A few days ago we received an email from Aaron, a primary school teacher in South Australia. Apparently motivated by some of our posts, and our recent thumping of PISA in particular, Aaron wrote on his confusion on what type of mathematics teaching was valuable and asked for our opinion. Though we are less familiar with primary teaching, of course we intend to respond to Aaron. (As readers of this blog should know by now, we’re happy to give our opinion on any topic, at any time, whether or not there has been a request to do so, and whether or not we have a clue about the topic. We’re generous that way.) It seemed to us, however, that some of the commenters on this blog may be better placed to respond, and also that any resulting discussion may be of general interest.

With Aaron’s permission, we have reprinted his email, below, and readers are invited to comment. Note that Aaron’s query is on primary school teaching, and commenters may wish to keep that in mind, but the issues are clearly broader and all relevant discussion is welcome.

*Good afternoon, my name is Aaron and I am a primary teacher based in South Australia. I have both suffered at the hands of terrible maths teachers in my life and had to line manage awful maths teachers in the past. I have returned to the classroom and am now responsible for turning students who loathe maths and have big challenges with it, into stimulated, curious and adventure seeking mathematicians. *

*Upon commencing following your blog some time ago I have become increasingly concerned I may not know what it is students need to do in maths after all! *

*I am a believer that desperately seeking to make maths “contextual and relevant” is a waste, and that learning maths for the sake of advancing intellectual curiosity and a capacity to analyse and solve problems should be reason enough to do maths. I had not recognised the dumbing-down affect of renaming maths as numeracy, and its attendant repurposing of school as a job-skills training ground (similarly with STEM!) until I started reading your work. Your recent post on PISA crap highlighting how the questions were only testing low level mathematics but disguising that with lots of words was also really important in terms of helping me assess my readiness to teach. I have to admit I thought having students uncover the maths in word problems was important and have done a lot of work around that in the past. *

*I would like to know what practices you believe constitutes great practice for teaching in the primary classroom. I get the sense it involves not much word-problem work, but rather operating from the gradual release of responsibility (I do – we do – you do) explicit teaching model. *

*I would really value your thoughts around this.*

*Warm regards,*

*Aaron*

Hello Aaron. I’m a U.S. resident, as you might guess. My advice is not to be in a rush, but to learn from the best people and organizations. Then your natural creativity will help you design your own activities and worksheets. So read the books in chronological order by Martin Gardner. Look at the materials published by The Art of Problem Solving and the Mathematical Circle Library books published by the AMS-MAA. Look at the Brilliant website. Play with the best manipulatives like the Cuisenaire Rods. Turn every subtraction problem into a money problem where you are cashing in tens or hundreds instead of “borrowing”. Have the students learn to move around the Monopoly board without counting. This will help them add and subtract without counting. Have the students look closely at the times table to notice patterns. This will help with memorization. For example, 7 times 8 is 56. Make the smaller number 1 number smaller and the bigger number 1 bigger. The new product will be smaller by 2. Notice that 6 times 9 is 54 (2 smaller than 56). Teach students to multiply on their fingers. Each fist is 5. Lift up two fingers from each balled fist. Now you are ready to multiply 7 times 7. The 4 fingers that are up are tens, so you have 40. There are 3 fingers down in each hand, so multiply 3 times 3, which is 9. Combine 40 and 9 for 49. So 7 times 7 is 49. But why does this work? Learn how John Conway can calculate rapidly the day of the week you were born in less than 5 seconds. Then teach this to the students by having them work with a circle with 7 points evenly spaced on the circumference, one for each day of the week. Learn about the Kaprekar constant 6174. And on and on.

Marc Roth

Thanks, Marc. I have serious reservations about your third and fourth sentences. (And the second: get out while you can.) After that, you point to some great people/resources, which are indeed undervalued or are plain unknown by many teachers. But I’m not sure, except perhaps by implication, that you’re suggesting an overarching philosophy or approach, which is how I was interpreting Aaron’s question.

Hi Marty. My third and fourth sentences are what worked for me. But my situation was atypical. I was single and had more time than most to study and try to be creative. I gave students juvenile detention centers a lot of drill, but it was rich drill with a lot of tactile and visual support. I used what’s now called open middle problems and multi-step drill. Also, I worked backward in the sense of knowing where I wanted students to get to and to take them there step by step. I made some original discoveries about the harmonic mean and quadratic functions and incorporated them in the sequence. I also made use of modular arithmetic and finite fields for teaching operations with polynomials.

