A question from frequent commenter, RF:
If you were teaching a (very) mixed ability Year 7 class in their first term of secondary school and had COMPLETE control over the curriculum, what would you start with as the first topic/lesson sequence?
A question from frequent commenter, RF:
If you were teaching a (very) mixed ability Year 7 class in their first term of secondary school and had COMPLETE control over the curriculum, what would you start with as the first topic/lesson sequence?
Algebra.
And the only problem with that plan is …
Algebra requires a knowledge of numbers and operations.
Which requires a knowledge of what a number is.
Which requires a knowledge of what counting entails.
Which requires a desire to learn.
I can go on and on…
Thanks Terry. Algebra was the idea floated by my leader. I don’t think it has really worked the three years it has been tried, so looking for some inspiration for an alternative.
I do agree that Algebra is really THE big idea that separates primary from secondary school in this regard; I’m just not convinced it is the ideal starting point for VERY mixed ability classes (some of the better students are already pretty good with algebraic manipulation and others came from primary schools that “didn’t believe in learning times tables” – quote from a parent)
I also think that the official curriculum documents are pretty useless in this regard, hence my throwing out the question.
I’ve not taught Year 7 maths, but my initial reaction is that a thorough review of arithmetic with natural numbers would be very useful (standard algorithms for addition / subtraction / multiplication / division with remainder; place value; factors / multiples; commutative / associative / distributive laws)
I would be curious to try out James Tanton’s exploding dots (https://gdaymath.com/courses/exploding-dots/) approach to teaching binary and other number systems – has anyone tried that?
Actually the first thing we did when I started high school (many years ago) was different number systems – and I guess it was memorable because I remember. I think it could be good for mixed ability classes because if students are really good at doing decimal arithmetic, the different base might make it tricky for them? But if they’re aren’t then it might help them with it.
Hi, S-T. I think someone showed me exploding dots previously but I can’t remember much. A quick look now, it looks pretty over-egged to me, and definitely it’s annoyingly over-hyped in the video you linked. But, I can’t with any authority suggest you ignore it as a possibility.
Oh, I forgot to say, I have a lot of time for your number systems idea. Actually, there’s a couple different ideas in that idea, which I’ll write about in a separate comment soon.
OK… this is a question mainly to S-T who raised it, but I’m interested in all ideas. Let’s play with the number systems idea for a moment. Would you proceed as follows or differently:
Explore different number systems with the big idea being “what is a number, really?”
Then pose the big idea: “without numbers, what is mathematics about?”
Thoughts?
I’m not sure if Year 7 students would have a clear sense of what mathematics is exactly, but it could be interesting as a way of finding out the impression they have been given from their previous schooling.
I’m still studying to be a teacher (hence my user name). I have found the process of training to be a mathematics teacher in Victoria a bit of a shock. I thought I knew roughly what mathematics is about, having majored in it and done a PhD. But then the education department seems to have a different understanding of what the subject is, and my knowledge is not even particularly relevant. For example, “proofs” was described by my lecturer as a “topic”, separate from the rest of mathematics. I’m still trying to make sense of it, but what mathematics is about seems to vary depending on the context.
Firstly, having worked for a university in the training of pre-service teachers, I can say without hesitation that their idea of what Mathematics is about doesn’t warrant consideration by anyone with any tertiary education in Mathematics (you are right, they are wrong).
Secondly, I don’t mean trying to teach students what Mathematics is (that is just impossible) but rather give them a sense that there is more to it than what they have experienced up to this point.
I like the idea of learning by discovery to an extent, try to get some excitement happening. It is an ongoing struggle, but (as you may find, but I hope not) it is not the students that are the problem.
There you have it in a nutshell: “proofs” as a “topic” says all you need to know. And, it’s much worse than that.
S-T, I’ll confess I peeked at your email address to see who you were, which makes it difficult to now pretend that I haven’t. But, an honest question I would have asked anyway: do you really doubt so much your own sense of mathematics, and your own sense of plain old sense, that these fifth rate clowns are confusing you?
Coming from the university-research world there is plenty for you to learn about teaching in a secondary school. These idiots, however, have not a single thing to teach you, and the sooner you understand that, or simply trust me and accept it as an axiom, the quicker you’ll be on the road to meaningful teaching.
Jump through their idiot hoops and otherwise ignore them.
