Math Humor.

Ability Grouping, Equity and Survival of the Quickest

The use of ability grouping is certainly a contentious issue in education.  Should students be taught in groups where they are with students of similar ability( homogeneous groupings), being taught at a rate that is designed for their needs.  Or, should students be taught in mixed-ability (heterogeneous groupings) where some will find the pace and material challenging but will allow them more opportunities for post-secondary education? In Newfoundland and Labrador all students are taught in heterogeneous groups until they reach junior high, where, for some subjects such as mathematics, they are tracked into more homogeneous groupings based on ability.  This topic has come up several times in our discussions as a group, as an elementary teacher it never occurred to me to question what 'tracking' really means to a child's future. Why would education systems track students according to ability?  What exactly is at stake when we use ability sets to teach students, how are their needs best met, how can they be motivated to learn, what are the implications for their future, what is the overall effect of being tracked?  The importance of the answers to these questions cannot be overstated, Boaler's findings in this chapter go a long way to answering these questions for me.

One of the main issues involved is what are the advantages/disadvantages of being placed in 'like' ability groups for mathematics or in other words, why do we use this type of grouping at all?  According to Boaler the main reason is "so that they [the teachers] can reduce the spread of ability within the class, enabling them to teach mathematical methods an procedures to the group as a unit. They "cannot see how "bright" students may may be stretched or "weak" students supported" (p. 155) when a mixed-ability grouping system is used.  Tracking students by ability allows the teacher to give the form and speed of instruction that is most useful for those students appears to be the basis of the argument for this type of education system.   Students expressed two main complaints or concerns with this type of instruction they received, one was that 'fixed pace diminished understanding' and the other was "the pressure they felt was created by the existence and form of their setted environments".   The need to work at a paced fixed by the teacher was incredibly frustrating for students, particularly in the higher sets where the pace was the fastest, " I don't mind maths but when he goes ahead and you're left behind, that's when I start dreading going to maths lessons (Helen, AH, Year 10, Set 1)".    When in homogeneous groupings students also felt they were constantly being compared to and judged by peers.   While the basis of using homogeneous groupings is that it would motivate students and raise achievement this was found by Boaler to be true for a very small number of students.
 

The use of homogeneous grouping has some inherent components which are detrimental to the learning of students in the lower sets. What is at stake when we track students according to ability is perhaps the most disturbing finding of all in this chapter.  Boaler refers to this as " Restricted Opportunities" students were angered by 'what they felt to be unfair restrictions on their potential mathematical achievement".  Students were if anything dis-motivated to learn by tracking restrictions "they could not see the point in working in mathematics for the grades available to them" (p. 166).  Students often felt that the set they were put into was not fair that the decision was based on items other than ability such as personality clashes between them and teacher  or on their behaviour.  Boaler uses information from Tomlinson (1987) which provided evidence that student behaviour does in fact play a role in the decision about which set they were assigned to.  To me,subjectivity of the teachers feelings towards a student should not be allowed to play a part in the decision behind where that student ends up!  In her summary Boaler states that 'setting' did not affect all students the same, some were indeed given an advanatage but others were negatively impacted.  The consequences of being setted, especially those in lower sets was important to students and was discussed in greater detail later in the chapter.

When Boaler looked at the mixed -ability groupings at Phoenix Park and compared them to Amber Hill she found that AH students had something to react against that the PP students did not because grouping was not an issue at the latter school.  The teachers at PP chose activities that gave access to students "at multiple levels" of ability, this is of course related to the newest train driving education in Newfoundland and Labrador ( and elsewhere) otherwise known as Differentiated Instruction(DI).  I have to admit I don't have much experience with DI , I attended two workshops on this topic both of which were way above my level of ability as they were designed for teachers who had been working with this system and had experience using it components. These experiences left me feeling frustrated and overwhelmed and so I just used what I could glean from them in my own fashion.  After reading this chapter, I am once again reminded how useful DI can be and will probably delve into a little more being careful to begin at a place where my needs are met.  Obviously students at AH who, in a set , are all being taught the same material at the same pace are not having their needs met. When Boaler compared the achievement levels between AH and PP students her research found that mixed ability grouping increased achievement opportunities for all students, particularly girls and low socioeconomic status students, which lends itself to the issue of equity in mathematics.


 I was profoundly disturbed by Boalers  revelation that "students are rarely told the implications of the group withing which they are placed."  This is shocking to me, if the knowledge of the group they are placed in must be kept secret because it is detrimental to motivation why wasn't that a flag that perhaps a closer look at this system was in order.  The short term and long term effects of tracking are enormous.  In the short term students go through school being compared or comparing themselves to others in an unfavorable light.  Students who work at a lower level and don't know it and who are achieving well are shocked ( and understandably so) that they haven't achieved what they thought. In the long term, where you are placed at 13 determines for many the outcome of their professional or adult lives!   Imagine thinking you were doing really well in math only to find out that you weren't really learning what was needed to give you the greatest options for post-secondary education, yikes!  I can't believe this has been allowed.  This was yet another eye-opening chapter and one I am glad to have met.

