H. Wu is a professor at UC Berkeley and is the author of the books on which this course is based. He coined the phrase “Textbook School Mathematics,” or TSM, to describe the mathematics curriculum commonly taught in grades K-12 in this country.
[T]here has been a de facto national mathematics curriculum for decades: the curriculum defined by the school mathematics textbooks. There are several widely used textbooks, but mathematically they are very much alike. Letâs call this de facto mathematics curriculum Textbook School Mathematics (TSM). In TSM, precise definitions usually are not given and logical reasoning is hardly ever provided (except in high school geometry texts) because the publishers mistakenly believe that intuitive arguments and analogies suffice. Thus, fractions are simultaneously (and incomprehensibly) parts of a whole, a division, and a ratio; decimals are taught independently from fractions by appealing to the analogy with whole numbers; negative numbers are taught by using patterns and metaphors; the central idea of beginning algebra is the introduction of the concept of a variable (which implies, wrongly, that something is going to vary), when it ought to be becoming fluent in using symbols so as to do generalized arithmetic; solving equations is explained by the use of a balance to weigh variables on the weighing platforms; etc.
H. Wu
Assignment, due Wednesday, 9/6/23: Watch the following video of Wu speaking about TSM and the way we prepare teachers (16 min). Respond to the questions below by leaving a comment on this post.
- What do you think is Wu’s main objective in this talk?
- What is something Wu said that you agree with? Explain.
- What is something Wu said that you disagree with? Explain.
- How do you think you can you help students learn mathematics, especially students who have spent their lives learning “Textbook School Mathematics”?
- Extra Credit. Respond in some way to one of your fellow students’ comments. Do you agree? Disagree? Do you have a response to their question? Did their comments make you think or provoke additional questions? Reminder: Be respectful, be kind.
Video
Wu on K-12 math teacher preparation:
“Stop selling what you have; start selling what they need.” (16min)
BONUS: If you’re interested in more detail, here are the slides from a longer version of this talk that Wu gave around the same time.
Wu’s main objective is to explain why the way we currently teach mathematics is infective. As someone with a history degree, I can say this isn’t just a problem in mathematics; what he mentions also happens in humanities courses. We, as teachers, are drilled to teach students how to memorize instead of think critically. The public school system fails to see where math and English teachers should collaborate.
I remember being in Algebra I and having difficulty translating a word problem into an algebraic expression. I will compare this example to a significant history paper I had to write. We were required to use 4-5 primary and secondary sources. Naturally, this required us to synthesize all these sources to make a cohesive argument. Working through proof requires a cohesive argument with a beginning, middle, and end in mathematics.
I disagree that revamping the methodology would change our system. This may be my cynicism talking, but mostly I know it won’t be immediate and will face much political resistance. It will take time and financial support from upper management. The other reason I believe this will be a gradual change is that many people still are under the impression that there are two groups of strengths; ‘math people and ‘English people.’
I still need to test my methodology, but I believe a project-based curriculum with the standard curriculum can gradually help us shift from a TSM methodology to the methodology that Wu describes. I am researching and slowly developing a skeleton of a few financial management projects. One of them is a project that applies compounded growth. So far, the skeleton includes vocabulary like monthly, annually, and quarterly, and I include readings about Roth IRAs and high-yield savings accounts.
This project aims for students to apply the compounded interest to visualize how their money can grow. The project also includes the difference between a long-term vs. short-term financial goal. It lets them choose one to apply the formula to see how long it will take them to save up for their goal or how long it would take them to pay off a loan. While such a project is tedious, I know realistically they won’t remember the formula after they graduate. Still, experiencing a project with a practical, direct application, I hope it will reinforce the concept versus memorizing the formula.
*I should also note outside of mathematical standards, my hope is such projects that apply these standards to financial management will help students from falling prey to predatory companies when opening their first credit card or taking out a loan
Hi Rachel – thanks for getting the ball rolling on responses. I really like your point about “math people” vs “English people” — this seems to be such a deeply-ingrained way of splitting people up in our culture, and not very useful in the end (in fact, often harmful!).
I think Wu’s main point of the conversation was to provoke thought to changing teachers approach to teaching math without heavily leaning toward textbooks or stop allowing the textbook to be the curriculum.
I disagreed with him saying “They need the correct version of the mathematics” that we have to teach. Also showing us based off study students getting questions wrong because they wasn’t taught in a particular way. There is no such thing as the correct version in my opinion. We know there are multiple ways to solve almost all math problems without there being any correct version to solving. Why would we as teachers mirror a particular way or version. Mastering math topic come with a journey to understanding. On your journey as a math student discoveries and connections are made despite any version of math taught to you, through self practice and study. There you find the understanding he wants. I think the version of understanding he wants may take a different approach another group of students may not understand correctly.