…I would hasten to add that not all hoops need to be jumped through. P makes for a Degree as a wise man (my father) once said when I complained about such nonsense…
That’s okay, and thank you! It’s just when they kept going on about how the most important thing to teach is being strategic and efficient use of technology and knowing what steps to write down it seemed like what they call mathematics is not the same as what I (or a lot of people) call mathematics, and maybe the VCE is not something I’d be particularly qualified to teach because it’s just a different subject with the same name.
On the plus side, when I did my placement last year, it was actually okay, and the students really did seem to want to learn about mathematics as well (I taught Years 9-10). So in any case I can teach Years 7-10! I think that’s my plan now.
You’re still being way too polite. What they call mathematics is not mathematics, full stop.
And yes, many students do want to learn, even if their discipline is not what it should be (pun intended). In that regard, my advice is to avoid VCE, and Methods in particular. You can have much more influence by teaching 7,8,9 and 10 (in decreasing order).
With the bar where it is for VCE teachers you are already going to be fine. Do note what Marty said in the previous comment, because Methods (especially at Year 11) can be a soul-crushing experience as a teacher, experienced or not, at times.
Ugh. “Being strategic” I can get on board with – although in VCE context, that’s generally code for “know how to maximise marks”, rather than “know how to work your way through a difficult problem”.
Advising teachers to teach “knowing what steps to write down” is hilarious. Even if we want to play the maximising-marks game, VCAA are notoriously vague about what counts as sufficient working. Not to mention their infuriating and anti-mathematical standards on the amount of working that needs to be written down as a response to a “show that” question.
Having taught Year 11 Methods for many years in succession now, I can confirm Marty’s and RF’s comments above. It is a race against the clock to cram a whole lot of “do this process to answer this kind question” down a student’s throat. You will have underprepared students (taking the subject because they need a 25 study score for their intended tertiary course) struggling to keep up, and you have no time to address their difficulties. You will have strong students (taking the subject because it is a prerequisite for Specialist) who would benefit from a more critical approach to the content (with more focus on proof and generalisation), but again you have no time to address their needs. And for the reasonably competent students, they receive an impoverished and narrow education. In the end, it grinds students down into just wanting to know what they need to do to “get the marks”. And of course, unless you are comfortable teaching a class with low morale, you will adjust to teaching the students what they want.
I’m not sure it’s productive to frame what is taught / learned, at this stage, quite so philosophically. Any approach will necessarily be rather naive, and I don’t think much is gained.
Without having thought through exactly what would be taught / learned by having Year 7s consider different “number systems”, my initial thought is that the benefits would revolve more around these students gaining greater awareness of different properties that number systems can have, and how those properties play a role in the number systems with which they are more familiar.
Maybe it might be better to frame the topic as “What are the important features of the integers and how do they help us count / calculate?” Or “What are the important features of positional notation and how does that help us count / calculate?”
Hi, SRK. I’m not sure the intention is to frame the lessons so philosophically, but rather the lessons be implicitly guided by a deeper conceptual understanding. So, the question is whether lessons about “number systems” might assist in developing an intuition that will help with the core arithmetic-algebra topics.
Oh thanks! I would be interested to read your thoughts about it.
To be honest, I’m not sure I could do the theatrical aspect like James Tanton does! How silly do you need to be to succeed as a teacher?
But I love diagrams for explaining and thinking about mathematics. So that’s why I thought it might be worth teaching the diagrams at least – to help anyone who says they can’t do things by bringing it back to a simple process of counting dots, while at the same time challenging those who can to think about what they are actually doing and why it works. My six year old liked the explosion aspect (with added sound effects) but how would that work with older kids?
I think having different bases separates the representations of numbers from the numbers themselves, so it could help develop a sense of detachment, that could lead to the abstraction and generalization that Red Five says is the aim of secondary school maths.
But I’m not sure I entirely agree with those aims. I think there’s more to maths than that. Abstraction is a way of relating different concepts, but it can also make things harder to think about. I think there’s a danger in rushing to making things harder to think about, in that it trains students to make things harder than necessary for themselves (eschewing diagrams and simple strategies).
Plus, there’s a certain puzzle-solving aspect of maths that occurs once concepts are familiar – when people know what they are trying to do and can concentrate on actually doing it. I think that’s also important and fun. Where does that fit in?
Thanks, S-T. Lots to respond to there. The number systems thing I’ll definitely talk about, but will maybe give others a chance to comment first. I am skeptical about exploding dots, but I will look more closely. The other points you touch upon I’ll respond to later in the day.
Hi, S-T. Just a few responses to some of the points you raised (but not exploding dots and numbers and number systems).
