I believe that maths (generally) is a damaged subject. What other subject provokes a lifelong fear in learners? Teachers often focus on the wrong things and teach in inaccessible ways. Good maths teachers know that mathematics is a wonderful subject for all learners, it’s deeply conceptual and lends itself perfectly to exciting learning engagements. It’s actually my favourite subject to teach, but I have also been guilty of these bad practices.
Carol Dweck, the psychologist behind growth mindsets/fixed mindsets, agrees with Jo Boaler that mathematics is a subject that is “most in need of a mindset makeover”. Traditional mathematics classes promote fixed mindsets in all students and are driven by grades, scores, and test performances.
“Mathematics, more than any other subject, has the power to crush students’ spirits.”
This post is inspired by Jo Boaler’s must-read book. Anyone who teaches mathematics needs to buy a copy immediately (click the above image). The book outlines the myths about mathematics that are still widespread and damaging. Of course, the book goes into a lot more detail and offers many innovative ideas. To summarise though, I just want to outline some of the bad practices that are still prevalent. If you haven’t already, will you join me in ditching these in 2017?
There is a widespread misconception that good mathematicians answer questions quickly. As a result, the emphasis has been on memorisation and recalling more than conceptual understanding. To deeply understanding something often means to think about it slowly. Mathematician Laurent Schwartz famously recalled feeling stupid at school due to his slow processing.
“Rapidity does not have a precise relationship to intelligence.”
Consider the ability groups in your class (this is discussed below). Is it possible that your ‘low’ kids are deep mathematical thinkers? Is it possible that these children are highly capable mathematicians who, just like Schwatz, feel stupid because education has always valued the wrong thing? Likewise, is it possible that your ‘top’ kids can answer quickly but lack understanding? The emphasis on speed and the comparison that students make to each other are damaging. Rapid recall of facts is not as important as we think. In regard to speed tests, it is often argued that students are only in competition with themselves. Their self-esteem is raised as they see their own increased scores. Even if this is true, the emphasis is still put on speed and this mistaken mindset is reinforced.
Children are very aware of who the ‘top’ and ‘bottom’ kids are. However subtle you think you’re being, students are aware of how their ability group ranks within the class. This is obviously damaging to ‘lower’ students’ self-esteem but we rarely discuss how equally damaging it is to other groups. ‘Middle’, or ‘average’ groups are famously ignored in comparison to the two extremes. Furthermore, ability groups enforce a fixed mindset in all students. If students believe that their maths ability is fixed (as they often do), then we promote a classwide notion that nobody needs to try. This affects the high-achieving students as well, explaining why many of them ‘average out’ as they proceed through schooling.
“Research shows that a high number of high-achieving students drop mathematics, and a decline in conceptual understanding when they are pushed into higher level mathematics classes and tracks.”
Instead, I prefer rich tasks and/or optional tasks. Rich tasks are sometimes called ‘low threshold, high ceiling’ tasks because they are accessible by all students but can be taken to any level. With one task (as long as it’s the right task) you can meet the needs of all students. With optional tasks, I prepare a few different learning engagements, varying in difficulty, but give the ownership to the students. As long as they are in their stretch zones, I don’t mind which they choose. As evidence of their stretch zone, I expect struggles, perseverance and mistakes. In both alternatives to ability groups, ‘lower’ students will often surprise you. It’s difficult to balance high expectations and realistic expectations, so our group assigned tasks often fail to meet the students’ needs. Your students are usually more aware of their stretch zones than you are, so give the responsibility to them.
“Countries as different as Finland and China top the world in mathematics performance, and both countries reject ability grouping, teaching all students high-level content.”
Facts and procedures
Ok, these should not be ‘scrapped’ as the title suggests, but neither should they be the end point. Mathematical facts and procedures should be the foundation for understanding. We must take our students’ learning much deeper. For example, my student recently decided to create a pie chart from percentage data. He recalled the process that he had learnt from his tutor (the percentage divided by 100 x 360 = the degrees in the pie chart). I asked him to explain the process. Why divide by 100? Why multiply by 360? He had no idea! This wasn’t a problem, but his known procedure was simply the starting point for deeper thinking. By the end of the lesson, he suggested that we could just multiply the percentage by 3.6 instead. I had to think about this, but he was absolutely right! Stretching your students’ thinking is surprisingly easy, using questions such as ‘why?’, ‘why not?’ and ‘what makes you say that?’ One of my favourite ways to see their understanding is with true or false statements. E.g. 0.25 is bigger than 0.5. True or false? PROVE IT!
