Math in 1950: A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price. What is his profit?
Math in 1960: A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price - or $80. What is his profit?
Math in 1970: A logger sells a truckload of lumber for $100. His cost of production is $80. Did he make a profit?
Math in 1980: A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Underline the number 20.
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Math in 2015: Five loggers sell their truckloads of lumber. The log counts for each logger are 10, 22, 18, 16, and 12. Use a stem-leaf plot to find the difference between the highest and lowest log count.
Math – past, present and future. I know very little about teaching math but became intrigued after reading two recent articles on the subject and listening to my colleagues. Society and technology have forever changed how we must teach — and learn math.
An NPR article, The Way You Learned Math Is So Old School, compares two ways to solve a multiplication problem (36x24). The traditional approach asks students to add the two products — 36x4 to 36x2 (+ adding a zero on the end). Today’s approach involves creating and adding 4 products - 20x30 + 20x6 + 4x30 + 4x6.
Why? NPR spoke to Keith Devlin who said today’s math reflects the different needs of society .First, very few people today actually need to do calculations themselves — computers and calculators do the arithmetic for us. Second, in order to make computers do the things we want them to do, we need to think in algebraic terms. Yet to master algebra we must first master arithmetic. This has made arithmetic more of an on-ramp than a road or destination. Teaching mathematics today is about getting people to become sophisticated, algebraic thinkers.
Devlin says that this change doesn’t mean that kids will bypass the ritual multiplications table tasks — it’s just that the way it’s taught is changing so that kids get a better understanding of the number system and what numbers mean.Check out Khan academy and run a stem-leaf plot video to see how kids today are being asked to solve age old questions– like how many points a basketball team earned during a game.
Stanford University mathematician James Milgram says the math standards are too advanced for younger students and not nearly rigorous enough in the upper grades. He contends that educators are being asked to teach math in a way that is incredibly complicated for kids who aren’t ready for it.
In a NY Times blog, A Better Way To Teach Math, David Bornstein interviews John Mighton, founder of the nonprofit organization Jump Math. Mighton says that most students label themselves based on how they perform in math. Differences in background knowledge, confidence, focus and speed – all of which are palpable in the classroom – multiply the advantages. When students start losing their confidence, that’s when the hierarchies develop. Doing well in math is omnipotent.
Michael Rubinkam examines this further in his NBCWashington.com article 2+2=What? Parents Rail Against Common Core Math. He says Common Core learning standards have made adding, subtracting, multiplying and dividing as complicated as calculus. In many cases, there’s nothing elementary about today’s elementary mathematics education. Not only do elementary math teachers face daunting tasks in the classroom, they can no longer count on parental support for homework guidance because today’s math homework often appears absurd without context - and most parents lack the context.
Teaching methods are not aligned with what cognitive science says about the brain and how learning happens. Mighton says that math educators need to consider the differences in their students’ working memory and remember that extensive practice - repetition, is needed before mastery is achieved. Although the educational system supports “problem-based” or “discovery-based” learning, current teaching strategies underestimate the amount of explicit guidance, scaffolding and practice children need to consolidate new concepts. He says that asking children to make their own discoveries before they solidify the basics is akin to asking them to compose a guitar song before they can form a C chord.
Rote vs Meaning
Grover Whitehurst shares how math skills eventually become limited when instruction focuses exclusively on learning facts and procedures. The research message is strong: teach math for meaning or risk never getting students beyond a superficial understanding that will leave them unprepared to apply their learning.
Gretchen Vierstra, in her TeachingChannel.org article, What Does Math Look Like in Today’s Classroom?, says that many math teachers are using formative assessment to create classrooms that encourage perseverance, collaboration, and deep mathematical thinking. These teachers suggest the following to make math education successful:
Make math hands-on, encourage collaboration (have students work in small groups most of the time)provide a variety of ways to experience math and make math connections (using print, photos, presentations, videos, interactive sites) so that students have multiple representations of concepts and alternative paths to developing understanding
create mathematical thinkers (a classroom doesn’t always have to have an answer at first, but employs talking, resources, and problem solving as a means to the solution)focus on the meaning of things by posing open-ended questions: Why do you think that? Can you explain your reasoning? How do you know that?
