# Algebra is the wrong direction for math instruction

I received an announcement about a board sponsored teacher work shop to increase teacher competency in teaching algebra; it prompted this little rant^{1}.

I suspect the workshop is part of our system’s reaction to the PISA test scores and its mistreatment for political or economic concerns in the media (or just plan inadequate and incompetent) which has lead us to believe that math scores are falling compared to the rest of the world and that this is a problem. While I strongly disagree with those tenants, that’s not the issue I’m having. Algebra is part of the curriculum and sure, there are teachers who could stand to build competency in this area. Teachers who don’t know what they’re doing in math, as in any subject, are likely less effective and therefore waste the time and efforts of their students. The problem I am having is we are focusing our resources in the wrong stand of math; we are focusing on a strand that will have little impact on our student’s lives and we are neglecting a strand that teachers, as a group, need to build competency in.

Many people frustratingly argue or express dissatisfaction with math because they fail to see its relevance in their life. “When am I going to use this?” and the like, are questions we uncomfortably endure during the teaching of algebra. In Real Life, most situations can be done using the same sort of subtraction that identifies number families in primary without “let statements” or other algebraic strategies. True, algebra does get amazingly complicated where proper procedure is necessary, but not in real life. People can live their lives without algebra without disadvantage. One reason why they never solidify their understanding or that skills are allowed to degrade is there are so few practical applications. This is part of a problem with our school curriculum. Our system is a sort of pre-calculus model; we teach math focused on, and heading towards, calculus as a mathematical goal. The problem with that is, beyond some engineers, very few of us will end up learning/needing/applying calculus in our lives. It has been 21 years since I took calculus—I don’t remember how to do it, but I have never had an opportunity where I needed it; I have not suffered for want of calculus. Calculus may be a stunning example of human brilliance; it may be invaluable to engineers, but to the rest us, it is impractical and unnecessary.

That’s not necessarily a bad thing. Aristotle taught that education is both a good unto itself and what you can do with it. Esoteric knowledge is not necessarily a bad thing, but in this case, it might be as it has a serious negative consequence. The math we need in life; the math we need daily to understand or solve our current problems is the math our system least values even though it is most relevant. The negative consequence is our students are short changed on relevant math when we, as a system, focus on pre-calculus. We should be focusing on Data-management and probability as a system. Understanding the importance of standard deviation would be so much more useful to people than calculating the area under a curve.

Many teachers and students think that these 2 units are easy and of little value; this is mainly the case because that’s how our curriculum treats them. Each year they make their surveys and walk around the school collecting data (and interrupting classes) so they can graph their “authentic experience” in whatever graphing style they are learning. In probability, students spin spinners, roll dice, and pull marbles from imaginary bags—boring and pointless. As a result, Psychology students struggle unprepared through their Stats course in first year university; teachers nod dutifully when Sir Ken Robinson tells them a test resulted in 98% of kindergarteners scoring at genius level, we fill in our school plans for continuous improvement without discussing significant difference, mean or standard deviation; we don’t understand in what ways the ambiguities of “average” and how it’s being used to mislead us; in short, we don’t know what we’re missing, ignoring and short changing our students on. I was lucky; I met a teacher that loved both…he taught me! I learned.

For data:

Hand out a diagnostic test and realize that by grade 7 and 8 everyone can make a bar graph—fix the odd problem with scale and move on!

Now you can:

1) Present David McCandless on TED.com and the artistic/beauty of graphing. Combine graphing to tell a different story…a picto-scatter graph, a three dimensional bar graph, a bar graph were the width of the bar displays other data, etc.

2) Present Chris Jordan…graphically display something that so large we can’t deal with intellectually—only emotionally.

3) Present Dan Ariely on TED.com and see the predictable mistakes we make with data everyday…advertisers know them…so should your kids

4) Watch Hans Rosling tell a story with data in “the river of myth” or “200 countries over 200 years.” Turn your data into stories.

5) Teach them graphic manipulation techniques. It’s not wrong when a magazine or politician starts their graph at 50 rather than 0…its strategic–learn the strategies. Stop telling them it’s wrong to not start at zero and start telling them when they might want to (remember, “there are three types of lies: lies, damn lies, and statistics”…teach them how they are being lied to)

6) When should you use mean or mode or median….depends on the story you want to tell

7) How to use biases to your advantage…advertisers and politicians don’t avoid them…neither should you

8) Have them explain the math behind jokes (3 mathematicians go hunting; a duck flies by. The applied mathematician takes a shot but misses—2 feet too high. The Abstract mathematician takes a shot; he misses—2 feet too low. The statistician starts jumping…”we got him! We got him!)

