# Making progress

Maybe I miss teaching maths as I feel compelled to use some mathematical terms and concepts in this post about how my (non-Maths) PBLs mentioned in the previous post, are progressing. Oh, and this is inspired by this video on dissociating learning from performance, linked to me by the wonderful Kelli McGraw (@kmcg2375) who constantly pushes my academic thinking, among others.

Anyway, I had to watch that video lots and lots and lots of times. What really hooked me was this notion of variability as aid to learning and transference, even if the performance gain (observable stuff we teachers measure as evidence of learning) is non-existent or slow. See folks, this is why I love Algebra! And in fact, that’s how Algebra should be taught, i.e. change those variables so students can see that the relationships expressed in an equation will yield different values as variables change. This is transference in number terms, literally.

I’m not going to pretend I understand the video completely. I don’t. What does spacing even means? I’m guessing interleaving means making connections. Interleaving vs blocking new things to be learned. whoa!

This is exactly how my PBLs are progressing. Variability. Interleaving vs blocking new things. Conditions are neither constant nor predictable (these terms are from the video, ok?).

Precisely because of how the Year 9 Digital Media Jedi Academy is set up, there is so much variability. I’ve got kids learning to write HTML code, writing ebooks, creating wikis, typography and critically analysing their process…yep, writing their applications to level up. And they’re excited about what they’re doing that invariably (haha) I have to boot them out when bell goes. Comments heard today: “I’ve done so much”, “I’ve learned so much”, “This is exciting”.  They’re collaborating, giving peer feedback and affirmations and best of all, learning how to help themselves.

Their applications to level up are done in Word, submitted to our virtual classroom (a Sharepoint site). I annotate these. Then it hit me that I had no idea of checking if they’ve really read these annotations – we’re talking individualised feedback here that took time and effort! Bianca Hewes (@biancaH80) to the rescue. More specifically, her post on feedback (a must read so go there, will you?) that mentioned the Goals, Medals, Missions framework. I told the students that they had to hunt down the medals and missions in my feedback. This had the added bonus of student feedback on my own annotations. It was clear that I was rather austere on the medals department. haha. I’ll fix that. I wish Sharepoint has notifications like edmodo.

My Year 10 PBL on the school purpose has taken twists and turns I could not have predicted. These kids are getting so engrossed on making sense of the school purpose and want to take the rest of the school with them. I’m actually rather flabbergasted though obviously proud of them taking ownership.  They designed surveys for staff and students and we had amazing discussions on the art of writing surveys and the challenges of collation…we were optimistic we’d get heaps of responses. Now they’re talking about making it a game and what do I know really of Game Design. Well, I’m learning along with them. Like my year 9s, I have to dismiss them a few times before they actually leave the room.

This post is long enough methinks. Anyway, I’m feeling good about the progress. Yeah, I still feel lost but I think I might get used to it and welcome it. That’s a good thing, right?

And just to end on the idea of abstraction: neither Bianca nor Kelli teach maths or computing; yet, see how I’ve abstracted from their work and applied to my context (steal like an artist – go on, check it out….interleaving, see?). This abstraction is Algebra IRL. really.

# Multiplication, Multiples and Factors

My daughter is in Year 5 and yes it is NAPLAN year; even if I’ve forgotten this fact, her homework would have reminded me.  She has just finished worksheets on multiples and factors – including concepts of highest common factor, lowest common multiple.  She told me they haven’t been taught in class – which, if it boils down to it, is going to be her word against the teacher’s.  And that’s not really what I’m posting about.  Nor is this post about worksheets and homework.

This post is about multiplication and using the times table.

As a high school teacher, I know that some kids struggle multiples and factors conceptually.  Once applied in fractions and Algebra, the confusion mounts.  A good foundation built up through primary/elementary school years cannot be under-rated.

One of the problems I think is that kids really don’t know, i.e. memorised, their times tables.  Though they conceptually know of multiplication, or even have several models of it, many still have to calculate often by repeated addition through to college/university as @angrymath tells us (and I can believe it).  If you think Cognitive Load Theory makes sense, it follows that knowing one’s times tables reduces the cognitive load when doing maths.  There are also strategies – even ‘tricks’ – to help.  One of the strategies I teach my kids (my daughters and students alike) is what I call the “goalpost multiplication” approach.  Briefly, estimate to the nearest goalpost (x2, x5, x10) and work up or down.  For example, 8×7 = 8×5 + 7 + 7.  Note that even in this strategy, there is a need for conceptual understanding (and review) of certain concepts such as associative property and number sense via estimation.

