Here’s what math curriculum looks like in most schools.

The computations we want students to be able to do are chosen, and then we find problems that match these computations, and if we are able, we find some real life connections to the computations. Generally the real world connections are an after thought, and many times teachers are responsible for finding these connections when the textbook problems they are given are really examples of pseudo-context rather than a real connection. It can be difficult to attach real world context to the curriculum we are expected to teach and many times teachers aren’t able to do it.

There is a serious flaw in how this curriculum is constructed. The mathematics that is chosen has no motivation in the minds of the people learning it. If you have wondered why so many people hated mathematics class, it was because they couldn’t see the point of it. That’s because there was no point. People who hate mathematics were learning apparently meaningless algorithms for the sake of the algorithms themselves rather than for the processes in our world where the algorithms describe.

What if we flipped this process? What would happen if we changed the diagram so the real world problems were in the middle of the diagram, and we chose what mathematics to look at based directly on how it helped us understand the real world? The diagram would look like this:

I need to be careful here to define “real world.” I do not mean that students will spend their time learning how to solve the math problems that may arise in their life like an endless stream of super-market math. What I mean by “real world” is that the problems students would work on would have a shared context to which students will understand. This context might be a real problem that students or their teachers find, it might be part of an interesting challenge given to them by a mentor, or it might be simply exploring ‘what if’ in a puzzle.

The thing is, when you place context at the centre of the curriculum an immediate shift happens. Now the mathematics itself has immediate relevance, since the applications are focused on something which has meaning for the students. You also gain the ability to shift away from a computational focus (since that’s what should be for) and look at problem formulating and solving. Gone is pseudo-context. Gone is mathematics which has no relevance in the lives of our students.

Formulating the curriculum like this also makes finding connections to other areas where the students are studying much easier. Everyone can talk about the real world, and it can happen in every subject area. It becomes easier to turn our curriculum from caged subject areas into an open dialogue about life.

It also becomes easier to update the curriculum. The topics of a typical mathematics curriculum have changed very little in the past 100 years, and are nearly universal across the globe. However, in this curriculum the focus is on what is happening in real life, and it becomes easier to select what is important to teach as all of us experience the real world daily. In fact the curriculum itself could easily change from year to year and the actual mathematics that is taught could be different in each school. After all, what is important in mathematics is the process that students go through, not the end result.

There is also room in this model of curriculum for fairly advanced topics. You can see that calculus could be one of the branches of this type of curriculum since it deals intimately with understanding many complex phenomena in our world. If a student wants to extend themselves and get excited by the mathematics itself, we can still give them that opportunity. We can be more flexible in how we plan our lessons and give students more choice in how they approach problem solving.

What is not easy to represent in this diagram is what the arrows, which are common to all of the different possible branches of mathematics and their representation to the real world content. In my mind, these arrows represent part of the process of translating a real world problem into mathematics. Students have to know how to formulate problems, develop criteria for establishing what pieces of information they have are useful, and determine if their solution makes sense. Since they know for certain that the solution represents a real world phenomena, it will be easier to judge a correct solution from an obvious false one.

This shift also makes it easier to talk about the big ideas of mathematics. Most of the time we spend our classroom time so focused on the minutia that we forget that there are some **powerful ideas in mathematics** that are useful tools for thinking for students.

Here is a presentation to explain some of what this shift looks like for me.

Mathematics education has to change. I have spent my life either feeling defensive about my love of mathematics, or commiserating with people who agree with me. People say that they hate mathematics because they do not see how it is relevant. Let’s change mathematics curriculum so that context (which does not necessarily have to be “real world” but should be meaningful) is king.

**Update:** In my current work as a curriculum developer, I’ve been working on an alternative to the proposal above in our curriculum work which embeds contexts where they are meaningful but does not flip the arrangement as above. The key shift in our curriculum work is viewing students as sense-makers and fore-fronting student thinking as much as possible while still making mathematics accessible to all students (through shifts in instructional practices rather than shifts in the mathematics itself).

## Linda Pilkington says:

I am also passionate about this. I struggled to understand mathematics at school and once asked my teacher, ‘What’s this for, then I might understand it?’ He told me it was just to sort out who could go to university or not. I left school the next day with my new found knowledge that I obviously couldn’t ‘do’ university. I eventually went to University to become a teacher as a mature aged student. I took lots of mathematics subjects and electives when I was a pre-service teacher because I had no confidence in my own ability. I got nothing less than High Distinction for every maths subject I took. I love maths now. I’m still not completely confident, but I can relate those students of mine who don’t just get it right away. I’m always on the lookout for real-world connections to mathematics and science to help my students see the relevance and to fall in love with mathematics.

