One of the nice things about the IBO MYP curriculum is that it’s completely untracked and all students have access to it. Except for math…in the last two years of the program math is split into Math Standard and Math Extended. The idea is that Math Extended is supposed to cater to the top students to prepare them for DP Higher Level Math and a future STEM career, but in our school things have worked out differently.

Instead any student that has any inclination at all towards math takes Math Extended in an increasingly packed (and diverse) class, leaving the students who despise math to wallow in what’s almost become slow math instead of the *standard* thing. And so the rigor of the class has also declined in response…I hate that!!!

This year I have my usual MYP4 Math Extended (9th grade) but I also have MYP5 Math Standard, and I’m so excited to try to really get these kids doing math!

Our first unit is on quadratic equations and functions and their first big investigation is on how the form of the equation affects the graph of the function. I decided to jump right in on the first day with a formative assessment of their knowledge on graphing linear functions as a way to introduce several things. It started off nice and slow (so I thought):

See the entire formative assessment here.I gave them each two pieces of paper and asked them to keep track of their “noticings” on one and their “wonderings” on the other with these prompts up on the beamer:

“I notice/see that _____” “I realize that _______” “I’m confused by _______” “I wonder if/how/why ______” “How do/can you _______” “How/Why does ____ work?” “What would happen if ______”

I had them work individually for 6 minutes — I intended it to be 10 minutes, but I noticed right away that one or two students were unable to complete even that first question. I stopped them and asked them each to jot down at least one noticing or wondering and then begin to discuss in their groups.

I was extremely pleased with how much they started helping each other and working together! In one group there was even a disagreement and a debate over how a line with a gradient of -3 should look.

I had let them pick their own groups and three students who are particularly down on math had sat together and they really struggled to get any of the problems done, but in each case someone had an idea and they were able to be the group leader for that particular problem, which was nice for them (though I don’t know that it’s tenable to have them remain as a group).

They all worked *really* diligently for about 30 minutes and which point I asked them to consider as a group the following questions:

- What was the point of this activity?
- What do you think I, the teacher, should learn from it?
- What do you think you, the student, should learn from it?
- How do you feel about what you did?

Here’s the list of what they came up with as answers to the first three:

- To refresh their memory and get them ready for the new year
- To check what level they are at so I know how to help them
- To work together in a group
- “Math”

To which I said, “math isn’t specific enough! Be specific.” And then they were — they said graphing, equations and how it relates to the graph (BINGO!), linear equations (YES, so we can next distinguish between linear and quadratic!), gradient. I love that *they* came up with everything I wanted them to learn for themselves!

Then I asked them to flip to the last problem in the formative assessment (no one had gotten close to it), which was meant as an introductory exploration to quadratics:

We managed to go through the first two parts of the problem together (though it took a lot of prodding they were able to discover the answers for themselves, leading one student to say, “I feel smart!”) and then I told them about their HW, which I said would be different than what they’re used to and difficult but to not give up and if they don’t know one question, to write their thoughts anyway and then go on to the next.

Here’s that HW, which might be way optimistic, but I’m excited with how I hope it helps them develop (we shall see and I will report back!).

As they left, I collected their noticings and wonderings. A handful wondered what the connection between an equation and a graph was (yes!) and a full 11 of them noticed that they’d forgotten a lot over the summer. But my favorite, even though it’s not about the math itself, is this one:

I think I’ve gotten off to the right start with them…I hope!

I really admire the way you have approached the “revision”. Very bold!

One thing regarding rigor, however:

Your paragraph 4 uses “function” and “equation” interchangeably.

You are definitely dealing with equations here, otherwise they will not be able to find the intersection of the line x = 2 with the curve.

hi howard, i’m not sure exactly what you’re referring to, could you clarify?

I am nitpicking, that’s me, but a graph is picture of the relationship between two quantities (variables if you like) showing points whose coordinates satisfy the relationship. There is no restriction on the points. An equation has a graph. However, a function is a process whose job is to convert an input into an output. f(x) = 2x+3 is a representation of a function, as it tells you what to do to the input (x) to get the output f(x). If for each x I call the output y then I have an equation

y=2x+3, that is, a relationship between the quantities x and y, so I can have a graph.

This may look like being very fussy, but using the term “function” without the idea of a process is not going to lay the best groundwork for later study.

There is no problem so long as the words “Graph of a function” is understood to mean “Graph of the equation of a function”.

Of course, the whole business is muddied by using the words “Quadratic equation” only to refer to x^2+2x+7=0 (for example, and everybody does it). Strictly what is called quadratic function is really a particular case of a second degree equation, so maybe the damage is already done!!!!!!!!!!!!!

And all this because you used the word “rigor”, and it’s time somebody did!

Hi Howard, thanks for clarifying. I’m not sure I agree with you, though admittedly my background is in mathematics teaching not mathematics. I’ve just double-checked what all my various textbooks have to say to make sure I’m not saying anything incorrect though of course they are not infallible either.

We teach the students function machines early on where x is the input and y is the output, but they’ll only learn actual function notation later this year. I do not see that there is a real difference beyond semantics between y=2x+3 and f(x)=2x+3. We refer to y=2x+3 as the function equation or the equation of the function (as does our international MYP textbook, which is Australian). You are just as able to graph the line whether you begin with f(x) or y. Our international DP textbook, which is from Oxford University Press, says things like “consider the function y=x^2” and “Draw the graph of the function y=2x-4.” The Dutch, by the way, refer to this is a linear formula and would not call y=2x+3 an equation.

Typically when we introduce quadratics and lines, we make a distinction between an expression (no equal sign so no solution), an equation (an equal sign so solution(s) for x), and a function (an equal sign with two variables, so infinite pairs of solutions which can be represented on a graph). Later on we delve into what is and is not a function through set mapping etc and the various ways to represent a function notationally.

I do want to have rigor in my teaching, but I’m not sure that using the word function here compromises that?

Hi Kate. Many thanks for reading my stuff!

There is nothing wrong with your approach at all. Very thorough.

One of the real problems with the learning and understanding of mathematics is that students think that what is written down IS the thing that is being looked at, investigated, etcetera. They are often not encouraged to see the written form as a representation of the thing. It is mainly due to the fact that mathematicians have learned to do this automatically, and so for example “The function y=x^2” and “The equation y=x^2” are likely to be seen as the same by the students.

Here is a lovely example I came across, from a computer based test:

Question: A number (call it n) plus seven equals ten.

a) Write the sentence using mathematical symbols.

(correct answer n+7=10)

b) What number is n ?

( the answer expected was n=3 )

My rigorous answer is “The number whose name is n”

This is a simple case of confusing the name of a number and its value.

Hi Kate, again.

Firstly, I love your blog title and your wordpress name.

Now for a bit of history.

The “New Math” (USA) and the “School Mathematics Project” (UK) of the 60’s tried to do the function thing properly, starting with mappings from one set to another, and relations as sets of ordered pairs (a,b) with a from one set and b from another. This is the “Modern Algebra” approach, and provides a sound (rigorous) basis for functions, graphs, equations. It all proved too much! Now we have a somewhat watered down approach (as in the new Common Core in the USA), which has interesting holes.

Here is some not too light, not too heavy, reading for your next vacation : “A Concrete Approach to Abstract Algebra” by W W Sawyer. Most of this guy’s other books are written for School level maths, and are all worth reading.