# ‘The Points System’: One approach to differentiation

When I was in school I remember that I found nothing more boring than the instruction to ‘do questions number 1 to 5, parts a) to h)’. I would get my head around the concept by question b or c of each section, then have to spend what seemed like an eternity working through a whole bunch of exactly the same question with different numbers. With this knowledge of my own experience in mind, I’ve always known that I wanted to try to avoid this same boredom for my own students.

This approach to differentiation and boring repetition reduction was brought to my attention by one of my Masters of Teaching lecturers Nicky Dulfer, who said that one of her friends had implemented an ‘earn points’ system in their own mathematics class. I thought it sounded like a nice idea. Here’s how I approached it…

The slide I showed to students:

For those unfamiliar with the proficiency strands of fluency, understanding, reasoning and problem solving (‘open-ended’ in this case), here’s an example of a few example Q’s for context (from the Pearson Mathematics 9 Textbook).

Did students like it?

This was implemented with a year 9 mathematics class.

Mid placement feedback: In response to the question ‘What should Ollie keep on doing?’ five students nominated the points system. In response to the question ‘What should Ollie stop doing or modify?’ no students nominated the point system. (Highest number of votes for any one category was 8, which was to suggest that I continue making videos of content).

End of placement feedback: In response to the question ‘What did Ollie do that most helped you learn?’, four  students nominated the points system. In response to the question ‘What are some things Ollie shouldn’t do in future, or things to modify’, one student said ‘Point system (15 in 30 minutes was too rushed and stressful)’. (Highest number of votes for any one category was 12, which was to suggest that micro-revisions (post on micro-revisions to come…) were what most helped students to learn).

So, on the whole, the points system was well received by students.

Did it help the students to learn?

source: http://store.discovery.com/img/product/catl/00275198-962183.jpg

Unfortunately I didn’t conduct this points system long enough to be able to tell if it increased student learning. But I can say that it appeared to increase engagement, and as ‘a major precursor to learning is engagement’ (Hattie, 2012, Chapter 8, Section 2, para. 1), it’s plausible that it increased learning.

I do however think that it’s fair to say that this task helped the higher achieving students to learn more. After they had gained their 15 points (some of them would finish this in under 10 minutes), students were able to move onto ‘challenge questions’, like the pythagoras challenge questions that I’ve written about previously. The traditional approach for these students was to get them to do a set of questions from each of the proficiency strands (fluency, understanding… etc), which would take up the whole lesson, and was pointless in many cases as these high achieving students could easily complete the task and weren’t being challenged at all.

The idea of students self-differentiating was also intended to help promote metacognition. Fostering metacognition is a key step to helping students to ‘become their own teachers, which is the core attribute of lifelong learning or self-regulation, and of the love of learning’ (Hattie, 2012, loc 168). But actually scaffolding this metacognition is something that I need to do better in this task and in the classroom more generally. I feel that it’s unreasonable to assume, as I implicitly did, that a year 9 student will be able to select a question at the appropriate level without any help.

Conclusion:

I feel that this task was a step in the right direction and, coupled with the power of spacing repetition of content for students (Carpenter, Cepeda, Rohrer et al., 2012), has great potential to be expanded and improved upon in future. One immediate improvement to make would be to have ‘challenge questions’ (for students who work through the 15 points) of different levels of difficulty, rather than just the one ‘challenge question’ that wasn’t so easily accessible to some students.

Note: Sunshine College has taken a similar approach where students select for themselves a ‘just right’ task and form small groups to solve it. This is an approach that I’ll be exploring more in the near future. You can read a brief paper about it here.

References:

Hattie, J. (2012). Visible learning for teachers. [Kindle version]. Retrieved from Amazon.com.au

Carpenter, S. K., Cepeda , N. J., Rohrer, D., Kang, S. H., & Pashler, H. (2012). Using Spacing to Enhance Diverse Forms of Learning: Review of Recent Research and Implications for Instruction. Education Psychology Review , (24), 369-378.

# Metacognition: Can it help students problem solve?

I was recently doing a little reading into metacognition and began to wonder if it could be used as a tool in the Maths classroom to help students, particularly with problems solving. I got hunting in the research and found the following paper.

Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A Multidimensional Method for Teaching Mathematics in Heterogeneous Classrooms. American Educational Research Journal, (2). 365.

I got a lot out of it, and thought some others might like to hear how the IMPROVE method works.

This study was done in the late 90’s in Israel. It was to test a modified teaching model based on the incorporation of three elements that aren’t always seen in the classroom

• Metacognitive Training
• cooperative learning
• systematic provision of feedback-corrective-enrichment.