1) Of course you are correct, above: no way can Year 7 students have a proper sense of mathematics. (Most teachers lack any such understanding.) But what you can do, which is RF’s point, is to try, always, to give students a sense that there is more to mathematics than they think. It is more than just numbers and arithmetic and symbol-chasing and button-pushing. So, every topic at every year level, you want to be thinking, at minimum, of how to give hints and teasers, so that the students are wondering what mathematics is.
2) In terms of “abstraction and generalisation” being the “goal” of secondary mathematics, I don’t think RF is advocating a pure maths curriculum. But RF’s underlying point, I believe, is that one must try to counter the insane, anti-pure ideology that drives Australian teaching. So, abstraction and generalisation (and justification and proof) are not necessarily the ultimate goals, but they are the approach that one tries to foster, at least in part, whenever considering a concept or puzzle or problem.
3) Yes, puzzles and problems can be fun, and a meaningful end in themselves. But they best come when students have the understanding and the tools and the techniques to properly tackle decent problems. So, don’t be tricked by the snake-oilers, and let fake puzzling supplant the proper and necessary teaching of fundamentals.
4) No, you don’t not have to be theatrical whatsoever to be a brilliant teacher. And, in fact, most theatrical teachers are shithouse. The students may well love them, but are the students learning a damn thing? Usually not. Usually theatricality and flamboyance is simply a distraction from real learning, and usually a deliberate distraction. A little gimmick goes a long way and is a good thing; a lot of gimmick is a disaster.
Agreed on all points.
Hi, S-T. I finally got to look a little at Exploding Dots. Here are my thoughts, still tentative but not that tentative.
1) I can’t stand the videos. They seem to me the archetypal examples of pointless and distracting theatricality.
2) Even if I could tolerate (or ignore) the videos, I really can’t see the point. It seems a hell of a journey for what amounts to, at best, a (admittedly cute) gimmick, and is probably worse. If the idea really is to replace standard positional understanding computations with dot boxes, that is a very very bad idea.
3) Even if I could see the point, I don’t see how one would work this in practice.
Positional notation is not an easy thing for kids to pick up. But I don’t think ED makes it any easier. And if the idea is to use ED to explore after the positional understanding is there, I don’t buy it. I don’t buy that it is giving kids insight rather than just a (purportedly) entertaining gimmick.
I’m willing to be corrected. So, anyone can feel free to argue why they think ED might be worthwhile, and when and how. I’ll then consider it further. But, failing such argument, I’ll assume ED is just another nothing.
I missed this at first. Thank you for looking into it. I feel bad that I made you watch the videos now, because it seems like it was quite unpleasant for you. Sorry!
No, don’t be silly. I obviously volunteer myself for these things, and it is interesting to see a little of what these things are. You are also welcome to argue/conjecture for why you think ED might be useful: plenty of people seem to think so.
Hi, S-T. Sorry to be so slow to reply on number systems, but here we go.
First of all “number systems” is ambiguous. It can refer both to cultural systems – Roman and Babylonian and so on – as well as to different bases. The former is standard to include, although typically in a quick, thoughtless and pointless manner. I took a quick look at the current Cambridge Year 7 text, for example, and it was trivial and awful afterthought. As for the latter, there are undoubtedly a few teachers who do some base arithmetic, but it seems to be mostly ignored or actively frowned upon. I don’t think the Australian Curriculum has anything, and all I could see in the Cambridge texts was a single, unexplained reference to “binary numbers”. (Kaye Stacey would approve, I guess.)
Part of, and maybe a lot of, the reason for this hostility is a lingering contempt for the “New Math” of the 60s. There are definite reasons for some contempt, as highlighted by Tom Lehrer. It is easy to overdo base arithmetic, and it was typically overdone. But there was plenty of value in the New Math, including base arithmetic, and its entire disappearance is ridiculous and damaging.
As it happens I bought a Year 7 (Form 1) book a month or so ago: Modern Mathematics 1, printed in 1966. I’ll write more about the book in a separate post, but here I’ll briefly note its approach to number systems. Chapter 1 is titled Numbers and Numerals, and it is a very, very nice chapter. Totalling 24 pages, it works very hard to explain in a gentle and engaging manner the critical difference between a number and various representations of that number. This includes some good history and detail of Egyptian, Babylonian and Roman systems, as well as the abacus (which is how the Romans, for instance, actually performed arithmetic). This leads to a discussion of Hindu-Arabic numerals, place value and the zero. As a whole, the chapter is as nice and natural an introduction to high school mathematics as I have seen.
