What are pseudocontexts, and should K-6 math teachers be concerned about them?
I came across the term in Dan Meyer’s excellent blog, in which he explores better ways of engaging students in learning math, and calls out “fake math” and poor teaching. You should go check out Dan’s work, I find it really challenging and encouraging.
As a publisher of educational content for K-6 math, I am deeply concerned about students’ recognition of the importance and the usefulness, the utility of the math they learn.
This means that I am always looking for ways to show students examples of math in real life. But does that mean I have to restrict my examples to ones where someone is asking a specific math question that matches the topic students are learning, or can I ask questions which someone might ask, but probably didn’t?
What do you think? Leave a comment below if you’d like to share your thoughts.
Recent examples of “math in real life contexts” are found in my Where’s the Math? video series, which you can see here. Am I guilty of introducing pseudocontexts?
In the busyness of classroom teaching, do you find math lessons becoming a bit stale? Are textbook lessons getting you and your students down a bit?
I believe that students crave interesting, relevant lessons, especially in math. How can we provide such lessons?
It’s a simple idea: find real math going on in the world around you and bring it to the children’s attention.
In the video I talk a bit about the new kitchen we’re having put in. There’s a heap of math that the builders have to get right, including measurement, simple arithmetic and three-dimensional geometry.
What can you talk about with your students? Here are some ideas:
Real-Life Contexts for K-6 Math: Some Suggestions
Take photos with your smartphone, add them to PowerPoint slides, ask students “What do you notice?” or “What questions are you thinking of?”
Talk about shopping experiences where you had to figure out a best buy, someone gave you the wrong change or something cool happened
Explain how you adapted a recipe for a different number of servings
Talk about your favourite sport and how rankings work, and how many points teams have to win to come out on top this season
Talk about designing something cool, such as a garden, a craft project, a greeting card or a decorated cake
What ideas have you used to bring classroom math to life? Leave a comment below.
Have you ever felt like banging your head on the wall when trying to get kids to learn something? Do you feel you’re not up to the task of “getting through to these kids” and making them learn something?
Here’s a simple suggestion from me: trust the kids’ natural abilities to make sense of their world.
Don’t sit on your hands and leave it all in the kids’ hands, of course: that isn’t what I mean.
But allow the capabilities built into every human being to “kick in” in response to your input. It’s your job to create the best atmosphere for learning to take place; you just don’t have to make it happen due to your force of character, cajoling, pleading or other desperate tactics.
Do you wonder about the wisdom of sending math homework home? Have you ever had a sneaky suspicion that parents may actually not be helping their kids learn math?
Well, now there’s evidence that supports your caution.
Research: Do Parents Help, or Hinder Their Kids’ Math Learning?
If you have any interest in helping kids with their math learning, I urge you to watch the video, then go and read this article:
Briefly, the findings of this year-long research using children in grades 1 and 2 and their parents were:
Parents who reported math anxiety themselves, and who helped their children with math homework resulted in children who:
were more anxious about math themselves
learned less math over the year
Reading scores were not affected
Parents who did not help with math homework had no significant effect
This is a hugely important set of findings. I believe all K-6 math teachers should address the issues here, for the sake of their students’ learning.
Suggestions for Every K-6 Math Teacher:
Don’t set math homework that is likely to cause stress or anxiety in parents who themselves are not confident in math
Talk to parents about the messages they send (often unwittingly) to their children about math and math learning
Make suggestions to parents about low-stress ways to help their kids with math
Point out everyday activities that parents could use as springboards for incidental math conversations:
Shopping
Cooking
Budgeting
Deciding what to buy
Keeping track of sports scores, race times, etc.
Playing board games and dice games
Urge parents to let their children know that they believe in their children’s future success in math and to talk positively about how useful math is to all adults
Teach times tables to children in grades K-6: this is arguably one of the most important jobs a K-6 teacher has.
Why Teach Times Tables? Surely Calculators Make Memorization Redundant?