I’ll expand on each of these a little below, then talk about the results of using them in tandem (as was done by Mevarech and Kramarski).

## Metacognitive Training

I like to think bout metacognition as stepping back from a situation and asking ‘how is my brain reacting to the stimulus here? And how would I like it to react?’. The IMPROVE method got students to begin to do this by introducing three new questions to the mathematics classroom. These three questions were made into cards and passed around when problems got challenging. The questions were

Comprehension Questions: What’s the problem actually saying. Students were asked to ‘read the problem aloud, describe concepts in their own words, and discuss what the concepts meant or into which category the problem could be classified’ (p. 374)

Connection Questions: “How does this question relate to things that you’ve seen before?”

Strategic Questions: All about how you’re going to attack a problem. Ask “What strategy/tactic/principle can be used in order to solve this problem?”, “Why” and “How will you carry this out?” (p. 376)

The idea of these questions was to help students to differentiate between equivalent problems (Qs with the same structure and ‘story context’), similar problems (different structure but same ‘story complex), isomorphic problems (same structure, different ‘story context’), and unrelated problems (very little in common). These categories are from Weaver and Kintsch (1992) and the terminology wasn’t taught to students.

## Cooperative Learning

The method used followed Brown and Palincsar’s method (1989) in which students were put into teams of 4 students of 1 high, 2 middle and 1 low achieving student (p. 377). As students progressed teams were changed to maintain this structure. It was stated at this point in the paper that the question-answering technique based on that of Marx and Walsh (1988) was used following a brief teacher exposition of approximately 5 minutes. I plan to further explore this mentioned questioning technique.

## Feedback-Corrective-Enrichment

At the end of each 10 or so lessons (constituting a unit) students took a formative test to check their comprehension of the unit’s main ideas. Tests were based on the work of Bloom (1976). Students who didn’t achieve ‘Mastery’ (taken as 80% correct) were given extra support to solidify the basics, students who did went on to enrichment activities. Essentially a form of differentiation.

# The Studies

The paper detailed 2 studies that were undertaken. The first included 247 year 7 students split into an experimental (n=99) and control (n=148)  group and the second study consisted of 265 students (experimental n=164, control n=101)). The first study was completed in a region of heterogeneous classrooms (ie: students weren’t split into classes based on ability, ‘tracked’. These classes spanned more than 5 ability years) whilst the second was undertaken in a district where ‘tracking’ was the norm. The second study was undertaken in order to see if the IMPROVE method applied for an entire year would yield encouraging results as it did over the shorter period as in Study 1, as well as to expand the topics to which the method was applied.

Study 1 applied IMPROVE to the topics of rational numbers, identification of rational numbers on the number axis, operations with rational numbers, order of operations, and the basic laws of mathematics operations.

Study 2 applied IMPROVE to the topics of numerals and rational numbers, variables and algebraic expressions, substitutions in algebraic expressions, linear equations with one variable, converting words into symbols, and using equations to solve problems of different kinds.

Tests were composed of computational questions (25 items) and reasoning questions with no computational requirements (11 items).  The reasoning questions were marked based on the following progressive marking scheme: 1 point: Justification by use of 1 specific example, 2 points: reference to a mathematical law but imprecise, 3 points: references mathematical law correctly but conflict incompletely resolved, 4 points: question completely resolved with correct reference to relevant mathematical laws.

Based on pre-intervention test scores students were classified as low, middle or high achieving with pre and post test results compared within these groups.

# Results

p. 381

Study 1: No difference existed between control and experimental group prior to the intervention but IMPROVE students significantly outperformed those in the control post-intervention. Overall mean scores were 68.03 (control) vs. 74.72 (treatment) post-intervention (p<0.05) with means scores on the reasoning component 53.15 (control) vs. 62.56 (treatment). Improvements were seen at all achievement levels.

p. 384

Study 2: As with study 1 mean scores for the experimental group increase significantly more than those in the control group with 2 important points to note. Firstly, only the gains to the high achievers group were statistically significant, with the medium achievers group being milldly significant (p-0.052). Low achieving treatment scores>Low achieving control scores in all cases but this result wasn’t statistically significant. Secondly, these trends held for all topics except for ‘operations with algebraic expressions’. It was suggested this was due to the fact that this unit required more practice than other units, thus, being a more of a procedural topic the benefits of metacognitive approaches weren’t as impactful.