The chapter doesn’t include any discussion of number bases, except implicitly in the discussion of Babylonian numerals. A later chapter, on indices, includes a discussion of bases and base arithmetic. Though it’s a hell of a lot better than nothing, which is the current standard, this material seems much less satisfactory. There is some decent depth to the binary arithmetic, but all in all the material is unmotivated and it seems out of place. It’s not quite Tom Leherish, but it has that flavour.
The question is, what would be the purpose of showing a year 7 student other bases and base arithmetic? The New Math idea was to encourage students to think of numbers and the arithmetic of numbers independent of their representation. The goal is important, especially now when calculators are making students’ ideas of numbers moronically deci-centric, but I’m not convinced “we can also do multiplication in base seven” has ever achieved this goal, for anyone.
So, I think if you want to show other bases, it has to be with an explicit and sellable purpose. That can be naturally done with binary and similar systems. My own favourite is base three, which gives the easiest to accept irrationality proof (of √2). Of course that means also selling the idea that irrationals are important, and weird, but that is also a hugely important and typically ignored goal.
Vaguely similar to my experience with 1st year Science/ and Engineering at Swinburne. I think it’s a bad idea to just do revision. The students are keyed up for a new experience and you need to catch the moment. My solution was to start with functions of 2 variables – graphs, derivatives etc. which involved revision of much of Year 12. THINKS What is a new topic in Year 7 that revises Primary arithmetic? I was going to say other base arithmetic but I see that student-teacher has already mentioned it …
Currently I am (or was before Covid 19) volunteering at the Broady homework club. The most persistent student there is a Year 9 lad who struggles with “two times a half”. If we ever resume I will move to “three times a third”. When it comes to say “five times two” he is quite quick with the fingers. But I guess this group all have similar horror stories.
Oh, and Hi Terry.
Hi Tom; long time no see. I agree with your suggestion about a new experience.
I recall that when Monash University started, it was a radical place. Not only radical about politics. First year mathematics started with Feller’s Introduction to Probability, vol. 1.
Even at Sydney University, we started with lectures on geometry by Professor Room FRS that were nothing like anything I had ever seen before. I also recall sitting in the lecture theatre saying to myself “Terry – never forget how confused you are” – and I haven’t!
I am flying off at a tangent, yet again- sorry Red Five. The new experience needs to be do-able for even the weaker students. My former colleague John Iacono was in the first main intake at Monash doing first year maths, maybe with you Terry? John already had a graduate diploma from RMIT. He relates how was surprised that he could not understand a single word after the first few minutes. After several lectures he summoned the courage to ask a question. The lecturer replied by asking where he lost the thread. I paraphrase: John “right at the start of the first lecture”. Lecturer “Anyone else lost there?” Then slowly a sea of hand went up. To the lecturer’s credit he stopped there and restarted the course.
That’s an ancient memory and a second hand account. Ring a bell Terry?
The restarted approach worked. So the moral for me was that you can teach difficult things if you know your students and tailor accordingly. As Terry was able to do with his Easter Algebra.
I was not a student at Monash; I have only been there twice; however, I recall being told that they starting with Feller.
As for being lost, when I was in 3rd year mathematical statistics at Sydney, the professor gave us a few lectures on Ramsey’s theorem (which every statistician must know?); anyway we didn’t understand; so we asked for repeat lectures and sure enough he gave them to us again and still we didn’t understand; so we asked the tutor who devoted a tutorial to Ramsey’s theorem; he probably used the professor’s notes but still we did not understand.
I recall that it involved triple induction on
. Fortunately the examination fixed the values of 
and asked for a proof using induction on 
.
Strangely enough, just recently, I have become interested in Ramsey and his theorem; quite a character.
OK. So to try to move things along a bit, I’m of the attitude that we really want to work on two aims in secondary school Mathematics: abstraction and generalisation. Of course, these ideas are (perhaps ironically to some) both general and abstract; too much to be of much help to a person designing a unit or course.
And to answer the question that hasn’t been asked, I have been given guidance as to what is meant to happen, but I think it sucks.
Very interesting, let’s work with this. Introducing algebra as just a summary of what is learnt – going from the concrete to the abstract? As an example you could start by revising some aspects of fractions. Two times a half (in words and in symbols
), three times a third etc, all done cutting up pies or whatever. (My experience at the homework club shows me that this will be new to some students in Year 7.) Then, after lots of examples, summarise with 
explaining that 
can stand for any integer. This will be new to them all. Note I would keep the multiplication sign in the summary.