So... did I miss anything?

Just kidding!

Girls, Boys and Learning Styles
Let me start by saying I can 't believe how out of touch I was with this issue (gender differences/inequity), I hadn't heard much about it of late and thought we'd moved past this, (if I thought I about it at all that is.) boy was I wrong.   This was such an interesting chapter, I couldn't help but think about the whole "Men are from Mars, and Women are from Venus" thing. The differences in gender are striking in many ways! However, it is so important to come to understand why this is so in mathematics, understanding it can help us change it. I can't personally see how males are able to put aside understanding for the sake of what amounts to competitiveness" relative performance", but then they are from Mars after all!

I was glad to read that "In England, girls now attain the same proportion of the top grades in the GCSE ... as boys,and stereotyped attitudes about the irrelevance of mathematics for girls are largely disappearing." (p.137-138). The news about the top 5% of highest attaining students being largely male wasn't as encouraging, and was frankly sort of disappointing. Women valuing "connected knowing characterized by intuition,creativity and experience" is meaningful, I think. Still females do not rise to the highest levels of achievement in the same numbers as males,one has to ask what is about mathematics that causes this? Becker (1995) makes the suggestion that "girls have traditionally been denied access to success in mathematics because they tend to be connected knowers and traditional models of mathematics teaching have encouraged separate ways of working." this resonates with me, I believe Becker may be on to something here. Girls certainly seem to be at a disadvantage.

Take Amber Hill girls for example, they having been taught with the traditional approach to mathematics instruction and Boaler found that 11 out of 15 of the underachievers were females. With boys being more comfortable, playing the game, willing "..to abandon their desire for understanding and race through questions at a high speed", those (boys) from Amber Hill are at less of a disadvantage then the girls due to the differences identified. In a school where students of mathematics are operating at a disadvantage due to instructional style and speed of instruction this added layer is patently unfair to the girls.

It is disconcerting to read that the Amber Hill girls achieved less than cohorts from Phoenix Park. Think about your daughter going to this school once having been a high achiever in math and watch as her grades slip. Boaler indicates that this was the case at Amber Hill, in the group of 11 of 15 underachievers.  She specifically points out that two of the girls "Carly and Lorna" attained the highest and second highest grades on the NFER respectively; then they both came out at the lowest in GCSE!  To add insult to injury the 'attribution theory" findings and suggestions tend to "blame the victim".  Thankfully Boaler makes the point that the girls themselves attribute their difficulties in mathematics with outside forces and not with themselves, belying attribution theory claims. Still I believe Boaler is correct when she says that "it is important not to lay the blame for their [girls] disaffection on mathematics per se because the fault lies not with the intrinsic nature of mathematics, but with school mathematics as it is commonly constructed" Gender equity in mathematics is certainly a critical issue and one that I will be paying closer attention to in the future!

Connections

I have been thinking about the Amber Hill students a lot lately, in relation to my own teaching. I believed I had made significant changes to my instructional methodology but, since starting this course, a sneaky little voice inside my head ( that won't go away now no matter how much I try to ignore it) has been telling me differently. The connection between, the lack of true understanding exhibited by the students of Amber Hill, the style of teaching by AH teachers and my own practices has been in the fore front of my thoughts. The fact that I have introduced journals and portfolios, that I use a 'problem solving' approach for some lessons and to introduce concepts do not mean I am giving my students any greater understanding then the students of AH received. This has to stop! The realization came when I read some of what was contained in the"teachingmathematics4understanding' blog. This teacher has really got it going on. She has made a sweeping change and is in my opinion doing it the right way.

I don't know if my standard of "make changes at the rate I'm most comfortable with" stance is going to cut it for much longer. That said, the idea of making wholesale change in instructional style in the middle of a year, while doing a graduate course is daunting to me. Can I wait until this summer to plan in advance for how I want to begin? The answer is, I really shouldn't, but (and its a big one) can I do justice to the kind of changes needed at this point in time? I will continue to look for ways to make the mathematics in my classroom more inquiry based. I have already been doing that. Without Jo Boaler's work, this course and our discussions I would not have been able to see that I need to do more, and I will!

I have to add that the experiences in the classroom when the students are involved in discovery and exploration of mathematics are extremely fulfilling. I have never had such enthusiasm displayed for geometry before! I do know that I need to have greater knowledge of not only the big ideas behind the mathematics myself but greater ability to analyze student thinking. My inquiry project on professional development has become an exploration of what is out there and what I need. I am now feeling as though I must take over control of my own PD and find ways to support my own learning instead of passively waiting for whatever the next session offered by district or department is. The quality of the PD sessions delivered by the district are not in question, it is just that I have needs that are not being met and I am responsible for myself. Learn Sharon Learn!