# Discussion

It’s clear that the IMPROVE intervention aided in student achievement. It increased their ability to draw on prior knowledge to solve problems and to establish mathematical mental schemata to increase their ease of access to this prior knowledge. One challenges with this study (as outlined by Mevarech and Kramarski themselves) was that the three elements; metacognitive training, co-operative learning, and feedback-corrective-enrichment were all applied simultaneously making it impossible to distinguish which of these was contributing by how much to the observed effects. Another question surrounds how this method appeared to facilitate gains to students proportional to their ability starting point, with higher achievers improving relatively more than middle and low achievers.

The authors suggest the program was successful in the following ways. it:

• made it necessary for participants to use formal mathematical language accurately
• made students aware that problems can be solved in multiple ways
• encouraged students to see different aspects of problems.
• gave high achievers opportunities to articulate and further develop their thinking processes at the same time as letting lower achievers see these thinking processes modelled

Post-intervention the IMPROVE method was implemented in all classes of all schools that the trials were performed in.

Notes: I initially found this article via the article:  Schneider, W., & Artelt, C. (2010). Metacognition and mathematics education. Zdm, 42(2), 149-161. doi:10.1007/s11858-010-0240-2 . Schneider and Artelt’s article also outlined various other metacognitive training strategies that have been trialled in different classrooms. I chose to focus on Mevarech and Kramarski’s IMPROVE model here as it was a rigorous study and was cross referenced in several other papers also.

IMPROVE is an acronym for: Introducing new concepts, Metacognitive questioning, Practicing, Reviewing and reducing difficulties, Obtaining mastery, Verification, and Enrichment.

References:

Brown, A., & Palincsar, A. (1989). Guided cooperative learning: An individual knowledge acquisition. In L. Resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert Glaser (pp. 393-451). Hillsdale, NJ: Erlbaum.

Marx, R. W., & Walsh, J. (1988). Learning from academic tasks. The ElementarySchool Journal, 88, 207-219.

Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A Multidimensional Method for Teaching Mathematics in Heterogeneous Classrooms. American Educational Research Journal, (2). 365.

Weaver, C. A., III,& Kintsch,W. (1992). Enhancing students’ comprehension of the conceptual structure of algebra word problems. Journal of Educational Psychology,84, 419-428.

# Key take-homes from Matt Skoss at MAT2014 conference

This post is part of an ongoing “Wot-I-Got” series. This series acts as a way for me to share Wot-I-Got out of a book or presentation, and to whet your appetite for enquiry. It also forces me to finish books that I start, and to review and summarise my conference notes!

These are my notes (and annotations)  from Day 1 of the Mathematics Association of Tasmania conference, 2014. MAT Website.

• mathspathway. Software for differentiation. Richard Wilson. richard@mathspathway.com, 0403787349, www.mathspathway.com . These guys have developed differentiation software that assesses a child’s ability, finds their knowledge gaps, then designs a personalised learning plan and delivers the associated content. Worth having a chat to!

Day 1, Friday, 4pm. Matt Skoss: Becoming a ‘Top Draw Teacher’… What might that look like? See Matt’s Wiki , Maths-no-fear, here.

• #lessonstarter as a twitter hashtag-Matt created this as a hashtag as a place for people to share inspiring lesson/conversation starting pictures.

• Vi Hart on youtube, 3 to 8 minute doodles on maths ideas.

• “It’s ok to re-visit a similar lesson several times.”-if the content is interesting/complicated enough, follow up lessons can lead to deeper learning.

• http://danah.org, danah boyd, book: I’ts complicated, the social lives of networked teens.

• ipad triple click function, locks students onto the current app that they’re working on.-find Matt and ask him how to do this.

• Sorting Mat: Demonstrates how a simple operation can lead to complicated and ordered results-image to come

• discovr people appUse it for social networking. Find good connections through people that you trust.

• Tech: todaysmeet.com. A great  backchannel that can substitute for twitter.-The idea with this is that it’s a feed that you can use to promote discussion in your class. Set up a virtual classroom on this website and have your students make comments on a current task. Other benefits:

• More simple than twitter with less distractions

• Can be a great way to get students to respond to a question, or to brain dump everything they already know about a concept

• Keep in mind, your backchannel dissappears when the time runs out, so ensure that you back it up if you want to keep the content (copy paste all to a word doc for example)

• Don’t just give up if it doesn’t work the first time. When Matt first used it some kids put up some really inappropriate stuff. So push through it.

• “I want to try something out with you so I can try it on younger kids and I need your feedback”-a great feedback prompter!

• Great task: “Think of a fraction between ½ and ¾ . Try to think of one that no-one else can”–engages students in a more active rather than passive way

• Evernote: A great tool-Matt uses it to keep students work ordered and can quickly draw on images and comment associated with different tags (a particular student, a particular class, a particular learning module) in situations such as teacher meetings.