In probability:

Think it’s easy? Try this: if 72% of people prefer milk chocolate to dark chocolate, what is the probability of at least 8 out of a random 10 people survived prefer milk chocolate? Probability is great, because, like data we’re not very good at it and it is often counter intuitive–we have lots to learn

1) Look up “Linda bank teller” on Google and explore the conjunction fallacy with your students. How many chose option 2…was it the standard 85% even though it’s wrong?

2) Have fun even…look up Donald Duck and Flipism

3) If 5 friends are drawing straws or picking numbers is it better to go first or last? Prove it (it doesn’t matter—a beautiful little pattern emerges)!

4) Have you explored the Monty Hall problem…I like it because so many mathematicians were wrong…Why?

5) Penny’s game: try it! Explain it!

6) Try using black jack

7) Explain the birthday problem—why in a room of 23 people is the chance of 2 sharing a birthday 50%

8) Go to NLVM.com and find the coin flipper…talk about the law of large numbers–more chance of being close to the mean but less chance of being exactly the mean…cool!

9) Probability has multiple modes to solve problems which create multiple points of entry for different learning styles. There’s diagrams like tree diagrams, some formulas and calculation techniques, tables and charts, and experimental. What is the chance of getting a value of 7 rolling 3 dice is a great question because people will approach it differently

10) Then, if you’re really adventurous, try Bayesian logic…if your doctor gives you 3 months to live why will you likely live much longer than that?

While we hardly use algebra, we are constantly running data management and probability software in our heads but it needs constant upgrades to remain useful as we grow into more complicated situations. Algebra is great; it is one of my favourite units because I like the symmetry—it’s beautiful; however, I’m not fooling myself. Algebra gets them ready for high school; data and probability get them ready for life.

^{1}While I came about these conclusions on my own and in discussion with a colleague, we were both delighted to be vindicated by Arthur Benjamin: “Teach statistics before calculus!” on TED.com when we found it. Some of the phrasing in this post is inspired by his presentation.

Patrick, you hit a few chords in this post, and I’m glad that you are taking a stand. It’s not a wishy-washy position which means that there are some that will agree, and some that will disagree.

At the risk of sounding ambivalent myself, I will say that there are a couple of your points that resonate. The most obvious is the desire to have teachers who have the appropriate level of expertise in the subjects that they teach. That’s just makes sense. I fear that too many teachers (and parents) implicitly and explicitly communicate their own negative experience of mathematics education to their students. With most elementary math programs being taught by generalist teachers, the chances of a student running into a few along the way that have a resistance to, if not a fear of, math coursing through their veins is pretty good.

But that’s not really what you’re talking about here, is it?

If I read you correctly, you are arguing for and against the study of certain branches of mathematics from a place of utility. For me, this is a perspective that ignores the aesthetic qualities of mathematics and the sheer joy of discovering and exploring math as a way to “play with the world” in a very unique way.

Some of the first mathematical puzzles that my sons and I solved were, it could be argued, grounded in elementary algebra: “I’m thinking of a number and when I add 3 to that number, the answer is 12″. Not only were games like this helpful for me in discovering how they thought, but they also helped to strengthen their ability to think about relationships and patterns in a very unique manner.

As a student, some of my first math worksheets were algebraic in nature: “Box” + 2 = 3

I’m not sure, but I think grounding students in early algebraic thinking allows them to think about numbers in a different way.

On a completely different track, I was talking (via email) with a colleague who is very passionate about mathematics and we were discussing your blog piece. He made the excellent point (well…I thought it was excellent) that the argument from utility would put many things in danger of elimination. Should students not learn about musical structure. Is the fact that most students will never write a symphony a good enough reason to remove a study of symphonic form from our Arts curriculum? What might happen to our curriculum if we applied the same line of thinking to the study of dance, drama or visual arts? What would disappear if we admitted that, once they leave school, most people would never read or write a piece of poetry?

I think that we need to teach things that are useful, but I fear that if the judge of a discipline’s value is the degree to which we explicitly apply it once our schooling is over, we stand to miss out on a huge piece of what it means to be an educated person. A system of curriculum based primarily on utility widens the gap, I believe, between what it means to be schooled, and what it means to be educated.

I look forward to some lively conversation on your post. I wanted to jump in with some “points of resistance” in order to fuel the fire a little more!