January 13, 2011 — 6:22 pm

## NorthTOmom says:

I agree that a lot of students hate math because they don’t see a connection to their world, but there are many other reasons why kids come to dislike math. In my own case, I disliked concrete applications, and only began to enjoy math when it became more abstract–in middle school (algebra). My kids hate math, despite being good at it, because they are constantly frustrated by the way it’s taught: not only is there no real-life connection for them, but the contrived problems in the text often simply don’t make sense (even to my computer scientist husband!). It doesn’t help that my daughters’ (non-specialist) teacher clearly neither likes nor fully understands math. I think that if a teacher possesses a deep understanding and love of math, specific pedagogy may not matter as much as one might think: s/he will find a way to pass on his/her enthusiasm for math to students. (And it may be non-real-world math that students come to love. One of my daughters gets more excited about the bit of formal algebra we’ve taught her than about any math she’s learned in school.)

Nonetheless: interesting diagrams, interesting post.

January 13, 2011 — 9:47 pm

## David Wees says:

I don’t think that making kids not hate math is the primary focus of my post. I do think it is about a basic revision of mathematics to be more useful and focused on real world applications than what we really do.

I also recognize that some people are just going to love the abstract nature of math and that we need to design a model of instruction which is flexible enough to encompass both types of learners.

January 13, 2011 — 9:56 pm

## NorthTOmom says:

I know it’s not the focus of your post, but you mention people disliking math, and I do think it’s important to get kids not to hate math. (I probably zeroed in on this because it bothers me so much that my kids say they hate math.) I also think the revision you propose would be a huge improvement over the way the curriculum is currently structured. The irony for me is that our board uses a purportedly reformist/”constructivist” curriculum, yet my daughters’ math program is structured in the manner illustrated by your first diagram.

January 13, 2011 — 10:16 pm

## @mmeveilleux (Ingrid) says:

I wish I had thought of this myself. I couldn’t agree more. The shift you describe is the one that needs to happen in all mathematics teaching so that the starting point is the context.

These diagrams remind of John VandeWalle’s dots. He sees learning as a series of dots (metaphorically) and the more tenuous the learning, the more discrete the dots. But as learning deepens the dots become increasingly connected.

Great food for thought!

January 13, 2011 — 11:17 pm

## David Wees says:

You know, I wanted to make some of the arrows dashed, but I couldn’t figure out how to do it in Google Drawings, which I thought would be the most efficient way of creating these diagrams. In particular, I wanted to make the arrow linking to the real world examples dotted to show the tenious connection between real world applications and much computational mathematics.

January 13, 2011 — 11:23 pm

## Paul Morris (aka P. Mo) says:

I love diagrams like yours, which is excellent. You have depicted a fairly major weakness in mathematics curricula around the world!

Mathematics does represent “computation” for most students. Ask them what they did for homework and they will answer :#1-25, odds. Even if they could tell you the chapter, they couldn’t much tell you “why” they were doing the hw. The math-is-computation is being re-inforced when they answer like that. Relevance is gone; number crunching prevails. Problem solving is gone; repetitive work is the norm. Changing this would have a huge impact on how people view problems, which are viewed as “problems” or items that get in the way, and not as challenges, to expand perception or awareness.

Love this line: “Since they know for certain that the solution represents a real world phenomena, it will be easier to judge a correct solution from an obvious false one.” This is a nice benchmark. Students have no way of knowing if their solution is correct since the context is not real world. Since feedback and reflection are important for learning, we are not maximizing our students’ potential when we don’t connect to the outside world.

January 15, 2011 — 1:53 am

## David Wees says:

I actually know of a mathematics curriculum which is constructed with the questions one might have about the real world as the organizer, I don’t know if anyone is using it, but at least it means that this idea is feasible. I’ll look up the name of the mathematics curriculum, and post it here when I find it.

January 15, 2011 — 9:11 am

## Anonymous says:

As homeshoolers have some flexibility in defining their methods and approaches putting the world experience at the centre of the lesson and extending out from there (as described above) is not unusual and some even choose home education for that reason.

I think our reticence to begin from experience or real world problems could be due to the cultural value placed on abstract thought over experience, likely stemming from the mind/body dualism in our history of knowlege/thought. Obviously knowledge must be grounded in reality and empirical evidence/experience but there is also something of a distrust of experience over what is considered abstract principles and there are many abstract principles in maths.

The hit TV show ‘Numbers’ was interesting & unique in that it showed the applications of math in fictional real world contexts. It seems to me that the idea of ‘learning by doing’ is considered more appropriate to younger children and by the time kids get into grade school they are relegated to desks and a more intellectualised abstract approach. Again probably the mind/body dualism seeing young children as the appropriate developmental stage for experiential learning as opposed to ‘grown-up’ abastract thinking in a context isolated from the real world.