What the research shows
In their ASCD article, Research Matters - Teach Mathematics Right the First Time, Steve Leinwand, and Steve Fleischman, comment on the gap that exists between research findings on developing students’ conceptual mathematics understanding and what is actually done in the classroom.
Instrumental practices and relational practices (Skemp 1987) are two major approaches to teaching and learning. Instrumental practices involve memorizing and routinely applying procedures and formulas - that is, what to do and how to get answers. Relational practices emphasize the why of learning - explaining, reasoning, and relying on multiple representations—that is, teaching for meaning.
Research suggests that if initial instruction focuses exclusively on procedural skills, then students may have difficulty with applying those same math concepts.One example is listening to 7th graders defining perimeter as “adding up all the numbers,” while the teacher struggles to move them toward the understanding that perimeter is actually the distance around an object.
While studies on studentsexposed to instrumental instruction prior to relational instruction showedless conceptual mastery than students exposed only to the relational component, the jury may be out, as all students learn differently.
Implications for the real world - Quantitative Literacy
In his NY Times article, The Wrong Way to Teach Math, Andrew Hackerfeb says that while most Americans have taken high school mathematics, including geometry and algebra, a national survey found that 82% of these adults could not compute the cost of a carpet when given its dimensions and square-yard price.
In a multi-country test on basic numeracy skills, the Organization for Economic Cooperation and Development asked questions involving odometer readings and produce sell-by tags. The US came in 22nd - behind Estonia and Cyprus. Are we not teaching enough math or just not the right way?
While calculus and higher math have their place, the majority of citizens need to be able to read graphs and charts and be capable of calculating simple figures in their heads. We live in a quantitative century where decimals and ratios are now as crucial as nouns and verbs.
Hackerfeb believes that students need math that reflects real world situations - situations they can relate to. Like how many households in the United States have telephones, land and cell or how to analyze changing trends. He asks his students to evaluate the January National Center for Health Statistics Births: Final Data and scan the various columns, looking for patterns. Doing this, they found that women in Nebraska averaged 2.2 children, while the Vermont’s ratio is 1.6. From this they create and discuss relevant theories.
Thinking that fancy math will make us numerically literate is misleading. In fact in the real world, where we constantly settle for estimates, advanced math training does not necessarily ensure high levels of quantitative literacy. Real math — like what kids face on the SAT — demands that you get the answer precisely right.
In her GreatSchools.org article, Does our approach to teaching math fail even the smartest kids?, Carol Lloyd says that despite the concern of politicians and educators over lagging American math and science scores (compared to Shanghai and Japan), research shows that, as many as 60% of all college students in STEM (science, technology, engineering, math) based tracks end up transferring out. This finding - which has spawned a new field of research on “STEM drop-out,” - cites various etiologies from gender and race, to GPAs and peer relationships.
Lloyd proposes that the STEM exodus in American schools is because kids don’t have a good foundation in math. One third of American high school seniors don’t score proficient in math. Yet university STEM attrition rates are even higher- even at the most selective colleges where kids had the grades/courses to get in.
Math experts say it’s not that kids aren’t getting enough math, they say, but that we’re teaching K-12 math all wrong. Accomplished high school students are being faced with the fact that university math requires more than rote learning — it requires creativity, grit, and strenuous mental gymnastics. Many students learn that math is simply a set of destinations with rules that will lead them there - however, most are never taught how to read the roadmap.
Traditional math curriculum teaches discrete algorithms which elicit a correct answer, like how to do long division or how to apply the Pythagorean theorem. Students “learn” the material by doing a large quantity of similar problems. Some call it drill and kill. The fact is, students are rarely asked to apply those algorithms to solve a problem they are unfamiliar with.
My math colleague Shanna Nazinitsky, reminds us that math shouldn’t be just about following rules - but more focused on the application of those rules to solve relevant situations.