This sounds a little plausible, but I encourage you to stop and imagine this scenario: first, you have to imagine that you’re a child, around 10 years old. You haven’t had the experiences that your future adult self will have. You’ve been told by teachers that you don’t have to learn math facts by heart. You have a calculator in your desk, and you are encouraged to use it.
Now, picture this: you are working out the perimeter of a 6 by 8 rectangle using the formula “P = 2x(L + W)”.
Imagine This: You are a Child Whose Teachers Did Not Teach Times Tables
You remember you should add the length and width first, but you don’t know what six plus eight equals, since you never learned the addition facts by heart either. You look around in your desk and find the calculator, switch it on, look at the question again, press “8”, “+”, “6”, “=” and see “14” in the display. “What does that mean?” you think. Oh yes, that’s what “L + W” equals. Somehow you figure out the next step is to multiply 2 by the number you just found. You pick up the calculator, press “2”, “x”, then ask “What do I times this by?”.
You have forgotten the answer and you didn’t write it down, so you start again: “8”, “+”, “6”, “=”. This time you take note of the answer, “14”. You look back at the formula again, and press “2”, “x”, recall the previous answer again, “1”, “4”, “=”, and see the display shows “28”. You quickly write “28” in the space for the answer and move on to the next question. Oh look, it’s another perimeter question – it will be quicker this time, because you know the sequence of steps you have to take.
This is what happens if no-one takes the time to teach times tables. Notice that not only does this imaginary child take much longer to complete this simple question than it would have been if tables were memorized, the child is repeatedly interrupted in working through the question to carry out mechanical actions, mostly the pressing of calculator buttons, with little or no thought of why he or she is carrying out the process in the first place.
I promise you this: from today onward, I will not apologize for expecting students to memorize the times tables.
I’ve had enough. Students who don’t know the multiplication facts by heart will not achieve much success, if any, in their future math studies. So why don’t we do more to get our students to memorize facts?
Teach Times Tables: A True But Sad Story
In the video: The experience I had recently watching Year 5 kids trying to do a multiplication test without knowing the times tables illustrated this perfectly for me.
Students simply had no idea of the times tables, and so most of them resorted to drawing arrays of dots and then counting the dots from 1. Of course, this was much too slow, and in addition students frequently made mistakes.
Who knew? Math is important. Like, really, really important.
We shouldn’t even need to say this, surely? Our kids need to grow up being good at math.
Watch the video:
Western Nations Spend More on Math, and Get Lower Results
And yet, despite most ordinary folks agreeing that math is a vital part of kids’ schooling, in western democratic nations we seem to be spending less time on math teaching, and getting lower results.
If you’re a teacher I’m confident that you believe in the value of a good mathematical education. But at the same time, you’re far from being alone if you find students seem to be less engaged and less proficient at math as time goes on.
Do you need ammunition to make math excellence a priority for your students?
Think on this: in 2016, terrorists are as likely to carry a laptop as a bomb. And they are probably in a basement somewhere, not risking being found out in the open.
Want to try that line “We don’t really need to be all that good at math, now we all have smartphones” again? I thought not.
Math is Important: We Should Spend More Time on Math in School, Not Less
Lastly, it distresses me to hear people say that modern technology has taken the place of people good at math, and that we can cut back on hours spent in school learning math, to make room for more trendy subjects such as coding and technology. I believe that educators as a group should spread the message that math is important; and if anything, it’s more important than it was in the past, given the spread of technology into nearly every part of our lives.
Infants as young as six months recognize interesting shapes. And babies who show higher spatial reasoning skills do better in math at age four. This is good news for parents and carers who purposefully try to help their children understand the world around them in explicitly mathematical ways: developing better spatial reasoning in infancy should result in better understanding of math generally when they start school.
Watch the video for more:
In this study, babies between six and thirteen months were shown images on two screens, where on one screen pairs of images appeared in a symmetrical arrangement. This caught the babies’ attention, and using eye-tracking technology the researchers were able to measure how often and for how long they focused on the more interesting images, and so deduce relative levels of spatial reasoning.
