+1 to what Gale said!

Break a leg (or the dissertation hearing equivalent)!

@Brock: I remember someone doing an awesome inequality lesson on the point spread in sports books, but I can’t for the life of me find it.

For instance, in the upcoming Super Bowl, some places have the Seahawks are a 2.5 to win, which means if you bet on the Seahawks you only win if

Seahawks Score > Patriots Score + 2.5

@G I think the problem is selling only the hook as engaging. We have to sell the pure maths as well, which should not be hard given the ubiquity of maths in decent professions (let alone the sciences that make our lives safer, easier, more entertaining).

If I were back in the classroom I would follow the hook with “OK, the related of is “XYZ”. Let’s grow our maths “fu” by mastering quadratics by Friday. Now moving on…”.

The sell here is on the pride of mastering a mildly complex process that is key to one of the most important humans arts: maths. Looked at another way, we cannot for a second buy into “why do we need this?”. Maths in and of itself simply rocks, and we maths folks should not apologise for our enthusiasm: we are right!

The beauty of a good blended tool is that the kids can pretty much solo on those, leaving class time for applications and other enticements/motivations. On-line tools are extremely efficient because they are checked as they happen — no overnight turnaround waiting to find out how we did.

@Brock @G
Throwing a few ideas on the table — discussion welcomed.
I see both questions as related. On the one hand we are seeking ways of making aspects of mathematics interesting. Typically these are algebraic in nature (I sense an abstract/concrete discussion in here). I found G’s question particularly important, as it’s something we math educators all struggle with: “If I have not taught the skill (eg factorising quadratics) then the students can not do…If I teach the skill first then I am missing the hook.”

I think it’s worth thinking about what we mean by the students able to “do?”

If it’s factoring, and it’s framed as “factor this polynomial” then the likelihood of the students being able to do this without prompting is fairly low. But what if it’s, for example, getting 2 students to sit in pairs — one student has a graphical representation of a parabola, and the other is not allowed to see it. Their task is basically Pictionary — getting the other one to draw a parabola that’s exactly the same. Then through this we get them to recognize some of the key features of the parabola that perhaps is helpful for getting started.

We can also consider the fact that in a factored form, we have basically a multiplication between the two. Then maybe starting them with the idea of areas of rectangles and have them play with tiles or any other forms of representations. E.g. give them algebra tiles which represent some sort of quadratics, and they have to put the pieces in a rectangle. Then talk about what they’ve effectively done….etc

What do we mean by whether students can or cannot do a question? If they can’t do it without acquiring some sort of skill from us first, then maybe it isn’t a great question. What if it’s a question they come up with on their own? If we go with @Desmo’s Penny circle, they are doing lots of wonderful things with quadratics. If we frame it differently, these would be questions that they would come up with on their own, which we can then direct and explore.

Besides, what use is any of these “skills” anyway, if not an exercise on being able to think better and reason effectively?

I don’t think there is an easy answer. Like I said, I’m just throwing on some ideas for discussion purposes.

Dare I say that factoring quadratics is just plain fun?
Try looking at it like a little “box challenge” puzzle – this is an example from 5th grade, but the challenge problem is thought provoking. With these students it’s not looked at as “factoring quadratics,” but I think you will see the connection. http://www.base-ten.com/tm/wp-content/uploads/2015/01/BoxChallenge.png

Hey Dan, I wish you success on your dissertation! Let us know when we need to call you Dr. Meyer. :)

To be honest, *I’m* not sure anymore why we factor quadratics.

There are plenty of countries that don’t even bother.

Jason,

I would like to know the name of one such country.

I spoke to a woman at a Chicago selective enrollment school who had recently emigrated to the US. She let one factor guide her search for a school to teach at when she landed in Chicago… how much time the schools curriculum devoted to quadratic functions.

Paraphrasing … what percentage of mathematical models and meaningful functions are quadratic? And among those what percentage are factorable?

There’s a fun mathematical question.

Reply to G: I enjoy using the 3-act task precisely because it gets students involved in the lesson – act two is the teaching of the material, but the students are listening, responding, making connections (many times connections that are unexpected and creative!) and building their own knowledge. It’s messy and very high energy, the pain of learning is tangible, and the importance of not giving them too much information is an important part. You comment that you either do the problem for them or you teach the lesson. Stick that lesson in the middle! The third act is also important, when, just as you clean up your workspace, you clean up their ideas and sum up the findings, and allow your kids to “debrief”, to organize their thoughts.
One thing I do try to do before using a three act is thinking about what my kids already know and what I am hoping they will observe. Sometimes they are not ready for the ideas in the 3-act, at least not what I wanted them to see or wonder about. I am prepared to assist them from whatever starting point they pick. That is the teachable moment – the second act… It’s shouldn’t be one or the other. Be careful not to show them; let them struggle. That’s the beauty and the value of the 3-act task.