What I agree with was, when he mention the revolving door of the textbook ways. With his previous statement of teaching the correct version, this is the very conundrum he’s speaking of about the textbook school mathematics way. I believe we need to take more of an artistic approach than scientific and stop the revolving door.
My approach to helping students learn math is always mapping the journey, understanding them individually in the subject, catering to their perspectives as a collective, and using the material that best fit them. And I don’t see anything wrong with utilizing TSM. For students taught
Jquan, I think the point you make about multiple ways to solve math problems is absolutely right on — it reminds me of some of the BIG IDEAS we talked about on the first day of class, that everyone knows stuff about math, and that different perspectives on math concepts are important. But also, while there may not be one right way to approach a problem, there are often many wrong ways – helping students see when their approach has wrong assumptions or faulty logic is a part of our job (but can we do it while lifting them up at the same time?)
As far as I can see, Wu’s main talking point is the need for change of the way mathematics is being taught in schools.
One of the main takeaways I’ve noticed from Wu’s talk about mathematics is that mathematics taught in schools follow the Textbook School Mathematics (TSM) which is rather ineffective as it contains fragmented or broken teachings. I personally agree with this statement as my experiences with textbooks frequently resulted in being lost, unsure of where to refer to, as with how Wu explains that textbooks have many missing definitions, fragmented presentations, and overall lack of guidance for the students. While textbooks can provide assistance, I believe that, at least for me, a great deal of learning comes from the guidance of a teacher/instructor/professor as students left without guidance and a mere textbook with no clear instructions will not perform as well as others who receive proper guidance to not only just mathematics, but any topic as a whole.
I can’t really say there’s much I will disagree on as like Jquan said in his post where Wu claims “They need the correct version of mathematics”, mathematics can be view, taught, and understood differently but still arrive at the same understanding of how mathematics work. Therefore, rather than disagree with what Wu talks about, I believe that it’s more of something along the lines of: Everyone has their own take of mathematics. As we can see, some people struggle with it. However, this does not mean they are necessarily bad at mathematics. Rather, they may need a different outlet or means to comprehend the subject at hand, and even if it varies from other individuals, if there is a form they may understand the same mathematical concepts and principles via a different method, I see no problem with it as they’ve come to understand it. Everyone can have a different perspective on how they perceive mathematics yet at the same time, either sides may be correct with their final results.
One way I can imagine helping students understand mathematics is to show them an example of mathematics applied in a real life setting. Most mathematics taught using TSM only explains definitions (to some extent), statements, and two-dimensional figures only seen used in textbooks. I believe that while this is enough for some students to understand, some other might not and will require a different example they can visualize or understand.
I really like your third paragraph, especially the passage beginning with “Everyone has their own take of mathematics…”. This is so important!
I think Wuâs main objective in this talk is to emphasize that mathematics does not need to be taught by textbook only and make the entire curriculum be just about that. There should be some type of change to teach mathematics better.
I agree with Wu with his objective, I feel teachers base the entire material off a textbook and memorization. Where as instead of attempting to understand concepts, we are memorizing steps or certain things but it does not mean weâre learning. I personally feel like I would memorize formulas and steps without even understanding when to use what, which at the end of the day does absolutely nothing because when will you know to use what you have memorized if you donât understand when to use it?
I do disagree with Wu when he says that they need to learn the correct version of math because I donât believe there is such thing. I grew up seeing everyone was different in a sense where certain people are stronger in different areas than others. Not everyone learns the same and not everyone will be good at the same thing. There is not only one way to learn mathematics and thatâs my favorite thing about it. There are so many ways you can approach a problem and you get to choose what you understand better and what works best for you. As a teacher you make the effort to work with your students weaknesses and strengths to help the understand the material.
The way I would try to help students learn mathematics is by mainly helping them be confident with themselves. A textbook is not completely useless however basing your lectures and materials only solely that can be confusing and not as helpful. Mistakes are always okay to make because thatâs how you learn. Showing all possible ways to approach a problem because all students will NOT learn the same, where we may have students who prefer a different approach from another. Lastly, I would definitely pay attention to each students weaknesses and strengths and have them work together where one can help another in an area that theyâre good at and where the other student may seem to be struggling.
I like that you bring up confidence – I think in a lot of ways our biggest job as math teachers is teaching students to have confidence in their own thinking (and, especially, that this confidence is not broken when they make a mistake)!
Wu’s main objective is to highlight that how math is being taught is not effectively and from what remember about my own high school experience I agree. I was mostly taught from a textbook and when students would something works they often would not a good answer. I’ve seen middle schools (more so) moving towards illuminating the underlying patterns and the definitions that affect how problems can be solved.