The really encouraging news for parents and carers who want to give their babies a head start before they start school is that higher abilities in spatial reasoning were associated with better math results at age four.
Since spatial reasoning is not a fixed ability but can be taught, I know what I will be doing when I get the chance with my grandchildren; all parents should do the same!
What do you think?
Does trying to expand young infants’ awareness of the world around them in targeted ways really help prepare them for school success? Share a comment below.
Indigenous students in Australia typically lag two years behind other kids in math: a disappointing statistic by any measure. Dr Chris Matthews of Griffith University has come up with a new approach that shows great promise for connecting indigenous kids with mathematics.
Going further, I believe that this new method would help any kids of a non-mainstream background to understand math. Watch the video for more:
The approach recommended by Dr Matthews is explained by Prof Tom Cooper of the YuMi Deadly Centre at QUT, using the acronym RAMR:
Reality: connect first of all with students’ existing culture and interests. In the case of indigenous kids, this includes story telling and dance
Abstraction: come up with ways to turn stories and concerns into mathematical problems, equations and so on
Mathematics: invent mathematics in standard symbolic format to capture the original question or scenario
Reflect: consider the result and match the mathematical results with the original source situation and consider how well the mathematics enabled the solution for the problem, or explained a story in mathematical terms
[Click the link below to watch Prof Cooper’s explanation]
But a teacher who teaches students of non-indigenous but also non-mainstream backgrounds could adopt the same basic pedagogy, starting with those students stories, culture and questions that interest them.
Mathematics has sadly often been presented as the product of a lot of “old white males”, which for some students immediately puts them offside and makes math irrelevant and boring, in those students’ minds. This approach deals with this problem by starting with examples from the students’ own culture and background.
What do you think?
How should we teach math to students from backgrounds other than our own? Share a comment below.
Education planners and commenters in the west seem to be looking enviously at Asia, especially China, Korea and Singapore, for inspiration to improve math test results in their own countries. But everything is not as it may first appear; let me explain why below the video:
Apparently, since students in developing countries did so poorly on international math tests compared to most Asian nations, we in the west should adopt Asian methods as soon as possible. This is surely a scenario that old-style communist dictators could dream of: China beats the United States (and most of the developed world), showing the superiority of the disciplined approach forced on their citizens by the central government.
I admit, that last paragraph sounds just a little over the top.
But consider this: the government of the UK is spending £41m to train teachers in 8000 English primary schools in so-called “mastery maths”, based on the approach in Shanghai, China.
So, can we deduce from this decision by Whitehall that “Shanghai Maths” is the secret to success in maths for schoolkids in the UK?
Not so fast.
The space here won’t allow for a detailed analysis of this issue, so let me instead list a few points for consideration:
First up, why copy Shanghai? It isn’t a country, but rather a city, albeit a very large one. Perhaps Shanghai is cited as the example to follow because out of all results in China, Shanghai’s were the pick of the bunch?
Do planners believe that merely training teachers and changing textbooks will provide equivalent results with today’s English schoolkids as teachers in Shanghai see?
Why should we assume that an educational system based on a highly-regimented repetition-based pedagogy is preferable to teaching for understanding, after abandoning rote teaching for English students in the 1960s?
Why should western democracies copy nations in which avoidance of failure and measuring a person’s worth is based on academic results?
While Shanghai did well on highly-structured math tests, statistics on the numbers of Nobel Prizes tell a very different story: the USA tops the list by a huge margin, whereas China is almost dead last.
What do you think?
How much rote learning should we allow? None? A little? As often as possible? Share a comment below.
Rain gauges measure rainfall by collecting a small sample and measuring how deep the water is. The trouble is, we are interested in very small units – in the metric system, rainfall is measured in millimetres/millimeters. How can you accurately measure such small amounts?
How can we use everyday examples to teach measurement?
Watch the video: I explain how a rain gauge amplifies the depth of water collected to make it easier to measure.
