Patterns, Mathematics and Statistics
11 February 2019
Multiplication facts or multiplicative thinking
12 March 2019

What do high school teachers want from their students when they arrive in Year 9?

This is an important question.

One of the biggest jumps in education in New Zealand is from primary/intermediate (years 1 to 8) to secondary (Years 9 to 13). In most cases children are taught by generalist teachers in primary/intermediate (which I will call primary from now on) and by specialist maths teachers at secondary school.

Please be clear that this is NOT a criticism of Primary teachers. Primary teachers do an amazing job teaching such a wide range of subjects in a crowded curriculum to increasingly challenging groups of children. The aim of this survey is to enable communication across the gap, in order to provide better for our learners.

There have been changes in the way mathematics is being taught, and some of them have been more successful than others. I wrote about this in the post: Numeracy Crisis in New Zealand. In 2018 I ran several workshops with high school teachers entitled “Mind the Gap”, where we examined what is being taught and how in primary schools and how the two sectors might work together better to reduce the gap.

In preparation for a workshop for primary teachers I decided to ask secondary school maths teachers what they would like from the arriving Year 9s. Here is a summary of the replies from thirty-three high school maths teachers from around New Zealand. Well over half of them had been teaching for more than ten years.

Short summary

High school teachers would like their Year 9 students to be fluent in multiplication tables, both multiplying and dividing. The students need to understand place value, including decimals. A positive attitude to mathematics and a willingness to persist was also highly desired. For some teachers the attitude was more important than the skills.

What really struck me, as I read the comments, was how much the teachers care that their students are successful at mathematics.

Results

The survey begins with an open-ended question. The thing mentioned most often was multiplication tables, with fourteen mentions, followed by a positive attitude or growth mindset (ten mentions), and place value, preferably including decimals.

Teachers were asked whether each of the following was Essential, Helpful, Not really needed or “Do not waste time. The topics were suggested in an earlier informal discussion.

What is most important?

This table lists the priorities teaches give to some topics, summarised here in descending order of importance:

EssentialHelpfulNot really neededDo not waste time on this
Place value27400
Multiplication facts23600
Order of Operations181110
A “growth mindset” around maths181400
Not afraid of algebra151331
Liking maths111920
Names of geometric shapes72130
Multiple strategies for adding71742
Long division algorithm213107

Other Essentials

The next question asked about what might have been left out of the previous question – “What would you add as essential?” One answer made me chuckle – “A pen.”

Many were attitudinal, which could be seen as similar to liking maths or a “growth mindset”.

  • Willingness to give it a go. Conceptual thinking, knowing or considering WHY they do something in maths eg knowing why they use a strategy, not just a whole lot of “tricks or cheats” you have to memorize to get the right answer
  • Willingness to learn and teach others
  • A positive attitude toward maths
  • Thinking skills and a willingness to try/persist with a difficult problem (2)

Measurement figured in the essentials. Comments included:

Measurement skills, Able to measure correctly with a ruler from zero in cm, mm. Some measurement concepts, such as using a ruler by starting at zero

And fractions

Equivalent fractions, Working with fractions, Fluent in the use of Fractions Decimals and Percentages, Understanding of decimal place value; decimals, choosing how to solve problems (how many 10cm lengths can be cut from a 138cm length – what strategy to use?)

Good things they are seeing in their students

The teachers were asked what good things they are seeing in their students. Many commented that the attitudes are good. Comments included:

  • Passion for Maths (sometimes), willing to learn, talking about why they use that strategy
  • Some have different ways or can see another way to solve to get same answer
  • Their eagerness to learn after the first term when they realise that Maths is not scary
  • They are getting more technology savvy and friendly.
  • Usually students from the ‘top’ maths group at primary school have a reasonable background.
  • Many are positive with simple problem solving and routine calculations
  • Willingness to engage in learning maths
  • Keen to learn. Have seen some algebra and advanced number.
  • Resilience, this comes with a growth mindset
  • Enthusiasm amongst most of them.
  • The range of strategies they bring to the classroom, their ability to voice what they are doing.
  • I like the ability to switch between and build strategies which the numeracy project has given (to some).
  • Year 9s coming into our school usually have reasonable understanding of place value and addition strategies, and have a good understanding of the basics of fractions/decimals/percentages, shape names, angle basics, statistics

What to leave out

In order for teachers to focus on what really matters, we may need to leave out material that does not matter, or which would be better left until later. The question was: What would you prefer Primary teachers NOT to teach? These responses were sometimes contradictory.

Algebra figured highly in this, with seven teachers specifically saying that formal algebra should be left to high school. Patterns are fine.

There were comments about strategies taught as algorithms. A representative comment is that they would prefer teachers not to teach “so many different strategies for adding that students can’t do one way properly. One method for adding that a student can apply every time would be more useful.”  But then another teacher did not want their students to use “algorithms as a first choice”.

Two other comments I have previously noticed as a problem even at university level – “The crocodile < ,  > idea”, and “Don’t teach = as meaning ‘put your answer after this symbol’. Do teach the meaning of equality.”

Then there were misinterpretations of the curriculum:

“Don’t teach mean, median and mode. It warps the teaching of statistics. Teach the statistics in the NZ curriculum L3 and L4 instead and leave the NZC level 5 statistical measures until secondary school.”

One comment:

“I’d always rather pupils came fully grasping the essentials rather than had a weak understanding of lots of things”

But conversely:

“Also, I think primary teachers spend a lot of time on numeracy, but students having some idea of other areas of maths (eg measurement – how many cm are in a m, how many degrees are in a triangle) would be really helpful, as it feels like we are starting from scratch in most of these areas.”