Hi Marc. I’m not arguing against what you do (which sounds very interesting), but the “atypical” may be the issue. My problem with your third sentence is that is impossible for the majority of teachers to determine the “best people and organizations”. The field is flooded with hucksters. My problem with your fourth sentence should be pretty obvious: teachers are insanely busy. Plus, the idea that each individual teacher should be required to be that creative is plainly ridiculous. There

mustbe good (I mean genuinely good, not officially good) class resources readily available. So, where?One solution may be to have more math specialists teaching math in the elementary school. Time should be carved out to provide career long learning. Right now I am recently retired and trying to share my worksheets and lessons. Getting that done is not easy. I speak at conferences and math circles, but getting things published and shared is not easy for someone like me with few technical skills.

Hi, Marc. I’m still not convinced. Unfortunately, most “math specialists” are not that special. I’m also not convinced that there is any shortage of very good materials. What I think is lacking is the ability of teachers to properly evaluate either specialists or resources. I don’t see how that problem is overcome without better training/retraining of teachers.

(Sorry for the delay in your comment being posted. It went to the spam folder for some reason.)

6174 is a great exercise. I do it a lot with early secondary students. Also is the write a 3 digit number twice, such as ABCABC then show them it can be divided by 7, 11 and 13 without leaving a remainder – this one really surprises at first, and is a really non-confronting way for me to find out quickly which students know division procedures.

As a secondary teacher, I really love it when students leave primary school with really solid understanding of

goodstrategies for adding, subtracting, multiplying and dividing. Knowing WHY things work can be done in secondary school (usually) but confidence can really be made amazingly strong with a competent primary teacher.Estimation is a bonus skill that I feel is a bit under-valued.

Thanks, RF. So, perhaps a 1-line summary, which you may or may not endorse is: the foundation of good primary school mathematics is arithmetic, and the foundation of good secondary school mathematics is algebra. I also really like your ABCABC test. (Can you do it as an exercise(s) in some manner rather than a demonstration? Or, is that too heavy in the Year 7 context?) As for estimation, I’m too pedantic in my nature to enjoy it, but I strongly agree that the value of estimation is underrated.

However, even if we agree that arithmetic is the main game in primary school, even if we agree to not be distracted by numeracy and data and statistics and bar charts and rotating houses and computer games, and all the other garbage the Curriculum mandates, I think the devil can slip into your proposal. Modern maths education is the source of our sickness, but I doubt there is any maths ed prof who would disagree with “really solid understanding of

goodstrategies …”. (Well, that’s probably wrong: some profs arereallydumb. But most would agree with your statement.)But what constitutes a “good” strategy, or a solid “understanding” of that strategy? Do written strategies matter, or just mental? Should some strategies have a central focus, or is it a backpack of approaches? And doesn’t “understanding” points towards the “why” that you suggest can mainly be left for secondary school?

Yes I endorse that summary. Algebra is to me the single most important idea in secondary school (you could add the word Mathematics at the end of that phrase, but I remain unapologetic for the omission) and arithmetic is the vital set of skills from primary school which make the idea possible.

Written strategies are important, but there comes a time when it is easier to simply know things (in primary school this is times tables and in secondary school this is exact values of common trigonometric ratios) to enable more difficult ideas to be explored unencumbered.

As to the ABCABC activity, I try to build it up to the point where students realise that 7x11x13=1001 and ABCx1001=ABCABC and hence it always works. The 6174 activity takes a bit longer to demonstrate as there are so many different starting numbers although I have seen some nice flow-charts drawn for this to show that 6174 is always the result.

Unfortunately, I feel that too many primary teachers (and I have met a lot of pre-service primary teachers in my work at a major university education department) seem to fear mathematics and this does not make for a class of students who love numbers.

As to why I feel the “why” can be left to secondary school – algebra.

There are a couple of axioms, one of which is the following:

1) You cannot teach fractions to students who need 10 second for working out 3 * 3. They will always struggle with elementary arithmetic instead of keeping their eyes on the new rules. As a consequence, basic arithmetic is mandatory. One of the worst things you can do is imitate the method currently en vogue in Germany: teach four different ways of adding numbers and let the students choose which they prefer. Instead make sure there is a

standardmethod.2) 80 % of 5th-graders in Germany compute 12 * 12 = (10 + 2) * (10 + 2) = 104 because they were taught how to multiply 8 * 12 this way. Either do distributivity properly or not at all.

3) Teach them to read exercises with open eyes, that is: throw in problems such as 12 + 19 – 12 etc. that become trivial by looking at the problem the right way. This involves talking about rules (without using big words such as commutativity, of course).

4) Estimation is difficult. Most kids first work out the precise result and then decide how to estimate. Estimation is important when you get to decimal fractions. If you do estimation, do it in connection with money.