So to me I see the model of maths to stem from historical cultural values.

January 17, 2011 — 8:45 pm

## Ian says:

Hi David,

Interesting post. I’ve been thinking about what you said and turning it around in my head for a few hours to try and distill exactly what’s been bothering me. I completely agree with your sentiments but I don’t think the central issue is really one of curriculum. I think what your describing is more a pedagogy problem. Curriculum documents are really there to tell teachers what topics have to be covered but how they cover them, (what methods they use) are open to the teacher. However I don’t think that really takes away from the point you make. The methods use to teach math only really instill confidence and interest in those students who would likely succeed without high level math instruction.

The issue facing educators in this area is more one of time. I think the bigger issue is to put into practice what you’ve described and still cover all the material the curriculum as its prescribed. To make real world connections with every topic and still have students complete enough practice to gain sufficient mastery of the material would require either reducing the number of topics covered or expanding the amount of time we have. Either option is possible (my preference being the former) but I suspect neither are likely.

I’m not sure I have a solution, sadly. I think the first step would be what you alluded to in the last part of your post; essentially stepping away from the calculations and dealing with issues of critical thinking. If students can puzzle through questions and sort of what information is relevant and necessary then the specific pedagogy or curriculum is largely irrelevant.

Thanks for an interesting topic!

January 17, 2011 — 10:50 pm

## David Wees says:

Right, that’s my point. Instead of thinking of a list of curriculum content as what is required, think of a list of skills you want students to be able to do. The really interested budding mathematicians should be encouraged to explore further, but the main goal is to ensure that everyone knows how to formulate a problem, choose some appropriate mathematics to apply to that problem, and check that their solution makes sense.

Pedagogy will take you only so far. There are some things for which there is no obvious link to the real world which are still being drilled into kids in some areas. I remember when I taught in NYC that I was required to teach kids about commutative groups. Why? Presumably someone on the curriculum planning committee for the state loved mathematical groups. To me, that’s not a good enough reason to be teaching that topic.

January 18, 2011 — 12:38 am

## Michael Paul Goldenberg says:

Read Derek Stolp’s MATHEMATICS MISEDUCATION and you’ll see a well-formed alternative to your diagram.

February 13, 2011 — 10:13 am

## David Wees says:

You are the second person to suggest I read it, so I’m taking a look around for it. Not in the library, so it seems I will have to buy a copy. I just subscribed to your blog by the way, very interesting stuff. Thanks for recommending Derek’s work to me.

February 13, 2011 — 2:44 pm

## John Gabriel says:

Naturally mathematics education has to change. Ever since the Jewish mathematician Cantor was immortalized, the entire study of mathematics has gone to the sewer.

Today mathematics is governed by Real Analysis which is descended from the rot of Cantor’s set theory. Unless this theory is completely abandoned, mathematics will become more difficult at an exponential rate only to be abandoned at some future time when it becomes clearly evident that the study thereof is no longer possible using this flawed theory (Cantorization of Mathematics).

April 29, 2011 — 1:01 am

## David Wees says:

I’m really not sure what the fact Cantor was Jewish has to do with his theory. It sounds like thinly veiled antisemniticism when you include his background with your arguments against his theory. Further, I read the article to which you linked, and in your second to last paragraph you indicate that limits are unnecessary to construct a theory of calculus, while then using a limit in your formula immediately following this statement.

More importantly, I don’t understand what your argument has to do with my argument. How does the need for a change of mathematics education to something more real world based have anything to do with an article trying to reframe calculus?

April 29, 2011 — 1:34 am

## John Gabriel says:

I’ll ignore your comments about anti-semiticism. These are irrelevant. If you bothered to read the link carefully, you would have noticed that the very next paragraph after the formula reads:

“The previous definition is presented in a new form without the use of limits and extended to multi-variable calculus in What you had to know…”

Now let me address your argument in more detail: Calculus is but one branch of mathematics that has been built on ill-defined concepts that make it difficult for most students to understand. In the same way, basic mathematics is hard to learn because most mathematicians don’t understand the difference between a magnitude and a number. Given this, how are their students expected to learn and understand mathematics?

Every concept must be well-defined. This is the crux of changing the way mathematics education is delivered.

June 6, 2011 — 10:40 pm

## Paul Morris (different from the one above) says:

I like what you are saying. In fact, your thoughts are the basis for many of the secondary math eduction courses being taught out there today. Mainly, problem based learning or inquiry based learning is the way to go. I have used PBL in my classes and it really does push kids to go beyond the simple, “what do we need to know for the test” mentality. So in essence I agree with what you are saying.