“Mathematics is a conceptual domain. It is not, as many people think, a list of facts and methods to be remembered,”
I almost don’t dare to write about homework on my blog. It is hugely controversial and stirs up many passionate opinions. This huge debate is ongoing. Regardless of our own opinions, many teachers have to assign it due to school policy. Jo Boaler, by the way, does not support homework. For teachers who do assign it, we can at least make the best of it. By ‘traditional homework’, I mean the kind of soul-destroying worksheets that I was given when I was a student. As an alternative, I am an advocate of flipped learning. Connecting to the facts and procedures point, consider ‘flipping’ these aspects as a homework task so that you can use class time more purposefully for deeper thinking, application and inquiry. For example, my teaching partner and I recently flipped some knowledge regarding 3D shapes. It is possible to explain what edges, faces and vertices are in a video, so we don’t want to waste precious class time on this. The video itself is not what enhances learning. Learning is enhanced by making better use of face-to-face class time. The flipped model also works well for introducing calculation procedures.
As I explained in The problems with school league tables, my issue is not with tests themselves, it is how they are often used. Tests reinforce the notion that maths is a performance subject of grades and scores. They should not be the only way that your students can demonstrate their understanding. Last year, my students did a pre-test about fractions. This was a way for us to gauge their prior knowledge. I also offered an additional task. I simply gave them some manipulatives and asked them to show me what they know about fractions. The same student who scored 0% on the test used the equipment to demonstrate equivalent fractions, calculating fractions and the connection to decimals. An unbelievably different outcome! Pencil and paper tests just don’t work for some students. In the quote below, Jo Boaler is talking about the USA, but this probably applies to most of the world!
“For many decades in the United States, tests have assessed what is easy to test instead of important and valuable mathematics.”
I don’t judge teachers who use these methods. I used to! I confess to every mistake mentioned above. We often teach how we were taught. Thankfully, times have changed. By changing what we focus on, how we teach and (most importantly) how students learn, we can instil a lifelong love of mathematics as opposed to a lifelong dread.
How do you make your maths lessons more joyful? How do you help all of your students to progress? Are there any ways that you taught in the past that you now regret? Please leave a comment below and keep the discussion going. Once again, buy Jo Boaler’s book! I can’t recommend it enough.
On a side note, my blog turns one this week! I have been publishing my musings for one year. To mark the occasion, I’ll use my next post to answer my most frequently asked question: “how do you find time to blog?”. Click ‘follow blog via email’ or like my Facebook page to be notified of new posts.
This will always be one of my favourite blogs as I can so relate to it as a predominately Maths primary school teacher in the first part of my teaching career. However for me it’s a bit like the chicken & the egg for some…. some children need the rote and process to then be able to unpick and realise the concepts involved (this was me), others need the concepts, break down, various methodologies to them determine a process and rote method that they can then hang on to…. no way is wrong…. the fundamental underlying point is that ALL these should be taught from the outset and children given the choice of what best fits them not what best fits the teacher.
Thanks again Adam for a thought provoking discussion topic.
Thanks for sharing your thoughts and experience. I agree with your summary. We need to make sure that we teach for conceptual understanding, including the facts and procedures, in whatever way brings them all together for the students. The point here is that maths should be much deeper than facts and skills alone (as all subjects should be).
I have been teaching for 20+ years, and could not agree with you more. When I was a student, I shut down in math class because I wanted to know the “Why?” not just memorize data.
Had math been taught the way you describe in your blog, I would have had a totally different mindset.
I LOVE watching my students discover the why’s of math, and do not assign traditional homework assignments.
This is great to hear! The shift is definitely gaining momentum and Jo Boaler’s research is reaching more teachers and students. I hope that we see big changes in maths education over the next few years. No student should shut down while learning this fascinating, engaging subject.
‘Why?’ is such a powerful question to ask in mathematics. It’s great that you knew this even as a student! Your students are lucky to have a maths teacher who understands what good maths education is really about. It’s not about facts.
If you enjoyed her book, take the course. I’ve really been enjoying it, and as it’s offered all the time, you can do it at your own pace! 😉
Our Head of Maths is trying to organise this for a group of us. If she is unable to, I’ll definitely do it myself. I’m glad you’re enjoying it. I was concerned that it would just be the same as the book, but I guess not from your reaction.
See you tomorrow night for the book study!
It’s interesting and set up well. I also love the go-at-your-own-pace system. There’s videos, some reading and lots of reflection.
“Mathematical facts and procedures should be the foundation for understanding.”
I would argue that this statement is upside-down: conceptual understanding and making meaning should come first, acting as the foundation for transferring meaning to symbols, developing fluency with facts, and understanding and using procedures. This is in line with PYP beliefs about the nature of learning mathematics. Imagine students working on the procedures for 2-digit addition facts without first understanding the concept of place value?
What do you think?
Thanks for the comment. Excellent point! I don’t disagree with you. But I do think that students often know how to do things (by memorising a method) before they actually understand why. I guess it could work both ways. What do you think?
For me facts, procedures and conceptual understanding are not linear but work together to achieve deep understanding. You still need facts to reach conceptual understanding. You can’t understand place value if you don’t know the names and symbols of the numbers and what they each represent. It depends on the experience and needs of each student.
I totally agree (and not just in a maths context). Problems arise however when teachers stop at facts and procedures and students don’t get the opportunity to reach deeper understanding. This issue is particularly prevalent in places where standardised tests come with high stakes and teachers feel rushed to cover content.
Thanks for the comment.