If no/limited direct application of quadratics justifies not teaching it, let’s try to remove all such topics from math teaching. Do we need to teach addition and subtraction when calculators are around?

In my mind, knowledge of a topic/skill is also an enabler for learning other topics that depend on the first one in some way.

(responding to Pawan Kumar): I don’t know if anyone is advocating against teaching quadratics at all (although personally, if I were designing US math standards, I’d strongly consider putting more about exponentials into the required-for-every-HS category and much of quadratics into pre-calculus).

I believe the objection was to teaching methods of finding factors of quadratics that are separate from using the quadratic formula. For instance, the Common Core algebra standards include: “Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring.” That’s an example of how in the US, we talk about “factoring a quadratic” as being distinct from “find the factors of a quadratic” (which could be covered using the QF).

I don’t teach high school algebra any more, but in my experience of it, we spent several weeks total on factoring quadratics; at the time, our state standards also insisted on doing it for equations whose x^2 coefficient was >1. Admittedly a lot of the time spent reinforced (or retaught) factors of numbers, +/- integers, etc., and factoring can also be a fun puzzle. But it drove me insane to see what a barrier this topic was, considering how seldom quadratics in the physical world are so easily factorable.

If the quadratic formula can solve any problem that factoring can, and if a student can demonstrate knowledge of what the roots of a polynomial are, how to get them, and what they mean in the context of a problem, why do they need to know how to factor? In particular, is this really something we should let high school graduation hinge upon? Because right now it is a big topic in required high school algebra classes, and it takes a ton of time because students don’t get it, partly because they don’t understand why they need to know it.

(The same arguments may apply to completing the square, but I haven’t thought that one out so well.)

Chester, New Zealand seems to have a good balance. I like how the emphasis is on how students would actually apply it with technology they’ll all have, and how they need to understand what the solution means and how to get it.

However, if I were teaching high school algebra here (which I am not), I’d think long and hard about implementing your solution, even though I’m not an “old-fashioned teacher”. I’d be knowingly exposing my students to the risk of doing poorly on Common Core test questions (the tests allow built-in calculators only, and provide them only for certain questions). Since I believe a certain Common Core test score will be required for the main path to high school graduation, this could reduce their chance of graduating. Also, I’d be going against what my district tells me to teach, and not in a minor way.

(None of that stops me from saying “I personally think factoring quadratics is great fun, but why on earth are we requiring it of all high schoolers?” at every opportunity, but nobody who could do anything about it is actually listening to me, so it’s more of a hobby discussion topic than an effective protest. I do keep hoping I’ll hear a good counterargument, but I haven’t yet.)

Julie, I wasn’t getting at you. It was more an aside that you can teach harder quadratics without pain if your local area allows you to.

If your exam doesn’t allow graphics calculators, then you have to teach without them. You’d be doing your children a disservice otherwise.

I find it more than annoying that school administrators and politicians in much of the world talk up a storm about getting technology into the classroom and preparing students for the modern world. Then don’t allow them to use relatively cheap purpose-built devices that allow that.

Instead try to shoe-horn in the likes of iPads, which are expensive and not designed with students in mind.

Thanks Clara, it is great to have some info and tips for 3 act tasks.

How long are your lessons? – I get 35 minutes and to fit everything in feels really rushed.

I do give them time to puzzle out their ideas but I find only the top end students actually doing this and only the very top get remotely close to the skill I want them to master. The lower end usually end up getting an answer quite quickly based on numerous misconceptions.

Once I have collected all the ideas, addressed as many misconceptions as I have time for and then built upon the ideas that were in the right direction I find the lesson has gone; I have now demonstrated the skill (but only once in a specific situation) and most of the students would not be able to answer a GCSE style question on it which ultimately they will need to do.

In terms of learning outcomes for this specific lesson, particularly at the lower end, not many have been achieved and certainly not any measurable ones. I recognise that the more I do these types of tasks the better at independent thinking my students will get but at the moment I feel that after a full lesson most of my students will still not be able to do a particular skill. Is this usual? Or, at the end of your 3 act lessons, can most of your students do the particular skill you chose the 3 act task to bring in? Are you seeing a measurable raise in the attainment for formal tests in your class for all students? Are you seeing a dip initially?

Thanks for all the help. I really want to develop the independent thinking in my students but at the moment it is coming at the cost of achievement in formal tests. Am I approaching things wrong? More help is very much appreciated.

David, thank you for finding mathematics to be a sufficient context to illuminate a mathematical idea.

I see two questions raised by Kevin Moore. Or at least there are two questions implicit in what he wrote.