5) The children need to be familiar with standard measures (money, time, lengths, weight or mass).

6) If the math book you are using mentions probability, then throw it away. The PISA-powers in Germany have made probability a topic in grades 3 and 4. Tar and feathers, please!

As a rule, children hate maths when the results they get are wrong most of the time. Your job is therefore not to entertain them but to show them how to do basic arithmetic.

Franz, I’m trying to act as facilitator and resist being the judge, but I think your last paragraph is brilliant. It’s worthy of Neil Postman.

I guess the question it raises is, how does a teacher deal with children (and their parents and principals and governments) who view education as a form of entertainment? Is there any place in primary (or secondary) school for exploration: Gardner stuff, open-ended problems and so forth? If so, how does one fit them in with the teaching of the fundamental skills?

Brief thoughts on your six ( = “couple” = “one”) axioms:

1) By “standard method” do you mean traditional written methods (modulo slight variation)? Or, are there other methods that are preferred or will suffice? Are mental methods (beyond automatic facts) important, and if so what should be taught, and how? Is there a danger that this can all become too syntactic, resulting in students manipulating symbols without concerning themselves with meaning?

2) Ugh! Should multiplication tables be taught to 12, or does 10 suffice? (Yes, it’s a Dorothy Dixer, but go for it.)

3) Important, not hard to do, and hard to argue against.

4) Why do you think estimation with money is the preferred scenario?

5) Sure. Don’t forget furlongs.

6) Why do you think probability is such a bad choice for middle primary school? (Another Dorothy Dixer.) Perhaps something to pursue on a PISA thread, but why do you attribute this probability push to PISA goons?

Go on QUORA and see the same stupid questions over and over: when do you multiply and when do you divide; when do we use fractions in real life? Why do we use pi when mathematicians don’t even know what it is, and why don’t we just define it to be 3.14? And for a while, endless discussion of BODMAS and PEMDAS, and which is better. And what is the best way to solve a quadratic equation?

I think it’s painfully obvious that a lot of kids are being badly treated by the education professionals.

Hi, Jack. Obviously things are in a bad way. But shooting those barrelled fish doesn’t really help Aaron. By “education professionals” do you mean teachers, or the education academics, the teachers of the teachers?

Jacks observation is of course sadly correct. I will admit really hating the BODMAS (what does the “O” even represent???) PEDMAS, BEDMAS, BIDMAS, PEMDAS and all other acronyms (except perhaps USBB which is genius) because in my (non-qualified) opinion, to use these acronyms is to miss the point entirely. It is a bit like a teacher telling a student “because I said so” when they ask “why?”.

I’m a high school teacher, so I’ll talk about teaching year 7 (and actually applies to any year). And have only been teaching for three years, so I’ll just briefly say somethings I do that work well for me (take it with a grain of salt, I’m still learning).

I guide students in developing the concept, not just state it to them. It involves lots of questions asked to the class to draw out their prior knowledge and build it towards understanding the concept. Visualisations help support my explanations.

I ask “what if” questions. I have a simple example and change small parts of it to see how that changes the solution. This is to support students understanding the “nature” of the concept / problem.

I scaffold how to apply their knowledge to solve problems. The scaffold is a series of questions students should be asking themselves (sometimes general, sometimes concept / topic specific). It prevents students getting too lost and being cognitively overloaded. Over time, hopefully, they will become internalised and not needed. Then the next level of scaffolds are needed.

I try to pose thoughtful problems. Theoretical ones (e.g. why do these differently shaped triangles with the same base and perpendicular height have the same area?) or practical ones (how do I find the volume of an irregularly shaped solid?). This is to get students to think about communicating clearly their solutions that require more than one line of work.

Potii, for an “only three years” teacher, you’ve posted some remarkably insightful comments on this blog.

I very much like the Socratic-ish feel of what you say you do. One question which may assist Aaron, and a question I

alwaysask of Year 7 teachers: what do you with students who come into secondary school without the necessary arithmetic skills? And, question 0, what do you regard as the necessary arithmetic (or other) skills?I find students who struggle with arithmetic struggle with abstraction. This goes the same for algebra, just those students can abstract to a better degree than students who struggle with arithmetic.

My strategy has been to link physical concrete examples to visual concrete examples then to the abstract notation / mental thinking. The challenge is to find the point in their understanding of arithmetic at which they “get it”. Then build from there.