My Year 7 algebra was just learning to manipulate algebraic expressions, no justification at all until some years later. So I am quite excited by this approach. But maybe this is old hat to this audience.
So the syllabus would require identification of the standard patterns in algebraic manipulation, such as
. In computing the “Gang of Four” codify patterns in computer code – so maybe we need a gang to do the same for arithmetic. Or would that just yield the axioms of algebra?
I do like where you’re heading with this. I’ve watched for a couple of years now some teachers try to use the phrase “inverse operations” in term 1 Year 7 with very mixed results.
Otherwise, your idea is fine.
Reality is that very few year 7 teachers know the difference between an axiom and a postulate (or that such things exist in the first place).
Not sure what you mean Red Five. I would have no intention of discussing “inverse operations” or “axioms” at this level. The rule
was just a pattern in arithmetic that most students learn, and in fact can explain why it makes sense. So we are introducing algebra covertly as a summary of the arithmetic rule. We would start with simpler patterns. Maybe by revising the zero times table! and end up with 
.
Again, it seems an obvious approach. Does anyone do it like this?
Sure, you mentioned axiom before and it brought back some bad memories of a staff meeting.
The idea of using algebra to explain patterns is a very strong one, so my follow-up question is how do you then move more deeply into what algebra is/does (or do you not worry about this at year 7 level) – both genuinely curiosities.
Hi, RF. I didn’t figure it was worth querying whether or not your Glorious Leader is an idiot.
You met them (all of them actually in different settings) it doesn’t take long to reach the conclusion. Some of course are more obvious than others!
RF,
Here is a link to the year 7 IB textbook published by Haesse used by Uni High in both their standard entry and seal programmes . I like the structure of each chapter with a critical question to encourage the students to think and collaborate.
At the diploma level there is no CAS in the final exams at both SL and HL
You can view a couple of chapters on this link
https://www.haesemathematics.com/books/mathematics-for-the-international-student-7-myp-2-2nd-edition
Steve R
As a rule, I condemn year-N books for mathematics with 500 pp to burn at the stake without a trial.
The Haese family will be relieved: their book is only 456 pages. (In truth, Haese texts are pretty good, at least by modern standards.)
Thanks Steve, I’m familiar with this publisher at the DP level but haven’t seen any of their MYP stuff so this will be interesting.
IB schools that only teach the DP and not the MYP have quite a bit of freedom with their 7 to 10 courses and finding a suitable text is too often an exercise in compromise. Maybe things have changed, I will have a look at the link and see.
So thanks.
When I was in a pre-service placement a couple of years ago, I was invited to talk to a Year 7 class about algebra. Like others above, I think that algebra is important and that the introduction has to be successful.
So I offered to the teacher that I would give a class on Easter algorithms. This was a Christian school so the theme was appropriate; the students had had only one lesson in algebra at this stage; they were not particularly strong students – the school had only one Year 7 class; although the teacher liked the idea, she could not see how you could predict the date of Easter from the year.
When preparing the class, I began to get cold feet. This could be a disaster. Still, a senior teacher in the school encouraged me to persist; “aim high” was his advice.
So I did, and, if I may say so, it worked. I can say this because the teacher subsequently surveyed the students to ask them to write down what worked and what didn’t.
Reference
Fitzpatrick, T., Martin, M., & Mills, T. (2009). Easter algorithms. Vinculum, vol. 46, no. 3 (2009), 16-17.
Thanks everyone for offering their ideas so far. I have plenty of (amateur) thoughts on what people have written, but at the moment it’s all hands on deck at home. So, I haven’t had time to comment, much less post. I hope to reappear in the next day or so.
My thanks too everyone. I do genuinely like the idea of algebra but really (genuinely!) worry that some teachers (not always those teaching out of their area of expertise either) can do some serious damage with this topic and if we ruin a student in the first weeks of Year 7…
Nonetheless, it does seem to be the big idea that everyone keeps returning to, so maybe the question is not WHAT would you teach but HOW (again, COMPLETE control of the curriculum is assumed as I am living in a fantasy world for the moment).
My wife has told me that she was good at mathematics until they started to use letters instead of numbers.
I think I’ve heard that from pretty much every student I’ve had that wasn’t in the “upper division” subjects.
Thanks to everyone. A very interesting discussion, to which I wish I could have contributed before now. I have many thoughts about this. For now, I want to reiterate and emphasise, and contest, some of the points made, and to perhaps add a little.