1. Should mathematics be in contexts?
2. Should students understand the mathematics they use?

There is some interesting evidence about the first question based on 100,000+ people’s performance on these three logic tasks: http://www.philosophyexperiments.com/wason/Default.aspx

If you don’t have the patience to do the tasks yourself, which I recommend, you’ll find that people found the first two tasks much more difficult than the third. However all three tasks are logically equivalent; the only thing that is different is the context. People reading this blog are more likely than the average population to be successful on these tasks, but if you tried out the tasks, you probably found the first two tasks took more mental energy than the third.

My suspicion is that this effect holds for more than just this one logic puzzle, it might even be true for all of mathematics.

Here’s another (potential) example. Which of the following two problems is easier?

1. 129 mod 24
2. If it is noon now, what time will it be in 129 hours?

My experience with teaching students is that they need opportunities to extend their knowledge from their existing knowledge. Yes, we want them to think abstractly and withholding opportunities for them to do this is not right. While it is true that younger children tend to think more concretely, they can think abstractly in areas where they have more knowledge. So let them have opportunities to think abstractly as they grow their knowledge, but balance this against the intellectual need for what they do to make sense.

It is possible to misuse contexts. If the ideas never leave the contexts, they never gain the generalization and power of mathematical reasoning. So while contexts might be good places to introduce mathematical ideas, they are probably terrible places to leave them.

As for the second question, I’m not really sure why we wouldn’t them to both be able to use the mathematics and to understand why it works? Mathematics is both a tool we use and a set of processes that we can extend. Both are important. If you only teach children a set of mathematical tools they can use to solve problems, then I feel like they are missing out on learning about the processes used to develop those tools. Why would we want to withhold opportunities to learn these?

I made their point perfectly: strained on first two but figured I had it and got them wrong, had no problem with the third because I spend a lot of time watching bouncers work and the context guided me. Nice!

Furthering David Wees’s point on development, it must be clear that to deem knowledge or skill as developmentally appropriate has a great deal more to do with an individual’s experiences than with his /her age.

Taking a reformed view of Piaget’s theory on intellectual development, this theory can be viewed as having emerged with Piaget heeding an era-specific social and cultural context along with presumptions of limitations present at biological stages. Given his observations within the context and considering biological limitations, he believed that it was highly unlikely for children to be confronted with critical ideas and pivotal situations at certain ages. And should they be, children do not possess a broad enough schema to make sense of them. Children’s narrow schema, he believed, was simply due to their lack of experience (or even perhaps negative experiences). He surmised that children’s schema is only broadened by a process of assimilation and accommodation. Thus, it can be extrapolated from Piaget’s theory that experience then, and the quality thereof, is of utmost importance to children’s intellectual growth, as many and varied experiences offers opportunity for a broadened schema.

Teachers are responsible for creating situations to engage children in ways that provide a strong foundation for their success as abstract thinkers, broadening their schemas. Children can develop deep, complete understandings of complex ideas, through highly structured, defining experiences. These experiences should provide an adequate base of knowledge upon which to build. That is, children should be able to internalize, categorize, and activate this knowledge to make sense of other novel situations. Once these experiences are had, children are capable of moving from the concrete to the abstract with relative ease. This attributes to the fact that some children enter school better prepared to access the content of higher-level texts and make critical connections that facilitate their understanding of ideas in complex subject matters. These children have simply had deliberately rich experiences that broadened their schema.

In the case of mathematics, the knowledge that students gained through their experiences must be principled and transferable in order for it to advance them as mathematicians. Building math ideas around restricting and distorting models does not meet this end. Such practices present mathematics as situational, fragmented, disjointed, completely stripping it of its coherence. How can students be expected to make sense of something that is incoherent? How do students assimilate fragmented notions into their stocks of knowledge? How is it accommodated? If learning to think mathematically, which is reasoning with mathematical ideas, is not the focus of mathematics instruction, what then is the aim of teachers of school math?

Indeed, we interpret ideas as best as we can, considering our understanding of language and our presumptions that those with whom we communicate understand and use language in similar fashion. Thus, we are always filtering information through our experiences and presumptions, hence the risk of distortion. Yet there is reflection, an attempt to mirror through language (symbolic or verbal) that which we believe is the intended message of something heard or observed by some other means. This allows for the minimization and/or clarity of distortion.

These children have simply had deliberately rich experiences that broadened their schema.

Or are born cleverer.

I have to say that I agree with almost none of your analysis Kevin.

I believe (as Piaget did before you “reformed” him to say the opposite) that children cannot think abstractly before they have the appropriate mental equipment. Merely giving more and better training will not improve a skill that is not present.