Unfortunately, in a classroom situation I don’t have a lot of time to carefully do this. I try to plan stuff for students to support them but I also have to go on with the syllabus as well as manage behaviour. Some schools have good intervention programs for groups of students or support teachers that help but not all. They also vary on quality depending on who is running them (e.g. support teachers don’t always have the best mathematical understanding and may just show procedures without meaning).

My first lesson of Year 7 (for what it is worth – I don’t teach below Year 7 under my current school structure) is underpinned by the following (which I state quite clearly and unashamedly):

Mathematics is

anythingwhich obeys the laws of logic. What you will be taught this year is only a tiny glimpse of what Mathematics really is.There are two key skills in this subject: abstraction and generalisation. They are not independent of each other.

Anyone who mentions the phrase BODMAS or anything similar will be politely asked to rephrase their question or statement. This request may become less polite as the year unfolds.

Before I offer my suggestions I totally agree with these comments which are insightful and should be obvious. Marty: ” the foundation of good primary school mathematics is arithmetic,”

Franz: “basic arithmetic is mandatory. One of the worst things you can do is imitate the method currently en vogue in Germany: teach four different ways of adding numbers and let the students choose which they prefer. Instead make sure there is a standard method……..As a rule, children hate maths when the results they get are wrong most of the time. Your job is therefore not to entertain them but to show them how to do basic arithmetic.”

My comments: You become excellent at arithmetic with practice, practice and more practice. As a teacher you design the practice that it does not become boring or overwhelming. The students see the improvement/success and have hope that they can do maths. A fundamental arithmetic skill is counting. I believe counting is not emphasized at the teacher training institutes; I ask preservice teachers at my school about this and they say no, it resonates straight away with them that this is a serious omission. Children enjoy counting, it is an excellent preparation for addition, subtraction, multiplication and division. 3, 6, 9, 12, 15, 18, 21…. is 3 +3 =6, 6+3 = 9, 9+3 =12. Counting backwards is subtraction 21 -3 =18, 18 -3 =15 ,15 -3 =12. Multiplication from counting 3 +3 = 6 which is 2 groups of 3, 3 + 3+ 3 =9 which is 3 groups of 3. This leads to division 9 divided by 3 =3. It is not difficult to vary the activities, eg what are the the missing numbers 3, 6, – , 12, 15, – or counting to 100 or beyond to see the pattern or read the count out aloud, say the count by memory and record in writing the count or have objects that represent the count or write in words three, six, nine and teach how the digits are written.

The importance of counting can never be downplayed or by not allocating enough lesson time.

I have used this approach from grade 2 to grade 6 with 100%success.

When I fill in for an absent grade 5 or 6 teacher I’m not surprised that the students are having difficulties with maths, they can not count.

As Franz has said teaching 4 ways to add or subtract, or multiply or divide is in vogue in Germany, it is also in vogue in Victoria. Yes teach more than one way but not at the same time. Teachers should dismiss this practice immediately as it confuses the students because of cognitive overload. The other ways can be introduced as each way is mastered.

Moving to the times tables which has been sic, renamed the multiplication facts.

You have to memorize the times tables like counting. The vast majority of students do not have automatic recall if they use strategies without chanting. I have found temporary learning occurs with strategies and permanent learning with chanting. Do both is best.

Two books I would recommend for Aaron to read are: “The woman who Changed her Brain” by Barbara Arrowsmith Young and “Soft-Wired” by Michael Merzenich. These will provide the connected link between learning and the brain.

I dunno, John. It sounds awfully like that Medieval rote learning, which people used to do.

Rote learning has a bad reputation. I believe the critics often take an extreme position against it.

Rote learning is necessary but it does not have to be boring. I apply it in the warm up part of the lesson. It has a purpose of automating arithmetic skills. There are many research and journal papers supporting its effectiveness.

I googled one of these.

186 THE JOURNAL OF SPECIAL EDUCATION VOL. 36/NO. 4/2003/PP. 186–205

Ten Faulty Notions About Teaching and Learning That Hinder the Effectiveness of Special Education William L. Heward, The Ohio State University

Drill and Practice Limits Students’ Deep Understanding and Dulls Creativity

Today’s teachers are told that drill-and-practice exercises on basic skills are not as important as was previously thought. Drill and practice, they are told, produces only rote memo rization. When did educators decide that memorizing things is undesirable? Although we can debate which facts, relation ships, and learning strategies are most useful to commit to memory, every educated person knows many things by mem ory. Rote, the word most frequently used to demean the out comes of drill and practice, means to do something in a routine or fixed way, to respond automatically by memory alone, without thought. It is good to know some things by rote. Prop erly conducted drill-and-practice exercises help students de velop fluency (the routine and automatic connotations of rote) in the knowledge and skills they already understand. For ex ample, students doing drill-and-practice activities for addition and subtraction facts should already know how to add and sub tract with accuracy; that is, they understand what they are doing. Drill-and-practice activities are designed to build stu dents’ fluency (accuracy and speed) with the math facts. Stu dents who can perform basic tool skills (e.g., simple math facts, letter–sound relationships) with fluency are then able to apply those skills as components of more complex tasks and problem solving (e.g., long division, reading). Executing tool skills in rote fashion—without having to stop and think about them—enables students to attend to and solve larger, more com plex tasks that require critical thinking (Johnson & Layng, 1994).