• A great question Genre: “Always true? Sometimes true?  or Never true?-Put up a sentence like “Squaring a number always makes it larger” or “Dividing one number by another makes it smaller” and get students to discuss whether it’s always, sometimes or never true and to come up with examples and counter examples.

• … What are some other good questions that suit this genre???

• Two questions that Dan Meyer leads with: “What would be an answer that’s way too big?, What’s an answer that would be way too small?”

• Dan Meyer’s questions: at http://101qs.com. Coke bottle pic, what questions can it launch? (See today’s meet)

• Find inspiration everywhere. Take a pic (eg: hexagonal plughole) and ask your students “what’s the mathematics behind this?

• Eventually you hope to get to the point where you only have to ask “what’s my next question going to be?”

• John Mason: “Mathematics hasn’t been done in a Mathematics lesson unless it has involved generalising.”-Encourage students to make predictions based on patterns that they’ve noticed from which they can make generalisations.

• Key take home message from Matt: …Get a key take home!!! – ie: Choose 1 thing from the conference that you’ve liked and put it into action in your classroom in the next 12 days. If you don’t enact it in the next 12 days you never will.
• #lessonstarter as a twitter hashtag. Matt created this as a hashtag as a place for people to share inspiring lesson/conversation starting pictures.

• Vi Hart on youtube, 3 to 8 minute doodles on maths ideas.

• “It’s ok to re-visit a similar lesson several times.”-if the content is interesting/complicated enough, follow up lessons can lead to deeper learning.

• http://danah.org, danah boyd, book: I’ts complicated, the social lives of networked teens.

• ipad triple click function, locks students onto the current app that they’re working on.-find Matt and ask him how to do this.

• Sorting Mat: Demonstrates how a simple operation can lead to complicated and ordered results-image to come

• discovr people app-Use it for social networking. Find good connections through people that you trust.

• Tech: todaysmeet.com. A great  backchannel that can substitute for twitter.-The idea with this is that it’s a feed that you can use to promote discussion in your class. Set up a virtual classroom on this website and have your students make comments on a current task. Other benefits:

• More simple than twitter with less distractions

• Can be a great way to get students to respond to a question, or to brain dump everything they already know about a concept

• Keep in mind, your backchannel dissappears when the time runs out, so ensure that you back it up if you want to keep the content (copy paste all to a word doc for example)

• Don’t just give up if it doesn’t work the first time. When Matt first used it some kids put up some really inappropriate stuff. So push through it.

• “I want to try something out with you so I can try it on younger kids and I need your feedback”-a great feedback prompter!

• Great task: “Think of a fraction between ½ and ¾ . Try to think of one that no-one else can”–engages students in a more active rather than passive way

• Evernote: A great tool-Matt uses it to keep students work ordered and can quickly draw on images and comment associated with different tags (a particular student, a particular class, a particular learning module) in situations such as teacher meetings.

• A great question Genre: “Always true? Sometimes true?  or Never true?-Put up a sentence like “Squaring a number always makes it larger” or “Dividing one number by another makes it smaller” and get students to discuss whether it’s always, sometimes or never true and to come up with examples and counter examples.

• … What are some other good questions that suit this genre???

• Two questions that Dan Meyer leads with: “What would be an answer that’s way too big?, What’s an answer that would be way too small?”

• Dan Meyer’s questions: at http://101qs.com. Coke bottle pic, what questions can it launch? (See today’s meet)

• Find inspiration everywhere. Take a pic (eg: hexagonal plughole) and ask your students “what’s the mathematics behind this?

• Eventually you hope to get to the point where you only have to ask “what’s my next question going to be?”

• John Mason: “Mathematics hasn’t been done in a Mathematics lesson unless it has involved generalising.”-Encourage students to make predictions based on patterns that they’ve noticed from which they can make generalisations.

Key take home message from Matt: …Get a key take home!!! – ie: Choose 1 thing from the conference that you’ve liked and put it into action in your classroom in the next 12 days. If you don’t enact it in the next 12 days you never will.-I’m going to use the ‘Always, Sometimes, Never true” question genre in class on Wednesday!

Dinner conversation…

• Computer Science Unplugged conference, July 2nd 2014. Campbelltown. Introducing digital technologies to the curriculum. More info here.
• Creative Mathematical Sciences Conference, Chennai, Dec 9-14 2014.
• Site sucker: An excellent app to suck a site onto your computer for use offline (or after a subscription expires!)
• Scratch: Drag-and-drop programming to get students (and teachers) familiar with programming processes
• Hour of Code: A 1 hour introduction to coding. Over 36 million people have tried this!