Another in the never-ending series on how ed tech can be used in positive ways.
This section was as far as I got (which admittedly is pretty much near the end of the article.) I have a strong stomach for this kind of stuff, but I have my limits:
“Exploration: The technology should provide opportunities for students to explore by conjecturing, testing out different ideas, and making mistakes. We should avoid digital learning programs that focus only on memorization or funnel students’ thinking.”
Nothing wrong with this idea per se but the disdain for memorization is quite apparent. Kids need to memorize their facts, period, and some programs actually help them do that. We should avoid such things unless it has the trappings of “conjecture” and other sound-good words?
“Multiple Solution Strategies. Identify technology applications that have more than one way to solve the problems. For example, rather than using digital flashcards such as 3+4 = ?, we can identify apps that ask students to find pairs of numbers that add to 7. The latter question has many solutions such as 1 & 6, 2 & 5, 0 & 7 and supports students to understand how one whole number (in this case 7) can be broken into parts in multiple ways.”
This is like those problems that reformers love to have TED talks about: “The number is 28; tell me everything about it.” Or “The rectangle has an area of 36; what are its dimensions.” Everything open-ended, nothing confined. You still have to know your math facts, no matter how you dress it up, and the open-ended approach serves as just another way to avoid that. In my opinion and no one else’s of course.
“Connections between concepts and procedures. Good educational technology supports students to focus on relationships, not discrete facts. Rather than choose a digital program that solely focuses on doing the same procedure over and over, identify a program that supports students to understand why the procedure works. For example, with regards to the earlier problem 3+4 = ?, a digital program that includes other representations, such as images of objects that students move around can better support to develop meaning of the procedure. Digital math games that focus solely on procedures should only be considered after students have strong understanding between concepts and procedures.”
Yes, and no article on education would be complete without the “procedures-bad, concepts-good” recitation. Reminds me of a boss I had who whenever he used the word “strength” when talking to the people working for him, as in “you have some very good strengths” he would be quick to add “but you have weaknesses too”, ostensibly to forestall any of us asking for a raise. In the above quote note the allegiance to “understanding must come first”. Then and only then can students do all the procedures you want them to do. How’s that been working out for the nation for the past 28+ years?
traditional math 
The more the world spins the more it stays the same. There have been articles written on this ad nauseum, and the same rules apply. Unless kids know their facts, and the basics back to front, and upside down, there is no path to critical thinking or higher knowledge for them. They will be left on the assembly line, making widgets, bemoaning the fact that the schools lied to them! Collaborative learning – wrong! Project based learning – who cares?!? What they sorely lacked, was a strong grasp of times tables and basic writing skills…all denied them in the pursuit of 21C learning.
It’s the reason why parents are now homeschooling their kids in droves. Education is a mess! We must do better – for our kids, and for our own nurses in our old folks home where we’ll end up one day.
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I am a divergent thinker. That is one thing. There is merit in divergent thinking, or so-called “lateral problem solving”. And it is a useful side-skill when doing math.
But that is NOT WHAT MATH IS. As a discipline it is fundamentally about convergent thought. Further, the LEARNING of math — something very different from the DOING — is necessarily convergent and focussed. Mathematics per se is an infinite domain. The learning of this subject must consist of familiarity with well-trodden pathways and well-established methods, before the learner is encouraged to wander from those paths and break their own ground.
This article and most of the edu-gurus wandering about on the landscape these days get this exactly backwards.
That’s what the denigration of “memorization” and “funneling students’ thinking” is about. Memory is the seat of learning and absolutely essential in laying the foundation in mathematics for later critical thinking skills. And focussed thought and action lie at the heart of effective mathematical practice.
On both counts educationists show their deep disdain for what is helpful in the education of novices in this discipline.
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Barry, you have a stronger stomach than I do, since I am unable to get through these articles. Families are beside themselves; they are dismissed and ignored and yet it is families, who must deal with confused and uneducated children and pay for outside tutoring. The people who support this type of math education are the very people who preach student voice, flexibility, multiple solutions and the need to listen. The reality is exclusion, rigidity and harm for thousands of families. I for one am weary of moms literally crying on my shoulder as they deal with the realities of this flawed approach.