Wu’s approach I agree with that you should know definitions and properties as a framework but there is no one “correct” way to teach just like there is no one way to learn. I agree that just using some formulas or doing a problem in a certain way that you memorized is not good. The moment that you are given a different variation of a question on the same topic that you might not be able to solve it if you only memorized. It’s much easier to solve problems if you something to go back to.
I believe that since a lot things have been gameified why not gameify math classes as well. The definitions and properties are your resources (map and ability screen), ways of solving problems (tools), differing toughness of problems (levels). At the end of a unit students get to do a fun activity (if behaved) that allows them to use each of the skills they gain throughout a unit and whatever framework is called for.
Your reference to gameify-ing caught my eye – there is a group at CUNY focused on this, the CUNY Games Network, which you might want to check out. And I agree, I think math is definitely a good fit for gameification!
The purpose of Wuâs presentation is about the ineffectiveness of the Textbooks School Mathematics (TSM) that are used by pre and service teachers and taught to students. Wuâs aiming on what to teach; the content of these TSMs, and how to teach; the pedagogy of teaching so that students can have a good understanding of what they are been taught.
I agree with the two sentences that Wu got through an ad by IBM in Londonâs Heathrow Airport (March 2008): Stop selling what you have; start selling what they need.
So, these TSMs should be revised and updated so that they have clear and adequate content with step-by-step resolution of problems with written words on how the steps flow; and have all different approaches to solve a problem including pictures, and links to videos for visual learners. Finding what teachers and students really need benefits both teachers to teach the course fully and professionally, and students to successfully learn the content materials.
There is no such thing that I disagree with Wuâs view of TSM, as he mentioned at 13â of the debate that changing the perspective of a teacher who has been teaching these TSMs for more than a decade is something difficult, and for students who stick to learning from the ineffective version of mathematics would take a long process to become familiarized with a new curriculum.
I think I could help students learn mathematics, especially students with these TSMs by first knowing their different backgrounds such as strengths and weaknesses. Also, knowing their learning styles to provide them with the necessary needed to learn and understand the key concepts of a topic. For example, for visual learners, after explaining the lesson and related problems, then show them some related videos, and have one-on-one and group work, practices, and hands-on activities for other types of learners.
I agree with Jquanâs comment âMy approach to helping studentsâŚâŚâŚ…………….using the material that best fits themâ.
It’s a great point about learning styles — you and Jquan both have an admirable focus on student-centered learning and meeting students where they are at.
Taspia Jannat:
I think Wu’s main objective is that math isn’t being taught in the correct way in the sense that textbook math isn’t helping. When he was showing how students are mostly solving the problems, he’s really highlighting that certain things aren’t clear to students and there are a lot of misconceptions. For example, when subtracting two fractions, you need to have a common denominator and not just subtract them. One thing that I most definitely agree with Wu said is how textbook school math TSM isnât effective enough for students to learn math. The reason I say this is because I could relate to this from a student perspective because whenever I read math textbooks, itâs like Iâm reading a foreign language. I canât see the math going on in a textbook and most of the time the explanations are even more confusing. I believe math should be taught one on one. I disagree a little bit with Wu about textbook math being not effective because without a proper format, nothing could properly be taught or executed. We need a certain guideline to follow so we donât lose track or miss out on anything important. I can help students learn mathematics, especially students like me who have spent their whole lives learning âtextbook school mathematics’ ‘ by giving them more of a one on one rather than following along in the textbook and reading that to them. From my personal experience, the classes where my teachers would read off from a textbook, I would do much worse than the teachers who explained a problem on the board. I believe math is a subject where you need to really show how itâs done rather than reading a script.
I want to agree with Rachel when she said âI remember being in Algebra I and having difficulty translating a word problem into an algebraic expression.â I could relate to that so much to the point that if I see a word problem I want to run away because that instant pressure kicks in. Now that we are doing higher math, we are seeing the way word problems are being translated into numbers and variables and with logical reasoning but an eighth grader doesn’t know that or have that knowledge.
I find your comment that math “should be taught one on one” to be such a great aspirational idea — some of the most effective learning happens in a one on one setting (although we don’t always have the freedom to work in this way, even if we want to). Of course, there is value in working with others, too (if nothing else, sometimes they are willing to ask the questions that you are not yet willing to ask) – finding ways to balance big group, small group, and individual instruction is a great challenge!
Yadira Vazquez
Metacognition and general strategies for thinking about your own thinking is one of the best ideas to come out of modern educational theory — I think it’s really powerful, and I’m glad to hear you are being exposed to it!