1) There seems general agreement (or at least not strong disagreement) that algebra is at the heart of secondary mathematics. Of course “algebra” means many things but, since this blog is not frequented by Boaleresque clowns, we probably don’t have to go into that too much here.
2) As such, the big natural approach Year 7, particularly early on, is to introduce and to emphasises algebra, both as technical skill to be mastered (solving for x or whatnot), but even more so as a general approach, as a means of capturing information and asking questions.
3) However, the essential foundation for learning algebra is a strong sense of number, a properly solid understanding of numbers and a proper facility with arithmetic techniques, both written and mental. And, unfortunately, criminally, this understanding/facility is simply not there for the majority of year 7 students. Worse, in general these students have no sense of what they don’t know. They have no inkling how weak their arithmetic is, or how that weakness will trip them up at every future stage.
4) So, there’s the problem. The natural first step is to make damn sure that all students are up on the arithmetic they should’ve been doing for the previous six years, and if it takes six months and there’s no time left for bar graphs and reflex angles, well, them’s the bloody breaks. But, as Tom wrote above, you can’t just do revision or, more accurately, “revision”. First, a few students genuinely won’t need it, and will object. Secondly, many more students will know they need it, but will object. Thirdly, many many more students will need it but will not know they need it, and will object.
So, what do you do? Damned if I know. I have thoughts, but not sufficiently well-formed to want to write much here. But I do think there are some clear principles:
a) Don’t lie to students. Ever. The students need the arithmetic, and they need to know they need it. Don’t ever let them imagine otherwise.
b) Don’t pander to students. Teachers are not in the entertainment industry. The students want new experiences? Sure, fine, and they should have them, if and only if, when and only when it is beneficial to their education. If your students first need to simplify a hundred fractions, and sum a hundred fractions, and multiply a hundred fractions, and divide a hundred fractions, and they almost certainly do need to, then, by hook or by crook or by cajoling or by trickery or by threats or by bribes or by getting them drunk, you get them to do it.
c) As such, I am really skeptical of simply going on with new topics (algebra/vector-calculus), hoping/assuming that the old topics (number/real-calculus) will be properly and sufficiently revised. There is no general reason to believe that this will work, I have seldom seen it work, and it has a snowflake’s chance in Hell of working in Year 7.
d) What it does leave room for, in part, is new topics/aspects that reinforce, rather than require, previous topics. So, as suggested by Tom, one can introduce algebra as a means of capturing and exploring arithmetic truth. And, as student-teacher suggested, different “number systems” (which is still ambiguous) can be explored with little understanding of number, and can reinforce that understanding. Both ideas seem to me in principle excellent, and could be part of genuinely engaging and, more importantly, educational lessons, which will reinforce a sense of numbers and how they work. BUT
e) Nothing in (d) negates (b). Suck it up. And, tell your students to suck it up.
One last point (for now). I think there is one aspect that is really, really important and which gives a Year 7 teacher great power (if the Glorious Leaders are not idiots): Year 7 students have absolutely no idea what to expect of or from secondary school. It’s a whole new world with a whole new culture, of which the incoming students know absolutely nothing. That means a Year 7 mathematics teacher has the authority to make Year 7 mathematics whatever they want. If the teacher wants the students to understand that no longer will the students be “exploring” and “playing”, and learning topics in some quarter-assed primary way, then they can arrange that. If a Year 7 teacher wants to be taken seriously and wants mathematics to be taken seriously, and if they don’t have idiot Glorious Leaders getting in the way, then they can do it.
And right there, in your final sentence lies the biggest problem of all time. Except the idiots think they are geniuses (think Boaler and co if you will accept my apologies for bringing them up) and if we dare suggest what we really think (or worse, try to demonstrate it in the classroom) we are labelled as trouble-makers.
True story, in a way.
I think you’re better off moving on to new content, even if the kids are weak.
I would do first 5 minutes (how grades are determined, scheduling issues, policies). Then a 20-30 minute applied arithmetic test. Then get started on pre-algebra. Very gentle (simple numbers). And be sympathetic with the kids. “This is where we start getting letters, not just numbers!” Show them
x +1 = 2. then tell them x is just like a box. Then show box +1 =2. So…what plus 1 gives me 2? Then do a little simple drill, in class.
That’s enough for an hour. It will get harder in the next few lessons. But at least they are dealing with the x. I would NOT do an arithmetic review. Yes, even if they are weak. Just at least give them a feeling of seeing the x in the algebra. (You may weave some review in later.)