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0+7=7 1+6=7 2+5=7 3+4=7 4+3=7 5+2=7 6+1=7 7+0=7
Practice, practice, practice. Everyone will understand what all of the additive pieces are of 7, along with all other parts of numbers. Their assumption is that study of a few examples causes conceptual ideas to turn into skills. Nope. There are too many variations to learn. How about 23 + 44? Everyone I know who had to practice and practice never had any problem with breaking apart pieces of numbers and adding them up in different ways. Understanding comes from skills. Back when I taught math, nobody could pass with rote application of a process to new problems, and that’s what they got on exams. Also, those who are taught concepts without practice are never able to apply those concepts to new problems. The key is practice. It would be one thing if this fuzzy talk was built on top of mastery of basic facts and skills, but it isn’t. Don’t directly memorize 3+4=7? Incredible. On one hand K-6 schools have now become NO-STEM zones, but on the other hand, they claim to provide some sort of high level and deep understanding. It’s just cover for low expectations and group learning in class. Meanwhile STEM parents are ensuring mastery of the basics at home with flash cards and worksheets. While it might seem that there is a lot of rote learning with basics, that’s not the case. They just don’t see the understanding that happens. As you get to fractions and algebra, it’s much easier to see how understanding comes from mastery of skills. Still, their arguments are built on the simplest level of problems.
Some students might need a more conceptual framework before studying a new topic, but all good textbooks do that. It’s quite another thing to drive all learning from concepts down to skills. The skills are never ensured because they think that everything can be derived from concepts. NO. Every time I tutor students and we talk about how and why things work – and they explain them back to me – I tell them they HAVE to do all of their homework again by themselves. It’s amazing how concepts fail when a new problem variation is tackled or even when they do the exact same problems over again. Unfortunately, almost all students who come to me have no appreciation for fully doing all of their homework problems by themselves. That is where the real learning takes place.
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I agree with the blog and its comments above, but there is a specific thing there that annoys me.
For example, with regards to the earlier problem 3+4 = ?, a digital program that includes other representations, such as images of objects that students move around can better support to develop meaning of the procedure.
This is just plain untrue IMO.
If you are trying to teach a new procedure you need to keep it as clean as possible. Factorising quadratics x^2 + ax + b is not hard — the target numbers add to a and multiply to b. There is nothing else needed to learn that skill, and in particular you have no need to introduce “meaning” into it. If you throw some visual representation into the mix then all you will do is confuse the students, who now have to try and link mentally the skill of finding the target numbers with the much harder task of identifying what it might mean in a physical way.
The people who propose these ideas that images help Maths are doing so because they implicitly understand the link between numbers and their representations in the real world. Our students largely do not. Anyone who teaches knows that problems are much harder when put into contexts rather than just left as Maths, yet they suggest that putting new concepts into contexts makes understanding easier?
I strongly suspect that the reason that students arrive at High School completely useless is because far too much emphasis is put on “understanding” before the students are ready to do that. If we just taught the mechanics, and only introduced what they might be used for afterwards, I think we would have students far less confused.
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I appreciate your comment. Putting things into context does indeed put extra burden on initial understanding. To that extent, requiring students in lower grades to explain problems so simple that they defy explanation is similarly self-defeating. I have nothing against “showing your work”, but requiring an explanation in writing seems to be a waste in my opinion for students in grades K-6. Yet, a physics professor took to Forbes to complain about an article that Katharine Beals and I wrote which was published in The Atlantic (online) in Nov 2015. I tried to comment on his article but couldn’t access the comment section. If you are so motivated and happen to break in, perhaps you might leave a comment. His article is here: https://www.forbes.com/sites/chadorzel/2017/02/21/why-writing-about-math-is-the-best-part-of-common-core/#19d64af2222d
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