That was a dense paragraph and there are things I’d like to qualify:

1. I don’t advocate rote learning at the expense of conceptual understanding
2. I think Cognitive Load Theory makes sense and I have noted that students who expend much effort (use of working memory) calculating relative to effort on mathematical thinking (e.g., application of calculations) tend to have lower self-concept with regards to their numeracy
3. Memorising the times table is not necessary but useful, if just to reduce cognitive load and corollary, improve self-concept with regards to maths
4. If you really cannot bear to memorise the times table, at least have some strategies to get you by.  As a teacher, help your students learn these strategies

Ok. so here’s what I did with my daughter.

I whipped out the Times Table (here’s one you can grab, too)

I used this multiplication grid as a visual aid to explain/highlight the following points:

1. Any number multiplied by 1 is itself;  one day, she’ll learn that this is called multiplicative identity property
2. The green line is the square of numbers showing, for example that 3 x 3 = 9 “boxes”
3. The green line is like a mirror, a line of symmetry, so one side is the reflection of the other showing 3 x 4 = 4 x 3; one day, she’ll learn that this is called the commutative property of multiplication
4. Rows or columns are multiples of that number (in orange)
5. The factors of a number are the numbers you multiply to get that number, i.e. the headings in orange
6. A number that only shows up in row/column 1 (numbers in blue) means it is a prime number, i.e. its factors are only 1 and itself
7. A number repeating across the table left-to-right, say, means it is a common multiple – a multiple shared by the numbers with it as a multiple  (e.g. 12 in purple is a common multiple of 1, 2, 3, 4, 6, 12)<most kids this age understand the concept of common – use this to connect to maths; this highlights literacy as well as help build confidence, i.e. common multiple is not such an alien concept after all>
8. A number repeating across the table means it has several factors (e.g. 12 in purple shows up in 1, 2, 3, 4, 6, and 12 and these are its list of factors)
9. A common factor is a factor that is shared by 2 or more numbers. <this is harder to show using the times table but build on concepts of common and factor list>

One could reasonably argue that teaching all points above is risking cognitive overload.  Teaching one-on-one, I could tell that she was keeping pace especially using the times table as visual aid.  My first attempt failed just using the definitions on the worksheets; the times table definitely helped.

I also found out that the tables they have at school where the other kind designed for rote learning: 2 x 2 = 4, 2 x 3 = 6, 2 x 4 = 8, etc.  She was quite pleased with this multiplication grid and using it for finding patterns (pre-Algebra skills) and not just as direct reference.

To be honest, I don’t mind if she forgets all of the “definitions” above if she can work to find them again.  Given the iterative nature of the Maths syllabus, she undoubtedly will have lots of opportunities to re-visit this.

As an aside,  creating a multiplication grid using a spreadsheet with formulas and auto-fill would make a good little tech-integration task and exercise of Algebraic skills; there’s more than 1 algorithm.

# Algebraic Equations Drill – virtual and real

This post is probably more of a reflection on my teaching than anything. However, there are some resources included that may be of some use to others.

Regarding algebraic equations, I’ve stuck to my teaching strategy mentioned in the previous post. That is, my focus is on hammering the points on equations with an additional 1; listing all 5 here every lesson.

1. There is an = sign (expressions don’t)
2. LHS = RHS (balancing strategy)
3. to solve for x, isolate it (make it the subject)
4. use inverse or opposite operations to help isolate x
5. during, check using points above; after, check by substitution

Once a fortnight, the class has ready access to computers. On that lesson, I got them to go to these resources:

1. Equation Buster – balance equations Levels 1 to 4
2. Balance LHS and RHS – balancing strategy
3. Connect Four
4. Solve equations in your head – good practice to get quicker

Then, we’ve done more work on the textbook to actually practice setting out equations.