Interesting considering our upcoming discussion on chapter 9.

As I was searching for something else I came across this piece. I'm not even sure if it is a reliable statistic but it is interesting to me that someone could make the claim that gender issue in mathematics is gone. Below find link to the article that this visual came from.

http://www.epi.org/economic_snapshots/entry/webfeatures_snapshots_20080820/

Knowledge, Beliefs, and Mathematical Identities

I can't say I was surprised by the findings outlined in this chapter, it was already apparent where the true learning was taking place. I did enjoy, as usual, our discussion of the chapter, it was a well organized presentation for sure, the flow chart was exactly the right touch! It was what I had attempted to do in my own presentation but I couldn't get to work out the way I wanted, great job!
The effect that teachers have on students mathematical beliefs and knowledge is in some ways a little scary. In their attempt to simplify content the AH teachers caused unintentional harm to their students abilities, they could not solve problems without cues and could not apply knowledge to novel situations which required them to make connections between concepts. This is what we don't 'see' as we teach, since the development of mathematical beliefs and identities are ongoing and cumulative. What we do take as student learning, when we say, "yes they've got this" ( like Michelle expressed with her story about integers and bedmas) is generally not true at all. The more I read and hear in this course the more I realize that what we see in the classroom is that some (but not all) students can follow what I've taught and replicate it at that moment in time, but that this is not knowledge and it is definitely not understanding. We have all expressed experience with the "what is the matter with them [students] they knew this yesterday (last week, two weeks ago). Yet, like the AH teachers we didn't stop to really ask why or to connect lack of success with our teaching, "I understand it, my steps were clear... so they should understand it".

We need to strive for more than what we have accomplished in the past with our instruction. I know that having had the discussions this course work and this group has allowed I now find it impossible to ignore what I know has to be changed. I say ignore because at some level isn't that what we've all done when we notice student frustration or lack of recall, ignore the real reason behind it? What we want is to produce students who are "flexible with mathematics knowledge and are able to adapt and change", we want them to take a stance of inquiry that makes them see that exploring mathematics and connecting concepts is a natural process.
Now how do we do that exactly?

Chapter 7 - Exploring the Differences

Krista, congratulations on a job well done! Great discussion provoking questions, they made me even more sure that I want to explore how Professional Development could be used to improve instruction.

Experiencing School Mathematics has taken us through introductions and explorations into the two schools (Amber Hill and Phoenix Park) and into discussions in detail of the results of assessments for both. Now in Chapter 7, Boaler lets us hear the voices of the students as we 'Explore the Differences'. Through a mix of analysis of assessment results and interviews with the students involved we are able to see the differences in the capabilities of the students in these schools.

This chapter brings together all we have learned about the differences in curriculum and teaching styles of AH and PP. It is the proverbial 'the proof is in the pudding' scenario. Amber Hill students were able to use math knowledge, in class, when questions contained cues which indicated which 'maths' to use and when questions (GCSE) were of the short answer type. However, these same students were unable to determine for themselves which math concept/rule/procedure was needed to find a solution unless directions were explicit. When Boaler says that the 'math competencies displayed in different situations reflected both their understanding of mathematics and the belief that the students had developed about mathematics" she is discussing the student mathematical capabilities and it is clearly evident that AH students had very few.

On the bottom of page 106 I was struck by the student comments as they reveal their own realizations of how difficult they found the exam questions. It was their first inkling that there was a problem with their understanding of mathematics. Although naturally they blamed the test questions and labeled them unfair. "It's stupid really ' cause when you're in the lesson,.... you get the odd one wrong... you think well when I go into the exam I'm gonna get most of the right,'cause you get all your chapters right, But you don't. ( Alan) To me as an observer Alan's comments are a poignant reminder of the insecurity and stress that students who have little understanding feel as they are being pushed through one grade/curriculum after another. My heart really did sink as I clearly remembered being in an exam and realizing I just didn't know how to proceed, it was not good for the self esteem.

Chapter 7 gives example after example of how damaging the traditional style of teaching is for students who participate in it. They are unable to make connections between math concepts and unable to connect math to the real world. All in all, I couldn't help feeling a little sad at what educators have done to generations of students. How can we stop this from continuing? Krista asked where do we start, I think we have to start with teachers own mathematical understanding, beliefs, knowledge and attitudes. I know this for sure, we can't continue to ignore what is evident and cling to our old way of making sense of mathematics, because it just promotes a false belief that understanding has been reached when in fact nothing could be farther from the truth.