Today’s teachers are also told that drill and practice dulls students’creativity. In fact, repeated practice leads to increased competence and confidence with the subject matter or skills being practiced, thereby providing students with the knowl edge and tools with which they can be creative. In a major re view of research on what teachers can do to influence student achievement, Brophy (1986) drew this conclusion on the re lationship between drill and practice and creative performance: Development of basic knowledge and skills to the necessary levels of automatic and errorless perfor mance requires a great deal of drill and practice, . . . drill and practice activities should not be slighted as “low level.” Carried out properly, they appear to be just as essential to complex and creative intel lectual performance as they are to the performance of a virtuoso violinist. (p. 1076) Compare the attitudes and approaches to drill and practice by many academic teachers with the attitudes of educators who are held accountable for the competence of their students. The basketball coach or the music teacher needs no convincing re garding the value of drill and practice on fundamental skills. No one questions the basketball coach’s insistence that his players shoot 100 free throws every day or wonders why the piano teacher has her pupils play scales over and over. It is well understood that these skills are critical to future perfor mance and that systematic practice is required to master them to the desired levels of automaticity and fluency. We would question the competence of the coach or music teacher who did not include drill and practice as a major component of his or her teaching. Some say that drill and practice of basic skills does not contribute to the achievement of literacy or higher-order think ing skills and that class time can be better spent in activities that are more enjoyable and will contribute to a deeper un derstanding. Kohn (1998) contended that “a growing facility with words and numbers derives from the process of finding answers to their own questions” (p. 211). In other words, it is unnecessary to provide students with drill and practice on basic academic tool skills such as multiplication facts and letter– sound correspondences; instead, teachers need only to en courage children to ask and to solve questions they may have about fun math problems and interesting stories. In the pro cess of constructing their own meanings from these activities, the students will become fluent readers and skilled calculators.

This sounds wonderful; I would welcome evidence of the phenomenon. It also places the cart before the horse, for it is facility with words and numbers that gives students the tools they need to solve problems and find answers to ques tions they or others may ask (Johnson & Layng, 1994; Sim mons, Kame’enui, Coyne, & Chard, 2002). Critics of drill-and-practice activities are so disdainful, one wonders what horrible malpractice they have witnessed. Kohn (1998) stated, “The educational crisis we are allegedly facing has occurred under a ‘drill-and-skill,’ test-driven sys tem in which students are treated as passive receptacles rather than active learners” (p. 197). He went on to say, “A sour ‘take your-medicine’traditionalism goes hand in hand with drill-and skill lessons (some of which are aptly named ‘worksheets’)” (p. 212). Kohn (1998) does not like that what students do in school is referred to as work. He does not like the terms seatwork and homework, and he has suggested that referring to schooling with the metaphor of work has “profound implications for the nature of schooling” (p. 210). The implications are indeed profound if students do not receive regular drill and practice of critical academic tool skills because their teachers regard such activities as unimportant, demeaning, or just too much work for students (Sewall, 2000). Of course, drill and practice can be conducted in ways that render it pointless, a waste of time, and frustrating for chil dren. Research has shown, however, that when properly con ducted, drill and practice is a consistently effective teaching method. For example, a recent meta-analysis of 85 academic intervention studies with students with learning disabilities found that regardless of the practical or theoretical orientation of the study, the largest effect sizes were obtained by interven tions that included systematic drill, repetition, practice, and review (Swanson & Sachse-Lee, 2000). For procedural guide lines and suggestions for conducting fluency-building prac tice for academic skills, see Binder, Haughton, and Van Eyk (1990); Johnson and Layng (1994); and Miller and Heward (1992)

John, I was being sarcastic …

Okay, got it.

No worries, and thanks, John. Just to be clear, I agree with you 100%. Your paper excerpt (and/or your own words – the formatting makes it difficult to discern) is excellent. The disdain for rhythm and practice and repetition indicates as well as anything the idiocy of modern mathematics education. The only useless chant is Kerry Chant.