I made a mistake in giving them a quiz where they can solve questions via Guess-and-Check (or Trial and Error), e.g. 3m=120. It’s a mistake in that the quiz did not really assess whether or not they can use algebraic techniques. It was not a good assessment of- nor for – learning …except mine….(no point sharing this quiz with you) …. but I digress….

Today, I gave them a worksheet and gave them the answers. This worksheet (PDF created using Exam Creator)  has 10 equations which included 1-step, 2-step, x on both sides and grouping symbols. I told the class to show using the ‘balancing strategy’ that the answer is right. This was a good tool for assessment for learning.

I heard comments like, ” I could do it on the computer but I’m confused now”.  I think this is because they get instant feedback every step of the way while on the computer. On paper, they have to get to the end. Herein lies one of the strengths of technology.

I also heard comments like, “Is that all? But it’s easy.”, while constantly repeating the 5 points above. For a class who still struggles with computational skills, they are doing rather well working with equations.

Anyway, my journey isn’t over.

Should I keep sticking to my strategy?

# Teaching Equations big-picture style

I started Equations with my year 8 maths. I love equations – probably why I love Algebra – and have decided to buck the textbook trend.

Lesson

Instead of teaching technique by technique, I decided to go with the big picture. And really, students in year 8 already know this big picture. I started the lesson by asking them “What do you already know about equations?” and listed these 4 main “principles” :

1. There’s an = sign (expressions don’t; it’s a number sentence)
2. Left hand side is the same as the Right hand side (LHS = RHS) in value, i.e. size and direction
3. Solve for the unknown (x) by isolating it or making it the subject (sentences have subjects) AND
4. Use inverse or opposite operations

After this discussion of less than 10 minutes, the students doing worksheets that spanned at least 3 sections of their textbook. I pointed out that the answers were on the back of the worksheet. Why? They replied that the focus was on the working out. (Conditioning works) 🙂

Using these principles, you don’t really need to distinguish whether or not it’s one-step or two-step or “what do you do to x to get x +3?”. I had a student who said she couldn’t answer the latter (from the textbook) when she’s already solving problems like x / 3 = 8, or even 2x + 15 = 6. That’s because the reason we use inverse operations is because we want to isolate x ; that’s the context.

I also had a student ask how they know if they’re on the right track. The textbook says – and rightly so – check by substitution. Instead, I said well check whether the principles still hold, i.e. is LHS = RHS? If not, then you’re definitely off-track. So really, check correctness through checking by principles and by substitution.

I purposely did not teach move it to the other side and change its direction (e.g. +5 in the LHS becomes -5 in the RHS). Sure, that’s what eventually happens after applying inverse operations but I insist in not doing this shortcut….not until I’m convinced that the students know why.

Rationale

Those who’ve been following me in this learning (and teaching) blog would recognise my constant battle with keeping maths relevant and meaningful (even fun) for those who are not naturally good at it. Maths has a lot of techniques and short-hand ways of doing things which, though can be followed step-by-step, often detracts from the beauty of maths itself and its many applications.

In doing this approach I’m hoping to have more time for practice (or drill) and (equation) fun (so many interactive games out there). I hate the term drill and use ‘practice’ because maths is a language too and fluency requires practice, not just understanding. For example, I know and understand the rudiments of Spanish and French grammar but nowhere near as fluent as I am in English (I am ESL) because I don’t practice those two as much as English.

I am hoping this will work, i.e. I don’t know if it will but I sure hope so.

# Problems vs Exercises

## Problems vs. Exercises

Inspired by a blogpost by David Cox (@dcox21), I decided to use the same problem and added an extra fraction. Anyone following me for some time would see that I value mathematical thinking and, as David said, this is a good one to show problem-solving skills. I think it’s also a good one to revise, reinforce and connect mathematical skills (or tools as David calls them).

Simplify:

I asked the class what their first thoughts were upon seeing this problem on the board:

1. “It looks complicated”
2. “It’s hard”
3. “It looks like a big problem with lots of little problems”

They were all correct, of course, and I told them so (affirmation is good). I then told them that, in fact, they already have all the skills to solve this problem and they looked at me to as if I’ve gone mad. I suggested that, because it did look complicated, we look at the little problems that make up the big problem to make it easier (note the use of their responses; affirmation is good – oops, said that already).

I then asked what the problem looked like, that is, what’s familiar about the problem.

1. Dividing fractions