Three people mentioned to avoid tricks and meaningless algorithms. Integers were also discouraged by two teachers. Advancement up the curriculum is another area that was discouraged. There are many ways for students to be enriched in their mathematics without pushing up into trigonometry or calculus. Discrete maths was suggested

Last words from the teachers

Several comments endorsed the efforts of primary teachers, and expressed a desire to work with them, asking me to send them the results so they can be better informed and supported.
Some teachers observed that primary teachers may feel less confident in their maths. One said, “ We need to support their own learning so they are comfortable with Maths too.”

(My own thought is that this needs to be done in a spirit of collaboration, for the good of the children. My experience in helping primary teachers with their mathematical understanding has really helped me in my other teaching. In an atmosphere of trust, adults can be good at articulating what they cannot understand or makes no sense to them, and can be less likely to just go with what you say. They are excellent at asking challenging questions. Between them, primary and secondary teachers can create some great learning.)

A rather telling comment was this: “There is such a range of what students have been taught – some consistency would be awesome! When we have such a big range of skills that students have and have not been exposed to, at High School we have to cover everything again.” This is likely to be a result of the way each school creates its own curriculum based on the New Zealand curriculum.

And this comment goes against some of the push towards multiplication facts:

“I don’t actually care what skills the students carry with them (INCLUDING table ‘facts’ etc) if only they believe in their capacity to learn. Actually while on table “facts”, I have quite strong feelings about them too. I don’t want them learned as facts (I could never do that at school myself, as it happens). I would rather students see the tables as a rich store of patterns to be explored – then they have built the capacity to become brilliant algebracists. If they learn them as facts then they have built the capacity of a cheap pocket calculator.”

Conclusion

There are a few universal ideas, but room for a lot more discussion. One does wonder why this is not specified in the New Zealand curriculum more specifically in the way it was previously. The question is, what do we do with this now? I would love to have some more comments with suggestions. I believe this is important work.

At the workshop I will be asking Primary teachers what they think High School teachers think is important, and what they, the primary teachers, think is important. Watch this space!

If people would like to add their comments and ratings to the survey, you can find it here:

https://goo.gl/forms/n6nCINXHpZ4AEwDo1

Or feel free to comment below and keep the conversation going.

 

7 Comments

  1. nigel says:

    It would be nice if the government in collaboration with teachers put together a resource book so all schools and students could work from the same page/book. Have everything that is needed or required in it!

    • Dr Nic says:

      That idea has merit. A good teaching method done well is better than an excellent teaching method done badly. Having a resource like that would support teachers who feel less confident. Maybe that is what NZMaths website is set up to do.

  2. Julie Small says:

    As a Year 8 teacher, I have spent many hours trying to work out what essential skills to teach students – in addition to engaging them in rich activities and mathematical inquiry. Over the last couple of years in particular, I attempted to unpack the Level 4 key mathematical ideas (https://nzmaths.co.nz/key-mathematical-ideas) and to compare these with the NZ Curriculum.
    From the Number Strategies and Knowledge key ideas, I concluded that students were expected to continue working with whole numbers, decimals, fractions and percentages (introduced at level 3), but that the mathematical content we are expected to cover at level 4 includes the knowledge of, and strategies to solve problems involving, negative integers. This reinforces the NZ Curriculum requirement that children working at level 4 should “Understand addition and subtraction of fractions, decimals, and integers”
    The key ideas for level 4 Equations and Expressions states “Linear equations take the form: y = mx + c … When linear equations are represented on a graph, they form a straight line with the value of “m” corresponding to the slope and “c” representing the point where the line crosses the y axis”. Once again, this is consistent with the NZ Curriculum expectation that students will learn to “Form and solve simple linear equations”.
    I therefore teach these new parts of the curriculum early in the year, and incorporate both negative integers and linear equations into as much of the maths we cover later in the year as I can. Much of the feedback you have received is contradictory to this however, which suggests a review of the curriculum and key mathematical ideas is overdue. Clarification on these issues would be extremely useful.

    • Dr Nic says:

      Hi Julie
      Thanks for that comment. There is so much that is unclear in what needs to be taught. The feeling I get is that high school teachers would like a really strong foundation, yet, as you say, there are topics in Level 4 of the curriculum that the high school teachers would rather were left for them.

  3. Sue Graham says:

    As a teacher of maths and an Across School Teacher within a Kāhui Ako I have really begun to appreciate the importance of effective transitions within the schooling sector, particularly regarding the dip in student progress as a result of transitions as well as the possible causes.
    Within our Kāhui Ako we have just begun a major focus on transitions from ECE to primary school, within the primary school and from primary to secondary school. The resulting discussions between all teachers involved is beginning to develop meaningful trusting relationships across the sectors and the basis for a shared understanding of the skills, concepts, values etc children/students might require as they make significant transitions. Discussions and visits to see programmes in action is also beginning to strengthen outcomes.

  4. Terry says:

    A fair survey – I have just moved to a new school where quite a few of my students feel like they are not good at maths, therefore don’t like it. One of the reasons I noticed (I did a survey with them to find out why) was that they can’t do their times tables – so I have given them a times table chart to use – because like the survey above says, we need to focus on creating a positive attitude, so if this is what is holding some students back, I am all for giving them a chart. Hopefully, whilst using the chart they will become increasingly confident in both their times tables facts and maths in general. As a primary school teacher we have far to much to cover, and with the ever changing society (where parents are working full time etc) it isn’t going to get any better – so we need to work smarter and focus on some of the basics, so that when they attend high school they still have a ‘positive’ attitude to school life.

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