this enhancement in attitude was probably because the use of calculators spreadsheets helped relieve students traditional dislike of problems expressed in words (by reducing the cognitive load of having to compute as well as problem solve)."
...
"Using manipulative materials and calculators spreadsheets helps to reduce students cognitive load and allows them to devote their attention to problem solving."

Adapted from Hattie J (2013). Visible learning, p146-147

Transforming Calculations & Mathematics

As powerful as spreadsheet applications like Excel and Numbers are, already leveraging SAMMS elements like access to the internet to gather data, clarify facts, strategies; move modes of operation from teacher centred/pushed data, to student centred/gathered data, and the the mutability afforded by the spreadsheet—what calculator has an undo key? What calculator allows you to grab aspects of data and literally tweak it, or even move it around the sheet, recalculating variables at the speed of thought?

But we can add even more SAMMS to the recipe. Moving the entire activity online, namely with a 'cloud' based spreadsheet application like Google Sheets, that allows us to add the transformative elements of ICT that are:

Situated:

Now students and teachers can work anywhere, any place, any space, on the same sheet. From real time feedback using comments, to transforming 'homework' into just an extension of classwork,

Social:

The sheet becomes a mini social network, a micro-community; allowing a group of students to collaborate on the same spreadsheet in real time, or to assign different aspect of a more complex problem to various member of the team, to be interconnected as the separate aspects of data are clarified, calculated and ratified by the team. At any point, any student can duplicate the entire sheet and continue work work separately, or move it to another tab within the same sheet.


All of the work you see here was completed with Grade 3 students, recently I have been able to continue this work successfully with students in Grade 2, using Numbers on the iPads. Here is one of our students to explain it for you:



If the only thing holding you back is lack of understanding of how spreadsheets work, don't panic, I have all the guidance you need right here:

Step by step, but to do the activity you see here really only requires a formula as simple as:

Answer (empty cell) = (click first number) + or any other operator, eg -,/,*) (click second number) Enter/return. Done.

Answer = 1+1. That's it.

See the Picasa Web Album below for some more examples:


Hattie J (2013). Visible learning: A synthesis of over 800 meta-analyses relating to achievement. Routledge. P 146-147

Visualising Data with Gapminder World

Data is an important quantitative tool in many subjects including Geography and Economics and it helps our students understand trends, patterns and the contrasts that exist in the world. Student's ability to visualise and therefore interpret data is becoming an important skill at the intersection of the arts, maths and humanities disciplines.

This blog post, looks at ways to help students interpret data through effective use of the Gapminder Tool. A subsequent post in this series "Creating Infographic Visuals with Easel.ly" explains how to create infographic visuals with Easel.ly which help them showcase their understanding.

Techniques for using Gapminder in the classroom

Gapminder World is the utopia of online statistical tools, that enables students to construct and interpret scatter diagrams with a range of human development statistics. The amazing aspect is that they can drag the data back in time, showing a 4th dimension to the pattern. Our students also have GapMinder Desktop installed on their MacBooks, enabling offline access.

Top Tips for using GapMinder

• Remember to change the indicator on each axis
• See this link for the full list of indicators, or to download the raw data.
• If you like a graph, save the URL into a Google Doc for future reference
• Click on two or three different countries to simplify the view.
• Drag the timeline back, and see the trails of data. 
• Ask students questions about the direction of change, what is the correlation, cause and effect.
• Hover the cursor over each circle to see the data and to see the axis statistical values.
• Need a lesson on correlation and causality? See the brilliant Khan Academy.

Going deeper with GapMinder - Skitch

• Try using programs such as Skitch with students to help them annotate the GapMinder diagram to highlight correlations and the main points. This technique forces students to think more deeply about the data, trends and patterns.
• Students can save these into their notes or assignment. 

Screen Recording and GapMinder - Quicktime

• To really push students, you need to make them think more deeply about the connections an trends in the data. Ideally you want to glimpse into their thinking processes (metacognition).
• You can do this by asking them to use the Screen Recording functions within Quicktime, and to answer a question such as "describe two countries, one that follows the correlation and another that does not and the relationship between the data"
• The student will record a short clip of them speaking, whilst also clicking and dragging the GapMinder gadget. Whilst you might not access this video explicitly the level of thinking required really pushes students who are comfortable writing something but find it harder to verbalise thier thinking. 

A little lovin' for Equations in Google Docs

I've been toying with equation editing in MS Word (it's been an uneventful but productive term break) and noticed a nice little nugget tucked away within Google Docs.

Not that this is anything new, but if you're like me, you're not having a fun time playing the mathematical equation version of Where's Wally.

I'm a text based thinker. So for those of you out there who, like me, type faster than we spot symbols, I present to you:

Automatic Equation Shortcuts in Google Docs

Sample:

	\times
	\div
	\pm
	\mp
	\dagger
	\ddagger

On a personal note, if someone explained where daggers come in to equations outside of medieval politics I would be deeply appreciative.

Diagnostic Assessment using Promethean Activotes

Early this term we did a small trial lesson with the Activotes. These are little voting devices that students can use to participate in assessments run though the teachers laptops and the Promethean Inspire software installed on the staff macbooks. Throughout the school there are several sets of these voting devices which you are able to borrow and use.

Alongside Christine Chaboyer from the Maths Department we used the devices as a start of topic diagnostic assessment tool with Grade 8 students. As Christine was a new teacher, who was taking over the this class she needed a fun tool to assess the students prior knowledge. She set up a simple flip-chart with 20 multi choice questions. These can either be typed on or copied and pasted into the flip-chart from existing resources. Her great teaching strategy was to create these questions with some common mistakes to act as trap multiple choice questions. These assessment can therefore become concrete evidence where the cognitive gaps in the students learning exists.

How do the Activotes work?

This is a nice tutorial developed by Gordon Hirons from the Maths Department - click here to download

1. Create a flipchart, and chose to insert questions. One per chart.
2. Once this is complete you need to choose the Tools menu and then Express Poll
3. A floating icon will pop up, select the icon, then Assign Devices to Students
4. You need to create a database of student names
5. Once this is complete, you need to use the floating icon again to register devices
6. Easiest way to register the devices is to give them out to the students and then choose the register with pin option. Students look at the projector and add the devices.
7. You need to check within preferences that the devices are set to Activotes and not Active Expressions. This is an option under the Inspire menu and then Preferences. See below
8. Once the devices are all registered you can click at top right to begin the assessment.

Some of the downsides are the on a screen everyone can see the students, who have yet to answer the question, therefore introducing an unintended element of peer pressure. (See 2nd picture below)

The Inspire program gave provided excellent feedback to both the students and the teacher. The class sees a simple bar chart after each question is complete and the teacher has private access to all of a particular students answers. Therefore the teacher can identify comprehension issues at a whole class level or for individual students.