But, and this is a really big but, to truly see this happen we as educators will have to embrace the idea that math is a tool to explain our world, not a subject that has no real meaning or that we can do without. Many teachers, even math teachers, do not see the connections between the material we are teaching and real world situations. Our teacher education programs fall short in prepping teachers to look at other fields and see how math is used there outside of academia.

May 21, 2011 — 3:10 am

## David Wees says:

I totally agree Paul, actually implementing this would be difficult because it would require a real shift from what we do now. I just finished Paul Lockhart’s "A Mathematician’s Lament" and he has some good suggestions for how to get started with transformation of our system, although he does deride the idea of "mathematics in the real world" a bit. The painful word problems that textbook authors construct are an attempt to situate the mathematics in a real world context, but they fail utterly.

Problem based learning is good, better than what we do now. So is just plain experimentation and playing with the numbers themselves, described in much more detail in Paul’s work.

May 21, 2011 — 10:07 am

## Anonymous says:

I have come to learn that I hate my math class more and more each and every week. First I am a mature student returning for a second degree. Graphic design, never used any math more complicated than pixel measurements or inches, mms and picas. Easy.

I did a catch up class to help me since algebra was over 20 years ago in high school. I did great in that class got a flying A but I worked to understand the concepts under a teacher who was highly stressed and made an average of 2 mistakes per day every single day for 7 weeks. Made it hard to trust her much.

Now I am stuck on an online class called Technical Math. This math is for deisel mechanics, medical students, construction people and architects. I am lost on it now half way in. I don’t get it, the book gives no break down instruction of what to do. I am asked to compute the number for piston displacement. I had to figure out piston displacement which I figured was how much space the piston takes in the hole it fires in. I get asked about steel rail expansion but never in what units to answer it in. Inches, fractions of feet what?

October 19, 2011 — 9:28 am

## Anonymous says:

sorry don’t know why it put it up without my finishing. Anyway, I am wondering, was I supposed to take some kind of algebra before this class? I have no idea what half the symbols are took me a long time to know what a delta was and meant and I still have no idea what a sub number or letter is for. I am having a horrible time figuring out just by pictures of numbers in the examples what to do with what anymore. And I find myself lost a bit on some of the concepts, centrefuge velocity vs particle balance, um what?

Basically nothing I am learning has any meaning for me. I have tried very hard to “get” the concepts but I feel as if i am trying to learn rocket science when I basically only had algebra 1 20+ years ago. I have learned that I am getting a nervous stomach and a load of hatred for all of this math. The bad thing is I can see the beauty in math and the joy if I could ever get someone ANYONE with intelligence to teach me any of the concepts. Not just shove some problems at me with little explaination as to what the math is doing. It’s all Greek to me and obviously will stay that way in this college. I am sick of people with the we are better than you are because we get the concepts nyahhhh. I don’t care, I can cook better than you, I am a fine artist who can paint like Norman Rockwell or name anyone in that league, I can speak foreign languages easily and learn them quickly. Math snobs I have not patience for anymore. I hate American Schools.

Monika

October 19, 2011 — 9:35 am

## Mark Tenney says:

The Peano Axioms and proofs of the basic laws of arithmetic are a way to give structure to the math in school, both arithmetic and algebra. I have written a book for learning the Peano Axioms and proofs at below college level, Pre-Algebra New Math Done Right Peano Axioms.

This has simple examples of addition facts and addition laws and a slow paced build up to the proofs that addition of natural numbers is associative and commutative. This is more interesting than the typical algebra one problems.

The Peano Axioms and proofs of addition laws for natural numbers connects to students own prior math learning. The 5 Peano Axioms in words are under 400 characters. 400 characters is within the ability of 6 year olds to memorize.

May 25, 2012 — 10:06 am

## David Wees says:

I’ve been following your blog, and I have not completely understood what you are talking about in some cases. It might be useful if you could share some of the specific content from your book (perhaps a few screen-shots)? You may have already done this, but I have only been following for a couple of weeks, so I haven’t seen it yet.

May 25, 2012 — 11:55 am

## Vlad Tepes says:

Seriously?? Memorizing and exploring proofs?? You think THAT is going to excite kids and help give them meaning?? I think your “new” direction could be described as orthogonal to what would help kids…

September 1, 2014 — 3:21 pm

## David Silverstein says:

Google for –

“JOHN GABRIEL” CRANK

and you will know all that you really need to know.

Google for –

“JOHN GABRIEL” JEW

and you will discover even more about this whacko.

October 24, 2015 — 3:57 am