Although we are no longer really allowed to say these things, everyone knows that some people are born with almost no ability at all for abstract thought. Others get it only very slowly, and even as adults struggle with abstract concepts. Whereas some children merely have to see a concept once and it is embedded.

Teaching to the strugglers can be rewarding, and I often enjoy teaching them Maths — slowly. They can enjoy Maths, provided it is pitched at their level. But I object strongly to being told that it is my failure as a teacher that prevents them from being able to see abstract patterns like more gifted can.

Your analysis effectively causes the blame for lack of progress in Maths to be laid completely at the feet of the teachers — since they have failed to provide the proper instruction. Politicians love this concept, because it allows them to brow-beat teachers for “failure”, when in reality there has been no failure.

We can seek to improve Maths teaching, and I imagine everyone here wants to, but we can’t improve the material we work with — people. No amount of wishing children were blank slates, onto which we can write at will, makes it so.

I do not only speak from a theoretical viewpoint regarding young children’s ability to learn abstract ideas, but I also speak as a practitioner. I have spent a number of years teaching young children mathematics; thus, I am not a novice in this regard.

I’d guess most of us here have taught Mathematics for a while. I teach older children, where the limits of their native ability become much more obvious, but it reaches back.

And in my experience, I have found that children — whether ill-prepared, struggling, lacking confidence, and/or unmotivated — are generally quite capable of understanding mathematics deeply.

I reckon I can teach almost any child almost any Maths skill. But getting them to understand it, no way!

I’ve seen it in myself. I cannot follow post-graduate level Mathematics. It’s not that I’m stupid. Nor ill-motivated. Nor badly taught in the basics. But at a certain point I cannot follow sufficiently difficult Mathematics.

For other people their level is lower. Much lower in many cases.

I am definitely not the first to take a more reformed view of Piaget’s ideas. His theory of cognitive development has been strongly contested since the 1960’s,

So about the time when the obvious — that some people are not very clever, by the misfortunes of genetics — became the unsayable?

Instead of algebra tiles for factoring quadratics, consider an arithmetic approach. Begin with the understanding that every whole number is a unit of itself, composed of a specific number of ones. As such, each number can be viewed as a specific array of units of one, revealing its dimensions, it factors. This supports the idea of the set of square numbers. And although there are whole numbers that are square numbers, not all whole numbers are squares. Yet all whole numbers can be compared to square numbers to better make sense of their dimensions.

Six, for example, being a composite of 6 units of one is also a single unit of 6^1 = 6 = 6 ● 1. Understanding the numbers as possessing dimension and being able to reason proportionally, more precisely inversely so, creates arrays of equal units of one and varying dimensions. Thus, 6 ● 1 = 1[6(1)] = (⅙)[6●1](6) = [(⅙)6][1(6)] = 1 ● 6, and:

6 ● 1 = 1[6(1)] = 2 (½)[6 ● 1] = [½(6)][1 ● 2] = 3 ● 2
6 ● 1 = 1[6(1)] = 3 (⅓)[6 ● 1] = [⅓(6)][1 ● 3] = 2 ● 3

However, when comparing six to 4, a square number,
4 (¼)[6 ● 1] = [¼(6)][1 ● 4] = 1 ½ ● 4,
six can be understood as 1(4) + ½(4) = 4(1 + ½) = ½(4 (1 + ½))2 = [½(4)][(1+ ½)2] = 2(2 + 1).

These are all important understandings that would aid students in factoring algebraic polynomials and especially quadratics. For example:

x^2 + 7x + 12; let x = 5 → (5)^2 + 7(5) + 12 = 25 + 35 + 12 = 72

Eight of nine is the expression of 72 that compromises whole numbers and presents 72 as close as it can be to a square. When 8(9) is compared with a square of 25 units, as 5 was substituted for x, is (5 + 3)(5 + 4) — the length of 5^2 is extended by three units and the width by 4 units. Thus, the factors/dimensions of x^2 + 7x + 12 are (x + 3)(x + 4).

This also works with negative integers:
x^2 + 7x + 12; let x = -2 → (-2)^2 + 7(-2) + 12 = 4 + -14 + 12 = 2; thus (-2 + 3)(-2 + 4) or (x + 3)(x + 4).

Another example:
2x^2 + 11x + 15; let x = 2 → 2(3)^2 + 11(3) + 15 = 18 +33 +15 = 66

Eleven of six is the expression of 66 that compromises whole numbers and presents 66 as close as it can be to a square. When 11(6) is compared with a square of 9 units, as 3 was substituted for x, is (2(3) + 5)(3 + 3) — 3^2 is doubled and the length extended by 5 units and the width extended by 3 units. Thus, the factors/dimension of x^2 + 11x + 15 are (2x + 5)(x + 3).